Paper Title: PDE formulation of some SABR/LIBOR market models and its numerical solution with a sparse grid combination technique.



























[mytitlenote]Partially financed by Spanish Grant MTM2013-47800-C2-1-P and by Xunta de Galicia (Grant CN2014/044). First author has also been founded by a FPU Spanish Grant.


auth]J. G. López-Salasjose.lsalas@udc.es

auth]C. Vázquezcor1carlosv@udc.es

[auth]Department of Mathematics, Faculty of Informatics, Campus Elviña s/n, 15071-A Coruña (Spain)

[cor1]Corresponding author. Tel.: +34 981167000; fax: +34 981167160.

§ ABSTRACT 

SABR models have been used to incorporate stochastic volatility to LIBOR market models (LMM) in order to describe interest rate dynamics and price interest rate derivatives. From the numerical point of view, the pricing of derivatives with SABR/LIBOR market models (SABR/LMMs) is mainly carried out with Monte Carlo simulation. However, this approach could involve excessively long computational times. For first time in the literature, in the present paper we propose an alternative pricing based on partial differential equations (PDEs). Thus, we pose original PDE formulations associated to the SABR/LMMs proposed by Hagan <cit.>, Mercurio & Morini <cit.> and Rebonato <cit.>. Moreover, as the PDEs associated to these SABR/LMMs are high dimensional in space, traditional full grid methods (like standard finite differences or finite elements) are not able to price derivatives over more than three or four underlying interest rates. In order to overcome this curse of dimensionality, a sparse grid combination technique is proposed. A comparison between Monte Carlo simulation results and the ones obtained with the sparse grid technique illustrates the performance of the method.









Stochastic volatility modelsSABR/LIBOR market models High dimensional PDEs Sparse grids Combination technique









§ INTRODUCTION


The LMM <cit.> has become the most popular interest rate model. The main reason is the agreement between this model and Black's formulas <cit.>. The standard LIBOR market model considers constant volatilities for the forward rates. However, this is a very limited hypothesis since it is impossible to reproduce market volatility smiles.

Among the different stochastic volatility models offered in the literature, the SABR model proposed by Hagan, Kumar, Lesniewski and Woodward <cit.> in the year 2002 stands out for becoming the market standard to reproduce the price of European options. SABR is the acronym for Stochastic, Alpha, Beta and Rho, three of the four model parameters. The SABR model can not be used to price derivatives whose payoff depends on several forward rates. In fact, SABR model works in the terminal measure, under which both the forward rate and its volatility are martingales. This can always be done if we work with one forward rate in isolation at a time. Under this same measure, however, the process for another forward rate and for its volatility would not be driftless.

In order to allow LMM to fit market volatility smiles, different extensions of the LMM that incorporate the volatility smile by means of the SABR model were proposed. These models are known as SABR/LIBOR market models (SABR/LMMs). In this article we will deal with the models proposed by Hagan <cit.>, Mercurio and Morini <cit.> and Rebonato <cit.>.

While Monte Carlo <cit.> simulation remains the common choice for pricing interest rate derivatives within SABR/LMM setting, several difficulties motivate to address alternative approaches based on PDE formulations. The first issue is that the convergence of Monte Carlo methods, although it depends only very weakly on the dimension of the problem, is very slow. Indeed, if the standard deviation of the result using a single simulation is ϵ then the standard deviation of the error after N simulations is ϵ/√(N). Therefore, to improve the accuracy of the solution by a factor of 10, 100 times as many simulations must be performed. The second drawback of Monte Carlo methods is the valuation of options with early-exercise, like in the case of the American options, due to the so-called “Monte Carlo on Monte Carlo” effect. Available Monte Carlo methods for American options are also quite costly, see <cit.> for example. In contrast, the modification of the PDE to a linear complementarity problem is usually straightforward. Finally, the weakest point of Monte Carlo methods appears to be the computation of the sensitivities of the solution with respect to the underlyings, the so-called “Greeks”, which are very used by traders, and are directly given by the partial derivatives of the PDE solution. Besides, path-dependent options, like barrier options, can be easily priced in the PDE context where only the boundary conditions need to be changed, in contrast to Monte Carlo methods, where Brownian bridge techniques <cit.> must be applied.

In view of previous arguments, in the present paper we pose equivalent PDE formulations for the three above mentioned SABR/LMMs. As far as we now, this is the first time in the literature that these PDE formulations are posed. From the numerical point of view, one main difficulty in these PDE formulations lies in their high dimensionality in space-like variables. In order to cope with this so-called curse of dimensionality several methods are available in the literature, see <cit.> for example, which can be put into three categories. The first group uses the Karhunen-Loeve transformation to reduce the stochastic differential equation to a lower dimensional equation, therefore this results in a lower dimensional PDE associated to the previously reduced SDE. The second category gathers those methods which try to reduce the dimension of the PDE itself, like for example dimension-wise decomposition algorithms. Finally, the third category groups the methods which reduce the complexity of the problem in the discretization layer, like for example the method of sparse grids, which we use in the present article.

The sparse grid method was originally developed by Smolyak <cit.>, who used it for numerical integration. It is mainly based on a hierarchical basis <cit.>, a representation of a discrete function space which is equivalent to the conventional nodal basis, and a sparse tensor product construction. Zenger <cit.> and Bungartz and Griebel <cit.> extended this idea and applied sparse grids to solve PDEs with finite elements, finite volumes and finite differences methods. Besides working directly in the hierarchical basis, the sparse grid can also be computed using the combination technique <cit.> by linearly combining solutions on traditional Cartesian grids with different mesh widths. This is the approach we follow in this article. Recently, this technique has been used for a financial application related to the pricing of basket options in <cit.>.

The paper is organized as follows. In Section <ref> some basic concepts related to interest rate derivatives and the corresponding terminology and notation are introduced. In Section <ref> we pose the PDE formulations for the SABR/LMMs. In Section <ref> we describe the use of a full grid finite differences scheme for the Mercurio and Morini model, the application of which is analogous for the other two SABR/LMMs. Numerical results show the limitations of the full grid method when the number of forward rates increases. Therefore, in Section <ref> we describe the sparse grid combination technique applied to the SABR/LMM and show numerical results that illustrate the behaviour of the method when the number of forward rates increases. For this purpose, a comparison with Monte Carlo simulation results is used when analytic expressions of the solution are not available, as it happens in most of the cases. Note that Monte Carlo techniques are the usual alternative to price with SABR/LMM.



§ INTEREST RATE DERIVATIVES. CAPLETS AND SWAPTIONS
 
This section provides a brief introduction to the interest rate derivatives we deal with in the present article, for a deeper study we refer the reader to <cit.>. Interest rate derivatives consist of financial contracts that depend on some interest rates.

A zero coupon bond with maturity at time T is a contract that pays its holder one unit of currency at time T. The value of this product at time t<T is denoted by P(t,T), and is called the discount factor from time T to time t. Note that P(T,T) = 1 for all T.

A tenor structure is a set of ordered payment dates {T_i,   i=0, …, N}, such that 
    T_0<T_1<…<T_N-1<T_N.
 The time between the payment dates is denoted by τ_i = T_i+1-T_i. In terms of the corresponding discount factor, a payment of x units at time T_i is worth xP(t,T_i) at time t<T_i.

A forward interest rate F_i(t) is an interest rate we can contract in order to borrow or lend money during the future time period [T_i,T_i+1], and can be expressed in terms of discount factors in the form: 
    F_i(t) = F(t;T_i,T_i+1) = 1τ_i(P(t,T_i)P(t,T_i+1)-1)  t≤ T_i.


Conversely, the price of a bond at time T_i that matures at T_j, P(T_i,T_j), can be expressed in terms of forward LIBOR rates as follows:

    P(T_i,T_j) = ∏_k=i^j-111+τ_k F_k(T_i).


A caplet is a European call option on a forward rate. If the caplet expires at time T_i+1, at that time we will receive the payoff τ_i (F_i(T_i)-K)^+, so that its discounted payoff at time t<T_i+1 is given by

    P(t,T_i+1)τ_i (F_i(T_i)-K)^+,
 where (·)^+ denotes the function max(·,0) and K is the strike of the contract, which is given by a fixed interest rate in the contract. If constant volatilities are assumed, the caplet pricing can be computed with Black's formula <cit.>

    P(t,T_i+1) τ_i (K, F_i(t),σ_Black√(T_i - t)),
 where

    (K, F, ν) = F (d_1(K,F,ν)) - K (d_2(K,F,ν)),


    d_1(K,F,ν) = ln(F/K)+ν^2/2ν,


    d_2(K,F,ν) = ln(F/K)-ν^2/2ν,

and σ_Black is the constant volatility of the forward rate which can be retrieved from market quotes.

An interest rate swap (IRS) is a contract to exchange interest payments at future fixed dates. At every time instant T_i+1 in a prescribed set of dates T_a+1,…,T_b the contract holder pays a fixed interest rate K and receives a floating interest rate at the LIBOR rate F_i(T_i) fixed at time T_i. The discounted payoff at time t<T_a of this swap can be expressed as

    (t;T_a,…,T_b) = ∑_i=a^b-1 P(t,T_i+1)τ_i (F_i(T_i)-K).


A European T_a × (T_b-T_a) swaption is an option giving the right (and not the obligation) to enter a swap at the future time T_a, called the swaption maturity. The underlying swap length T_b-T_a is referred as the tenor of the swaption. Therefore, the discounted swaption payoff to the current time t is equal to 
    P(t,T_a)((T_a;T_a,…,T_b))^+.


As indicated in the introduction, the main objective of this article is the pricing of the previously described interest rate derivatives in the framework of SABR/LMM which incorporates the stochastic volatility by means of PDE formulations. Note that in most cases there are no analytical formulas for the solution. Thus, the new models are posed and solved with suitable numerical methods that overcome the high dimension in space of the equations. Moreover, the proposed sparse grid technique is parallelized to make the approach computationally efficient. Although along the paper we concentrate on Mercurio and Morini SABR/LMM, the proposed methodology can be analogously applied to Rebonato and Hagan models.



§ DERIVATION OF THE PDE FROM THE STOCHASTIC PROCESSES
 

In <cit.> the authors analyzed the three SABR/LIBOR market models proposed by Hagan, Mercurio & Morini and Rebonato using Monte Carlo simulation and their implementation on GPUs in order to price several interest rate derivatives. They have concluded that the Mercurio & Morini model is the one with the best performance: it is the easiest to calibrate, it achieves the best fit to swaption market prices and it results the fastest one in the pricing with Monte Carlo simulation. Taking into account these reasons, in the present article we mainly choose this model to pose the PDE formulation and develop its numerical solution with the proposed methods. Nevertheless, at the end of this section we also pose the PDEs for the models of Hagan and Rebonato.

In order to describe the SABR/LMM setting, we first consider a set of N-1 LIBOR forward rates F_i, 1≤ i ≤ N-1, 𝐅=(F_1,…,F_N-1) on the tenor structure [T_0,T_1, …, T_N-1,T_N], the accruals being τ_i = T_i+1-T_i. The Mercurio & Morini model is defined by the following system of stochastic differential equations <cit.>:

    dF_i(t)    = μ_i(t) F_i(t)^β dt + α_i V(t) F_i(t)^β dW^𝒬_i(t),    F_i(0) , 
    
     dV(t)    = σ V(t) dZ^𝒬(t),    V(0) = α,

which are posed on a probability space {Ω, ℱ, 𝒬} with filtration {ℱ_t}, t∈[T_0,T_N]. In (<ref>) μ_i is the drift of the i-th forward rate, β∈ [0,1] is the local volatility coefficient, α_i is a deterministic (constant) instantaneous volatility coefficient, W_i^𝒬 are standard Brownian motions under the risk neutral measure 𝒬, ρ is the correlation matrix between the forward rates, i.e.

    <dW_i^𝒬(t),dW_j^𝒬(t)> = ρ_ijdt,   ∀ i,j ∈{1,…,N-1},

V is the stochastic volatility of the forward rates, dZ^𝒬 is a standard Brownian motion correlated with the Brownian motions of the forward rates and ϕ is the correlation vector between the forward rates and the stochastic volatility, i.e.

    <dW_i^𝒬(t),dZ^𝒬(t)>=ϕ_i dt,   ∀ i ∈{1, …,N-1}.

Due to the fact that the volatility process is lognormal, one can set the initial value of the volatility equal to one, i.e. α = 1 with no loss of generality, since any different initial value can be embedded in the model by adjusting the deterministic coefficients α_i. This is the choice we adopt in the following.

The drifts of the forward rates are determined by the chosen numeraire. Under the terminal probability measure 𝒬^T_N associated with choosing the bond P(t,T_N) as numeraire, the drifts of the forwards rates are given by
  
    μ_i(t) = {[ -α_i V(t)^2 ∑_j=i+1^N-1τ_j F_j(t)^β1+τ_j F_j(t)ρ_ijα_j                                                                                                      i < N-1,;                                                       ;                                                      0                                                                                                      i = N-1. ].








In order to price T_a×(T_b-T_a) swaptions we will consider the probability measure 𝒬^T_a associated with choosing the bond P(t,T_a) as numeraire. In this case the drifts of the forward rates are given by
  
    μ_i(t) = {[                                                   0                                                                                                  i = a,; α_i V(t)^2 ∑_j=a+1^iτ_j F_j(t)^β1+τ_j F_j(t)ρ_ijα_j                                                                                                  i > a. ].




Our model for the correlation structure is taken from Rebonato <cit.>, who suggests the time independent function

    ρ_ij = ^-λ|T_i-T_j|.

This function reflects the fact that the correlation increases as the time between the forward rates expiry decreases, so that two consecutive forward rates influence each other more than a forward rate in many years time.

A European option is characterized by its payoff function G, which determines the amount G(T,𝐅(T),V(T)) its holder receives at time t=T. The arbitrage-free value of the option relative to a numeraire 𝒩 is then given by

    u(t,𝐅(t),V(t))=U(t,𝐅(t),V(t))N(t) = E^𝒬( G(T,𝐅(T),V(T))𝒩(T)| ℱ_t ).

Closed-form solutions based on (<ref>) are rarely available due to the multi-asset feature of most LIBOR derivatives. In the next paragraphs we sketch the derivation of the PDE formulation associated to the Mercurio & Morini model.

By using Itô's formula, see <cit.> for example, the stochastic differential equation for u is given by

    du(t,𝐅(t),V(t))  =   ∂ u∂ tdt + ∑_i=1^N-1∂ u∂ F_i(t) dF_i(t) + ∂ u∂ V(t) dV(t) +  
       12∑_i,j=1^N-1∂^2 u∂ F_i(t) ∂ F_j(t) dF_i(t) dF_j(t) + 12∂^2 u∂ V(t)^2 (dV(t))^2 + 
       ∑_i=1^N-1∂^2 u∂ F_i(t) ∂ V(t) dF_i(t) dV(t),

with box algebra <cit.>: 




     dt     dW^𝒬_i     dW^𝒬_j     dZ^𝒬 

dt       0     0               0               0 

 dW^𝒬_i     0     dt              ρ_ijdt     ϕ_i dt 

 dW^𝒬_j     0     ρ_ijdt     dt              ϕ_j dt 

 dZ^𝒬       0     ϕ_i dt       ϕ_j dt       dt 

.


The interpretation of the box algebra is the following. In an expansion to terms of order dt, as dt → 0 higher order terms such as (dt)^j are all negligible for j>0. For example, (dt)^2 is of order 0 as dt→0, which is denoted as (dt) (dt) ∼ 0. Similarly, cross terms such as (dt)(dW_i^𝒬) are negligible because the increment dW^𝒬_i is normally distributed with mean 0 and standard deviation (dt)^1/2 and so (dt)(dW^𝒬_i) has standard deviation (dt)^3/2 which tends to 0 as dt→0.

Substituting equations (<ref>) in (<ref>) and using the box algebra, we get

    du(t,𝐅(t),V(t)) =    ( ∂ u∂ t + ∑_i=1^N-1μ_i(t) F_i(t)^β∂ u∂ F_i(t) + 
       12∑_i,j=1^N-1α_i α_j V(t)^2 F_i(t)^β F_j(t)^βρ_ij∂^2 u∂ F_i(t) ∂ F_j(t) + 
       12σ^2 V(t)^2 ∂^2 u∂ V(t)^2 + ∑_i=1^N-1σ V(t)^2 α_i F_i(t)^βϕ_i ∂^2 u∂ F_i(t) ∂ V(t)) dt + 
       ∑_i=1^N-1α_i V(t) F_i(t)^β∂ u∂ F_i(t) dW_i^𝒬 + σ V(t) ∂ u∂ V(t) dZ^𝒬.


In order to comply with the no-arbitrage conditions and (<ref>), the process du(t,𝐅,V) has to be martingale under the measure 𝒬. Thus, to satisfy this requirement, the drift term dt in (<ref>) must be equal to zero. The same result could be directly obtained by applying Feynman-Kac theorem, see <cit.>. The final parabolic PDE takes the following form:

    ∂ u∂ t + 12σ^2 V^2 ∂^2 u∂ V^2 + 12 V^2 ∑_i,j=1^N-1ρ_ijα_i α_j F_i^β F_j^β∂^2 u∂ F_i ∂ F_j +        
    σ V^2 ∑_i=1^N-1ϕ_iα_i F_i^β∂^2 u∂ F_i ∂ V + ∑_i=1^N-1μ_i(t) F_i^β∂ u∂ F_i   =    0,

with the terminal condition given by the derivative payoff, 
    u(T,𝐅,V) = g(T,𝐅,V),
 on ℝ^N-1×ℝ. For simplicity of notation, we have used the relative payoff g(·) = G(·)𝒩(T). The derivative price at time t<T is given by 𝒩(t)u(t,𝐅(t),V(t)).

Analytic solutions for (<ref>) can be only found for suitable simple specifications of the functionals forms of the PDE and for straightforward boundary conditions (e.g. simple caplets without stochastic volatility, i.e. σ = 0, see Section <ref>).

Finally, we are going to present the PDE for Hagan model, which is defined by the following system of stochastic differential equations <cit.>:

    dF_i(t)    = μ^F_i(t) F_i(t)^β_i dt + V_i(t) F_i(t)^β_i dW^𝒬_i(t),    F_i(0) , 
    
     dV_i(t)    = μ^V_i(t) V_i(t) dt + σ_i V_i(t) dZ_i^𝒬(t),    V_i(0) = α_i,

with the associated correlations denoted by

    <dW_i^𝒬(t),dW_j^𝒬(t)>    = ρ_ijdt,  
    
     <dW_i^𝒬(t),dZ_j^𝒬(t)>    = ϕ_ijdt,  
    
     <dZ_i^𝒬(t),dZ_j^𝒬(t)>    = θ_ijdt.

The PDE for this model is obtained in the same way as previously with the Mercurio & Morini model, thus obtaining:

    ∂ u∂ t   + 12∑_i,j=1^N-1θ_ijσ_i V_i σ_j V_j ∂^2 u∂ V_i ∂ V_j + 12∑_i,j=1^N-1ρ_ij V_i F_i^β_i V_j F_j^β_j∂^2 u∂ F_i ∂ F_j +  
       ∑_i,j=1^N-1ϕ_ij V_i F_i^β_iσ_j V_j ∂^2 u∂ F_i ∂ V_j + ∑_i=1^N-1μ^F_i(t) F_i^β_i∂ u∂ F_i + ∑_i=1^N-1μ^V_i(t) V_i ∂ u∂ V_i= 0.


Rebonato model <cit.> is analogous to Hagan one, therefore its PDE will be also quite similar to (<ref>).




§ FINITE DIFFERENCES METHOD WITH FULL GRIDS
 

Hereafter, as we have motivated in the previous section, we are going to just focus on the PDE (<ref>) of the Mercurio & Morini model. This backward parabolic PDE must be supplemented with a terminal condition, which describes the value of the variable u at the final time T. Moreover, appropriate boundary conditions are required, which prescribe how the function u, or its derivatives, behave at the boundaries of the necessarily bounded computational domain.

We are going to define a (N+1)-dimensional mesh with the time sampled from today (time 0) to the final expiry of the option (time T) at M+1 points uniformly spaced by the time step Δ t = TM.

The variables representing the forward rates F = (F_1, …, F_N-1) and their stochastic volatility V, often referred as the “space variables” will be sampled at M_i + 1 (i=1,…,N-1) and S+1 points spaced by h_i = F_i^max-F_i^minM_i and h_v = V^max-V^minS, respectively.

Notice that while the choice of the range of the time variable is totally unambiguous, [0,T], an a priori choice must be made about which values of the space variables are too high or too low to be of interest, so far we will denote them by [F_i^min,F_i^max] and [V^min,V^max]. Selecting boundary values such that the option of interest is too deeply in or out-of-the money is a common and reasonable choice.

For a given mesh, each point is uniquely determined by the time level m (m=0,…,M), the index vector of the N-1 forward rates 𝐟 = (f_1,…,f_i, …, f_N-1) (f_i=0,…,M_i) and the stochastic volatility level v (v=0,…,S). We seek approximations of the solution at these mesh points, which will be denoted by 
    U^m_𝐟,v≈ u(mΔ t, (f_i h_i)_1≤ i ≤ N-1, v h_v).


It is natural for this PDE to be solved backwards in time. We approximate the time derivative by the time-forward approximation 
    ∂ u∂ t|_t=mΔ t, 𝐅=(f_i h_i)_1≤ i ≤ N-1, V = v h_v = ∂ u∂ t|_m,f,v≈U^m+1_𝐟,v - U^m_𝐟,vΔ t.


For the space derivatives we have chosen second-order approximations. We will write 𝐟_i±1 to mean the forward rates index vector (f_1,…,f_i±1,…,f_N-1) which corresponds to the forward rates point (f_1 h_1, …, (f_i ± 1) h_i, …, f_N-1 h_N-1).

The first derivatives are approximated by central differences:

    ∂ u∂ F_i|_m,f,v≈U^m_𝐟_i+1,v - U^m_𝐟_i-1,v2h_i.


The second derivatives are approximated by:

 
  * ∂^2 u∂ F_i^2|_m,f,v≈U^m_𝐟_i+1,v - 2 U^m_𝐟_i,v + U^m_𝐟_i-1,vh_i^2,
 
  * ∂^2 u∂ V^2|_m,f,v≈U^m_𝐟,v+1 - 2 U^m_𝐟,v + U^m_𝐟,v-1h_v^2.


The cross derivatives terms are approximated by:

 
  * For i≠ j, ∂^2 u∂ F_i ∂ F_j|_m,f,v≈U^m_𝐟_i+1,j+1,v + U^m_𝐟_i-1,j-1,v - U^m_𝐟_i+1,j-1,v - U^m_𝐟_i-1,j+1,v4h_ih_j,
 
  * ∂^2 u∂ F_i ∂ V|_m,f,v≈U^m_𝐟_i+1,v+1 + U^m_𝐟_i-1,v-1 - U^m_𝐟_i+1,v-1 - U^m_𝐟_i-1,v+14h_ih_v.


The finite differences solution under the so-called θ-scheme is:

    U^m+1_𝐟,v - U^m_𝐟,vΔ t + θ W^m_𝐟,v + (1-θ) W^m+1_𝐟,v=0,
 where θ∈ [0,1] and W^m_𝐟,v is the discretization given by

    W^m_f,v =    12σ^2 V^2 U^m_𝐟,v+1 - 2 U^m_𝐟,v + U^m_𝐟,v-1h_v^2 + 
       12V^2 ∑_i,j=1^N-1_i≠ jρ_ijα_i α_j F_i^β F_j^βU^m_𝐟_i+1,j+1,v + U^m_𝐟_i-1,j-1,v - U^m_𝐟_i+1,j-1,v - U^m_𝐟_i-1,j+1,v4h_ih_j + 
       12 V^2 ∑_i=1^N-1α_i^2 F_i^2βU^m_𝐟_i+1,v - 2 U^m_𝐟_i,v + U^m_𝐟_i-1,vh_i^2 + 
       σ V^2 ∑_i=1^N-1ϕ_i α_i F_i^βU^m_𝐟_i+1,v+1 + U^m_𝐟_i-1,v-1 - U^m_𝐟_i+1,v-1 - U^m_𝐟_i-1,v+14h_i h_v +
       ∑_i=1^N-1μ_i(m Δ t) F_i^βU^m_𝐟_i+1,v - U^m_𝐟_i-1,v2h_i,

and with terminal condition U^M_𝐟,v = g(T,𝐅,V), where we have denoted 𝐅 = (F_i = f_i h_i)_1 ≤ i ≤ N-1 and V = v  h_v.

Three different θ values represent three canonical discretization schemes, θ=0 is the explicit scheme, θ=1 the fully implicit scheme and θ = 0.5 the Crank-Nicolson scheme. The fully implicit discretization is the best method with respect to stability, whereas the Crank-Nicolson timestepping provides the best convergence rate. Although the explicit method is the simplest to implement, it has the disadvantage of not being unconditionally stable.

We shall first discriminate explicit and implicit parts as follows:

    U_𝐟,v^mΔ t - θ W_𝐟,v^m = U_𝐟,v^m+1Δ t + (1-θ) W_𝐟,v^m+1 .


As a result of such discretization we arrive to the linear system of equations 𝐀𝐱 = 𝐛, where 𝐀 is the band matrix of known coefficients, 𝐱 is the vector of the unknown solutions U^m_𝐟,v and 𝐛 is the vector of known values corresponding to the right-hand side of (<ref>).

Equation (<ref>) can be rewritten as:

    d θ U_𝐟,v-1^m + d θ U_𝐟,v+1^m + ∑_i=1^N-1 (b_i - r_i) θ U_𝐟_i-1,v^m + ∑_i=1^N-1 (b_i + r_i) θ U_𝐟_i+1,v^m + 
       ∑_i=1^N-1(a_i θ U_𝐟_i-1,v-1^m + a_i θ U_𝐟_i+1,v+1^m - a_i θ U_𝐟_i-1,v+1^m - a_i θ U_𝐟_i+1,v-1^m) + 
       ∑_ij ∈ C( ψ_ijθ U_𝐟_i-1,j-1,v^m + ψ_ijθ U_𝐟_i+1,j+1,v^m - ψ_ijθ U_𝐟_i-1,j+1,v^m - ψ_ijθ U_𝐟_i+1,j-1,v^m ) + 
       ( -1 - 2 d θ - 2 θ∑_i=1^N-1b_i ) U_𝐟,v^m = 
        - d θ̂U_𝐟,v-1^m+1 - d θ̂U_𝐟,v+1^m+1 - ∑_i=1^N-1 (b_i - r_i) θ̂U_𝐟_i-1,v^m+1 - ∑_i=1^N-1 (b_i + r_i) θ̂U_𝐟_i+1,v^m+1
        - ∑_i=1^N-1(a_i θ̂U_𝐟_i-1,v-1^m + a_i θ̂U_𝐟_i+1,v+1^m+1 - a_i θ̂U_𝐟_i-1,v+1^m+1 - a_i θ̂U_𝐟_i+1,v-1^m+1)  
        -∑_ij ∈ C( ψ_ijθ̂U_𝐟_i-1,j-1,v^m+1 + ψ_ijθ̂U_𝐟_i+1,j+1,v^m+1 - ψ_ijθ̂U_𝐟_i-1,j+1,v^m+1 - ψ_ijθ̂U_𝐟_i+1,j-1,v^m+1) + 
       ( -1 + 2 d θ̂+ 2 θ̂∑_i=1^N-1b_i ) U_𝐟,v^m+1,

where θ̂= (1-θ), C is the set containing the combinations of numbers 1,2,…,N-1 taken two at a time without repetition (the number of elements in C is N-1  2 = 2^-1(N-1)(N-2)) and the known coefficients d, b_i, r_i, a_i and ψ_ij are defined as

    d = Δ tσ^2V^22h_v^2,    b_i = Δ t V^2 α_i^2 F_i^2β2h_i^2, 
        r_i = Δ t μ_i(t)F_i^β2h_i,    a_i = Δ t σ V^2 ϕ_i α_i F_i^β4h_ih_v, 
       ψ_ij = Δ t V^2 ρ_ijα_iα_j F_i^β F_j^β4h_ih_j.





 §.§ Boundary conditions
 

In order to specify boundary conditions, a combination of mathematical, financial and heuristic reasoning allows us to find consistent and acceptable ones. There are several possibilities, see <cit.> for example.

 We assume that forward rates and their stochastic volatility are non negative and hence take values in the range zero to infinity. We first truncate the unbounded interval to a bounded one and then we must specify conditions at the new boundary. Thus we will consider the truncated domain [F_i^min, F_i^max]×[V^min,V^max], with F_i^min = 0 and V^min=0.

For the forward rates we consider Dirichlet boundary conditions. Particularly, the terminal condition holds on the forward rates boundaries, i.e. 
    U^m_{𝐟 | ∃ f_i = 0 },v = U^M_𝐟,v,   ∀ m=0,…,M-1,
 
    U^m_{𝐟 | ∃ f_i = M_i },v = U^M_𝐟,v,   ∀ m=0,…,M-1.


At the stochastic volatility boundaries we consider the following conditions:

    ∂ u∂ t + ∑_i=1^N-1μ_i(t) F_i^β∂ u∂ F_i = 0,    V=0, 
       ∂ u∂ V = 0,    V=V_max.

Thus, when V=0 we require that the PDE itself must be satisfied on this boundary. When V approaches to infinity, the price of the derivative becomes independent of V. This is reflected by using Neumann conditions instead of the Dirichlet ones used for the forward rates boundaries.

At the boundary V=0, after discretizing the boundary condition (<ref>) we obtain (note that the coefficients d, b_i, a_i and ψ_ij of equation (<ref>) are zero):

    -∑_i=1^N-1r_i θ U^m_𝐟_i-1,0 + ∑_i=1^N-1r_i θ U^m_𝐟_i+1,0 - U^m_𝐟,0 = ∑_i=1^N-1r_i θ̂U^m+1_𝐟_i-1,0 - ∑_i=1^N-1r_i θ̂U^m+1_𝐟_i+1,0 + U^m+1_𝐟,0.


For the boundary V = V_max in order to maintain the second order accuracy in the discretization of the first derivative the ghost point method is considered. The ghost grid points U_𝐟,S+1 are added. Then, the finite differences scheme of equation (<ref>) can also be applied at the points U_𝐟,S. However, we now have more unknowns than equations. The additional equations come from the central finite differences discretization of the Neumann boundary condition (<ref>):

    U_𝐟,S+1-U_𝐟,S-12h_v = 0,
 which yields U_𝐟,S+1 = U_𝐟,S-1. Inserting this into the finite differences equation at V=V_max we achieve

    d̂θ U^m_𝐟,S-1 + ∑_i=1^N-1 (b_i-r_i)θ U^m_𝐟_i-1,S + ∑_i=1^N-1 (b_i+r_i)θ U^m_𝐟_i+1,S + 
       ∑_ij ∈ C( ψ_ijθ U_𝐟_i-1,j-1,S^m + ψ_ijθ U_𝐟_i+1,j+1,S^m - ψ_ijθ U_𝐟_i-1,j+1,S^m - ψ_ijθ U_𝐟_i+1,j-1,S^m ) + 
       ( -1 - d̂θ - 2 θ∑_i=1^N-1b_i ) U_𝐟,S^m = 
        -d̂θ̂U^m+1_𝐟,S-1 - ∑_i=1^N-1 (b_i-r_i)θ̂U^m+1_𝐟_i-1,S - ∑_i=1^N-1 (b_i+r_i)θ̂U^m+1_𝐟_i+1,S + 
        -∑_ij ∈ C( ψ_ijθ̂U_𝐟_i-1,j-1,S^m+1 + ψ_ijθ̂U_𝐟_i+1,j+1,S^m+1 - ψ_ijθ̂U_𝐟_i-1,j+1,S^m+1 - ψ_ijθ̂U_𝐟_i+1,j-1,S^m+1) + 
       ( -1 + d̂θ̂+ 2 θ̂∑_i=1^N-1b_i ) U_𝐟,S^m+1,

where d̂ = 2 d = Δ t σ^2 V_max^2h_v^2.



 §.§ Numerical results
 

It is not clear where to place F_i^max and V^max. On one hand, it is advantageous to place them far away of the initial forward rates. This reduces the error of the artificial boundary conditions. On the other hand a large computational domain requires a large discretization width. This increases the error of the approximation of the derivatives. In our experiments we will consider F_i^max = 0.1 and V^max = 3.5.

Some specifications of the financial product are given in Table <ref> and the employed market data, taken from <cit.>, are shown in Table <ref>. We will consider λ = 0.1 in the model for the correlation structure (<ref>). Besides, the Crank-Nicolson scheme will be used in (<ref>). For solving the system (<ref>) the Gauss-Seidel iterative solver has been employed using a tolerance of 10^-6.

The numerical experiments have been performed with the following hardware and software configurations: two recent multicore Intel Xeon CPUs E5-2620 v2 clocked at 2.10 GHz (6 cores per socket) with 62 GBytes of RAM, CentOS Linux, GNU C++ compiler 4.8.2.






We are going to value T_a × (T_b - T_a) European swaptions under the Mercurio & Morini model.

First of all, the results when pricing a 1× 1 European swaption are discussed. The value of this particular swaption is the same as the price of the corresponding caplet, and so it only depends on F_1. Hence, in one dimension a closed form expression for the price of a European swaption can be found by using Black's formula (<ref>) and is given by

    P(T_0,T_2) τ_1 (K, F_1(T_0),σ_Black√(T_1-T_0)).

This value is equal to 0.659096 basis points (one basis point is one hundredth of one percent, 1%/100=10^-4). As Black's formula for caplets considers constant volatility σ_Black, in this first test the volatility of the volatility parameter of Mercurio & Morini model is considered equal to zero, i.e., σ = 0, therefore a standard LIBOR market model is used. The solution was found on several levels and Table <ref> shows the convergence of the model when using 256 time steps. In all tables of this article, Level refers to the refinement level n, i.e., the mesh size is h_i = 2^-n· c_i in each coordinate direction, where c_i denotes the computational domain length in direction i, which is F_i^max in the case of the forward rates and V^max in the case of the stochastic volatility. Besides, the solution and the error with respect to the exact solution are also shown in basis points. Additionally, the execution time is measured in seconds and the column labeled as Grid points shows the number of grid points employed in the full grid used by the finite differences method without taking into account the time coordinate. The time discretization error decreases as increasing the number of time steps. As an example, in Table <ref> we show the achieved solutions and the associated errors using only 12 time steps for the higher resolution levels (n=9,  10), these results are to be compared with those of Table <ref>. From the results in these two Tables <ref> and <ref>, we can see that not only the time step Δ t = 1/256 is small enough but also the reported errors are dominated by the spatial discretization, at least up to level 10. Therefore, in the following numerical tests we will consider 256 time steps.

When the volatility of the volatility σ of the model is non zero or when the length of the underlying swap of the swaption being considered is greater than one, no closed form solutions are available. However, an estimate can be obtained from Monte Carlo simulations. On Table <ref> Monte Carlo values for the 1 × 1 European swaption with σ=0 are shown for several numbers of paths (#Paths). More details about Monte Carlo simulation of SABR/LMMs can be found in the article <cit.>.








In Table <ref> the pricing of the 1 × 1 European swaption with σ=0.3  for different resolution levels n are shown. In Table <ref> the results for the 1 × 2 swaption are given. Note that with this numerical method it was not feasible to price the swaption past refinement level n=8 due to the huge number of required grid points. In Table <ref> the results for the 1 × 3 swaption are given. Full grid pricing is only possible on low grid levels. It is not achievable to obtain a solution for a level greater than 6 in reasonable computational time due to the high number of grid points.







Theoretically, it is possible to solve the discrete system (<ref>) for a general number of dimensions. However, in computational science, a major problem occurs when the number of dimensions increases. A natural way to diminish the discretization error is to decrease the mesh step in each coordinate direction. However, then the number of grid points in the resulting full grid grows exponentially with the dimension, i.e. the size of the discrete solution increases drastically. This is called the curse of dimensionality <cit.>. Therefore, this procedure of improving the accuracy by decreasing the mesh step is mainly bounded by two factors, the storage and the computational complexity. Due to these limitations, using a full grid discretization method which achieves sufficiently accurate approximations is only possible for problems with up to three or four dimensions, even on the most powerful machines presently available <cit.>.




§ SPARSE GRIDS AND THE COMBINATION TECHNIQUE
 
Two approaches to try to overcome the curse of dimensionality are increasing the order of accuracy of the applied numerical approximation scheme or reducing the dimension of the problem by choosing suitable coordinates. Both approaches are not always possible for every option pricing problem. In this article we will take advantage of the sparse grid combination technique first introduced by Zenger and co-workers <cit.> in order to try to overcome the curse of dimensionality and allow to use the PDE formulation of SABR/LMM for the pricing problem we are dealing with. The combination technique replicates the structure of a so-called sparse grid by linearly combining solutions on coarser grids of the same dimensionality. This technique reduces the computational effort and the storage space involved with the mentioned traditional finite differences discretization methods. The number of sub-problems to solve increases, while the computational time per problem decreases drastically. This method can be implemented in parallel as each sub-grid is independent of the others. In the next two subsections we give a brief introduction to sparse grids and the combination technique. For a detailed discussion we refer to <cit.>.



 §.§ Sparse grids

First, we introduce some notations and definitions. Let 𝐥 = (l_1,l_2,…,l_d) ∈N_0^d denote a d-dimensional multi-index. Let |𝐥|_1 and |𝐥|_∞ denote the discrete L_1-norm and L_∞-norm of the multi-index 𝐥, respectively, that are defined as 
    |𝐥|_1 = ∑_k=1^d l_k      |𝐥|_∞ = max_1≤ k ≤ d l_k.
 We define the anisotropic grid Ω_𝐥 with mesh size 𝐡 = (h_1,h_2,…,h_d) = (2^-l_1 c_1, 2^-l_2 c_2, …, 2^-l_d c_d) with multi-index 𝐥 and grid length 𝐜 = (c_1,c_2,…,c_d).

Then, the full grid at refinement level n∈N and mesh size h_i = 2^-n· c_i for all i can be defined via the sequence of subgrids 
    Ω^n = Ω_(n,…,n) = ⋃_|𝐥|_∞≤ nΩ_𝐥.

Figure <ref> visualizes two dimensional full grids for levels n=0,…,4.


The number of grid points in each coordinate direction of the full grid is 2^n + 1 and therefore the number of grid nodes in the full grid increases with O(2^n · d), i.e. grows exponentially with the dimensionality d of the problem.

The sparse grid Ω_s^n at refinement level n consists of all anisotropic Cartesian grids Ω_𝐥, where the total sum of all refinement factors l_k in each coordinate direction equals the resolution n. Then, the sparse grid Ω_s^n is given by 
    Ω_s^n = ⋃_|𝐥|_1≤ nΩ_𝐥 = ⋃_|𝐥|_1= nΩ_𝐥.

Figure <ref> shows the two-dimensional grid hierarchy for levels n=0,…,4.




The total number of nodes in the grid Ω_𝐥 is ∏_k=1^d (2^l_k+1) = O(2^|𝐥|_1) = O(2^n). In addition, there exist exactly n+d-1  d-1 grids Ω_𝐥 with |𝐥|_1 = n,

    n+d-1  d-1   = (n+d-1)!(d-1)!n! = (n+d-1)·…·(n+1)n!(d-1)!n!
        = n+(d-1)d-1·n+(d-2)d-2·…·n+(d-(d-1))d-(d-1)
        = (1 + nd-1) ·(1 + nd-2) ·…·(1 + n2) ·(1 + n1) 
       ≤ (1+n)^d-1 = O(n^d-1).

Thus, the total number of grid points of the sparse grid Ω_s^n grows according to

    n+d-1  d-1·∏_k=1^d (2^l_k+1) = O(n^d-1) O(2^n) = O(n^d-1 2^n),

which is far less the size of the corresponding full grid with O(2^nd) grid points. Let h_n = 2^-n, therefore the sparse grid employs O(h_n^-1·log_2(h_n^-1)^d-1) grid points compared to O(h_n^-d) nodes in the full grid.


Bungartz and Griebel <cit.> show that the accuracy of the sparse grid using O(h_n^-1·log_2(h_n^-1)^d-1) nodes is of order O(h_n^2 log_2(h_n^-1)^d-1)) in the case of piecewise linear finite elements discretization and under the smoothness condition that the mixed derivatives are bounded. Thus, the accuracy of the sparse grid is only slightly deteriorated from the accuracy O(h_n^2) of conventional full grid methods which need O(h_n^-d) grid points. Therefore, sparse grids need much less points than regular full grids to achieve a similar approximation quality.

However, the structure of a sparse grid is more complicated than the one of a full grid. Common PDE solvers usually manage only full grid solutions. Existing sparse grid methods working directly in the hierarchical basis involve a challenging implementation <cit.>. This handicap can be circumvented with the help of the sparse grid combination technique which not only exploits the economical structure of the sparse grids but also allows for the use of traditional full grid PDE solvers.

Concerning finite differences, in <cit.> the authors obtain error bounds in terms of the Fourier transform coefficients when using a combination technique with a central difference scheme for the Laplace equation. The adaptive case with finite differences has been addressed in <cit.> for elliptic operators and the errors are obtained in L^2 and L^∞-norms under the assumption that the 2d-th mixed derivatives are bounded. In <cit.>, the smoothness condition is written in terms of Holder spaces, the continuity of the mixed derivatives and their associated semi-norm to be finite, also including more general finite differences schemes. More recently, in <cit.> a methodology to obtain error bounds for general finite differences schemes in any dimension is proposed. It is mainly based on an error correction scheme leading to an appropriate error expansion. The results are again based on the existence of bounded mixed derivatives of the solution in the L^∞ norm, which vanish at the boundary to avoid regularity problems. For this kind of functions, results about the expansion and interpolation are previously stated.

Concerning the smoothness of solution of the here treated PDEs, we note that the parabolic operator involves a smoothness effect (it is a kind of Black-Scholes operator in high spatial dimensions), so that the solution becomes C^∞ for t <T, and the usual payoff (final condition at t=T) is continuous. Note that sparse grids have been analyzed when applied to basket options in <cit.>, for example.

Finally, two and three dimensional sparse grids for several resolution levels n are shown in Figures <ref> and <ref>, respectively. Additionally, the growth of the grid points when increasing n can be observed.












 §.§ Combination technique


Similar to the Richardson extrapolation <cit.>, the so-called combination technique linearly combines the numerical solution on the sequence of anisotropic grids Ω_𝐥 where

    |𝐥|_1 = n-q,    q=0,…,d-1.


The combination technique reads

    U_s^n = ∑_q=0^d-1 (-1)^q ·d-1  q·∑_|𝐥|_1 = n-q U_𝐥,    l_k≥ 0,   ∀ k=1,…,d,

where U_𝐥 denotes the numerical solution on the grid Ω_𝐥 and  U_s^n the combined solution on the sparse grid Ω_s^n.

The grids employed by the combination technique of level n=4 in two dimensions are shown in Figure <ref>.


The idea of this technique is that the leading order errors from the dicretization on each grid cancel each other out in the combination solution.

The number of grid points involved in the approximation of U_s^n grows according to O(n^d-1· 2^n). In fact, from the formula (<ref>) we have to solve n+d-1 d-1 problems with O(2^n) unknowns, n+d-2  d-1 problems with O(2^n-1) unknowns, ... and n  d-1 problems with O(2^n-(d-1)) unknowns. This results in a total number of O(n^d-1· 2^n) grid points which is much less than the O(2^n· d) grid nodes used by traditional full grid methods. Thus, the efficient use of sparse grids greatly reduces the computing time and the storage requirements which allows for the treatment of problems with ten variables and even more <cit.>.

We have seen that the combination technique linearly combines the numerical solution on several traditional full grids. The solution can be calculated on each of these grids by using any existing PDE numerical method like finite differences, finite volume or finite elements. In addition, since all these sub-problems are independent the combination technique can be parallelized <cit.>.

The combination technique approach presumes the existence of a so-called error splitting. It requires for an associated numerical approximation method on the full grid Ω_𝐥 an error splitting of the form

    u(𝐱) - U_𝐥(𝐱) = ∑_k=1^d ∑_{j_1,…,j_k}
    ⊆{1,…,d} C_j_1,…,j_k(𝐱,h_j_1,…,h_j_k) ·  h^p_j_1·…· h^p_j_k,

at each grid point 𝐱∈Ω_𝐥. Here u denotes the exact solution of the partial differential equation under consideration, U_𝐥 the numerical solution on the grid Ω_𝐥, p>0 is the order of accuracy of the numerical approximation method with respect to each coordinate direction and the coefficient functions C_j_1,…,j_k of 𝐱 and the mesh sizes h_j_k, k=1,…,d are required to be bounded by a positive constant K such that 
    |C_j_1,…,j_k(𝐱,h_j_1,…,h_j_k)| ≤ K,   ∀ k, 1≤ k ≤ d,   ∀{j_1,…,j_m}⊆{1,…,d}.


The existence of the expansion (<ref>) is a crucial point to obtain the error bounds of the sparse grid recombination technique and usually requires the assumption of bounded mixed derivatives.

In <cit.> Griebel and Thurner showed that if the solution of the PDE is sufficiently smooth, the pointwise accuracy of the sparse grid combination technique is O(n^d-1· 2^-n· p) = O([log_2h_n^-1]^d-1 h_n^p), which is only slightly worse than O(2^-n· p) = O(h_n^p) obtained by the full grid solution.

The solution at points which do not belong to the sparse grid can be computed through interpolation. The applied interpolation method should provide at least the same order of accuracy of the numerical discretization scheme used to solve the PDE. Otherwise, the accuracy of the numerical scheme will be deteriorated.

Up to now we have assumed the existence of an error splitting of type (<ref>). However, such an error splitting has to be proven for each problem. Nevertheless, proving the existence of this error splitting is usually very complex. Bungartz et al. <cit.> showed the existence of such an error splitting for the finite differences discretization of the 2-d Laplace equation. Arciniega and Allen <cit.> proved the existence of this error splitting for the fully implicit as well as the Crank-Nicolson discretization scheme of the European call option. More recently, Reisinger <cit.> showed that such a splitting also holds for a wider class of linear PDEs, for example convection-diffusion equations. The author gives general conditions which need to be fulfilled to ensure the existence of the desired splitting structure: sufficiently smooth initial data and compatible boundary data, the existence of bounded mixed derivatives, a consistent numerical scheme which provides a truncation error of the desired splitting structure and stability of the discretization scheme. As a summary, we can say that the deduction of the error splitting formula is very complex and was until now only performed for some reference problems. However, we will see in the following Section <ref> that the numerical results for the sparse grid combination technique are promising, even for more complex financial products.



 §.§ Numerical results
 

Taking advantage of the previously described sparse grid combination technique, in this section we are pricing the same interest rate derivatives that have been valued in the former Section <ref> where traditional full grid finite differences methods were considered. In addition to those products, we are going to price interest rate derivatives with up to seven underlying LIBOR rates and their stochastic volatility, showing that the sparse grid combination technique is able to cope with the curse of dimensionality up to a certain extent. As in the previous Section <ref>, we will use Crank-Nicolson scheme, we will consider the Gauss-Seidel iterative solver and the same boundary conditions as in Section <ref>. In the present case, we are interested in the evaluation of the solution at a single point which corresponds with the value of the forward rates at time zero (see Table <ref>) and V(0)=1. The numerical solution on each grid handled by the combination technique is interpolated at this point using multilinear interpolation and then added up with the appropriate weights.

The sparse grid combination technique has been implemented to run on multicore CPUs. The program was optimized and parallelized using OpenMP <cit.>. CPU times, measured in seconds, correspond to executions using 24 threads, so as to take advantage of Intel Hyperthreading. The speedups of the parallel version with respect to the pure sequential code are around 16. To the best of our knowledge, graphic processor units (GPUs) are not well-suited to parallelize the combination technique, due to the fact that the different grids employed by the combination technique involve memory accesses patterns totally different, therefore, it is not possible to access the device memory in a coalesced way <cit.>, thus GPU global memory can not serve threads in parallel. In this scenario, the GPU code will be ill performing. In the work <cit.> the authors take advantage of GPUs to parallelize the solver of each full grid considered by the combination technique. However, they do not parallelize the combination technique itself.

In Table <ref> a 1× 1 European swaption is priced. The exact price of this derivative is 0.659096 basis points, as discussed in Section <ref>. These results are to be compared with those of Table <ref>, where it can be seen how the computational times and the grid points employed by the sparse grid combination technique have been substantially reduced.



Next, in Table <ref> a 1× 1 European swaption is priced considering stochastic volatility. These results are to be compared with those of Table <ref>.




In the following Tables <ref> and <ref>, the pricing of 1× 2 and 1× 3 European swaptions taking into account stochastic volatility is shown, as in Tables <ref> and <ref>, respectively. For the higher resolution levels, the full grid method became very slow, while the sparse grid combination technique results much faster. Note that the combination technique is able to price successfully the 1× 3 European swaption, this was not attainable in Table <ref>.








































Finally, in Tables from <ref> to <ref>, 1 × 4, ..., 1 × 7 European swaptions are priced considering stochastic volatility. The pricing of these interest rate derivatives was not viable with the full grid approach of Section <ref>. In order to be able to price derivatives with more than 8 underlyings, the combination technique method should be parallelized to run on a cluster of processors. In the Chapter 13 of the book <cit.>  Philipp Schröder et al. discuss the parallelization of the combination technique using MPI (Message Passing Interface) API. In <cit.> the authors parallelize the sparse grid combination technique taking advantage of a MapReduce framework, algorithms that are inherently fault tolerant.



§ CONCLUSION

In this work we have posed for first time in the literature the PDEs associated to the SABR/LIBOR market models proposed by Mercurio & Morini <cit.> and Hagan <cit.>. In order to price interest rate derivatives we have developed a traditional full grid finite difference method. This approach is able to successfully price derivatives up to two or three underlying forward rates in reasonable computational times. However, when the number of underlyings increases this scheme suffers from the well-known curse of dimensionality. In order to price derivatives over a moderately large number of forward rates we have proposed to use the sparse grid combination technique. Taking into account that this technique is embarrassingly parallel  we have parallelized it so as to drastically reduce computational times. Finally, we have tested the proposed method in two different ways. On one hand, using the analytical solution when it exists. On the other hand, when the exact solution is not known, we have used as reference solution the one computed with the Monte Carlo method, thus ensuring the correctness of the developed scheme.
















 
plain




 Paper Title: Optimal design problem with thermal radiation.







[mytitlenote]
K.K. is partially supported by JSPS KAKENHI Grant Numbers JP24K16955 and JP24KJ0010.
K.M. is partially supported by Mizuho Foundation for the Promotion of Sciences and JSPS KAKENHI Grant Numbers JP24K17191 and JP23H03413.
T.O. is partially supported by JSPS KAKENHI Grant Numbers
JP22K20331 and JP23K12997.
This work is also supported by the Research Institute for Mathematical Sciences, an International Joint Usage/Research Center located in Kyoto University.


mainaddress]Kosuke Kita
kosuke.kita.c3@tohoku.ac.jp
mymainaddress]Kei Matsushima
matsushima@mech.t.u-tokyo.ac.jp
mymainaddress2]Tomoyuki Okamycorrespondingauthor
[mycorrespondingauthor]Corresponding author
t-oka@fit.ac.jp
[mainaddress]Graduate School of Science, Tohoku University, Aoba-ku, Sendai 980-8578, Japan
[mymainaddress]Graduate School of Engineering, The University of Tokyo, Bunkyo-ku, Tokyo 113-8656,Japan
[mymainaddress2]Faculty of Engineering, Fukuoka Institute of Technology, Higashi-ku, Fukuoka 811-0295, Japan





 
Optimal design problem, 
Topology optimization, 
Nonlinear boundary condition, 
Thermal radiation, 
Homogenization, 
Level set method
[2020] Primary: 80M50; Secondary: 35J65, 80M40






§ ABSTRACT 
This paper is concerned with configurations of two-material thermal conductors that minimize the Dirichlet energy for steady-state diffusion equations with nonlinear boundary conditions described mainly by maximal monotone operators. 
To find such configurations, a homogenization theorem will be proved and applied to an existence theorem for minimizers of a relaxation problem whose minimum value is equivalent to an original design problem. 
As a typical example of nonlinear boundary conditions, thermal radiation boundary conditions will be the focus, and then the Fréchet derivative of the Dirichlet energy will be derived, which is used to estimate the minimum value.
Since optimal configurations of the relaxation problem involve the so-called grayscale domains that do not make sense in general, a perimeter constraint problem via the positive part of the level set function will be introduced as an approximation problem to avoid such domains, and moreover, the existence theorem for minimizers of the perimeter constraint problem will be proved.
In particular, it will also be proved that 
the limit of minimizers for the approximation problem becomes that of the relaxation problem in a specific case, and then candidates for minimizers of the approximation problem will be constructed by employing time-discrete versions of nonlinear diffusion equations. 
In this paper, it will be shown that optimized configurations deeply depend on force terms 
as a characteristic of nonlinear problems and will also be applied to real physical problems.








§ INTRODUCTION
 
Transfer of thermal energy can be classified into conduction, convection, and radiation. Generally, the first two are usually treated as linear Partial Differential Equations (PDEs)  as long as thermal conductivity, specific heat and heat source are independent of temperature. In the case of the steady-state, many problems determining distributions of composite materials such that energies (or objective functionals) are minimized have been studied in various fields such as mathematics, physics, engineering and computer science as optimal design problem (or shape and topology optimization problem).
Material distributions of two-material composites are represented by employing characteristic functions, and therefore, the optimal design problem is described by a minimization problem of functionals with respect to the characteristic function. 
To guarantee the existence of minimizers, homogenization theory plays a crucial role since the minimizing sequence of the characteristic functions usually oscillates (see, e.g., <cit.>).
Although there are no minimizers in general, it has been proved that there exists a pair of volume fractions (or densities) and homogenized coefficients that achieve the minimum value (see, e.g., <cit.>). 
Hence it is a crucial issue to construct volume fractions that are similar to characteristic functions and give values close to the minimum, and therefore, various numerical techniques that do not involve intermediate sets have been devised (see e.g., <cit.>).
As for the unsteady-state, the existence of minimizers for a true relaxation problem has been proved in <cit.> along with the result in <cit.> and clarified the long-time behavior of optimal configurations concerning volume fraction in <cit.>. 

On the other hand, thermal radiation depends on temperature due to Stefan–Boltzmann law (cf. <cit.>) and is stated as nonlinear PDEs (in particular, it is formulated by
PDEs with nonlinear boundary conditions). From a physical point of view, the Stefan–Boltzmann law implies that the intensity of electromagnetic radiation from a black body is nonlinearly dependent on its surface temperature. Thermal radiation is especially important in high-temperature environments or thermally isolated systems in terms of conduction and convection, e.g., satellites and spacecrafts in outer space <cit.>. 

Such potential applications motivate us to develop an optimal design method for nonlinear PDEs. As shape and topology optimization is widely used for enhancing static and dynamic properties of solid deformation, much effort has been devoted to shape and topology optimization for nonlinear elasticity, e.g., large deformation <cit.>, hyper-elasticity <cit.>, and elasto-plasticity <cit.>. Another important subject is the fluid dynamics with the aim of designing, e.g., aircraft wings <cit.>, microfluidic systems <cit.>, and heat-dissipating structures with natural convection <cit.>. See also <cit.> for optimal design or shape optimization problems of other nonlinear PDEs.


    Some earlier studies have presented shape-optimized designs of heat radiators. For example, Dems and Korycki calculated a shape derivative associated with steady radiative heat transfer and optimized some parameters representing the boundary of a heat radiator <cit.>. Transient heat transfer with radiation was also studied in <cit.>. Recent work by Liu and Hasegawa developed a level set-based method for shape optimization in radiative heat transfer <cit.>. However, rigorous mathematical theory has not yet been established for optimal design problems with thermal radiation, which is crucial to constructing a unified theory that involves radiation as a natural extension of linear problems such as conduction and convection. 




 §.§ Setting of the problem

In this paper, for a given composite material consisting of two materials with different diffusion coefficients, we consider the material distribution that minimizes the Dirichlet energy described by the solution to the steady-state diffusion equation with the maximal monotone operator to handle various thermal transfers in a unified way.
Let Ω be a bounded domain of ^d with 
Lipschitz 
boundary ∂Ω, d≥ 1 and
Ω_i⊂Ω (i=0,1) be such that Ω=Ω_0∩Ω_1 and Ω_0∩Ω_1=∅.
For α,β>0 such that β>α>0 and i,j∈{1,…,d}, the class of diffusion coefficients is defined by 

    ℳ(α,β):={A∈ [L^∞(Ω)]^d× d A_ij=A_ji and α|ξ|^2≤ A(x)ξ·ξ≤β|ξ|^2  for all ξ∈^d and a.e. x∈Ω}.

Let u_χ_Ω_1∈ K be a weak solution to 

    -(A_χ_Ω_1∇ u_χ_Ω_1)=f      in Ω,
    
    -A_χ_Ω_1∇ u_χ_Ω_1·ν∈β(u_χ_Ω_1)
          on ∂Ω,

where ν is the unit outward normal vector on ∂Ω, f∈(H^1(Ω))^∗, A_χ_Ω_1∈ℳ(α,β) is the matrix field depending on the characteristic function χ_Ω_1∈ L^∞(Ω;{0,1}) given as

    χ_Ω_1(x)=
        
            1,     x∈Ω_1, 
    
            0,     x∈Ω_0,

K is some closed convex subset of a Hilbert space (explained later), and β is a (possibly multi-valued) maximal monotone graph in × such that β(0)∋ 0. Thus β satisfies 


  (i) (Monotonicity)
Let G(β) be the graph of β, i.e., 
G(β)={[z,ξ]∈×ξ∈β(z)}. Then it holds that 

    ⟨ z-w,ξ-ζ⟩≥0   for all  [z,ξ], [w,ζ]∈ G(β).
    

  (ii) (Maximality)
Any monotone graph
F:→ 2^
whose graph G(F) involves G(β) coincides with β.

Since β is a maximal monotone graph on ℝ×ℝ, 
there exists a proper, convex and lower semicontinuous function j:ℝ→ (-∞,+∞] such that β=∂ j, where ∂ j denotes a subdifferential of j, i.e., for w∈ℝ,

    ∂ j(w):={ξ∈ℝ j(z)-j(w)≥ξ(z-w)  for all  z∈ℝ}.


Now, our target minimization problem is formulated as follows:

    inf_χ_Ω_1∈𝒞𝒟{ℰ(χ_Ω_1):=∫_Ω A_χ_Ω_1(x) ∇ u_χ_Ω_1(x)·∇ u_χ_Ω_1(x) x̣},

where 𝒞𝒟 denotes the classical design domain given as 

    𝒞𝒟={χ_Ω_1∈ L^∞(Ω;{0,1})χ_Ω_1_L^1(Ω)= γ |Ω|},

γ∈ (0,1) denotes the volume ratio, and |Ω| describes the Lebesgue measure of Ω⊂^d.

[Two-material composite with different isotropic diffusion coefficients]
Typically, 

    κ[χ_Ω_1]:=α(1-χ_Ω_1)+βχ_Ω_1

stands for the diffusion coefficient of a composite material consisting of Ω_0 with diffusion coefficient α>0 and Ω_1 with diffusion coefficient β>0, and κ[χ_Ω_1]𝕀 belongs to ℳ(α,β). Here 𝕀 is the identity matrix of ^d× d.



[Maximal monotone operator] 
By noting that every subdifferential operator is maximal monotone (see, e.g., <cit.>), the following thermal radiation boundary condition can be taken:
    
    β(u_χ_Ω_1)=σ|u_χ_Ω_1|^du_χ_Ω_1,

    where σ>0 stands for the Stefan–Boltzmann coefficient. 
    In particular, one can represent standard linear boundary conditions with maximal monotone graph β.
    Indeed, if β(w)≡ 0 (resp. β(w)=aw for some a>0), then the boundary condition of (<ref>) is nothing but the homogeneous Neumann boundary condition (resp. Robin boundary conditions).
    Moreover, setting
    
    β_ D (w)=
            {   ℝ,            w=0,
       ∅,            w≠ 0,
            .

    we can understand the boundary condition of (<ref>) as the homogeneous Dirichlet boundary condition with β=β_ D (for details see <cit.>).
    
    Hence describing the boundary conditions using the maximal monotone graph enables us to treat conduction, convection and radiation in a unified manner.






 §.§ Aims and plan of the paper

This paper aims to prove that there exist minimizers of a true relaxation problem for (<ref>) and
construct an approximated characteristic function concerning χ_Ω_1∈ L^∞(Ω;{0,1}) that achieves (<ref>) with (<ref>) and (<ref>) numerically as typical examples of optimal design problems with nonlinear boundary conditions.
To this end, we shall first discuss the existence of minimizers of (<ref>) by using the direct method. 
Thus we shall take a minimizing sequence (χ_Ω_1^n) in 𝒞𝒟, i.e.,

    lim_n→ +∞ℰ(χ_Ω_1^n)= inf_χ_Ω_1∈𝒞𝒟ℰ(χ_Ω_1).

Since 𝒞𝒟 involves functions that oscillate rapidly, we shall see that the minimization problem (<ref>) is ill-posed in general due to χ_Ω_1^n_k→θ weakly-∗ in L^∞(Ω) but θ∉𝒞𝒟 for some subsequence (n_k) and volume fraction θ∈ L^∞(Ω;[0,1]).
Hence showing that optimal volume fractions exist, we shall next consider how to numerically construct the optimal volume fraction θ∈ L^∞(Ω;[0,1]) with few intermediate set [0<θ<1]:={x∈Ω 0<θ(x)<1}.  
In this paper, we shall employ the method devised in <cit.> as one way to resolve the so-called grayscale problem, and
in particular, it will reveal numerically how the linearity and nonlinearity of the boundary conditions affect the material distribution. 
Furthermore, we shall present some applications of the proposed optimization algorithm and show that optimal design is superior to some physically reasonable but intuitive structures. 


This paper is composed of seven sections.
Before discussing the limit of (<ref>), the next section is devoted to proving the well-posedness of (<ref>) and a corresponding homogenization theorem. 
Section <ref> deals with the existence theorem for minimizers of a relaxation problem with respect to the volume fraction.
In particular, we shall prove that the minimum value of the relaxation problem coincides with that of the original design problem (<ref>) and discuss how to construct the volume fraction that achieves the minimum value numerically. In section <ref>, we shall consider a perimeter constraint problem via the positive part of the level set function as an approximate problem for the relaxation problem. In particular, the existence theorem for minimizers of the approximation problem will be proved, and then the energy convergence of minimizers with respect to the perturbation parameter (>0) will also be shown. Furthermore, we shall provide a numerical algorithm that does not raise the grayscale problem. 
Section 5 details the validity of the numerical calculations and the effect of the nonlinearity of boundary conditions for optimized configurations, and then numerical examples for thermal radiation problems as more physical settings will be presented in Section 6.
The final section concludes this paper. 



§ HOMOGENIZATION PROBLEM FOR (<REF>)

As a preliminary to discuss the existence of minimizers in (<ref>), we consider the homogenization problem for (<ref>) with A_χ_Ω_1 being replaced by A^n.
Here and henceforth, we only consider the single-valued case of β for simplicity. To define a weak solution to (<ref>),
we first prepare some notations.
For simplicity, we set V=H^1(Ω), A_χ_Ω_1=A and u_χ_Ω_1=u.
Let a closed convex subset K⊂ V be given by 
K={v∈ V  j(v)∈ L^1(∂Ω)}, and ⟨·,·⟩ denotes the V^*-V dual coupling.
We define some functionals

    a(u,v):= ∫_Ω A(x)∇ u(x)·∇ v(x) x̣       for all  u,v∈ V,
        J(u):= ∫_∂Ω j(u)(x) σ̣         for all  u∈ K.

In addition, we assume that there exist δ>0 and C_0∈ℝ such that

    J(v) ≥δv_L^2(∂Ω)^2 - C_0      for all  v∈ K.

The typical example of j:→ (-∞,∞] satisfying (<ref>) is j(w)=1r|w|^r (i.e., β(w)=|w|^r-2w) for r>1.
We first touch on an inequality, which plays an important role in dealing with nonlinear boundary conditions.


    Let Ω⊂ℝ^d be a bounded Lipschitz domain. Then
    there exists a constant C>0 such that 
    
    v_L^2(Ω)^2 ≤ C ( ∇ v _L^2(Ω)^2 + v_L^2(∂Ω)^2 )

    for all v∈ H^1(Ω). As a consequence, (∇ v _L^2(Ω)^2 + v_L^2(∂Ω)^2)^1/2 is a equivalent norm in H^1(Ω).


    Suppose that for any n∈ℕ there exist (v_n)⊂ H^1(Ω) such that 
    
    v_n_L^2(Ω)^2 > n ( ∇ v_n _L^2(Ω)^2 + v_n_L^2(∂Ω)^2 ).

    Setting w_n=v_nv_n_L^2(Ω)^-1, we have w_n_L^2(Ω)=1 and 
    
    ∇ w_n_L^2(Ω)^2 + w_n_L^2(∂Ω)^2 < 1/n.

    Hence, since Ω is bounded and (w_n) is bounded in H^1(Ω), we can deduce that there exists a subsequence of (w_n) (denoted by (w_n) again) and w∈ H^1(Ω) such that 
    
    w_n → w        H^1(Ω),
       w_n → w        L^2(Ω)  L^2(∂Ω).

    From (<ref>), we can see that w=0 a.e. on ∂Ω.
    Moreover, the lower semicontinuity leads ∇ w_L^2(Ω)≤lim inf_n→ +∞∇ w_n_L^2(Ω) = 0, which implies w is constant, especially w=0.
    However, this contradicts that w_L^2(Ω)=1.
    The equivalence of the norm follows from (<ref>) and the trace theorem. This completes the proof.




In the framework of the variational inequality,
the weak solution of (<ref>) is defined as follows:

[Weak solution of (<ref>)]
For given f∈ V^∗, a function u∈ K is said to be a weak solution of (<ref>) if and only if the following inequality holds:

    a(u,v-u) + J(v) - J(u) ≥⟨ f,v-u ⟩   for all  v∈ K.




[Weak form of (<ref>)]
    We note that the definition of weak solutions for (<ref>) is defined by employing the variational inequality (<ref>).
    From physical motivation, we deal exclusively with the case of β(u)=σ |u|^r-2u (r>1, σ>0). 
In this case, setting v=u±λφ∈ K for all φ∈ K and λ>0 and letting λ→ 0_+, we see by ⟨ J'(u), ϕ⟩_L^r(∂Ω)= ⟨σ|u|^r-2u, ϕ⟩_L^r(∂Ω) that 
 
    a(u,φ) + σ∫_∂Ω |u|^r-2u(x) φ(x) σ̣= 
        ⟨ f, φ⟩    φ∈ K.

Thus the weak solution u∈ K to (<ref>) satisfies (<ref>). In particular, by the monotonicity of β, uniqueness of solutions to (<ref>) also follows, and then 
 every weak solution of (<ref>) coincides with the solution of this weak form (<ref>).
The reason why weak solutions are defined by (<ref>) instead of (<ref>) will be explained in Remark <ref>. 





 §.§ Solvability of (<ref>)
 
Let us briefly touch on the existence and uniqueness of a weak solution to (<ref>).


There exists a unique weak solution to (<ref>).



We first seek a function satisfying the following minimizing problem: 

    For given f∈ V^∗, find u∈ K such that 
        E(u) = inf_v∈ K E(v),

where E V⊃ K→ (-∞,∞] is the functional defined by

    E(v) = 1/2 a(v,v) - ⟨ f,v ⟩ + J(v).

It is easy to see that E is convex, lower semicontinuous and E≢∞.
Moreover, it follows from (<ref>) and Lemma <ref> that

    lim_v∈ K, v_V→ +∞ E(v) = +∞,
 
which along with <cit.> ensures the existence of
u∈ K satisfying (<ref>).
Thus setting E_1 and E_2 as the first two terms and the third term of the right-hand side in (<ref>), respectively, we have E_1(u)+E_2(u)=inf_v∈ K(E_1+E_2), and hence, <cit.> yields

    ⟨ E_1'(u),v-u⟩+E_2(v)-E_2(u)≥ 0    for all  v∈ K,
 
which implies (<ref>). 
The uniqueness of weak solutions follows immediately from the strict convexity of E.
Indeed, let u_1 and u_2∈ K be two minimizers for (<ref>).
If u_1≠ u_2, then

    E(u_i) ≤ E(u_1+u_2/2) < 1/2{ E(u_1) + E(u_2) }     (i=1,2),

that is, E(u_1)<E(u_2) and E(u_2)<E(u_1).
This completes the proof.







 §.§ Homogenization problem for (<ref>)



As for the proof of the homogenization theorem for (<ref>) with A_χ_Ω_1 being replaced by A^n, the following notion developed by Murat-Tartar is very useful.  

[H-convergence, cf. <cit.>]
Let A∈ℳ(α,β).  
For n>0, a sequence A^n∈ℳ(α,β) H-converges to an element A_ hom∈ [L^∞(Ω)]^d× d (denoted by A^nH→ A_ hom) if and only if, for any ω⋐Ω and any f ∈ H^-1(ω), the weak solution u^n∈ H^1_0(ω) of

    -(A^n ∇ u^n)=f   in  H^-1(ω)

is such that 

    u^n   → u_ hom       weakly in   H^1_0(ω),
    
       A^n∇ u^n   → A_ hom∇ u_ hom       weakly in   [L^2(ω)]^d,

where u_ hom∈ H^1_0(ω) is the weak solution to

    -(A_ hom∇ u_ hom)=f   in  H^-1(ω).



Then we have the following 

Let u^n∈ K be the unique weak solution to

    -(A^n ∇ u^n)=f      in Ω,
    
    -A^n ∇ u^n·ν
    = β(u^n)
          on ∂Ω,

where A^n∈ℳ(α,β), f∈ V^* and β is a maximal monotone operator in .
Then there exist a (not relabeled) subsequence of (n), u_ hom∈ K, the homogenized matrix A_ hom∈ℳ(α',β') and β'>α'>0  such that  

    u^n   → u_ hom       weakly in   V,
    
       A^n∇ u^n   → A_ hom∇ u_ hom       weakly in   [L^2(Ω)]^d.

Moreover, u_ hom∈ K is a weak solution to the homogenized equation,

    -(A_ hom∇ u_ hom)=f      in Ω,
    
    -A_ hom∇ u_ hom·ν
    = β(u_ hom)
          on ∂Ω.




By (<ref>) and (<ref>), it holds that

    a^n(u^n,u^n) + δu^n_L^2(∂Ω)^2 ≤ C_0 + ⟨ f,u^n ⟩,

which along with Lemma <ref> yields u^n_V≤ C. 
By virtue of the boundedness of (u^n)_n>0 in K, there exist a (not relabeled) subsequence of (n) and u_ hom∈ V such that

    u^n → u_ hom     V,
       u^n → u_ hom     L^2(Ω)∩ L^2(∂Ω).

Furthermore, by the H-compactness <cit.>, A^nH→ A_ hom also holds.
In particular, since it is obvious that u^n±φ∈ K for any φ∈ H^1_0(Ω), by choosing v=u^n±φ in (<ref>), it follows that

    a^n(u^n,φ) = ⟨ f,φ⟩     for all φ∈ H^1_0(Ω),

that is, 

    -div( A^n ∇ u^n ) = f      H^-1(Ω).

Hence applying <cit.> to (<ref>), we obtain

    A^n∇ u^n → A_ hom∇ u_ hom     [L^2(Ω)]^d,
 

    ∫_Ω A^n(x)∇ u^n(x)·∇ u^n(x) φ(x) x̣→∫_Ω A_ hom(x)∇ u_ hom(x)·∇ u_ hom(x) φ(x)  x̣

for all φ∈ C_ c^∞(Ω). 
Choosing φ∈ C_ c^∞(Ω) such that 0 ≤φ≤ 1 in (<ref>), 
we can deduce that

    ∫_Ω A_ hom(x)∇ u_ hom(x)·∇ u_ hom(x) φ(x) x̣≤lim inf_n→ +∞ a^n(u^n,u^n)   for all  φ∈ C_ c^∞(Ω),

whence follows 
the lower bound inequality,

    a_ hom(u_ hom,u_ hom) 
     =
     sup_φ∈ C_ c^∞(Ω),
    φ_L^∞(Ω)≤ 1∫_ΩA_ hom(x)∇ u_ hom(x)·∇ u_ hom(x)φ(x) x̣≤lim inf_n→ +∞ a^n(u^n,u^n)

(see <cit.>). Here we 
note that the upper bound inequality is more delicate (see Remark <ref> below).
Since K is closed convex subset in V, u_ hom∈ K holds.
Therefore the weakly lower semicontinuous on L^2(∂Ω) of J ensures that

    J(u_ hom) ≤lim inf_n→ +∞ J(u^n),

which together with the definition of weak solutions, (<ref>), (<ref>) and (<ref>) yields

    a_ hom(u_ hom,v-u_ hom) + J(v) - J(u_ hom) ≥⟨ f,v-u_ hom⟩   for all  v∈ K.

Thus u_ hom∈ K turns out to be a weak solution to the homogenized equation.
This completes the proof.


[Energy convergence]
Under the usual definition via the weak form, the following weak convergence will be required in the proof of Theorem <ref>: 

    β(u^n)→β(u_ hom)
       weakly in  L^ℓ(∂Ω)

for some 1<ℓ<+∞ (e.g., ℓ=r/(r-1) with β(w)=|w|^r-2w).
However, this proof is more complicated in general.
Thus Definition <ref> is more reasonable than the usual definition using the weak form (<ref>).
Moreover, we note that even if the homogeneous Dirichlet boundary condition, the upper bound inequality,

    lim sup_n→ +∞ a^n(u^n,u^n)
    ≤
    a_ hom(u_ hom,u_ hom)

is derived with the aid of the weak form for the homogenized equation. 
On the other hand, as soon as u_ hom turns out to be a weak solution to the homogenized equation satisfying the weak form
(for instance, 
it suffices to assume that 

    lim_λ→ 0_+J(w+λ v)-J(w)/λ=∫_∂Ω∂ j(w)(x)v(x) σ̣   for all  v∈ K),

we readily obtain (<ref>) by noting that 

    lim sup_n→ +∞   a^n(u^n,u^n)
       =
    lim_n→ +∞⟨ f,u^n⟩- lim inf_n→ +∞ J(u^n)
       ≤⟨ f,u_ hom⟩ - J(u_ hom)
    =  a_ hom(u_ hom,u_ hom).

Therefore convergence of the energy ℰ(χ_Ω_1^n) in (<ref>) can be obtained, which is applied to the proof of Theorem <ref> below.  


[Qualitative properties of homogenized matrices]
It is noteworthy that the homogenized matrix A_ hom∈ [L^∞(Ω)]^d× d is completely unaffected by (nonlinear) boundary conditions, and then it is also characterized exactly as in the case of the homogeneous Dirichlet boundary condition. 
Hence, in case d=1, it can be written as the so-called harmonic mean. Conversely (i.e., d≥ 2), it cannot be written explicitly in general. 
However, since A is symmetric, the following upper and lower bounds can be obtained:

    Aξ·ξ≤ A_ homξ·ξ≤Aξ·ξ   for all ξ∈^d,

where A is the inverse of the weak limit of A^-1 and A is the weak limit of A (see, e.g., <cit.>).





§ RELAXATION PROBLEM FOR (<REF>)


This section is devoted to proving the existence theorem for minimizers of a relaxation problem (see Theorem <ref> below). Thanks to Theorem <ref>, most of the proof relies on the results in <cit.>; however, we shall show it for completeness and the reader's convenience. 
Furthermore, we shall describe how to construct a candidate for the minimizers of the relaxation problem numerically. 
In what follows, we set A_χ_Ω_1=κ[χ_Ω_1] as in (<ref>) and write κ[χ_Ω_1]=κ and χ_Ω_1=χ for simplicity.




 §.§ Existence theorem for minimizers

Since the limit of the minimizing sequence does not belong to 𝒞𝒟 in general, we consider the following relaxation problem: 

    inf_(θ, κ_ hom)∈ℛ𝒟ℰ_ hom(θ, κ_ hom),

where ℛ𝒟 is the relaxed design domain defined by 

    ℛ𝒟
    := {
    (θ, κ_ hom)∈ L^∞(Ω;[0,1]×^d× d)   there exists 
    (χ^n,κ^n)∈  L^∞(Ω;{0,1}×{α,β})
       such that κ^n=α(1-χ^n)+βχ^n,
       χ^n→θ weakly-∗ in  L^∞(Ω;[0,1]),
       κ^n 𝕀H→κ_ hom and θ_L^1(Ω)= γ |Ω|
    },

ℰ_ hom:L^∞(Ω;[0,1]×^d× d)→ is the relaxed Dirichlet energy given as 

    ℰ_ hom(θ,κ_ hom)
    =∫_Ωκ_ hom(x)∇ u_ hom(x)·∇ u_ hom(x)  dx

and u_ hom∈ K is a weak solution to the homogenized equation (<ref>) with A_ hom=κ_ hom.

Then we see that the relaxation problem (<ref>) is a true relaxation of the original design problem (<ref>) in the following sense:

Let u^n∈ K be a weak solution to (<ref>) satisfying the weak from.
Let u_ hom∈ K be a weak solution to the homogenized equation
(<ref>). 
Then there exists at least one minimizer of (<ref>).
Furthermore, it holds that

    inf_χ∈𝒞𝒟ℰ(χ)
    =
    min_(θ,κ_ hom)∈ℛ𝒟ℰ_ hom(θ, κ_ hom),

and every minimizer of (<ref>) is characterized as a limit of the minimizing sequence in (<ref>), that is, for any minimizing sequence (χ^n) in 𝒞𝒟, there exist a (not relabeled ) subsequence of (n) and (θ^∗,κ_ hom^∗)∈ L^∞(Ω;[0,1]×^d× d) such that

    χ^n→θ^∗ weakly-∗ in  L^∞(Ω;[0,1]),  κ^n𝕀H→κ_ hom^∗

and

    inf_χ∈𝒞𝒟ℰ(χ)
    =
    ℰ_ hom(θ^∗,κ_ hom^∗).

Conversely, every minimizer in (<ref>) is attained by a limit of the minimizing sequence (χ^n) in  (<ref>).



Let (χ^n) be a minimizing sequence in 𝒞𝒟 for (<ref>).
Due to χ^n_L^∞(Ω)≤ 1 and Theorem <ref>, there exist
a (not relabeled) subsequence of (n) and (θ^∗, κ_ hom^∗)∈ L^∞(Ω;[0,1]×^d× d) such that (<ref>)
and 

    f
    ∫_Ωθ^∗(x) x̣=lim_n→ +∞∫_Ωχ^n(x) x̣=γ|Ω|.

Moreover, one can derive by Remark <ref> that

    inf_χ∈𝒞𝒟ℰ(χ)
        =lim_n→ +∞ℰ(χ^n)   =ℰ_ hom (θ^∗,κ_ hom^∗).

In particular, the above continuity (<ref>) is valid for non-minimizing sequences.

Now, we show that (θ^∗,κ_ hom^∗)∈ℛ𝒟 is a minimizer of (<ref>). For any (θ,κ_ hom)∈ℛ𝒟, there exists (χ^n,κ^n)∈ L^∞(Ω;{0,1}×{α,β}) such that

    χ^n→θ   weakly-∗ in  L^∞(Ω;[0,1])   and   κ^n 𝕀H→κ_ hom.

In particular, we can construct the sequence (χ^n) in L^∞(Ω;{0,1}) such that χ^n_L^1(Ω)=γ|Ω|, i.e., χ^n∈𝒞𝒟. 
Indeed, 
let χ̂^n∈ L^∞(Ω;{0,1}) be such that χ̂^n_L^1(Ω)=γ|Ω|.
Combining χ̂^n_L^∞(Ω)≤ 1 with

    lim_n→∞∫_Ωχ̂^n(x) x̣
    =
    γ|Ω|=
    ∫_Ωθ(x) x̣,

we have

    χ̂^n→θ   weakly-∗ in  L^∞(Ω).

Then defining Ω^n⊂Ω by 

    Ω^n={ x∈Ωχ^n(x)≠χ̂^n(x)  for a.e.  x∈Ω},

one obtain |Ω^n|→ 0 as n→ +∞, which along with   
the locality of the H-convergence (see  <cit.>) yields κ[χ̂^n]𝕀H→κ_ hom, and therefore, (χ̃^n) turns out to be the desired sequence.
Hence the continuity of ℰ and (<ref>) ensure that

    ℰ_ hom(θ,κ_ hom)=lim_n→ +∞ℰ(χ^n)≥inf_χ∈𝒞𝒟ℰ(χ),

which along with (<ref>) yields (<ref>) and (<ref>).  

On the other hand, let (θ,κ_ hom)∈ℛ𝒟 be a minimizer of (<ref>).
As already mentioned the above, (<ref>) follows for some χ^n∈𝒞𝒟 and some κ^n∈ L^∞(Ω;{α,β}) such that κ^n=α(1-χ^n)+βχ^n, and moreover,
ℰ_ hom(θ,κ_ hom)=lim_n→ +∞ℰ(χ^n) also holds, which implies that (χ^n) is a minimizing sequence in (<ref>). 
This completes the proof. 


[Interpretation of Theorem <ref>]
We note that the following facts:


  (i)
Theorem <ref> asserts that at least one minimizer exists in the relaxation problem (<ref>), which gives the same minimum value as the original design problem (<ref>). However, there is no guarantee for the uniqueness of minimizers.
In particular, if we further add geometric constraints (see, e.g., <cit.> for perimeter constraints) such that χ^n converges some characteristic function a.e. in Ω, the above proof ensures the existence of minimizers of the original design problem (<ref>) (with geometric constraints). 


  (ii)
As for the relaxation problem (<ref>), the volume fraction θ∈ L^∞(Ω;[0,1]) takes a value other than {0,1} a.e. in Ω.
Thus there are intermediate sets that are neither the material with diffusion coefficient α>0 (i.e., Ω_0⊂Ω) nor the material with diffusion coefficient β>0 (i.e., Ω_1⊂Ω), and optimal volume fractions are characterized by using intermediate sets.
In terms of the original design problem (<ref>), it is necessary to construct (θ,κ_ hom)∈ℛ𝒟 that attains the value close to the minimum in (<ref>) such that the so-called gray-scale problem is rarely raised. 









 §.§ Numerical algorithm for optimization of volume fractions

In this section, we describe a method to construct candidates for optimal volume fractions in (<ref>) numerically such that the minimum value of (<ref>) is achieved.
In the rest of this paper, 
we adapt ·_V=(∇·_L^2(Ω)^2 + ·_L^2(∂Ω)^2)^1/2 which is a norm in V=H^1(Ω) equivalent to the usual one, and then
we set β(w)=σ|w|^r-2w (i.e., j(w)=σ/r|w|^r), r≥ 2 and write
κ_ hom=κ, u_ hom=u and  
ℰ_ hom(θ,κ_ hom)=ℰ_ hom(κ) for simplicity.
In addition, we assume that f∈ L^2(Ω;_+) to get the following 

Let u∈ K be a unique weak solution to the (homogenized) state equation,

    -(κ∇ u)=f      in Ω,
    
    -κ∇ u·ν=σ|u|^r-2u       on ∂Ω.

Suppose that f≥ 0, ≢0. Then it holds that u≥ 0 a.e. in Ω.



Multiplying u^-:=max(-u,0) by (<ref>) and using integration by parts, we can derive that 

    0 ≤∫_Ω f(x)u^-(x) x̣   = ∫_Ωκ(x) ∇ u(x) ·∇ u^-(x) x̣ + σ∫_∂Ω |u|^r-2 u(x) u^-(x) σ̣[0]
       =∫_{u≤ 0}κ(x) ∇ u(x) ·∇ (-u)(x) x̣ + σ∫_u≤ 0 |u|^r-2 u(x) (-u)(x) σ̣[0]
       = - ∫_{u≤ 0}κ(x) ∇ (-u)(x) ·∇ (-u)(x)  x̣ - σ∫_u≤ 0 |u(x)|^r-2 (-u(x))^2 σ̣[0]
       ≤ - α' ∇ u^-_L^2(Ω)^2 - σ∫_∂Ω |u(x)|^r-2|u^-(x)|^2 σ̣,

which implies
∇ u^- =  u^-|_∂Ω = 0,
and therefore, u≥ 0 a.e. in Ω. This completes the proof.


Thanks to Lemma <ref>, |u|^r-2u can be described as u^r-1 below. 
We first derive the Fréchet derivative of ℰ_ hom:ℳ(α',β')→.

 Let u∈ K be a nonnegative weak solution to (<ref>) and let v∈ K be a weak solution to the (homogenized) adjoint equation,

    -(κ∇ v)=f      in Ω,
    
    -κ∇ v·ν=σ((r-1)u^r-2v+ru^r-1)       on ∂Ω.

Then ℰ_ hom is differentiable at κ, and it holds that

    ⟨ℰ_ hom'(κ),h⟩_[L^∞(Ω)]^d× d
    =
    -∫_Ωh(x)∇ u(x)·∇ v(x) x̣

for any h∈ [L^∞(Ω)]^d× d. 



Define ℒ:ℳ(α',β')× K× K → by

    ℒ(κ,u,w_κ)
       =
    ∫_Ωκ(x)∇ u(x)·∇ w_κ(x) x̣
    -
    ∫_∂Ωσu^r-1(x)(u(x)-w_κ(x)) σ̣+
    ∫_Ω f(x)(u(x)-w_κ(x)) x̣,

where ℳ(α',β')∋κ↦ w_κ∈ K is a differentiable at κ. 
Note that, for any h∈ [L^∞(Ω)]^d× d,

    ⟨ℒ'(κ,u, w_κ),h⟩_[L^∞(Ω)]^d× d
    =
    ⟨∂_κℒ(κ,u,w_κ), h⟩_[L^∞(Ω)]^d× d
    +
    ⟨∂_uℒ(κ,u,w_κ),u'h ⟩_V
    +
    ⟨∂_w_κℒ(κ,u,w_κ),w'_κ h ⟩_V.

Here we used the fact that κ↦ u=u_κ is differentiable (see Lemma <ref> below). 
We derive by the symmetry of κ that, for any φ∈ K,

    ⟨∂_uℒ(κ,u,w_κ),φ⟩_V   =
    ∫_Ωκ(x)∇ w_κ(x)·∇φ(x) x̣
          -∫_∂Ωσ[ru^r-1(x)
    -(r-1)u^r-2(x)w_κ(x)]φ(x) σ̣+
    ∫_Ω f(x)φ(x) x̣,

whence follows 
⟨ℰ_ hom'(κ),h⟩_[L^∞(Ω)]^d× d
=
⟨∂_κℒ(κ,u,-v), h⟩_[L^∞(Ω)]^d× d due to the differentiability of κ↦ v=v_κ (see Lemma <ref> below),
ℰ_ hom(κ)=ℒ(κ,u,-v) and ∂_uℒ(κ,u,-v)=∂_w_κℒ(κ,u,-v)=0. Thus we obtain (<ref>).


[Existence and regularity of solutions to the adjoint equation]
The existence of a unique weak solution to (<ref>) is assured by some natural assumption.
    Indeed, if the weak solution u∈ K to (<ref>) satisfies 
    
    u∈ L^∞(Ω),      u(x)≥ C>0   (x∈Γ_0)

    for some C>0 and Γ_0⊂∂Ω with |Γ_0|>0, then we can deduce that (<ref>) possesses a unique weak solution v∈ H^1(Ω).
    This result comes from the following inequality;
    there exists C>0 such that
    
    v_L^2(Ω)^2 ≤ C (  ∇ v_L^2(Ω)^2 + ∫_Γ_0κ̃v^2 σ̣)

    for any v∈ H^1(Ω), where κ̃∈ L^∞(∂Ω) which satisfies κ̃(x)≥ C>0 (x∈Γ_0) for some C>0 and Γ_0∈∂Ω with |Γ_0|>0.
    The above inequality can be proved in a similar way to the proof of Lemma <ref> with slight modification.
    By virtue of the assumptions on u and this inequality, the usual method by Lax–Milgram theorem can be applied to (<ref>) in order to show the existence of a weak solution.
    Unfortunately, it is difficult to prove the above assumption (<ref>) on u rigorously.
    However, if Ω and κ are sufficiently smooth, the solution u belongs to H^2(Ω) (for detail, see <cit.>),
    and we can deduce that u satisfies the above conditions with Γ_0=∂Ω.
    Therefore, our assumptions (<ref>) are quite natural, and after this, we always impose (<ref>) on the solution of (<ref>) implicitly whenever we consider a solution v to (<ref>).
    Moreover, in this setting, we can derive v∈ H^1(Ω)∩ L^∞(Ω) and in particular v∈ K.



As for u'=u_κ', we have the following
 
Suppose that (<ref>).
Then the nonnegative weak solution κ↦ u=u_κ of (<ref>) is differentiable at κ and  
    u_κ'h= ũ
    for the direction h∈ [L^∞(Ω)]^d× d. Here ũ∈ V satisfies

    ∫_Ωκ(x)∇ũ(x)·∇φ(x) x̣
        +
        ∫_∂Ωσ(r-1)u^r-2(x)ũ(x)φ(x) σ̣=
        -
       ∫_Ωh(x)∇ u(x)·∇φ(x) x̣

for all φ∈ V.



Let κ[s]=κ+sh for s>0 and let u[s] be a solution to (<ref>) with κ=κ[s].  
In this proof, we set σ=1 for simplicity.
Differentiating with respect to s>0 in the weak form of (<ref>) with κ=κ[s], we have   

    ∫_Ωκ[s](x)∇ u'[s](x)·∇φ(x) x̣
        +
        ∫_∂Ω(r-1)u^r-2[s](x)u'[s](x)φ(x) σ̣=
        -
       ∫_Ωh∇ u[s](x)·∇φ(x) x̣,

which coincides with (<ref>) as ũ=u'[s] and s=0.
Noting that κ=κ[0]=κ[1]-h, u_κ+h=u[1] and u_κ=u[0], we observe that, for any φ∈ V,

    ∫_Ωκ(x)∇(u_κ+h(x)-u_κ(x)-ũ(x))·∇φ(x) x̣[0]
    
         +
        ∫_∂Ω
        (u_κ+h^r-1(x)-u_κ^r-1(x)-(r-1)u_κ^r-2(x)ũ(x))φ(x) σ̣[0]
       =
    ∫_Ω[(κ[1](x)-h(x))∇ u[1](x)-κ[0](x)∇ u[0](x)-κ(x)∇ũ(x)]·∇φ(x) x̣[0]
         +
        ∫_∂Ω
        (u[1]^r-1(x)-u[0]^r-1(x)-(r-1)u[0]^r-2(x)ũ(x))φ(x) σ̣[0]
        =
        -
       ∫_Ωh(x)∇ (u[1](x)-u[0](x))·∇φ(x) x̣≤h_L^∞(Ω)∇(u_κ+h-u_κ)_L^2(Ω)∇φ_L^2(Ω).

As for the upper bound of ∇(u_κ+h-u_κ)_L^2(Ω), we deduce from the same argument that 

    ∫_Ωκ(x)∇(u_κ+h(x)-u_κ(x))·∇ (u_κ+h(x)-u_κ(x)) x̣
    +
    ∫_∂Ω
      (u_κ+h^r-1(x)-u_κ^r-1(x))(u_κ+h(x)-u_κ(x)) σ̣
       =
      -
       ∫_Ωh(x)∇ u_κ+h(x)·∇ (u_κ+h(x)-u_κ(x)) x̣≤h_L^∞(Ω)∇ u_κ+h_L^2(Ω)∇ (u_κ+h-u_κ)_L^2(Ω),

which along with the boundedness of (u_k+h) in V and the uniform ellipticity of κ yields 

    ∇(u_κ+h-u_κ)_L^2(Ω)≤ Ch_L^∞(Ω).

Hence, setting φ=u_κ+h-u_κ-ũ in  (<ref>), we have

    ∇(u_κ+h-u_κ-ũ)_L^2(Ω)^2
        +
     ∫_∂Ω
        (u_κ+h^r-1(x)-u_κ^r-1(x)-(r-1)u_κ^r-2(x)ũ(x))(u_κ+h(x)-u_κ(x)-ũ(x)) σ̣
         ≤ Ch_L^∞(Ω)^2 ∇(u_κ+h-u_κ-ũ)_L^2(Ω).

By noting that the integrand of the second term of the left-hand side in (<ref>) is written as

    (u_κ+h^r-1-u_κ^r-1-(r-1)u_κ^r-2ũ)(u_κ+h-u_κ-ũ)
       =
    (u_κ+h^r-1-u_κ^r-1-(r-1)u_κ^r-2ũ-(u_κ+h-u_κ-ũ))(u_κ+h-u_κ-ũ)
    +
    (u_κ+h-u_κ-ũ)^2.

The integral of the first term in the last line can be estimated as follows:

    ∫_∂Ω
    (u_κ+h^r-1(x)-u_κ^r-1(x)-(r-1)u_κ^r-2(x)ũ(x)-(u_κ+h(x)-u_κ(x)-ũ(x)))(u_κ+h(x)-u_κ(x)-ũ(x)) σ̣
         ≤ 
    Ch_L^∞(Ω)^2 ∇(u_κ+h-u_κ-ũ)_L^2(Ω).

Hence one can derive that

    ∇(u_κ+h-u_κ-ũ)_L^2(Ω)≤ Ch_L^∞(Ω)^2,

which together with (<ref>), (<ref>) and (<ref>) yields 

    u_κ+h-u_κ-ũ_L^2(∂Ω)≤ Ch_L^∞(Ω)^2.

Combining (<ref>) with (<ref>), we conclude that

    lim_h_L^∞(Ω)→ 0_+u_k+h-u_k-ũ_V/h_L^∞(Ω)=0,

which completes the proof.


By the same argument as in the proof of Lemma <ref>, we have the following 
 
Suppose that (<ref>). 
Let u∈ K and ũ∈ V be weak solutions to (<ref>) and (<ref>), respectively.
Then the weak solution κ↦ v=v_κ∈ K of (<ref>) is differentiable at κ and  
    v_κ'h= ṽ
    for the direction h∈ [L^∞(Ω)]^d× d. Here ṽ∈ V satisfies
    
    ∫_Ωκ(x)∇ṽ(x)·∇φ(x) x̣
          +σ∫_∂Ω
        [
        r(r-1)u^r-2(x)ũ(x)
        +(r-1)(r-2)u^r-3(x)ũ(x)v(x)
        +(r-1)u^r-2ṽ(x)
        ]φ(x) σ̣
       =
        -
       ∫_Ωh∇ v(x)·∇φ(x) x̣

for all φ∈ V.



Let κ[s]=κ+sh for s>0 and σ=1 for simplicity. Let u[s] and v[s] be weak solutions to (<ref>) with κ=κ[s] and (<ref>) with κ=κ[s], respectively. Then, by differentiating with respect to s>0 in the weak form of (<ref>), we obtain (<ref>) as u'[s]=ũ, v'[s]=ṽ and s=0. 
Furthermore, we get 
∇(v_k+h-v_k)_L^2(Ω)≤ C h_L^∞(Ω)
as in (<ref>).
Here we used the fact that ∇ v_k+h_L^2(Ω)≤ C by virtue of (<ref>) (see Remark <ref>).
Then we observe that, for any φ∈ V,

    ∫_Ωκ(x)∇(v_κ+h(x)-v_κ(x)-ṽ(x))·∇(v_κ+h(x)-v_κ(x)-ṽ(x)) x̣[0]
         +
    (r-1)    ∫_∂Ω[u_κ+h^r-2(x)
        v_κ+h(x)-u_κ^r-2(x)v_κ(x)-u_κ^r-2(x)ṽ(x)](v_κ+h(x)-v_κ(x)-ṽ(x)) σ̣[0]
         -
    ∫_∂Ω[
     r(r-1)u_κ^r-2(x)ũ(x)
        +(r-1)(r-2)u_κ^r-3(x)ũ(x)v_κ(x)
    ](v_κ+h(x)-v_κ(x)-ṽ(x)) σ̣[0]
       =
        -
       ∫_Ωh(x)∇ (v_k+h(x)-v_k(x))·∇ (v_k+h(x)-v_k(x)-ṽ(x)) x̣
       ≤h_L^∞(Ω)∇(u_κ+h-u_κ)_L^2(Ω)∇ (v_k+h-v_k-ṽ)_L^2(Ω)≤ 
        Ch_L^∞(Ω)^2∇ (v_k+h-v_k-ṽ)_L^2(Ω).

Here we note that the integrand of the second line is written as

    [u_κ+h^r-2
        v_κ+h-u_κ^r-2v_κ-u_κ^r-2ṽ](v_κ+h-v_κ-ṽ)
         =
    [(u_κ+h^r-2
        v_κ+h-u_κ^r-2v_κ-u_κ^r-2ṽ)
    -(v_κ+h-v_κ-ṽ)](v_κ+h-v_κ-ṽ)
    +
        (v_κ+h-v_κ-ṽ)^2.

Thus we see by 
the same argument as in the proof of Lemma <ref> and 
the uniform ellipticity of κ that

    ∇(v_κ+h-v_κ-ṽ)_L^2(Ω)^2+
    v_κ+h-v_κ-ṽ_L^2(∂Ω)^2
    ≤ 
    Ch_L^∞(Ω)^2
    ∇ (v_k+h-v_k-ṽ)_L^2(Ω),

which implies that 
v_κ+h-v_κ-ṽ_V≤ 
Ch_L^∞(Ω)^2, and hence,

    lim_h_L^∞(Ω)→ 0_+v_k+h-v_k-ṽ_V/h_L^∞(Ω)=0.

This completes the proof.



We next observe the relation between u and v.


Assume that (<ref>).
Let u∈ K and v∈ V be weak solutions to (<ref>) and (<ref>), respectively.
Then it holds that

    ∇ (u-v)_L^2(Ω)^2≤r-1/α'√(2/α_0)f_L^2(Ω){1/2α_0f_L^2(Ω)^2 + r-2/2σ |∂Ω| }^1/2,

where α_0 := min(α',r/2σ). 




By (<ref>) and  (<ref>), we have

    ∫_Ωκ(x) ∇(u(x)-v(x)) ·∇(u(x)-v(x)) x̣ - ∫_∂Ωκ(x) ∇ (u(x)-v(x)) ·ν(x) (u(x)-v(x))  σ̣=0.

From the boundary conditions, the second term of the left-hand side implies that

    -∫_∂Ωκ(x) ∇ (u(x)-v(x)) ·ν(x) (u(x)-v(x)) σ̣
       =  
    ∫_∂Ωσ{ u^r-1(x) -(r-1) u^r-2(x)v(x) -ru^r-1(x) } (u(x)-v(x)) σ̣[0]
       = 
    - σ (r-1) ∫_∂Ω u^r-2(x)(u^2(x)-v^2(x)) σ̣≥ 
    - σ (r-1) ∫_∂Ω u^r(x) σ̣.

Here we used the nonnegativity of u≥0 in the last inequality.
Hence we have

    α' ∇(u-v)_L^2(Ω)^2 ≤σ (r-1) ∫_∂Ω u^r(x) σ̣(<ref>)≤ (r-1)∫_Ω fu x̣≤ (r-1)f_L^2(Ω)u_V.

In the rest of the proof, it suffices to show that

    u_V ≤√(2/α_0){1/2α_0f_L^2(Ω)^2 + r-2/2σ |∂Ω| }^1/2.

By (<ref>), it holds that

    α' ∇ u_L^2(Ω)^2 + σ∫_∂Ω u^r(x)  σ̣≤∫_Ω fu x̣.

Moreover, it follows from Hölder's inequality and Young's inequality that

    ∫_∂Ω u^2(x) σ̣≤( ∫_∂Ω u^r(x) σ̣)^2/r |∂Ω|^r-2/r≤2/r∫_∂Ω u^r(x)  σ̣+ r-2/r |∂Ω|.

Thus we obtain

    α'∇ u_L^2(Ω)^2 + r/2σu_L^2(∂Ω)^2 - r-2/2σ |∂Ω| ≤f_L^2(Ω)u_V.

Using Young's inequality, we can derive

    α_0 u_V^2 ≤α_0/2u_V^2 + 1/2α_0f_L^2(Ω)^2 + r-2/2σ |∂Ω|,

which implies (<ref>).
Therefore, by (<ref>) and (<ref>), we obtain 

    ∇ (u-v)_L^2(Ω)^2≤r-1/α'√(2/α_0)f_L^2(Ω){1/2α_0f_L^2(Ω)^2 + r-2/2σ |∂Ω| }^1/2,

which is the desired result.

Combining Propositions <ref> with <ref>, we see by f≢0 
that, for any κ, h∈ℳ(α',β'),

    ⟨ℰ_ hom'(κ),h⟩_[L^∞(Ω)]^d× d     =
      -  ∫_Ωh(x)∇ u(x)·∇ v(x) x̣
       = 
    1/2∫_Ωh(x)∇ (u(x)-v(x))·∇(u(x)- v(x)) x̣
         
    -
    1/2∫_Ωh(x)∇ u(x)·∇ u(x) x̣
    -
    1/2∫_Ωh(x)∇ v(x)·∇ v(x) x̣
       <
    β'/2∇ (u-v)_L^2(Ω)^2
       ≤β'/2r-1/α'√(2/α_0)f_L^2(Ω){1/2α_0f_L^2(Ω)^2 + r-2/2σ |∂Ω| }^1/2.

Thus one may expect  
ℰ_ hom'(κ)=-∇ u·∇ v≤ 0 
a.e. in Ω at least in the case where f≥ 0 is small (see Remark <ref> below). 
In this particular case, due to ℰ(θ)≤ℰ_ hom(κ) by (<ref>), one can estimate the minimum value by replacing (<ref>) with the following minimization problem: 

    inf_θ∈Θℰ(θ),

where Θ:={θ∈ L^∞(Ω;[0,1])θ_L^1(Ω)=γ|Ω|}[In the self adjoint problem (i.e., v=± u), there is a case where an optimal homogenized matrix can be characterized as κ^∗=κ[θ^∗]=α(1-θ^∗)+βθ^∗. Here θ^∗ is an optimal volume fraction (i.e., ℰ_ hom(κ^∗)=ℰ_ hom(θ^∗,κ_ hom^∗)=ℰ(θ^∗)). Hence it suffices to consider (<ref>) instead of (<ref>); however, nonlinear problems cause non-self adjoint problems in general (see <cit.> for an optimal homogenized flux). 
Thus the problem with χ_Ω_1 being replaced by θ as in (<ref>) is just a problem to estimate the infimum value in general. 
On the other hand, as in the homogeneous Dirichlet boundary condition, one can construct the self adjoint problem for the homogeneous Robin boundary condition (i.e., r=2) by setting ℰ(χ)=⟨ f,u_χ⟩. 
]

Now, we are in a position to describe a numerical algorithm for the volume fraction θ∈Θ.
Based on the (steepest gradient) descent method (or time-discrete version of the gradient flow) and Proposition <ref>, we set 

    θ_i+1=θ_i-τℰ'(θ_i)=θ_i-τ (β-α)∇ u_θ_i·∇ v_θ_i   for  i∈∪{0}.

Here θ_0∈ L^∞(Ω) is an initial volume fraction,
τ>0 stands for the step width (or time step, i.e., θ_i implies θ_i=θ(x,τ i)) and
u_θ∈ K and v_θ∈ K are unique weak solutions to (<ref>) with κ=κ[θ] and (<ref>) with κ=κ[θ], respectively. 
Repeating (<ref>) until θ_i+1_L^1(Ω)=γ|Ω| and θ_i+1-θ_i_L^1(Ω)≤η for η>0 small enough, one can estimate the minimum value of ℰ(χ_Ω_1) numerically by Theorem <ref>. The following is the numerical algorithm:





[Linearization of the thermal radiation boundary condition] 
      To solve (<ref>) with κ=κ[θ] numerically,  
we first approximate β(u_θ) as β(u_ old)+β'(u_ old)(u_θ-u_ old) as in the Newton–Raphson method. 
Here u_ old is an arbitrarily given function, and we choose r=d+2 in (<ref>) based on <cit.>, that is, 
β(u_θ)≈σu_ old^d+1+
σ(d+1)u_ old^d(u_θ-u_ old)
=
σ(d+1)u_ old^du_θ-σ du_ old^d+1.
We next solve the following linearized equation: 

    0   =-∫_Ω f(x) φ(x) x̣
    +
    ∫_Ωκ[θ](x) ∇ u_θ(x)·∇φ(x) x̣
         + 
    ∫_∂Ω[σ (d+1)u_ old^d (x)u_θ(x)
        -
        σ du_ old^d+1(x)]_linearized thermal radiation boundary conditionφ(x) σ̣   for all φ∈ V.

We finally check the following convergence condition:

    |
        ∫_Ωκ[θ](x) |∇ u_θ(x)|^2 x̣+ 
        σ∫_∂Ω |u_θ (x)|^d+2 σ̣-∫_Ω f(x) u_θ(x) x̣|≤η_3
       for some η_3>0.

If (<ref>) is not satisfied, we set u_ old← u_θ, and then we solve (<ref>) again. 
This procedure is repeated until (<ref>)
is satisfied. 




[Convergence condition]
By (<ref>), it holds that

    |θ_i+1-θ_i|=τ (β-α)|∇ u_θ_i·∇ v_θ_i|.

If θ_i+1 attains the critical point of ℰ(θ), then the right-hand side vanishes.  
Since it belongs to L^1(Ω) at least, the convergence condition (<ref>) is reasonable.
In this paper, we do not mention the regularization of sensitivity ℰ'(θ_i) to become θ_i+1∈ L^∞(Ω) since Algorithm <ref> is only used to estimate the minimum value of the original optimal design problem (<ref>) and inf_θ∈Θℰ(θ)= inf_θ∈Θ̃ℰ(θ),    
where Θ̃:={θ∈ L^1(Ω)θ(x)∈ [0,1]  and θ_L^1(Ω)=γ|Ω|}.


[Self adjointness and convexity of a linearized problem] 
Let u_ L be a solution to (<ref>) 
with |u|^r-2u being replaced by u_ old^r-1, and then consider the minimization problem (<ref>) 
with u_ hom being replaced by u_ L. Here u_ old is a function that appears in Remark <ref>.
Then it can be regarded as a self-adjoint problem by the same argument as in Proposition <ref>. Thus (<ref>) with u_ hom being replaced by u_ L turns out to be a true relaxation problem by Theorem <ref>, Proposition <ref> and <ref>, and moreover, it has only global minimizers in terms of double minimization; indeed, define ℰ̃ V→ by

    ℰ̃(w)=
     1/2∫_Ωκ[θ](x)|∇ w(x)|^2 x̣
        +
        ∫_∂Ωσu_ old^r-1(x) w(x) σ̣-∫_Ω f(x) w(x) x̣.

Then we see that u_ L= argmin_w∈ Kℰ̃(w) and  

    ∫_Ωκ[θ](x)|∇ u_ L(x)|^2 x̣   =
    2(∫_Ω f(x) u_ L(x) x̣    
    -∫_∂Ωσu_ old^r-1(x) u_ L(x) σ̣)
    -
    ∫_Ωκ[θ](x)|∇ u_ L(x)|^2 x̣
       = -2 inf_w∈ Kℰ̃(w).

Let ℰ̂:V× [L^2(Ω)]^d→ be such that ℰ̂(w,∇ w)=ℰ̃(w). 
Since ℰ̂ is convex, the dual energy yields 

    inf_w∈ Kℰ̃(w)=inf_P^∗∈ [L^2(Ω)]^d,
     - P^∗=f  in  Ω, 
    
    -P^∗·ν=σu_ old^r-1 on  ∂Ω∫_Ωκ[θ]^-1(x)|P^∗(x)|^2 x̣

(see, e.g., <cit.>).
Thus the minimization problem (<ref>) with u_ hom being replaced by u_ L is equivalent to the following double minimization problem:

    min_(θ,P^∗)∈𝒲∫_Ωκ[θ]^-1(x)|P^∗(x)|^2 x̣,

where 

    𝒲:={
    (θ,P)∈Θ× [L^2(Ω)]^d
    - P^∗=f  in  Ω and  
    -P^∗·ν=σu_ old^r-1 on  ∂Ω}.

Since 𝒲 is convex, and (θ, P^∗)↦κ[θ]^-1|P^∗|^2 is also convex, the assertion is obtained. 
Hence, if u_ old^r-1 sufficiently approximates u^r-1 on ∂Ω, the convergence value of energies via Algorithm <ref> also approximates the minimum value for (<ref>) with 
A_χ_Ω_1=κ[χ_Ω_1] and
β(u_χ_Ω_1)=σu_χ_Ω_1^r-1.  


In this paper, to estimate the minimum value of (<ref>) with A_χ_Ω_1=κ[χ_Ω_1] and β(u_Ω_1)=σu_Ω_1^d+1 numerically, we consider the state equation as an approximated equation with inhomogeneous Neumann boundary conditions in optimization of the volume fraction; in other words, ℰ'(θ_i) in (<ref>) is regarded as ℰ'(θ_i)=-(β-α)|∇ u_θ_i|^2.




§ APPROXIMATION PROBLEM FOR (<REF>) VIA POSITIVE PARTS OF LEVEL SET FUNCTIONS

In this section, we shall prepare a numerical analysis to find 
two-material distributions that give a value close to the minimum for (<ref>) with 
A_χ_Ω_1=κ[χ_Ω_1] and 
β(w)=σw^d+1. 
As already mentioned in (ii) of Remark <ref>, 
we need to construct the optimal volume fraction θ numerically such that
the intermediate set [0<θ<1] rarely appears due to non-existence of minimizers for (<ref>) in general. 
As one of the methods to avoid the so-called grayscale problem, 
level set methods (see, e.g., <cit.>) are known and employed to construct an approximated minimizer below.
In level set methods, the following level set function is introduced to represent two-material domains:  

    ϕ(x)
    
    >0,      x∈Ω_1,
    
    =0,      x∈∂Ω_1∩∂Ω_0,
    
    <0,      x∈Ω_0.

Based on <cit.>, we consider the following perimeter constraint problem via the positive part of the level set function as an approximation problem of (<ref>):

    inf_ϕ∈ U_ ad{J_(ϕ):=ℰ(ϕ_+)+/p∫_Ω |∇ϕ(x)|^p x̣},

where ϕ_+=max{0,ϕ}, U_ ad:={ϕ∈ W^1,p(Ω) |ϕ|≤ 1  and ϕ_+_L^1(Ω)=γ |Ω|}, 1<p<+∞ and >0.
In particular, the second term of J_ (i.e., the p-Dirichlet energy) plays a role of perimeter constraint (cf. <cit.>). 




 §.§ Characterization of minimizers for level set functions

In order to form the basis of numerical analysis for (<ref>), we first show the following

    There exists at least one minimizer of (<ref>).



Let (ϕ^n) be a minimizing sequence in U_ ad. Thus ϕ^n satisfies

    lim_n→ +∞ J_(ϕ^n)=inf_ϕ∈ U_ adJ_(ϕ).

Since (ϕ^n) is bounded in W^1,p(Ω) due to J_(ϕ^n)→ +∞ as ϕ^n_W^1,p(Ω)→ +∞, there exist a (not relabeled) subsequence of (n) and ϕ^∗∈ U_ ad
such that

    ϕ^n→ϕ^∗ weakly in  W^1,p(Ω)

and 

    ϕ^n_+→ϕ^∗_+  a.e. in Ω.

Hence Theorem <ref> ensures that

    lim_n→ +∞ℰ(ϕ^n_+)= ℰ(ϕ^∗_+).

Combining (<ref>) with the weak lower semicontinuity of norm, we obtain

    inf_ϕ∈ U_ ad  J_(ϕ)
    ≤
    J_(ϕ^∗)
    ≤lim inf_n→ +∞ J_(ϕ^n)
    =
    inf_ϕ∈ U_ ad  J_(ϕ),

which completes the proof.





Furthermore, we have the following

Let ϕ^ be a minimizer of (<ref>). 
Then there exist a (not relabeled) subsequence of () and ϕ^∗∈ U_ ad such that
ϕ^→ϕ^∗ weakly in W^1,p(Ω) and

    ℰ(ϕ^∗_+)=
    lim_→ 0_+J_(ϕ^)=inf_ϕ∈ U_ adℰ(ϕ_+).




We first note that a_:=J_(ϕ^) has a limit; indeed, if s≥ t for s,t>0, we have

    a_s≥ J_t(ϕ^s)≥ a_t≥inf_θ∈Θℰ(θ)≥ 0,

which yields the assertion. 
Furthermore, due to a_→ +∞ as ϕ^_W^1,p(Ω)→ +∞, we see that (ϕ^) is bounded in W^1,p(Ω). Thus, as in the proof of Theorem <ref>, there exist a (not relabeled) subsequence of () and ϕ^∗∈ U_ ad such that (<ref>) and (<ref>) with n=. 
Therefore, it follows that

    lim_→ 0_+ a_=ℰ(ϕ^∗_+).

Noting that, for any ϕ∈ U_ ad, 

    a_   ≤∫_Ωκ[ϕ_+](x)|∇ u_ϕ(x)|^2 x̣
        +/p∫_Ω|∇ϕ(x)|^p x̣→ℰ(ϕ_+)   as   → 0_+,

one obtains
ℰ(ϕ^∗_+)≤ℰ(ϕ_+) for all ϕ∈ U_ ad. This completes the proof.



[Approximate solutions for (<ref>)]
By Theorem <ref>, it holds that

    ℰ(ϕ^∗_+)≤ℰ(θ)   for all   θ∈Θ∩ W^1,p(Ω).

Thus ϕ_+^∗ turns out to be a minimizer of (<ref>) under
 inf_θ∈Θℰ(θ)
=
inf_θ∈Θ∩ W^1,p(Ω)ℰ(θ). 
In this case, Theorem <ref> ensures that ϕ_+^ for >0 small enough can be regarded as an approximate solution for (<ref>) under the optimal homogenized matrix can be written as the upper bound.
Since χ_Ω_1∈ BV(Ω) is required at least in the perimeter constraint problem for (<ref>), it is reasonable to assume additional regularity as a setting that avoids the grayscale problem. 
In particular,
the optimal volume fraction θ^∗ of (<ref>) is weakly differentiable in the direction of ∇ u_θ^∗ under u_θ^∗∈ H^2(Ω); indeed, we observe that, for any φ∈ C^∞_ c(Ω),

    ( F,φ)_L^2(Ω)   :=-∫_Ωθ^∗(x) [(∇ u_θ^∗(x))φ(x)] x̣
        =
    -∫_Ωθ^∗(x)Δ  u_θ^∗(x) φ(x) x̣
    -∫_Ωθ^∗(x) ∇ u_θ^∗(x)·∇φ(x) x̣.

Then the second term in the last line is written as

    -∫_Ωθ^∗(x) ∇ u_θ^∗(x)·∇φ(x) x̣
    =
    1/β-α∫_Ω [α∇ u_θ^∗(x)·∇φ(x)-f(x)φ(x)] x̣,

and therefore, (∇θ^∗·∇ u_θ^∗,φ)_L^2(Ω) makes sense
by noting that

    |( F,φ)_L^2(Ω)|
    =
    |-∫_Ω[(θ^∗(x)+α/β-α)Δ  u_θ^∗(x)+ 
    1/β-αf(x)]φ(x) x̣|≤ C_θ^∗, fφ_L^2(Ω)

(see <cit.> for the homogeneous Dirichlet boundary condition).



[Extension from ϕ_+ to ϕ_+^m]
In Theorems <ref> and <ref>, one can replace ϕ_+ in (<ref>) with ϕ_+^m for all m≥ 1. Indeed, let A:L^m+1(Ω)→ L^(m+1)/m(Ω) be an operator defined by A(w)= w_+^m.
Then A is maximal monotone in L^m+1(Ω)× L^(m+1)/m(Ω). Noting that

    ϕ^n   →ϕ^∗       weakly in  L^m+1(Ω),
               strongly in  L^p(Ω),
    
     A(ϕ^n)    →ξ         weakly in  L^p/(p-1)(Ω)∩ L^(m+1)/m(Ω)

for some ξ∈ L^p/(p-1)(Ω)∩ L^(m+1)/m(Ω) and n=n_k, we have

    ⟨ A(ϕ^n),ϕ^n⟩_L^m+1(Ω)=
    ⟨ A(ϕ^n),ϕ^n⟩_L^p(Ω)→⟨ξ,ϕ^∗⟩_L^p(Ω)
    =
    ⟨ξ,ϕ^∗⟩_L^m+1(Ω)   as  n→ +∞ .

Hence Minty’s trick (see e.g., <cit.>) ensures that ξ=A(ϕ^∗)=(ϕ_+^∗)^m. 
Furthermore, 
since (ϕ_+^n) is also bounded in W^1,p(Ω), 
one can extract a subsequence of (n_k) (still denoted by n_k) such that ϕ_+^n_k→ϕ_+^∗ a.e. in Ω, which along with the boundedness of (ϕ_+^n) in L^∞(Ω) yields A(ϕ^n_k)→ A(ϕ^∗) strongly in L^ℓ(Ω) for all ℓ≥ 1.
The rest of the proofs runs as before. 

In this paper, we select m=1 to compare the results in <cit.> (cf. <cit.> for m>1). 





 §.§ Numerical algorithm for optimization of level set functions

Before describing the numerical algorithm, we derive the equation to update the level set function. 
As in (<ref>), we introduce the following (gradient) descent method:

    ϕ_i+1=ϕ_i-τ J_'(ϕ_i)   for  i∈∪{0}.

Since the Fréchet derivative of p-Dirichet energy for the level set function is -Δ_pϕ=-(|∇ϕ|^p-2∇ϕ) under homogeneous Dirichlet/Neumann boundary condition, we have

    ϕ_i+1=ϕ_i-τ (∂_ϕ_iℰ((ϕ_i)_+)-Δ_p ϕ_i+1)   for  i∈∪{0}.

Here we note that Δ_p ϕ_i is replaced with Δ_p ϕ_i+1 in order to satisfy ϕ_i+1∈ W^1,p(Ω)
(see, e.g. <cit.>) for forward-backward splitting schemes).
Furthermore, by Remark <ref> and Proposition <ref>, the above update equation
(<ref>) with (ϕ_i)_+ being replaced by (ϕ_i)_+^m can be written as 

    ϕ_i+1=ϕ_i-ζ[ϕ_i](-m(β-α)(ϕ_i)_+^m-1χ_ϕ_i∇ u_ϕ_i·∇ v_ϕ_i -Δ_p ϕ_i+1)
       for  i∈∪{0},

where u_ϕ_i and v_ϕ_i are the unique weak solutions to (<ref>) with κ=κ[(ϕ_i)_+] and (<ref>) with κ=κ[(ϕ_i)_+], respectively.
Here we note that τ>0 in (<ref>) is extended to 
ζ[ϕ_i]≥ 0, which is the generalized step width such that ζ[0]=0. Indeed,
in our setting (<ref>) (i.e., m=1),
although ϕ_+ is not differentiable at [ϕ=0], thanks to  
ζ:U_ ad→ L^∞(Ω), the sensitivity with the weight can be denoted by ζ[ϕ_i]∂_ϕℰ((ϕ_i)_+)=-ζ[ϕ_i](β-α)χ_ϕ_i∇ u_ϕ_i·∇ v_ϕ_i formally.

Now, as in <cit.>, we characterized the level set function by a solution to the time discrete version of the following doubly nonlinear diffusion equation <cit.>:  

    ∂_tϕ^q-Δ_p ϕ=
        (β-α)χ_ϕ∇ u_ϕ·∇ v_ϕ   in Ω× (0,+∞).

If we set q∈ (0,1) such that q≈ 1 for simplicity of linearization, one has |ϕ_i+1|^q-1≈ |ϕ_i|^q-1, and then the time discrete equation of (<ref>) is described as follows:  

    |ϕ_i|^q-1ϕ_i+1-ϕ_i/τ-Δ_p ϕ_i+1=
        (β-α)χ_ϕ_i∇ u_ϕ_i·∇ v_ϕ_i   in Ω.

In particular, multiplying it by τ|ϕ_i|^1-q, we obtain (<ref>) with ζ[ϕ_i]=τ|ϕ_i|^1-q. 
In this paper, we choose p=2 since one expects that the positive parts of optimal level set functions belong to V from Remark <ref>.
Thus ϕ_i+1 satisfies

    ∫_Ω   |ϕ_i|^q-1(x)ϕ_i+1(x)-ϕ_i(x)/τφ(x) x̣
    +
    ∫_Ω∇ϕ_i+1(x)·∇φ(x) x̣
          =∫_Ω (β-α)χ_ϕ_i(x)∇ u_ϕ_i(x)·∇ v_ϕ_i(x)φ(x) x̣   for all  φ∈ V∩ L^∞(Ω).

As in Algorithm <ref>, the following algorithm is proposed:






§ NUMERICAL RESULTS FOR (<REF>)


Based on the previous sections, we shall numerically construct the material distribution of two materials with diffusion coefficients of α>0 and β>0 such that the Dirichlet energy is minimized by using FreeFEM++ <cit.> with piecewise linear Lagrange elements on a triangular mesh. 
Throughout this section, we set σ=1, d=2, Ω=(0,1)^2, α=1, β=10 and θ_0≡ϕ_0≡γ for γ∈ (0,1).



 §.§ Numerical validity

We first check the numerical validity. 
Based on Algorithms <ref> and <ref>, we set γ=0.6 and f=0.001.
As for (<ref>), the characteristic function χ_ϕ_i is treated approximately as 0.5tanh (ϕ_i/0.1)+0.5. 
Then we obtain Figures <ref> and <ref>. From Figures <ref>–<ref>, it is confirmed that Algorithm <ref> makes Ω⊂^2 that almost consists of materials with diffusion coefficients of α>0 (the blue domain) and β (the red domain). 
In particular, it is noteworthy that [ϕ_+=1]:={x∈Ωϕ_+(x)=1} and [ϕ_+=0] involve [θ=1] and [θ=0], respectively. 
Furthermore, Figure <ref> shows that the convergence value of the Dirichlet energy ℰ is monotonically decreasing with respect to >0, which means that 
the necessary condition (<ref>) is satisfied, and 
ℰ(ϕ_+) asymptotically tends to ℰ(θ) constructed by the optimized volume fraction.
As a qualitative property of (locally) optimal configurations, it is suggested that the family of optimal configurations contain two-phase configurations, and then we see that the Dirichlet energy decreases by increasing the perimeter of the interface.
This completes the confirmation of the validity of the proposed method (see also Remark <ref> below).





       


[Worst conductor] 
    The minimization problem (<ref>) corresponds to the problem for determining the so-called best two-material thermal conductor. Conversely, as for the worst case, we obtain Figure <ref>. 
    Here we set the objective functional and the diffusion coefficient as -ℰ (χ_Ω_1) and κ[χ_Ω_1]=αχ_Ω_1+(1-χ_Ω_1)β, respectively. Comparing Figure <ref> with Figure <ref>, we see that similar configurations are obtained, and moreover, it can be confirmed that the convergence values are almost equivalent in Figure <ref>. These results suggest the effectiveness of the proposed method.  






 §.§ Characteristics of nonlinear problems




This subsection focuses on how the optimized configurations vary for different heat sources to see the characteristics of nonlinear problems.
Here we set =10^-6 and γ=0.5.
In particular, we compare the cases with the thermal radiation and the Robin boundary conditions. 
Since (<ref>) with the Robin boundary condition is linear, the solution is a constant multiple of the original if the heat source is multiplied by a constant, and therefore, 
optimized configurations do not vary by multiplied by a constant of the heat source. 
This is confirmed by Figure <ref>. 
In contrast, with the radiation boundary condition, the solution is different from the original solution multiplied by a constant even if the heat source is multiplied by a constant, and therefore, we see by Figure <ref> that optimized configurations deeply depend on variations in heat sources, which implies that one of the characteristics of nonlinear problems can be obtained.
Furthermore, based on Figure <ref>, the following physical interpretation can be made:



  (i)
 Since both solutions with convection and with radiation asymptotically reach the trivial solution as |f|→ 0_+, similar optimized configurations are obtained
 in the case where the value of the heat source is small,
 and therefore, the convergence values of the objective functionals are almost the same; in other words, the contributions of convection and radiation to minimize the energy are almost the same (see (a)).

  (ii)
On the other hand, it can be confirmed that, in the process of increasing the value of the heat source, the optimized configurations with radiation asymptotically tend to be the same as those with the homogeneous Dirichlet boundary condition (see, e.g., <cit.>). 
In some cases, convection seems to contribute more to energy minimization than radiation (see (b)). 

  (iii)
Figures (c)–(d) suggest that the contribution of radiation to energy minimization increases with increasing temperature.













§ APPLICATION TO THERMAL RADIATION PROBLEMS

This section is devoted to the application of the optimal design theory and level set-based optimization algorithm to some practical engineering design problems. As the nonlinear boundary condition under consideration describes the thermal radiation, one of the most straightforward but yet important applications is the design of heat radiators. 

Here we assume that a two-phase heat conductor occupying the domain Ω is situated in a vacuum. The domain Ω is convex so that its view factor is zero, i.e., no radiating waves can hit the surface ∂Ω. Our aim is to find a piecewise-constant distribution of the coefficient κ in Ω such that it efficiently emits the thermal energy into the ambient vacuum.


As shown in Figure <ref> (a), let Ω = (-L/2,L/2)^3 be the cube with side length L>0. The cube contains a heat source f.
The boundary of the cube Ω comprises a radiative surface Γ_R and thermally insulated one ∂Ω∖Γ_R.
Then the temperature u in Ω solves

    -(κ∇ u)=f      in Ω,
            
    
            -κ∇ u·ν=σu^4       on Γ_R,
            
    
            -κ∇ u·ν=0       on ∂Ω∖Γ_R.


As in Section <ref>, we seek the distribution of diffusion coefficients (thermal conductivities) α and β such that the Dirichlet (internal) energy is minimized under the volume constraint. Throughout this section, the thermal conductivities are set as α = 15 W m^-1 K^-1 (nichrome) and β=400 W m^-1 K^-1 (copper), respectively.

Let us start with the case of Γ_R = ∂Ω, i.e., all the surfaces are radiative. In this numerical experiment, the volume constraint is set to γ = 0.15, and the heat source f is uniformly distributed in the ball of radius L/10 located at the center of Ω, i.e., f= f_0 χ_B_L/10(0) with positive constant f_0>0.

Unlike usual conductivity problems with linear boundary conditions, the constants L and f_0 may affect the optimizer of the best-conductor problem. This can be confirmed from the results in Figure <ref>, where the optimal designs are shown for two parameter pairs (L,f_0). Figure <ref> (b) and (c) show the optimized configuration of the conductivity κ and convergence history of the objective functional. From the results, we observe some clear differences between the two shapes, e.g., the number of spikes. 
    
Another numerical example is shown in Figure <ref>. As in the previous example, the fixed design domain is a cube Ω=(-L/2,L/2)^3 with the length L>0. We give a heat source f in the bottom part of Ω as f=f_0χ_Ω_source with constant f_0>0 and Ω_source = (-L/4,L/4)× (-L/4,L/4)× (-L/2,-9L/20). Unlike the previous example, the radiative surface is set to Γ_R = { x_3 = L/2 : x∈∂Ω}, i.e., only the upper surface is radiative. In terms of physics, our aim is to enhance the radiation of heat energy generated in the bulk.

As in the previous example, we consider two cases: (L,f_0) = (0.1 m, 8× 10^5 W/m^3) and (L,f_0) = (10 m, 3.3× 10^3 W/m^3). The optimized configurations and temperature fields are shown in <ref> (b) and (c) along with convergence history of the objective functional. While both results attain convergence, the obtained design and corresponding temperature fields are quite distinct. The significant difference originates from the nonlinearity in terms of u. To see this, let us consider the energies

    E_in =   ℰ(ϕ_+),
        
    
        E_rad =   σ∫_∂Ω u^5 (x) dσ,
        
    
        E_source =   ∫_Ω f(x)u(x) x̣.

From the weak form, it immediately follows the energy conservation E_in + E_rad = E_source. From a physical point of view, this ratio E_in / E_source represents the amount of internal energy stored inside the structure. We calculated the ratio E_in / E_source for the two optimized designs (Figure <ref> (b) and (c)) and obtained the values 4.11× 10^-3 and 0.179, respectively. This indicates that the usual scaling law does not hold due to the thermal radiation effect.


We finally discuss the performance of the designed heat radiators. As shown in Figure <ref>, let us consider a fin-like structure with N× N tilted pillars (cylinders) inside the cube Ω=(-L/2,L/2)^3. The conductivity takes κ=β inside the pillars and κ=α elsewhere. 
The fin-like structure is a reasonable design of a heat radiator as it conducts heat from the bottom to the top radiative surface via the highly conductive pillars. We wish to check that the optimal design is superior to this non-optimized radiator in terms of the objective functional. 


To this end, we calculate the value of the energy ∫_Ωκ(x)|∇ u(x)|^2dx for various N using the same finite element analysis with quadratic Lagrange elements on a body-fitted tetrahedral mesh. Note that the radius R of each pillar is determined such that the fin-like design satisfies the same volume constraint with γ=0.15 for fair comparison. The calculated values are shown in Table <ref>. The results indicate that the energy decreases as the number of pillars increases in both the cases of L=0.1 m and L=10 m. These values are, however, greater than the optimal values 15.1 W/m^3 and 2.63× 10^6 W/m^3 for L=0.1 m and L=10 m, respectively. These results suggest that the optimal designs yield much more efficient heat radiation than a physically reasonable but non-optimized radiator.



§ CONCLUSION

In this paper, we considered the optimal design problem for the steady-state diffusion equation with nonlinear boundary conditions described by the maximal monotone operator. 
The main target was to analyze the distribution (or shape and topology) of the two-material composite that minimizes the Dirichlet energy with thermal radiation. 
The results obtained in this paper are as follows:


  * We proved that there exists at least a pair of the optimal volume fraction and the optimal homogenized matrix for a true relaxation problem such that the value of the relaxed Dirichlet energy coincides with the minimum value of the original design problem. To this end, we also proved the existence and uniqueness of the weak solution to the state equation with nonlinear boundary conditions described by the maximal monotone operator and the corresponding homogenization theorem.

  * In order to estimate the minimum value of the original design problem, the sensitivity of the relaxed Dirichlet energy was derived rigorously, at least under the smoothness assumptions for the domain and the two-material diffusion coefficient.

  * We considered the perimeter constraint problem via the positive part of the level set function as an approximation problem for the relaxation problem and proved the existence of minimizers. In particular, it was shown that the limit of the sequence of minimizers with respect to the perturbation parameter becomes a minimizer of the restricted relaxation problem in the Sobolev space.

  * By deriving the so-called weighted sensitivity, the level set function was updated by employing the time-discrete version of the nonlinear diffusion equation, and optimized configurations with almost no intermediate sets were obtained. Furthermore, it was numerically verified that the convergence value of the Dirichlet energy is asymptotic to a minimum value if the perturbation parameter is sufficiently small.

  * As one of the characteristics of the nonlinear problem, it was confirmed that the optimized configuration deeply depends on the value of the heat source. In particular, the contribution of radiation to energy minimization seems to increase with increasing temperature.

  * Three-dimensional numerical examples were also provided. We designed the distribution of thermal conductivity such that it minimizes internal energy due to an external heat source. The performance of the designed radiators was tested via comparison with a simple fin-like structure.





§ DECLARATION OF COMPETING INTEREST

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.






§ DATA AVAILABILITY

Data will be made available on request.


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 Paper Title: Spreading properties of the Fisher--KPP equation when the intrinsic growth rate is maximal in a moving patch of bounded size.




[2010]35K57, 92D25.

Operator-based semantics for choice programs: is choosing losing? (full version)
    
Jesse Heyninck


Open Universiteit, the Netherlands

University of Cape Town, South-Africa

    August 5, 2024
================================================================================================



§ ABSTRACT 
This paper is concerned with spreading properties of space-time heterogeneous Fisher–KPP equations in one space dimension. We focus on the case of everywhere favorable environment with three different zones, a left half-line with slow or intermediate growth, a central patch with fast growth and a right half-line with slow or intermediate growth. The central patch

moves at various speeds. The behavior of the front changes drastically depending on the speed of the central patch. Among other things, intriguing phenomena such as nonlocal pulling and locking may occur, which would make the behavior of the front further complicated. The problem we discuss here is closely related to questions in biomathematical modelling. 


By considering several special cases, 
we illustrate the remarkable diversity of possible behaviors. In particular, the case of a central patch with constant size and constant speed is entirely settled.






§ INTRODUCTION


This paper is concerned with the Cauchy problem associated with the reaction–diffusion equation

    KPP∂_t u = ∂_xx u + f(t,x,u),

where t>0 is a time variable, x∈ is a one-dimensional space variable, and u is a population density function of time and space. The function f is a reaction term which is assumed to be globally bounded with respect to the variables t and x, and of class 𝒞^2 with
respect to the variable u. Moreover, it is of the so-called KPP type with respect to u, namely:

    A1∀(t,x)∈^2 ,   
    f(t,x,0)=0, 
    ∂_u f(t,x,0)>0, 
    ∀ u≥ 0  ∂_u f(t,x,0)u≥ f(t,x,u) ≥∂_u f(t,x,0) u - M u^2, 
    ∀ u>1   f(t,x,u)<0,

where M is a given positive constant.

Denoting 
    r:(t,x)↦∂_u f(t,x,0),
 we assume it has the following form:

    A2
    r:(t,x)↦
    r_1    if x<A(t), 
    
    r_2    if A(t)≤ x<A(t)+L, 
    
    r_3    if A(t)+L≤ x,

with r_1,r_2,r_3, L positive constants and A∈𝒞(,[0,+∞)).

The prototypical example of function f we have in mind comes from ecology and has the form f(t,x,u)=r(t,x) ( u-u^2/K(t,x) ), where r is an intrinsic growth rate and K a carrying capacity. In order for (<ref>) to be satisfied, K must then satisfy that 0 < inf K ≤sup K ≤ 1 (notice that 0<inf K≤sup K < + ∞ would actually be enough up to some rescaling). The assumption (<ref>) implies that there is a time-dependent zone Ω_0(t) of size L where the intrinsic growth rate is r_2, while the intrinsic growth rate is r_1 (resp. r_3) on the left (resp. right) side of Ω_0(t). This situation is illustrated in Figure <ref>, though the actual ordering of the values r_1,r_2,r_3 may vary throughout this paper.



The equation (<ref>) is supplemented in the Cauchy problem with initial data of the form

    A3
    u_0∈𝒞_c(),   u_0≥ 0,   u_0≠ 0,

where 𝒞_c () denotes the set of compactly supported continuous functions on . We will investigate the large time behavior of solutions, and more specifically their spreading properties. What exactly is meant here by spreading will be clarified below.




 §.§ Some known results
 

Consider the above-mentioned prototypical example f(t,x,u)=r(t,x)(u-u^2 /K) with K=1.  When r_1=r_2=r_3, the equation (<ref>) then reduces to the classical Fisher–KPP equation in homogeneous media:

    ∂_t u = ∂_xx u + ru(1-u),

for which it is well known <cit.> that solutions of the Cauchy problem emanating from compactly supported initial data 
asymptotically spread at speed 2√(r) in the following sense:

    [                                         2√(r)                                             = sup{ c≥ 0 | lim_t→+∞sup_|x|≤ ct|1-u(t,x)|=0 };                                                                                           =    inf{ c≥ 0 | lim_t→+∞sup_ct≤ |x|u(t,x)=0 }. ]


However, much less is known when r_1, r_2 and r_3 differ. Yet heterogeneous growth rates are natural to consider, for at least two reasons. First, in the last few years many studies have been devoted to
the so-called “climate change problem” 






<cit.>, which deals with the situation where the favorable zone moves at a certain speed. Our second motivation comes from the analysis of multi-species models. More precisely, when one considers population models that take the form of reaction-diffusion systems involving multiple species, it is often possible to characterize the spreading speed of the fastest species, but the speeds of the second, third and other slower species generally remain elusive. This is because, although the position of the fronts of slower species lie much behind that of the fastest species, their thin leading edge stretches far beyond the front of the fastest species; therefore it is not immediately clear how the environmental change caused by the spreading front of the fastest species affects the speeds of the slower species. Are the slower species fully adapted to the new environment generated by the front of the fastest species? Or can they somewhat benefit from the environment ahead of the first front despite the increasing distance? 
Quantitative sharp answers are known only in the simplest cases 
<cit.> 
and their proofs typically rely on a very specific structure in the system (directions of instability, comparison principle, 
decoupled equation, etc.). 
A better understanding of the heterogeneous scalar equation (<ref>) should help to obtain more general results. See Section <ref> below where the relation between the equation (<ref>) for the case r_2>max(r_1,r_3) and some three-species competition or two-prey-one-predator population models is explained.

Now let us recall some known results on the heterogeneous scalar equation (<ref>).  When the heterogeneity remains confined (A(t) being globally bounded), only the asymptotic space-time growth rate matters and determines the spreading speed <cit.>, which in our case means that the population spreads toward the left at speed 2√(r_1) and toward the right at speed 2√(r_3).



Next, in the special case where r_1 ≠ r_2 = r_3, only a single transition appears in the intrinsic growth rate; see Figure <ref>. One may speculate that if A(t) moves to the right too fast, then the spreading front is located far to the left of A(t), where the intrinsic growth rate equals r_1. Thus the spreading speed is likely to be 2√(r_1), as in the homogeneous KPP equation. On the other hand, if A(t) moves too slowly, then the spreading front is located far ahead of A(t), where r = r_3. Thus the spreading speed is likely to be r_3. 
These speculations turn out to be correct, but what if the speed of A(t) is neither too fast nor too slow?  Actually the answer is not so simple: totally different types of phenomena, locking and nonlocal pulling, are observed, depending on whether r_1 > r_3 or r_1 < r_3. 
These two terms, locking and nonlocal pulling,  were first introduced in <cit.> and <cit.>,  respectively[In <cit.>, nonlocally pulled fronts were referred to as accelerated fronts. But, in the Fisher–KPP literature, acceleration also refers to superlinear spreading. We believe that nonlocal pulling better highlights the underlying mechanism, which is that a small population in a moving frame strictly faster than the actual front may still contribute to the propagation due to variations in the intrinsic growth rate.].

In the simplest case when the transition moves with constant speed, i.e., A(t) is a linear function, the known results (that were proved in possibly more complex contexts
in <cit.>) 

can be summarized as follows.


Assume r_1≠ r_2=r_3 and the existence of c_A≥ 0 such that, for all t≥ 0, 

    A(t)=c_A t.

Then the asymptotic spreading speed c^⋆>0 of u is well-defined as 

    c^⋆=sup{ c≥ 0 | lim inf_t→+∞inf_0≤ x≤ ct u(t,x)>0 }=inf{ c≥ 0 | lim_t→+∞sup_ct≤ xu(t,x)=0 }

and satisfies:


  * if r_1>r_3:


  * c^⋆=2√(r_3)  if  c_A≤ 2√(r_3),

  * c^⋆=c_A  if  2√(r_3)<c_A≤ 2√(r_1) (locked front),

  * c^⋆=2√(r_1)  if  2√(r_1)<c_A;


  * if r_1<r_3:


  * c^⋆=2√(r_3)  if  c_A≤ 2√(r_3),

  * c^⋆=c_A-2√(r_3-r_1)/2+2r_1/c_A-2√(r_3-r_1)  if  2√(r_3)<c_A≤ 2√(r_1)+2√(r_3-r_1) (nonlocally pulled front),

  * c^⋆=2√(r_1)  if  2√(r_1)+2√(r_3-r_1)<c_A.



Since these six cases form the basis for our intuition, let us describe briefly what happens in
each case. The statement is also summarized in Figure <ref>.

Cases (a)(i) and (b)(i) correspond to a situation outlined above where the transition A(t) moves so slow that the population moves ahead of it at speed 2√(r_3). In cases (a)(iii) and (b)(iii), the transition is so fast that the spreading speed 2 √(r_1) is determined only by the left side zone. The first intermediate case (a)(ii) is the locking case, where the population spreads exactly 
at the speed c_A of the growth rate transition.





Finally, case (b)(ii) is the nonlocal pulling case. Here, the population in the moving frame of the transition decays exponentially in time, yet it is still able to contribute positively to the speed of the front that comes afterward. Indeed, in this case the speed c^⋆ is strictly larger than 2√(r_1), which corresponds to the spreading speed in the left side zone if the other two zones are not present.






Note that the persistence behind the propagating front is described as 

    lim inf_t → +∞inf_0≤ x≤ ctu(t,x)>0.

This generalizes appropriately the classical picture

    lim_t→+∞sup_|x|≤ ct|1-u(t,x)|=0

to our more general setting where the local behavior of the solution is typically not a convergence to a constant
steady state, which may not exist here, but rather complicated oscillations.

The more general case (more than one unconfined interface) remained elusive until very recently. In 2021, Lam and Yu 
<cit.> managed to characterize the spreading speed in many cases thanks to a Hamilton–Jacobi framework. 
Although the formula is in general complicated and implicit, in some cases it can be made explicit. In particular, 


















 
























in the case of two interfaces A(t) and A(t)+L with constant successive values 
r_1, r_2 and r_3, all subcases can be solved provided the central patch
does not have a positive effect on the growth rate from both sides (i.e., provided r_2<max(r_1,r_3)).








This assumption is not a technical issue but rather a true limitation of the Hamilton–Jacobi approach, which involves a hyperbolic rescaling. Basically, this approach cannot deal with nonlocal pulling exerted from areas that are too small. This is why we will use a different approach here, based on super- and sub-solutions, to study this more delicate type of nonlocal pulling.

To conclude this general introduction, we point out a very recent work by Bovier and Hartung <cit.> where 
a probabilistic method was used to quantify rigorously the nonlocal pulling in a model for the spread of a trait in a population, originally introduced in <cit.>. 
It is not clear to us whether this third method of proof could be used to deal with the case we consider here.




 §.§ Organization of the paper

In Section 2 we present our main results, which mainly deal with the previously open problem when the intrinsic growth rate is larger in the intermediate zone between the two transitions.

The proofs of our results are done in Section 3, using a construction of sub- and super-solutions together with an eigenvalue problem in the moving frame of the heterogeneity. In Section 4 we collect a few interesting properties
about this eigenvalue and discuss possible extensions of our work.





§ MAIN RESULTS


We now consider the equation (<ref>), supplemented with the assumptions (<ref>), (<ref>), (<ref>). Throughout the rest of this paper, we assume that

    r_2>max(r_1,r_3).




This means that the intrinsic growth rate is at its highest in the central patch of positive and bounded length between the two moving transitions. As we explained in the previous section, such a situation has so far remained unaddressed in the mathematical literature, partly due (among other reasons) to the difficulty of estimating the aforementioned nonlocal pulling effects for this case.

From now on, we only consider the rightward spreading properties. 
Since 0≤ A(t)≤ A(t)+L, it is indeed standard to prove that the leftward
spreading speed is well-defined and equals 2√(r_1).

Before stating the results, we define two convenient quantities, the minimal spreading speed c
and the maximal spreading speed c:

    c=sup{ c≥ 0 | lim inf_t→+∞inf_0≤ x≤ ct u(t,x)>0 },


    c=inf{ c≥ 0 | lim_t→+∞sup_ct≤ xu(t,x)=0 }.

These two quantities are well-defined since the initial value u_0 satisfies (<ref>). By comparison with solutions of homogeneous problems, one may verify that

    2√(min(r_1,r_3))≤c≤c≤ 2√(r_2).

The equality between c and c is a very natural question. For more general initial data u_0,
it is known that the equality can fail even if r_1=r_2=r_3 <cit.>. When there is indeed equality between c 
and c, the quantity c=c is referred to as the spreading speed; 
in other words, the spreading speed is well-defined if, and only if, c=c.
Our results will in particular assert that, when r_2>max(r_1,r_3) and u_0 is compactly supported, the
spreading speed can be well-defined or ill-defined, depending on the variations of A. 
In fact it will 
become clear that the range of possible outcomes is very wide and that a completely general result is hopeless. 
Therefore our goal will only be to solve extreme cases and to illustrate the variety of outcomes.

We begin with the case where A is a linear function of time. 
This is our main result.


Assume the existence of c_A >0 such that, for all t≥ 0, 

    A(t)=c_At .

Let 

    L=
    
    0    if r_1=r_3, 
    1/√(r_2-max(r_1,r_3))arccot(√(r_2-max(r_1,r_3)/|r_1-r_3|))    if r_1≠ r_3,

where arccot denotes the inverse of _|(0,π). Then:

    c=c=
    
    2√(r_3)   if c_A<2√(r_3), 
    
    c_A    if 2√(r_3)≤ c_A≤ 2√(-λ_1)  (locked front), 
    
    F(c_A)    if 2√(-λ_1)<c_A<2√(r_1)+2√(-λ_1-r_1)
         (nonlocally pulled front), 
    
    2√(r_1)   if  2√(r_1)+2√(-λ_1-r_1)≤ c_A,

where


  * λ_1 = - max (r_1,r_3) if L ≤L;



  * λ_1∈(-r_2,-max(r_1,r_3)) if L>L, and in this case λ_1
is characterized as the unique solution in (-r_2,min(-max(r_1,r_3),π^2/L^2-r_2)) of:

    (L√(r_2+λ_1))=r_2+λ_1-√((r_1+λ_1)(r_3+λ_1))/√(r_2+λ_1)(√(-r_1-λ_1)+√(-r_3-λ_1));


  * the function F is defined by:

    F:c∈(2√(-λ_1-r_1),+∞)↦c-2√(-λ_1-r_1)/2+2r_1/c-2√(-λ_1-r_1);




The intervals (2√(r_3),2√(-λ_1)) and (2√(-λ_1),2√(r_1)+2√(-λ_1-r_1)) correspond to a locking situation and a nonlocally pulled situation, respectively. As is easily seen, the former interval is empty if and only if L≤L and r_3 > r_1, while the latter is empty if and only if L≤L and r_3 < r_1. Therefore the two intervals cannot be both empty. 
There are five possible cases, which are illustrated on Figure <ref>.



We remark in particular that if r_1=r_3, then L=0<L, therefore the above two intervals are both non-empty. Consequently, the patch can induce locking or nonlocal pulling, depending on the value of the
speed c_A, even if the patch size L is arbitrarily small. This shows that even a small perturbation of the homogeneous KPP equation may have a substantial impact on the large time behavior of solutions by substantially altering the spreading speed.

As shown in Lemma <ref>, the quantity λ_1 will turn out to be the generalized principal eigenvalue of the problem (<ref>). Its properties,
as a function of the parameters L or r_2, will be studied in Section <ref>. It is directly related to the leading eigenvalue λ in <cit.>, as illustrated by the correspondence between Figure <ref> and <cit.>. In particular, one may replace (<ref>) by the more general assumption that x↦ r (t,x- A(t)) is a compact perturbation of the Heaviside type function 

    ϕ : x ↦
    r_1    if x<0, 
    
    r_3    if  0 ≤ x.

Then, when A(t) = c_A t, one may redefine the principal eigenvalue λ_1 appropriately in this context and recover the formula (<ref>) for the spreading speed. We refer to Section <ref> for more details.


Regarding regularity issues, we note that the spreading speed c^⋆ in
Theorem <ref> is
only Lipschitz-continuous as a function of c_A and λ_1 (see
also Figures <ref> and <ref>).

We continue with two corollaries confirming that if the patch [A(t),A(t)+L] moves either too slowly or too fast, with
explicit thresholds given by the preceding theorem (accounting for lim_L→+∞λ_1(L)=-r_2), 
then it has no effect and the front is locally pulled, even in the presence of arbitrary oscillations.


Assume 

    sup_t≥ 0A(t)+L/t≤ 2√(r_3).

Then the front is locally pulled: c=c=2√(r_3).



Assume 

    2√(r_1)+2√(r_2-r_1)≤inf_t≥ 0A(t)/t.

Then the front is locally pulled: c=c=2√(r_1). 

These two corollaries remain true if the constant L is replaced by some (positive and continuous) function of time L(t). We also expect that sup and inf may respectively be replaced by lim sup and lim inf. 
For the sake of brevity, we leave the latter generalization as an open problem.

Finally, we turn to the interesting case where the patch is neither too slow nor too fast but oscillates slowly. For the sake of brevity, 
we deliberately do not prove a sharp result but rather construct a telling counter-example.


Let λ_1 be as in Theorem <ref>, and assume that λ_1≠ -r_1. 
Let c_A,1, c_A,2 be positive numbers satisfying  2√(-λ_1)<c_A,1<c_A,2<2√(r_1)+2√(-λ_1-r_1)

and let (t_n)_n∈⊂ be an increasing sequence such that:

    t_0=0,   t_n+1/t_n→+∞ (n→ +∞).

Let A(t) be a function defined iteratively as follows:

    A(t)=
     0     t=t_0=0,
    
    A(t_2n)+c_A,1(t-t_2n)     t∈[t_2n,t_2n+1)  (n=0,1,2,…),
    
    A(t_2n+1)+c_A,2(t-t_2n+1)     t∈[t_2n+1,t_2n+2)  (n=0,1,2,…).

Then:

    c≤1/2(c_A,2-2√(-λ_1-r_1)+4r_1/c_A,2-2√(-λ_1-r_1)),


    c≥1/2(c_A,1-2√(-λ_1-r_1)+4r_1/c_A,1-2√(-λ_1-r_1)).

In particular, c<c.



The condition λ_1 -r_1 is satisfied if and only if either r_1< r_3, or r_1 ≥ r_3 and L >L.


In the above example, the speed of the patch oscillates slower and slower between two values either of which would lead to nonlocal pulling in the absence of oscillations. The speed of the front then slowly fluctuates forever, thus making the spreading speed ill-defined. 
This result is inspired by <cit.>, where temporally constant environments with slower and slower spatial oscillations lead similarly to a gap between the minimal and the maximal spreading speeds. 
It will be clear from the proof that the result can be extended straightforwardly to the case c_A,1>c_A,2 or to the case where L or r_2 oscillates instead of c_A. 
The general conclusion we draw from this result is that oscillations of the patch, in height, width or speed, can be enough to break the uniqueness of the spreading speed, even if r_1=r_3. Furthermore, this result holds for an arbitrarily small L, which implies that a small perturbation of the homogeneous KPP equation may not only change the spreading speed (as we already outlined above) but even make the speed ill-defined.




 §.§ Relation with multi-species models


We end this section with a discussion related to the study of spreading properties of multi-species reaction–diffusion systems, which is our main motivation 
for starting the present work. The results of the present paper show how a traveling patch of a highly favorable zone 
can strongly impact the speed of the invasion fronts, even if they travel far behind the patch. 
In particular, if a nonlocal pulling occurs, finding the precise front speeds would become a much more intricate task.   
This suggests that, when one studies the behavior of invading fronts in multi-species reaction–diffusion models, determining the speeds 
of the slower fronts could be much harder than determining those of the faster fronts, since the environmental changes created by the 
faster fronts may induce such effects as nonlocal pulling or locking on the slower fronts. 
For example, consider the following (conceptual) reaction–diffusion population models:


  (A) 3-species competition model;

  (B) 2-prey & 1-predator model.


First, in the model (A), let u,v, w denote the populations of mutually competing species that invade a region at the speeds c_u>c_v>c_w. More precisely, 


  * u invades the open space at speed c_u as if it was alone in the environment; 

  * then the slower species v replaces u by competition at speed c_v; 

  * finally, the slowest species w propagates at speed c_w by replacing v. 

Let us focus on the speed of w. 
If the interspecific competition between u and v is very strong, then at the interface between the two species 
(which travels at speed c_v), 
there appears a zone where u, v are both very small. This zone can be seen as a favorable patch for the third species w. Therefore, although this patch lies far ahead of the front of w, it may still have a strong effect on the speed of w, just like the central patch in our equation (<ref>). 
At the moment, the precise effect of this patch is not fully understood, but at least it is known, though in a different context, that a strong competition between u and v can create a favorable patch for w; see <cit.>.

On the other hand, if the interspecific competition between u and v is not very strong, the superposition of the two populations would have higher density around the interface between u and v, thus creating an unfavorable zone for w, contrary to the above case. The paper of Lam, Liu and Liu 
<cit.> precisely identifies a parameter regime where this would indeed occur. In such a situation, the interface between u and v may not have much effect on the speed of w.

Next we consider the model (B). Let u, v denote the prey populations and w the predator that invade a region at speeds c_u>c_v>c_w, basically in the same manner as (1), (2), (3) above, except that the slowest species w is a predator. We assume that the interspecific competition between u and v is relatively mild, so that the superposition of the two populations have higher density around the interface between u and v, as in <cit.>. Then this zone may serve as a favorable patch for the predator w, therefore it may have a non-negligible effect on the speed of w, just as in the case (A) with strong inter-specific competition between u an v.  
In such parameter regimes, the study of the speed of the predator would require a sharp quantification of the height, width and speed of the corresponding patch, which is currently out of reach in general. The same discussion may be held in the case of one prey and two predators, for which we refer to <cit.> where among other things the nonlocal pulling phenomenon was also observed.










§ PROOFS





 §.§ Technical preliminaries




  §.§.§ A change of variables and a parabolic eigenproblem in the whole space-time


In what follows we assume in addition that A is of class 𝒞^2. 

Following Allwright <cit.>, we can transform the linearized equation

    ∂_t u - ∂_xx u = r(t,x)u

into

    ∂_t v - 1/L^2∂_yyv = ( m(y)+y A”(t)L/2)v,

where

    m(y)=r_1 1_y<0 + r_2 1_0≤ y <1 + r_3 1_1≤ y,

by means of the change of variable

    v(t,y)=u(t,Ly+A(t))exp(∫_0^tA'(τ)^2/4τ +yA'(t)L/2).


Similarly, considering an eigenproblem where r(t,x) is replaced by r(t,x)+λ
changes in the new equation m(y) into m(y)+λ. More precisely, inspired by 
<cit.>, we are led to considering the following eigenproblem:

    𝒫φ_1=λφ_1    in ×, 
    φ_1>0    in ×, 
    φ_1∈{ψ∈𝒟'(^2) | ∂_tψ, ∂_yψ, ∂_yyψ∈ L^1_loc(^2)},

with

    𝒫=∂_t - 1/L^2∂_yy - ( m(y)+y A”(t)L/2).


Let us emphasize that, to the best of our knowledge, there is no known general theory
for such eigenproblems. In particular, if A”(t)≢0, keeping φ, ψ within a bounded range is already an issue difficult to overcome. This is why from now on we focus on a more feasible special case. Other special cases where this eigenproblem makes sense might be investigated in future sequels.




  §.§.§ An elliptic eigenproblem in the whole space


If A is linear, written as A(t)=c_At, then the above transformation reduces to

    ∂_t v -1/L^2∂_yy v = m(y)v,

with the same change of variable

    v(t,y)=u(t,L y +c_At)exp(c_A^2 t/4 +c_A L y/2).


To study the above equation, the following eigenproblem for the operator

    ℒ=L^-2^2/ y^2+m,

where m is as defined in (<ref>), will play a crucial role:

    -ℒφ_1 =λφ_1    in , 
    φ_1>0    in , 
    φ_1∈ W^2,1_loc().

This problem can be solved explicitly, piece by piece with a global 𝒞^1 regularity[Recall
that W^2,1_loc() is continuously embedded into C^0,α_loc() for any α∈(0,1). 
After a standard bootstrap procedure, we find that the weak solution φ_1 of
-L^-2φ_1”-mφ_1=λφ_1 is
in C^1,1_loc()≃ W^2,∞_loc(), namely it is of class C^1 and its derivative
is locally Lipschitz-continuous.]. 

Unlike a similar eigenvalue problem on a finite interval, an eigenvalue λ of -ℒ with a positive eigenfunction is not necessarily unique. Therefore the right concept of principal eigenvalue has to be defined carefully. According to <cit.>, the second-order self-adjoint operator -ℒ admits a generalized principal eigenvalue λ_1, which can be defined as the supremum of the set of values of λ for which (ℒ+ λ) φ≤ 0 holds for some positive function φ. 
Equivalently, it can be defined as the limit of the principal eigenvalues of truncated problems with the Dirichlet boundary condition. By analogy with many previous studies on KPP heterogeneous problems, we expect this generalized eigenvalue to be crucial to accurately predict the spreading speed of solutions for compactly supported initial data. Therefore, this subsection is devoted to the computation of this λ_1.

For later use and convenience, we sum up some of the main results from <cit.>.


Define

    λ_1 := sup{λ  |  ∃φ∈ W^2,1_loc () ,  φ >0   and  (ℒ+ λ) φ≤ 0 }∈ℝ.

Then λ_1 satisfies

    λ_1     =    max{λ  |  ∃φ∈ W^2,1_loc () ,  φ >0   and  - ℒφ = λφ}
        =    inf{λ  |  ∃φ∈ W^2,1_loc () ∩ L^∞ () ,  φ >0   and  (ℒ + λ) φ≥ 0 }.

In particular, if there exists a positive and bounded solution φ of -ℒφ = λφ on , then one must have λ = λ_1.
Furthermore, 
    λ_1 = lim_R → +∞λ_1^R,

where λ_1^R denotes the unique solution of the eigenvalue problem

    -ℒφ  = λ_1^R φ   in  (-R,R), 
    φ_1>0    in  (-R,R), 
    φ_1 (± R) = 0 .



As a preliminary step, we note that

    -r_2 ≤λ_1≤-max(r_1,r_3).

Indeed, on the one hand, from our assumption that r_2 = max m > max (r_1,r_3), we get by taking φ =1 as a test function that

    ( ℒ - r_2) φ≤ 0.

Hence, by its definition, λ_1 ≥ -r_2. On the other hand, for any eigenvalue λ>-max(r_1,r_3), any eigenfunction is a sinusoidal function at least on a half-line, and therefore necessarily changes sign. Since Proposition <ref> implies that λ_1 is associated with a positive eigenfunction, we infer that λ_1 ≤ -max (r_1,r_3).


Hereafter, we will compute the eigenpair (λ_1,φ_1) in three complementary cases:





  * L>L;

  * r_1<r_3 and L≤L;

  * r_1>r_3 and L≤L.

Recall that the threshold L is defined by:

    L=
    
    0    if r_1=r_3, 
    1/√(r_2-max(r_1,r_3))arccot(√(r_2-max(r_1,r_3)/|r_1-r_3|))    if r_1≠ r_3.

This definition will become natural in a moment. Among other properties of the eigenpair that will be summarized at the end of this subsection, we will establish the following lemma.
 The generalized principal eigenvalue λ_1 satisfies:


  * if L ≤L, then 

    λ_1 = - max (r_1, r_3 );


  * if L > L, then λ_1 is the unique zero of the function

    λ∈(-r_2,λ)↦(L√(r_2+λ))-r_2+λ-√((r_1+λ)(r_3+λ))/√(r_2+λ)(√(-r_1-λ)+√(-r_3-λ))

where λ = min( -max (r_1,r_3) , π^2/L^2 - r_2 ).


The constructions below will involve real constants C_1,C_2,C_3,C_4,C_5
with different meanings in each case.




  §.§.§ Construction of the generalized principal eigenpair in the case L>L


In the first case we look for an eigenfunction of the form:

    φ_1(y)=
    
    C_1exp(L√(-r_1-λ_1)y)    if  y≤ 0, 
    
    
    C_2sin(L√(r_2+λ_1)y+C_3)    if 0<y<1, 
    
    C_4exp(-L√(-r_3-λ_1)y)    if  1≤ y,

with C_1 >0,C_2,C_3,C_4 satisfying

    C_1=C_2sin(C_3),
    
    C_2sin(L√(r_2+λ_1)+C_3)=C_4exp(-L√(-r_3-λ_1)),
    
    C_1√(-r_1-λ_1)=C_2√(r_2+λ_1)cos(C_3),
    
    C_2√(r_2+λ_1)cos(L√(r_2+λ_1)+C_3)=-C_4√(-r_3-λ_1)exp(-L√(-r_3-λ_1)),
    
    0<C_3<L√(r_2+λ_1)+C_3<π.

These conditions ensure that the function φ_1 is C^1 and positive. If such an eigenfunction exists, and since it is bounded by construction, then its corresponding eigenvalue λ_1 indeed coincides with the one from Proposition <ref>, hence our use of the same notation.

Fixing without loss of generality C_1=1 (which implies φ_1(0)=1) and rearranging terms thanks to classical trigonometric identities, we end up with:

    C_2=1/sin C_3,
     C_3 =√(-r_1-λ_1/r_2+λ_1) ,
    
    C_4=^L√(-r_3-λ_1)sin(L√(r_2+λ_1)+C_3)/sin C_3,
    (L√(r_2+λ_1))=r_2+λ_1-√((r_1+λ_1)(r_3+λ_1))/√(r_2+λ_1)(√(-r_1-λ_1)+√(-r_3-λ_1)) ,
    
    0<C_3<L√(r_2+λ_1)+C_3<π.

Note that the last inequality implies L√(r_2+λ_1)<π, so that (L√(r_2+λ_1)) is well-defined indeed. 

It only remains to show that the equation defining implicitly λ_1 admits indeed a solution smaller than 
λ=min(-max(r_1,r_3),π^2/L^2-r_2). Note that the value
of λ depends on the sign of L-π/√(r_2-max(r_1,r_3)). Even with the restriction 
L>L, this quantity can have any sign, since the arccot function maps (0,+∞) onto (0,π/2),
so that the threshold π/√(r_2-max(r_1,r_3)) is larger than 2L.

The function

    λ∈(-r_2,λ)↦(L√(r_2+λ))-r_2+λ-√((r_1+λ)(r_3+λ))/√(r_2+λ)(√(-r_1-λ)+√(-r_3-λ))

is the sum of two smooth decreasing functions, which can be checked after a lengthy computation which we omit here. When λ→-r_2, it tends to +∞.
When λ→λ, it tends to:


  * -∞ if one of the following two conditions hold true:


  * r_1=r_3, 

  * r_1≠ r_3 and L≥π/√(r_2-max(r_1,r_3));


  * (L√(r_2-max(r_1,r_3)))-√(r_2-max(r_1,r_3)/|r_1-r_3|) if r_1≠ r_3 and
L<π/√(r_2-max(r_1,r_3)).

The finite limit in the second case has exactly the sign of
L-L, by decreasing monotonicity of the arccot function, and is consequently negative.
Therefore, in all cases, the continuous decreasing function

    λ∈(-r_2,λ)↦(L√(r_2+λ))-r_2+λ-√((r_1+λ)(r_3+λ))/√(r_2+λ)(√(-r_1-λ)+√(-r_3-λ))

admits indeed a unique zero.

We conclude that in the case L > L, we have found a unique positive and bounded eigenfunction of the form (<ref>), associated with the eigenvalue λ_1 defined both as the unique zero of (<ref>), and as the generalized principal eigenvalue from Proposition <ref>.




  §.§.§ Construction of the generalized principal eigenpair in the case r_1<r_3 and L≤L

In the second case we look for an eigenpair (λ_1, φ_1) of the form:

    λ_1 = - r_3,

and

    φ_1(y)=
    
    C_1exp(L√(-r_1-λ_1)y)    if y ≤ 0, 
    
    C_2sin(L√(r_2+λ_1)y+C_3)    if 0<y<1, 
    
    C_4 L (y-1)+C_5    if 1 ≤ y,

where C_1>0,C_2,C_3,C_4,C_5 satisfy

    C_1=C_2sin(C_3),
    
    C_2sin(L√(r_2+λ_1)+C_3)=C_5,
    
    C_1√(-r_1-λ_1)=C_2√(r_2+λ_1)cos(C_3),
    
    C_2√(r_2+λ_1)cos(L√(r_2+λ_1)+C_3)=C_4,
    
    0<C_3<L√(r_2+λ_1)+C_3≤π/2.

The last condition of (<ref>) ensures the nonnegativity of C_4, and the positivity of C_5, hence of φ_1 for all y > 1. On the other hand, notice that the eigenfunction φ_1 is no longer bounded here. Still, we have already pointed out that there is no positive eigenfunction associated with some λ > -r_3. Therefore, if an eigenfunction of the above type (<ref>) exists, then the corresponding eigenvalue -r_3 = λ_1 is the largest eigenvalue associated with a positive eigenfunction,  hence it again coincides with the generalized principal eigenvalue from Proposition <ref>.

Fixing without loss of generality C_1=1 (i.e., φ_1(0)=1) and rearranging terms as in the previous case, we end up with:

    C_2=1/sin C_3,
     C_3 = √(r_3-r_1/r_2-r_3).
    
    C_4=√(r_2-r_3)cos(L√(r_2-r_3)+C_3)/sin C_3,
    
    C_5=sin(L√(r_2-r_3)+C_3)/sin C_3,
    
    0<C_3<L√(r_2-r_3)+C_3≤π/2.

The value of C_3∈(0,π/2) is uniquely defined.
It only remains to verify that L√(r_2-r_3)+C_3≤π/2. By assumption, L≤L<π/√(r_2-r_3). Moreover, π/2-C_3∈(0,π/2). 
Hence both L√(r_2-r_3) and π/2-C_3 are in (0,π), where the function  is well-defined and decreasing, and 
therefore L√(r_2-r_3)+C_3≤π/2 is equivalent to 

    (L√(r_2-r_3))≥(π/2-C_3)=1/ C_3=√(r_2-r_3/r_3-r_1),

where classical trigonometric identities have been used. This inequality is true by virtue of the assumption
L≤L. Note that in the equality case, C_4=0.



  §.§.§ Construction of the generalized principal eigenpair in the case r_1>r_3 and L≤L

The construction in the third case is done exactly as in the second case, interchanging the roles of r_1 and r_3.



  §.§.§ Important properties of the generalized principal eigenpairs

To summarize, the elliptic problem (<ref>) admits a generalized principal eigenvalue λ_1 characterized in Proposition <ref> and computed in Lemma <ref>. The eigenpair (λ_1 ,  φ_1) also satisfies the following:


  * if L>L (which in particular is always the case when r_1= r_3), then λ_1∈(-r_2,-max(r_1,r_3)), φ_1∈ L^∞(), φ_1'(0)=L√(-r_1-λ_1)>0, φ_1 decays exponentially at ±∞;

  * if r_1<r_3 and L≤L, then λ_1=-r_3, φ_1'(0)=L√(r_3-r_1)>0, φ_1 grows linearly at +∞ and decays exponentially at -∞;

  * if r_1>r_3 and L≤L, then λ_1=-r_1, φ_1'(0) ≤ 0, φ_1 grows linearly at -∞ and decays exponentially at +∞;

  * the definition of λ_1 when L>L implies, by virtue of 
the implicit function theorem, that (L,r_1,r_2,r_3)↦λ_1 is smooth in the parameter set {L>L};
the definition of λ_1 when L≤L implies that (L,r_1,r_2,r_3)↦λ_1 is also smooth in the parameter set {L≤L, r_1<r_3} and in the parameter set 
{L≤L, r_1>r_3}; it is therefore at least 
a piecewise-smooth function globally, with at this point an unclear regularity at the interface {L=L}.





 §.§ Proof of Theorem <ref>

In this section, for the sake of brevity and clarity, c^⋆ is defined by anticipation as the quantity on the right-hand side of (<ref>) in Theorem <ref>, i.e.,

    c^⋆ =
    
    2√(r_3)   if c_A<2√(r_3), 
    
    c_A    if 2√(r_3)≤ c_A≤ 2√(-λ_1) ,
    
    F(c_A)    if 2√(-λ_1)<c_A<2√(r_1)+2√(-λ_1-r_1),
    
    2√(r_1)   if 2√(r_1)+2√(-λ_1-r_1)≤ c_A

where the value of λ_1 is given in Lemma <ref> and F is defined by (<ref>). 
Consequently, c^⋆ is not, at this point, a notation for a spreading speed. With
this slight abuse of notation, the proof of Theorem <ref> reduces to the proof of two inequalities: 
c≤ c^⋆ and c≥ c^⋆.

The proof will be done in four steps:

    
  (Step 1) construct a family of super-solutions showing that c≤ c^⋆ if c_A≤ 2√(-λ_1);
    
  (Step 2) construct a family of super-solutions showing that c≤ c^⋆ if c_A>2√(-λ_1);
    
  (Step 3) construct a family of sub-solutions showing that c≥ c^⋆ if c_A≤ 2√(max(r_1,r_3)) or c_A≥ 2√(r_1)+2√(-λ_1-r_1));
    
  (Step 4) construct a family of sub-solutions showing that c≥ c^⋆ if finally c_A∈(2√(max(r_1,r_3)),2√(r_1)+2√(-λ_1 - r_1)).

Bringing all four steps together will immediately end the proof. We note that the open interval (2√(max(r_1,r_3)),2√(r_1)+2√(-λ_1 - r_1)) is empty if, and only if, r_3 < r_1=-λ_1. In such a case, the proof ends at Step 3.

 
Assume c_A≤ 2√(-λ_1). Define

    u(t,x)=2min(1,^-λ(max(2√(r_3),c_A))(x-max(2√(r_3),c_A)t-L)),

where λ(c)=1/2(c-√(c^2-4r_3)). 
This function u(t,x) decays exponentially as x→ +∞ and its front propagates at the speed max(2√(r_3),c_A). 
Let us show that u is a super-solution of (<ref>). Note first that u(t,x) satisfies the inequality

    ∂_tu-∂_xxu≥ f(t,x,u(t,x))

in {x-max(2√(r_3),c_A)t<L}∪{x-max(2√(r_3),c_A)t>L}. 
Indeed, if x-max(2√(r_3),c_A)t<L, then since u=2>1, we have

    ∂_tu-∂_xxu=0> f(t,x,u(t,x)),

while, if  x-max(2√(r_3),c_A)t>L, then since λ(c)(c-λ(c))=r_3, we have

    ∂_tu-∂_xxu=r_3u=r(t,x)u≥ f(t,x,u(t,x)).

Note also that the “correct angle condition” (i.e., negative derivative gap)  at x-max(2√(r_3),c_A)t=L is clearly satisfied. Therefore u is a super-solution of (<ref>).

Note that Cu is also a super-solution of (<ref>) for any C≥ 1. Indeed, if x-max(2√(r_3),c_A)t<L, then since Cu=2C>1, we have

    ∂_t(Cu)-∂_xx(Cu)=0> f(t,x,Cu(t,x)),

while, if  x-max(2√(r_3),c_A)t>L, then  

    ∂_t(Cu)-∂_xx(Cu)=r(t,x)(Cu)≥ f(t,x,Cu(t,x))

by the assumption (<ref>). Also the correct angle condition at x= max(2√(r_3),c_A)t+L clearly remains to hold after multiplication by a positive constant C.
Now choose C sufficiently large so that Cu is initially above the initial condition u_0. Then u(t,x)≤ Cu(t,x) for all t≥ 0, x∈. 

Recall that u decays exponentially as x→ +∞ and its front propagates at the speed max(2√(r_3),c_A). As we are assuming c_A≤ 2√(-λ_1), we have c^⋆ = max (2 √(r_3), c_A ).  Hence the inequality u(t,x)≤ Cu(t,x) proves that c≤ c^⋆.


 Assume now that c_A > 2 √(-λ_1), so that in particular c_A > 2 √(r_1) by (<ref>). Now, following <cit.>, we consider the following continuous function:

    u(t,x)=
    
    2    if x≤ ct-ln 2/λ(c) , 
    ^-λ(c)(x-ct)   if x∈(ct-ln 2/λ(c),c_At) ,
    ^-λ(c)(c_A-c)t^-c_A(x-c_At)/2φ_1(x-c_At/L)    if x≥ c_At ,

where c is a constant satisfying 2 √(r_1)≤ c<c_A, whose value will be specified later, and in this step
λ(c)=1/2(c-√(c^2-4r_1)). The principal eigenfunction φ_1, which is constructed as in Section <ref>, 
is appropriately normalized so that φ_1(0)=1. Since φ_1 is in both cases λ_1=-r_3 or λ_1>-r_3 bounded above by a 
linear function as x→+∞, it is clear that u decays to 0 as x→ +∞
and that the position of its front propagates at the speed c. Let us show that u is a super-solution of (<ref>). 

First, in {x<ct-ln 2/λ(c)}, u satisfies, by virtue of the assumption (<ref>):

    ∂_t u-∂_xxu = 0≥ f(t,x,u(t,x)).

In {ct-ln 2/λ(c)<x<c_At}, u satisfies:

    ∂_t u-∂_xxu = r_1u(t,x)≥ f(t,x,u(t,x)).

In {x>c_At}, it is more convenient to change the viewpoint by defining

    v(t,y)=u(t,L y +c_At)√(L)^c_A^2 t/4 +c_A L y/2=√(L)^-λ(c)(c_A-c)t^c_A^2t/4φ_1(y).

It satisfies by construction

    ∂_tv-1/L^2∂_yyv-mv = (-λ(c)(c_A-c)+c_A^2/4)v+λ_1v.

Assuming λ_1-λ(c)(c_A-c)+c_A^2/4≥ 0, we obtain ∂_tv-1/L^2∂_yyv-mv≥ 0, which gives (back to the original variables)

    ∂_tu-∂_xxu≥ r(t,x)u(t,x)≥ f(t,x,u(t,x)).

Summarizing, under the assumption that λ_1-λ(c)(c_A-c)+c_A^2/4≥ 0, u satisfies

    ∂_t u-∂_xxu≥ f(t,x,u(t,x))

on {x<ct-ln 2/λ(c)}∪{ct-ln 2/λ(c)<x<c_At}∪{c_At<x }.

In order for u to be a super-solution, we also need to verify the angle conditions  (i.e., negative derivative gap) at x=ct-ln 2/λ(c) and at x=c_A t. The former is easy to check by the definition of u in (<ref>), so we only need to check

    lim_x-c_At→ 0^-∂_x u(t,x)≥lim_x-c_At→ 0^+∂_x u(t,x),

which reads:

    -λ(c)≥-c_A/2+φ_1'(0)/L,

or equivalently

    φ_1'(0)/L≤c_A-c+√(c^2-4r_1)/2.

Thus, for u to be a super-solution, it suffices to verify the following two conditions:

    φ_1'(0)/L≤c_A-c+√(c^2-4r_1)/2, 
    λ_1-λ(c)(c_A-c)+c_A^2/4≥ 0.


Using the properties summarized in Section <ref>, we distinguish two cases:


  * if λ_1=-r_1, then by construction of φ_1, φ_1'(0)≤ 0,
whence φ_1'(0)/L≤ 0<(c_A-c+√(c^2-4r_1))/2, so that
the first condition is always satisfied;

  * if λ_1≠ -r_1, then by construction of φ_1, 
φ_1'(0)/L=√(-r_1-λ_1) and the first condition rewrites as

    √(-r_1 - λ_1)≤c_A/2 - λ (c).

Squaring both sides, and using that (c_A/2-λ(c))^2=c_A^2/4-r_1-(c_A-c)λ(c) thanks to the equality λ(c)^2-cλ(c)+r_1=0, we find that the first condition implies the second one.

In all cases, in order to show that u is a super-solution, all we have to verify is (<ref>), or equivalently

    √(c^2 - 4 r_1)≥ c - c_A   + 2 √(-λ_1 - r_1).


Let us first consider the subcase when c_A ≥ 2 √(r_1) +  2 √(-λ_1 - r_1). Then the above inequality is automatically satisfied with c = 2 √(r_1). 
Thus u is a super-solution if we choose  c = 2 √(r_1). As in the previous case, Cu is also a super-solution for any C>1. Choose C appropriately large so that Cu(0,t)≥ u_0(x). Then u(t,x)≤ Cu(t,x) for all t≥ 0, x∈. Since u decays to 0 as x→ +∞ and propagates at the speed c=2√(r_1), the inequality u(t,x)≤ Cu(t,x) proves that c≤ 2 √(r_1), where 2 √(r_1) precisely coincides with c^⋆ in this subcase. Hence c≤ c^⋆ in this subcase.

Next assume that c_A < 2 √(r_1) + 2 √(-λ_1 - r_1). 
Since c ≥ 2 √(r_1), we have c - c_A   + 2 √(-λ_1 - r_1)>0. Hence 
we can square both sides of (<ref>) and rewrite it as

    c≥1/2(c_A-2√(-λ_1-r_1)+4r_1/c_A-2√(-λ_1-r_1))=F(c_A)

where the function F is the one defined in (<ref>). It is a smooth and decreasing function. It maps the interval 
(2√(-λ_1-r_1),2√(r_1)+2√(-λ_1-r_1)] onto the interval
[2√(r_1),+∞). Moreover its unique fixed point is F(2√(-λ_1))=2√(-λ_1). Therefore
it satisfies 2√(r_1)≤ F(c')<c' for any c'∈(2√(-λ_1),2√(r_1)+2√(-λ_1-r_1)]. Notice also that c^⋆ coincides with F (c_A) in this same interval.

Finally, the partial differential inequality and the angle condition are again satisfied whenever 
2√(-λ_1) < c_A < 2 √(r_1) + 2 √(-λ_1 -r_1) and for any c∈ [F(c_A), c_A), this last interval
being nonempty. Therefore u is a super-solution in this case. Arguing precisely as in the previous case, we see that Cu is also a super-solution for any C≥ 1. By choosing C sufficiently large, we have u(t,x)≤ Cu(t,x) as in the previous case, which proves that c≤ c^⋆ if c_A> 2√(-λ_1).


Before we proceed, for later use we summarize our analysis in this step in the form of the following lemma, which is presented in a slightly generalized setting:


Assume c_A>2√(-λ_1) and that A(t) = c_A t for t∈[0,T) for some 0<T≤ +∞. 
Let u be the function defined by (<ref>), with c = c^⋆ from (<ref>). Then for any constant C≥ 1, Cu is a super-solution of (<ref>) 
for t ∈ [0,T) and x ∈ℝ.



Define r̃(t,x) by (<ref>) with r_2 replaced by r_3, 
and let f̃(t,x,u) be a function satisfying (<ref>), ∂_uf̃(t,x,0)=r̃(t,x) and f̃(t,x,u)≤ f(t,x,u). 
Denote by ũ(t,x) the solution of (<ref>) with f replaced by f̃ for the same initial data u_0. 
Then ũ is a subsolution of the original (<ref>), therefore ũ(t,x)≤ u(t,x) for all t≥ 0, x∈. 
Thus the speed of ũ gives a lower bound for the speed of u.
If r_1=r_3, then r̃(t,x) is identical to r_1, therefore the spreading speed of ũ is clearly 2√(r_1). 
On the other hand, if r_1 r_3, then r̃(t,x) satisfies the assumption of Theorem <ref>, therefore 
the spreading speed of ũ is again well understood. 
Note that the assertions of Theorem <ref> remain valid even in the case r_1=r_3. 
Indeed, if r_1=r_3, the cases (a) (ii) and (b) (ii) become empty and all other cases give the value c^⋆=2√(r_1)=2√(r_3). 
Therefore, in what follows we do not deal with the case r_1=r_3 separately. 

First, we consider the case where c_A ≤ 2 √(max (r_1,r_3)).  
If r_1≤  r_3, then by (b) (i) of Theorem <ref>, the speed of ũ is 2√(r_3), which coincides with the value of c^⋆ in (<ref>). On the other hand, if r_1>r_3, then (a) (i), (ii) of Theorem <ref> apply, which show that the speed of ũ equals max ( 2 √(r_3), c_A). 
Again this value coincides with c^⋆ in (<ref>), since 2√(max(r_1,r_3))≤ 2√(-λ_1) by virtue of (<ref>). 
Thus we get in this case that c≥ c^⋆.

Next, in the case where c_A ≥ 2 √(r_1 )+2 √(-λ_1 - r_1), then by (<ref>) we have c_A ≥ 2√(r_1) + 2 √(max(r_3 - r_1 ,0)). 
Hence, by Theorem <ref> (a) (iii) and (b) (iii), the spreading speed of ũ is 2 √(r_1). This gives 
c≥ 2 √(r_1), which again coincides with c^⋆.

To summarize this step, we have found as announced that c≥ c^⋆ if either c_A ≤ 2 √(max (r_1 ,r_3)) or c_A ≥ 2√(r_1) + 2 √(-λ_1 - r_1).


 Here we assume that c_A∈(2√(max(r_1,r_3)),2√(r_1)+2√(-λ_1 - r_1)). Therefore, throughout this step we have that c^⋆ = min(c_A, F (c_A) )> 2 √(r_1), where F is given by (<ref>). 
We refer to Step 2 above for related computations on the function F, including the fact that F (c') > c' for any c'<2√(-λ_1).

Now we consider, for large values of R>0,

    -L^-2(φ^R_1)”-mφ_1^R=λ_1^Rφ_1^R    in (-R,R), 
    φ_1^R(± R)=0,    
    φ_1^R>0    in (-R,R), 
    φ_1^R∈ W^2,1((-R,R)).

By the Krein–Rutman theorem, 
the above problem possesses the principal eigenpair (λ_1^R, φ_1^R), with φ_1^R>0, which is unique up to multiplication of φ_1^R by a constant.
Moreover, according to <cit.> (see also Proposition <ref> above), we have that

    λ_1=lim_R→+∞λ_1^R.

Below, we extend φ_1^R in  by setting φ_1^R=0 in \(-R,R). We consider the following continuous function:

    u(t,x)=
    
    0    if x<ln S/η+x_0-2R', 
    ι O(x)    if x∈[ln S/η+x_0-2R',ln S/η+x_0-R'/3), 
    ισ/2   if x∈[ln S/η+x_0-R'/3,ln S/η+x_0+ct+x_1), 
    ι P(t,x)    if x∈[ln S/η+x_0+ct+x_1,X(t)), 
    ι Q(t,x)    if x≥ X(t),

where

    O(x)     = σsin(π/2R'(x-ln S/η-x_0+2R')), 
    
    P(t,x)     = ^-λ(c)(x-ct-x_0)-S^-(λ(c)+η)(x-ct-x_0), 
    
    Q(t,x)     = γ^-λ(c)(c_A-c)t^-c_A(x-c_At)/2φ_1^R(x-c_At/L),

with η>0, S>0, R'>π/2√(r_1), ι∈(0,1], 
σ∈(0,2max_x∈P(0,x)), where λ(c)=1/2(c-√(c^2-4r_1)) and  c∈(2√(r_1),c_A), which is a nonempty interval here, x_0∈, γ>0, R>1, 
with the principal eigenfunction φ_1^R appropriately normalized so that φ_1^R(0)=1. 
The values of x_1, X(t) are chosen to make u continuous at x=ln S/η+x_0+ct+x_1 and at x=X(t). More precisely:

    
  * x_1>0 is the well-defined smallest zero of x↦σ/2-P(0,x+x_0+ln S/η) (note that it satisfies indeed, for
    any t≥ 0, σ/2=P(t,ln S/η+x_0+ct+x_1) and note moreover that x_1→0 as σ→ 0);
    
  * X(t)∈(c_At-RL,c_At) is, for each t≥ 0, the smallest zero of x↦ P(t,x)-Q(t,x).

The values of ι, σ, η, S, x_0, R and γ will be specified later on. 
Typically, 1/ι, 1/σ, 1/η, 1/γ, S, -x_0 and R are large positive numbers.  We will also check later that c can be chosen arbitrarily close to c^⋆.

The existence and uniqueness of X(t), provided the parameters are appropriately chosen, will be verified in Lemma 
<ref>. In particular, we point out that r(t,x)=r_1 if x≤ X(t) due to X(t) ≤ c_A t. Lemma <ref> will also establish that ln S/η+x_0 < inf_t ≥ 0 X(t) - ct. Then, choosing σ small enough so that ln S/η + x_0 + x_1 ≤inf_t ≥ 0 X(t) -ct, we get that each interval in the above definition of u is nonempty, which in turn means that u is continuous.

Note that, for each t≥ 0, the support of u(t,x) is the interval [ln S/η+x_0-2R', c_At+RL], and that the main front of u(t,x), which is represented by the above function P(t,x), propagates at the speed c.
Behind the front, u(t,x)=ισ/2 except near the left endpoint of its support. Moreover, u is globally
proportional to ι and this parameter can be interpreted as an amplitude parameter.

In order for u to be a potential sub-solution, first the following angle condition 
(i.e., positive derivative gap) needs to be satisfied:

    lim_x→ X(t)^-∂_x u(t,x)≤lim_x→ X(t)^+∂_x u(t,x).

This will be a consequence of Lemma <ref> below, whose first two statements imply

    lim_x → X(t)^-∂_x u (t,x) < 0 < lim_x → X(t)^+∂_x u (t,x).

We point out that the other angle conditions at the interfaces of the other subintervals are more straightforward, hence we skip their proof for the sake of brevity.

Next, we check that u satifies the desired differential inequalities on each subdomain. First, in the region {x<ln S/η+x_0-2R'}, u is identical to 0, which directly yields ∂_tu-∂_xxu≤ f(t,x,u(t,x)).

In {x∈[ln S/η+x_0-2R',ln S/η+x_0-R'/3)}, the function u satisfies:

    ∂_tu-∂_xxu
    =π^2/4R'^2u.

Recall that ln S/η + x_0 - R'/3 < 0 ≤ c_A t for t ≥ 0 by Lemma <ref>, and thus we have that ∂_u f (t,x,0) = r_1 here. By virtue of (<ref>) and π^2/4R'^2<r_1, we can assume that σ is so small that

    f(t,x,v) ≥π^2/4 R'^2 v,

for any x≤ln S/η+x_0-R'/3 and v ∈ [0, σ]. Remarking that u≤ισ≤σ in  { x ∈[ln S/η+x_0-2R',ln S/η+x_0-R'/3) } (by virtue of ι≤ 1), we deduce:

    ∂_tu-∂_xxu≤ f(t,x,u(t,x)).


In {x∈[ln S/η+x_0-R'/3,ln S/η+x_0+ct+x_1)}, again and quite similarly,

    ∂_tu-∂_xxu =0 ≤ f(t,x,u(t,x)),

provided σ is small enough, uniformly in ι≤ 1.

Next, in {ln S/η+x_0+ct+x_1<x<X(t), t>0}, the function u satisfies:

    ∂_tu-∂_xxu
        =ι r_1^-λ(c)(x-ct-x_0)-ι S(r_1-η^2+η√(c^2-4r_1))×^-(λ(c)+η)(x-ct-x_0)
        = r_1u(t,x)-ι Sη(√(c^2-4r_1)-η) ×^-(λ(c)+η)(x-ct-x_0)
       ≤u(t,x)^-η(x-ct-x_0)((r_1-f(t,x,u(t,x))/u(t,x))^η(x-ct-x_0)-Sη(√(c^2-4r_1)-η)) 
          +f(t,x,u(t,x)).

We claim that, denoting g(t,x,u)=r_1-f(t,x,u)/u, the function

    (t,x)↦ g(t,x,u(t,x))^η(x-ct-x_0) - S η( √(c^2 - 4 r_1 ) - η)

is negative in {ln S/η+x_0+ct<x<X(t), t>0} for η small enough and S large enough (depending on η). If this claim holds true, then u indeed satisfies the desired differential inequality in this subdomain.

Let us therefore verify this claim. First recall that X(t) < c_A t. Then, by the assumption (<ref>),

    r_1 u ≥ f(t,x,u) ≥ r_1 u - M u^2  for all (t,x)∈{x<c_At} and all u≥ 0,

so that

    0 ≤ g(t,x,u) ≤ M u  for all  (t,x) ∈{x < c_A t},

for some M>0. It follows that

    g(t,x,u(t,x))^η(x-ct-x_0) - S η( √(c^2 - 4 r_1 ) - η) 
         ≤ M u (t,x) ^η(x-ct-x_0) - S η( √(c^2 - 4 r_1 ) - η) 
         ≤ M ι^-(λ(c) - η) (x -ct-x_0)  - S η(√(c^2 - 4 r_1) - η)
         ≤ M ι^-(λ(c) - η) ln S/η  - S η(√(c^2 - 4 r_1) - η)

in {ln S/η+x_0+ct<x<X(t), t>0}. Therefore, provided that S>1 and η<λ(c), we get that

    g(t,x,u(t,x))^η(x-ct-x_0) - S η ( √(c^2 - 4 r_1 ) - η) 
       ≤ S ( Mι S^- λ(c)/η  -  η (√(c^2 - 4 r_1) - η) ) 
       ≤ S ( Mι S^- 1  -  η (√(c^2 - 4 r_1) - η) ) .

We deduce that if 

    η<min(λ(c),√(c^2-4r_1))  and  
    S>max(1,M/η(√(c^2-4r_1)-η)),

then u is a sub-solution in {ln S/η+x_0+ct<x<X(t), t>0},
independently of the exact values of ι∈ (0,1], R>1 and x_0∈.

Finally, for {x>X(t)}, we introduce a new function

    v(t,y) =u(t,Ly+c_At)√(L)^c_A^2 t/4 +c_A L y/2
    = ιγ√(L)^-λ(c)(c_A-c)t^c_A^2t/4φ_1^R(y).

It satisfies by construction

    ∂_tv-1/L^2∂_yyv-mv=(-λ(c)(c_A-c)+c_A^2/4+λ_1^R)v.

Assume that

    λ_1-λ(c)(c_A-c)+c_A^2/4< 0,

set δ=1/2|λ_1-λ(c)(c_A-c)+c_A^2/4| and recall lim_R→+∞λ_1^R=λ_1.
Assume now that R is so large that λ_1^R-λ(c)(c_A-c)+c_A^2/4<-δ, whence
∂_tv-1/L^2∂_yyv-mv≤ -δv, 
which gives (back to the original variables)

    ∂_tu-∂_xxu≤ (r(t,x)-δ)u(t,x).

Using again (<ref>) which ensures the continuity of (t,x,u)↦ f(t,x,u)/u, and all other parameters being fixed, we can assume that γ>0 is so small that 

    r(t,x)-δ≤inf_(t,x,v)∈××[0,γ e^c_A R L/2maxφ_1^R ]f(t,x,v)/v .

Remarking that u≤γ e^c_A R L/2maxφ_1^R in {x>X(t)} (by virtue of ι≤1 and X(t) ≥ c_A  t - R L), we deduce:

    ∂_tu-∂_xxu≤ f(t,x,u(t,x)).

We observe that, c<c_A ∈(2√(max(r_1,r_3)),2√(r_1)+2√(-λ_1 - r_1)) being given, the above necessary condition (<ref>), i.e., λ_1-λ(c)(c_A-c)+c_A^2/4< 0, is equivalent to c<F(c_A). Finally, we have shown that the function defined in (<ref>) is a sub-solution, and that its speed c can be chosen arbitrarily close to min( c_A , F (c_A) ), which coincides with c^⋆ in this parameter range.


To conclude Step 4, it remains to prove important properties of X(t).



Let P(t,x),Q(t,x) be as in (<ref>). 
For any positive values of c,η,S,R,γ with c<c_A, there exists x_0∈ such that 
the equation P(t,x)=Q(t,x), or more precisely, 

    ^-λ(c)(x-ct-x_0)-S^-(λ(c)+η)(x-ct-x_0)
        = γ^-λ(c)(c_A-c)t^-c_A(x-c_At)/2φ_1^R(x-c_At/L)

admits for all t≥ 0 an isolated solution X(t)∈ such that:


  * ∂_x P(t,X(t))<0; 

  * ∂_x Q(t,X(t))>0;

  * c_At-RL<X(t)<c_At;

  * ln S/η< inf_ t ≥ 0  X(t)-ct-x_0.

Moreover, the trajectory t↦ X(t) satisfies:

  
 

  * X∈𝒞^1([0,+∞),(-RL,+∞));

  * X(t)=c_At+O(1) as t→+∞.




For any t≥ 0, the function 
P:x↦^-λ(c)(x-ct-x_0)-S^-(λ(c)+η)(x-ct-x_0) is unimodal, increasing on the left of
x=ct+x_0+1/ηln(S(λ(c)+η)/λ(c)) and decreasing on its right.
Therefore, if 

    x_0<-RL-1/ηln(S(λ(c)+η)/λ(c)),

then, since c<c_A, x↦ P(t,x) is, at any t≥ 0, decreasing in [c_At-RL,c_At]. 
In particular, statement (a) will be a consequence of (c).

For any t≥ 0, the function

    y∈[-R,+R]↦γ^-λ(c)(c_A-c)t^-c_A L y/2φ_1^R(y)

admits as derivative:

    y∈[-R,+R]↦γ^-λ(c)(c_A-c)t^-c_A Ly/2(-c_A L/2φ_1^R(y)+ ( φ_1^R ) '(y)).

At y=-R, by virtue of the Hopf lemma (i.e., ( φ_1^R ) '(-R)>0), this derivative is positive.
Similarly, at y=+R, the derivative is negative.
Therefore, by virtue of the intermediate value theorem, there exists r∈(-R,R) such that 
y↦γ^-λ(c)(c_A-c)t^-c_ALy/2φ_1^R(y)
is increasing in (-R,-R+r). Without loss of generality, up to reducing r we may assume that -R+r <0.

Consequently, the function 

    x∈[-R L +c_At,-(R-r)L+c_At]↦ Q(t,x)
 
is increasing. 
Statement (b) of Lemma <ref> follows from this monotonicity property, along with the fact to be established below that X(t) belongs to the interval [-R L + c_A t, -(R-r) L + c_A t].

In view of the monotonicities in x of each side of the equality (<ref>) (decreasing on the left, increasing on the right), if -x_0 is so large that

    ^λ(c)((R-r)L+x_0)<γ^c_A/2(R-r)Lφ_1^R(-R+r),

then by the intermediate value theorem, at t=0 there is a (unique) solution X(0) of (<ref>) in (-RL,-(R-r)L).
By the implicit function theorem, this solution can be extended in a continuously differentiable way in an open time interval around t=0. 
To show that X(t) can be extended globally, for all t≥ 0, it suffices to prove that X(t)<c_At-(R-r)L for any t such that X(t) is well-defined. 
In order to verify this inequality, recall that X(t) satisfies P(t,X(t))=Q(t,X(t)) and that P(t,x) (respectively Q(t,x)) is monotone decreasing (respectively increasing) in x. Therefore all we need to show is that Q(c_At-(R-r)L,t)>P(c_At-(R-r)L,t) for such t, or equivalently,

    γ^-λ(c)(c_A-c)t^c_A/2(R-r)Lφ_1^R(-R+r)  
        >    ^-λ(c)(c_At-(R-r)L-ct-x_0)-S^-(λ(c)+η)(c_At-(R-r)L-ct-x_0) .

This follows directly from:

    γ^c_A/2(R-r)Lφ_1^R(-R+r)     >^λ(c)((R-r)L+x_0)
        >^λ(c)((R-r)L+x_0)-S^(λ(c)+η)((R-r)L+x_0)-η(c_A-c)t,

where we used (<ref>).

By construction, we have that c_A t - R L < X (t) < c_A t, i.e., (c) holds true. As explained above, statement (a) and (b) follow, and so does (f). The differentiability property (e) follows from the implicit function theorem. Finally, (d) i.e the estimate ln S/η< inf_t ≥ 0  X(t)-ct-x_0 results from a direct calculation:

    X(t)>c_At-RL>c_At+x_0+ln S/η+ln(1+η/λ(c))/η>ct+x_0+ln S/η,

where we used (<ref>). This completes the proof of Lemma <ref>.


To summarize Step 4, thanks to Lemma <ref>, the key partial differential inequality and the angle condition (<ref>) are verified for any 
c∈(2√(r_1),min(c_A,F(c_A))), with an amplitude parameter
ι∈(0,1] whose value can be chosen arbitrarily small. 

Since u is compactly supported, choosing ι small enough, u is below the solution u at time t=1.
Hence u is a sub-solution for all t≥ 1 and this proves that c≥ c^⋆ if c_A∈(2√(max(r_1,r_3)),2√(r_1)+2√(-λ_1-r_1)).


Again, for later use we state the following lemma which sums up our construction of a sub-solution in this last step.


Assume 
that c_A∈(2√(max(r_1,r_3)),2√(r_1)+2√(-λ_1 - r_1)) and 
that A(t) = c_A t. Let c^⋆ and u respectively be defined by (<ref>) and (<ref>) with c= c^⋆ - ε. 
Then for any ε >0 sufficiently small, there exist positive constants R', σ, η, S, R, x_0, γ such that, for all ι∈ (0,1], the function u is a sub-solution of (<ref>) 

for t >0 and x ∈ℝ whose front propagates at the speed c. For each t≥ 0, the support of u(t,x) is compact, and 
u is proportional to the amplitude parameter ι.  





 §.§ Proof of Corollaries <ref> and <ref>


The results follow from simple comparison arguments. 
As the proof proceeds exactly the same way, we replace the constant L in (<ref>) by an arbitrary positive and continuous function L(t).

 
Assuming that sup_t ≥ 0A(t) + L(t)/L(t)≤ 2 √(r_3),
we have that

    r (t,x) ≤ r(t,x) ≤r (t,x),

where

    r :(t,x)↦min( r_1, r_2, r_3 )    if x< 2 √(r_3) t, 
    
    r_3    if  2 √(r_3) t ≤ x,


    r :(t,x)↦max( r_1, r_2, r_3 )    if x< 2 √(r_3) t, 
    
    r_3    if  2 √(r_3) t ≤ x .

Let u and u denote the solutions of (<ref>) corresponding to r and r, respectively. Then by the comparison principle, 

    u(t,x)≤ u(t,x)≤u(t,x).

According to Theorem <ref>, u and u both spread with speed 2 √(r_3). Consequently, we have c = c = 2√(r_3) under the assumptions of Corollary <ref>.


 The proof proceeds similarly. Notice that if inf_t ≥ 0A(t)/t≥ 2 √(r_1) + 2 √(r_2 -r_1), then

    r (t,x) ≤ r(t,x) ≤r (t,x),

where

    r :(t,x)↦
    r_1     if x< 2 √(r_1) + 2 √(r_2 -r_1), 
    min( r_1, r_2, r_3 )     if  2 √(r_1) + 2 √(r_2 -r_1)≤ x,


    r :(t,x)↦
    r_1    if x<  2 √(r_1) + 2 √(r_2 -r_1), 
    max( r_1, r_2, r_3 )    if  2 √(r_1) + 2 √(r_2 -r_1)≤ x .

Recall that max( r_1, r_2,r_3 ) = r_2. Applying Theorem <ref> and using the comparison principle as in the proof of Corollary <ref>, we find that c = c =  2√(r_1).















 §.§ Proof of Theorem <ref>
 

Here we prove that if the moving speed of the heterogeneity alternates between two values very slowly, then the minimal and maximal spreading speeds c and c may differ. We will prove this result by constructing appropriate super- and sub-solutions.
 
The values of r_1,r_2,r_3,L being fixed, we deduce from the statement of Theorem <ref> a value
for λ_1 which does not depend on c_A. By assumption, λ_1≠-r_1, and 2√(-λ_1)<c_A,1<c_A,2<2√(r_1)+2√(-λ_1-r_1). For each i∈{1,2}, we define

    c^⋆_i = F(c_A,i) = 1/2(c_A,i-2√(-λ_1-r_1)+4r_1/c_A,i-2√(-λ_1-r_1)).

As one can check, we have

    2√(r_1)<c^⋆_2<c^⋆_1 < 2√(-λ_1)<c_A,1<c_A,2.

Note that the above definition of c^⋆_i (i=1,2) agrees with that of c^⋆ in (<ref>) for the case 2√(-λ_1)<c_A<2√(r_1)+2√(-λ_1-r_1), where c_A will be replaced by c_A,1 or c_A,2 in the later arguments. This enables us to apply Lemma <ref> and Lemma <ref> in the construction of super- and sub-solutions.

Hereafter, we define 

    r̃ : z ↦
    r_1    if  z< 0, 
    
    r_2    if  0 ≤ z < L , 
    
    r_3    if  L ≤ z .

Thus the function r(t,x) in (<ref>) is expressed as r(t,x)=r̃(x-A(t)).

The aim of this section is to prove that, under the assumptions of Theorem <ref>, the minimal and maximal spreading speeds of the solution u of (<ref>) satisfy

    c≤ c^⋆_2 + 2 ε < c^⋆_1 - 2 ε≤c,

for all sufficiently small ε >0, which implies c≤ c^⋆_2 < c^⋆_1  ≤c.

Before starting the proof, we note that the function A(t) defined in (<ref>) satisfies

    c_A,1t ≤ A(t)<c_A,2t  for all  t≥ 0.

In particular, we have A(t_2n+1)<c_A,2 t_2n+1 (n=0,1,2,3,…).

We will use basically the same super- and sub-solutions constructed in the proof of Theorem <ref> with minor modifications. Since we are assuming 2√(-λ_1)<c_A,1<c_A,2<2√(r_1)+2√(-λ_1-r_1), the functions u defined in (<ref>) in Step 2 of Section <ref> and u defined in (<ref>)  in Step 4 will be relevant. 

First, in the time interval I_2n+1=[t_2n+1,t_2n+2), we use the function u defined in (<ref>) with c_A replaced by c_A,2 and with c=c^⋆_2. 




It gives an upper bound for c. 
In order to make this function to serve as a super-solution in our later argument, we present a slightly modified version of Lemma <ref> as follows:


Assume c_A>2√(-λ_1) and that A(t) = c_A (t-τ) + B for t∈[τ, τ+T) for some constants B, τ≥ 0 and T>0. 
Let u be the function defined by (<ref>), with c = c^⋆ from (<ref>). Then for any constant C≥ 1, Cu(t-τ, x-B) is a super-solution of (<ref>) for t ∈ [τ, τ+T) and x ∈ℝ.



By the change of variables s=t-τ, y=x-B, the above function is converted to Cu(s,y) and (<ref>) is converted to

    ∂_s u = ∂_yy u + f̂(s,y,u)   (0<s<T, y∈),

where f̂(s,y,u):=f(s+τ,y+B,u). Since f̂ satisfies the same assumption as (<ref>) and (<ref>) for τ∈[0,T), Cu(s,y) is a super-solution of the above equation by Lemma <ref>. Coming back to the original variables proves the lemma.



Next, in the time interval I_2n=[t_2n,t_2n+1), we will modify the sub-solution u in (<ref>). More precisely, we will consider a function u_1,δ that satisfies

    ∂_t u_1,δ≤∂_xxu_1,δ +  (r̃ (x-c_A,1 t) - δ) u_1,δ - M u_1,δ^2,

and propagates with some speed c  ≥   c^⋆_1 - ε. Notice that the spreading speed c^⋆ depends continuously on the values r_1,r_2,r_3, so that the above speed inequality holds true if δ is sufficiently small. This will give a lower bound for c.

With these notations, we will proceed in two steps to prove Theorem <ref>. 



Let u_2 denote the function u in (<ref>) with c=c^⋆_2 as in (<ref>), which 
coincides with c^⋆ in (<ref>) for the case 2√(-λ_1)<c_A<2√(r_1)+2√(-λ_1-r_1), where c_A is replaced by c_A,2. 
For each n∈, since A(t)=A(t_2n+1)+c_A,2(t-t_2n+1) for t∈[t_2n+1,t_2n+2), by Lemma <ref>, Cu_2(t-t_2n+1,x-A(t_2n+1) is a super-solution of (<ref>) for t∈[t_2n+1,t_2n+2) and for any constant C≥ 1.
Recall also that u_2(x,t) decays exponentially as x→ +∞, and that its front propagates at the speed c=c^⋆_2. 

Let us also introduce an auxiliary super-solution u^#. 
Let η>0 and let

    λ^#=c_A,2/2+√(-r_3-λ_1)+η,   c^#=λ^#+r_2/λ^#.

Choose K>1 such that K min(1,  ^-λ^# x)≥ u_0(x) for all x∈. 
By construction, c^#≥ 2√(r_2) and 

    (λ^#)^2 - c^#λ^#  = -r_2 ≤ -r(t,x),

for all t >0 and x ∈ℝ. Thus

    u^#:(t,x)↦min(1,^-λ^#(x-c^# t))

is a super-solution of (<ref>) satisfying u^#≥ u globally in time and space.
Moreover, we can choose η large enough so that (c^# -c_A,2)t_1>L, which implies

    (c^# -c_A,2)t_2n+1>L  for all n∈.

Thanks to this auxiliary super-solution, we have a rough control of the decay of u(t,x) as x→+∞ for any t>0. 

Next, with u_2 as defined above, we construct a sequence (K_n)_n∈ such that

    u^#(t_2n+1,x)≤ K_nu_2(0,x-A(t_2n+1))  for all x∈, all large  n∈.

In order to show that such (K_n)_n∈ exists, it suffices to show that

    sup_x∈ u^#(t_2n+1,x)/u_2(0,x-A(t_2n+1)) < +∞  for all large n∈.

Let us estimate the above quantity.

First, in the region x≥ c^# t_2n+1, since (c^# -c_A,2)t_2n+1>L, we have

    x ≥ c_A,2 t_2n+1 + L ≥ A (t_2n+1) + L .

Therefore, by the definition of u_2 in (<ref>) for the case x≥ c_A t,

    u_2 (0, x- A (t_2n+1) )     =    e^- c_A,2/2 (x- A (t_2n+1))φ_1 ( x- A (t_2n+1)/L) 
       ≥    C e^- c_A,2/2 (x- A (t_2n+1)) e^-√(-r_3 - λ_1) (x- A (t_2n+1))  ,

for some C>0. Here we used the fact that either r_1 < r_3, or r_1 ≥ r_3 and L > L (see Remark <ref>), so that φ_1 is defined by either (<ref>) or (<ref>). Moreover, from our assumptions on the sequence (t_n)_n ∈ we have that

    A (t_2n+1) = c_A,1 t_2n+1 +  o (t_2n+1),

as n → +∞.
It follows that, by our choice of λ^# and η >0,

    u^#(t_2n+1,x)/u_2(0,x-A(t_2n+1))   ≤   K/C^-λ^# (x - c^# t_2n+1 ) e^( c_A,2/2 + √(-r_3 - λ_1))(x- A (t_2n+1))
       ≤   K/C  e^( c_A,2/2 + √(-r_3 - λ_1))(c^# t_2n+1- A (t_2n+1))
       ≤   K/C^(c_A,2/2+√(-r_3-λ_1))(c^#-c_A,1 +1)t_2n+1,

for all x ≥ c^# t_2n+1.

Next, in the region x≤ c^# t_2n+1, we have u^# (t_2n+1,x) = K, and, by (<ref>), 

    u_2 (0,x - A (t_2n+1)) = min( 2 , e^-λ (c^⋆_2) (x- A (t_2n+1))) ≥ 1,

if x ≤ A (t_2n+1), while using also the definition of φ_1 in either (<ref>) or (<ref>),

    u_2 (0, x - A (t_2n+1)) ≥^-c_A,2(x - A(t_2n+1) )/2min_[0,1]φ_1 × e^-√(-r_3 - λ_1) (x - A (t_2n+1))

if A (t_2n+1)≤ x≤ c^# t_2n+1. Thus

    inf_x ≤ c^# t_2n+1u_2 (0,x-A(t_2n+1)) ≥min_[0,1]φ_1  ×  e^-( c_A,2/2 + √(-r_3 - λ_1)) (c^# t_2n+1 - A (t_2n+1)) >0.
 
Hence

    u^#(t_2n+1,x)/u_2(0,x-A(t_2n+1))   ≤   K/min_[0,1]φ_1×  e^( c_A,2/2 + √(-r_3 - λ_1))(c^# t_2n+1 - A (t_2n+1))
       ≤   K/min_[0,1]φ_1×  e^( c_A,2/2 + √(-r_3 - λ_1)) (c^#- c_A,1 + 1) t_2n+1
 
for x ≤ c^# t_2n+1.

Therefore, for (<ref>) to hold, it suffices to define the sequence (K_n) by

    K_n= K  max( 1/C, 1/min_[0,1]φ_1 ) ^(c_A,2/2+√(-r_3-λ_1))(c^#-c_A,1+1)t_2n+1,

for all large n. Combining (<ref>) and the inequality u^#≥ u, we obtain

    u(t_2n+1,x)≤ u^#(t_2n+1,x)≤ K_n u_2(0,x-A(t_2n+1)).

As mentioned before, K_n u_2(t-t_2n+1,x-A(t_2n+1)) is a super-solution in t∈ I_2n+1, hence, by the comparison principle, we have for n large enough that

    u(t,x)≤ K_nu_2(t-t_2n+1,x-A(t_2n+1))  for all x∈, t∈ I_2n+1.

In particular, setting t=t_2n+2, we obtain

    u(t_2n+2,x)≤ K_nu_2(|I_2n+1|,x-A(t_2n+1))  for all x∈.

Let B_n denote the value of the right-hand side of (<ref>) at x=x_n: =(c^⋆_2+2ε) t_2n+2, that is,

    B_n:=K_nu_2(|I_2n+1|,t_2n+2(c^⋆_2+2ε-A(t_2n+1)/t_2n+2)).

Then, by (<ref>) we have that

    B_n    = K_n e^-λ (c^⋆_2 )  ( 2 ε t_2n+2 - A (t_2n+1) + c^⋆_2   t_2n+1)
        =K_n^-2λ(c^⋆_2)ε t_2n+2^λ(c^⋆_2)(A(t_2n+1)- c^⋆_2 t_2n+1),

provided that

    c^⋆_2 (t_2n+2-t_2n+1)-ln 2/λ(c^⋆_2)≤ t_2n+2(c^⋆_2+2ε-A(t_2n+1)/t_2n+2)≤ c_A,2(t_2n+2-t_2n+1),

which is clearly true for n large enough and ε >0 small, since t_2n+1=o(t_2n+2) as n→+∞.
Using (<ref>) and again the assumption t_2n+1=o(t_2n+2), we see that the right-hand side of (<ref>) tends to 0 as n→+∞. It follows that

    max_x≥ x_n K_nu_2(|I_2n+1|,x-A(t_2n+1))→ 0  as  n→ +∞,

since u_2(|I_2n+1|,x-A(t_2n+1)) is monotone decreasing in x. This and (<ref>) imply

    max_x≥ x_n u(t_2n+2,x)→ 0  as  n→ +∞,

which proves c≤ c^⋆_2+2ε.



By virtue of the global boundedness of the solution u of (<ref>), there exists ρ > 0 
such that 
f(t,x,u(t,x))≥ -ρ u(t,x) for any (t,x)∈[0,+∞)×. Therefore, by the comparison principle, for any t>0 and x∈,

    u(t,x)≥e^-ρ t/√(4π t)∫_^-(x-y)^2/4tu_0(y) y.

Then, there exists β>0 and x_0∈ such that u_0≥β1_[x_0-β,x_0+β]. 
It follows that, for any c>0 and t>0,

    u(t,ct)≥β^-ρ t - c^2/4 t/√(4π t)∫_x_0-β^x_0+β^cy/2-y^2/4t y
        ≥β^2^-ρ t -c^2/4 t/√(π t)^c(x_0-β)/2^-x_0^2+β^2+2|x_0|β/4t.

Recall that u_1,δ is a subsolution of (<ref>), as constructed in (<ref>) with c ≥ c^*_1 - ε; see also Lemma <ref>. For any n∈, the support of u_1,δ (0,x-A(t_2n)) is exactly

    A(t_2n)+[ln S/η+x_0 -2R',L R].

Consequently, for any n∈,
the support of u_1,δ(0,x-A(t_2n)) is included in [c_A,1t_2n+ln S/η+x_0  -2R',c_A,2t_2n+L R].
Provided n is sufficiently large, say n≥ n_0, the support is included in [(c_A,1-ε)t_2n,(c_A,2+ε)t_2n].
In such an interval, the decay of x↦ u(t_2n,x) can therefore be estimated
as follows (up to increasing n_0):

    min_x∈[(c_A,1-ε)t_2n,(c_A,2+ε)t_2n]u(t_2n,x)
        ≥ C^-C' t_2n,

where C,C'>0 are constants that only depend on β, x_0, ρ, δ, c_A,1, c_A,2, ε.

Now, defining a sequence (κ_n)_n∈ by 

    κ_n = C^-C' t_2n/max_x∈u_1,δ(0,x)  for all n∈,

we deduce

    κ_nu_1,δ(0,x-A(t_2n))≤ u(t_2n,x)  for all n≥ n_0, x∈,

and then

    κ_nu_1,δ(t,x-A(t_2n))≤ u(t+t_2n,x)  for all n≥ n_0, x∈,  t >0,

by the parabolic comparison principle.

At this point of the proof, we remind that u_1,δ travels with speed c ≥ c_1^⋆ - ε and is associated, not to the reaction term f, but to (t,x,v)↦ (r- δ) v - M v^2. More precisely, it satisfies (<ref>), i.e.,

    ∂_t u_1,δ≤∂_xxu_1,δ + (r̃(x-c_A,1 t) - δ) u_1,δ - M u_1,δ^2,

where r̃ is defined in (<ref>). Therefore, if we multiply the function κ_nu_1,δ (t,x-A(t_2n))
by ^δ/2t, we should still obtain a sub-solution, at least as long as this sub-solution is small enough.

Let us verify this last claim. First define 

    g:(t,x,v)↦r̃(x-c_A,1 t) v - M v^2,

so that

    g (t - t_2n ,x - A (t_2n),v) ≤ f (t , x ,v),

for any t ∈ [t_2n, t_2n+1 ], x ∈ℝ and v ≥ 0. 
Also we define

    u_1,δ^n:(t,x)↦u_1,δ (t-t_2n,x-A(t_2n)).

Using (<ref>), the function
(t,x)↦κ_n^δ/2(t-t_2n)u_1,δ^n(t,x)
is a sub-solution of (<ref>) as long as t ≤ t_2n+1 and

    M  ( κ_n e^δ/2 (t-t_2n) -1 ) u_1,δ^n (t,x) ≤δ/2 .

Now recall that u_1,δ is globally bounded by a constant C”>0 that depends on δ (hence on ε) but not on n. 
This constant is also an upper bound, independent of n, for each function u_1,δ^n.
Then the function
(t,x)↦κ_n^δ/2(t-t_2n)u_1,δ^n(t,x)
is a sub-solution of (<ref>) provided that
y

    t-t_2n≤ T_n =2/δln(1/κ_n(1+δ/2 M C”)) < t_2n+1 - t_2n,

where the latter inequality follows from the fact that t_2n+1/t_2n→ +∞ and | lnκ_n | = O (t_2n) by (<ref>).

By virtue of the comparison principle and (<ref>),

    κ_n^δ/2(t-t_2n)u_1,δ^n(t,x)≤ u(t,x)  for all n≥ n_0, x∈, t∈ [t_2n,t_2n+T_n].

By a similar sub-solution construction, not growing in time but with an appropriately chosen amplitude (the algebra
is exactly the same), we can then prove that

    κ_n^δ/2T_nu_1,δ^n(t,x)≤ u(t,x)  for all n≥ n_0, x∈, t∈ [t_2n+T_n,t_2n+1].


Noting that κ_n^δ/2T_n = 1+δ/2 M C”, we deduce

    ( 1+δ/2 M C”)u_1,δ^n(t_2n+1,(c^⋆_1-2ε)t_2n+1)≤ u(t_2n+1,(c^⋆_1-2ε)t_2n+1)  for all n≥ n_0.

Using t_2n=o(t_2n+1) together with the detailed formula (<ref>) where c ≥ c^⋆_1 - ε, it follows that

    u_1,δ^n(t_2n+1,(c^⋆_1-2ε)t_2n+1)= u_1,δ (t_2n+1 - t_2n, (c^⋆_1 - 2 ε) t_2n+1 - A (t_2n) ) = ισ/2>0.

Finally we conclude that c≥ c^⋆_1 - 2 ε. This ends the proof of Step 2. As ε can be chosen arbitrarily small, the proof of Theorem <ref> is complete.






§ PROPERTIES OF Λ_1


Let λ_1 denote, as before, the principal eigenvalue of (<ref>) characterized by Proposition <ref>. 
In this section, we uncover new properties of the map (L,r_1,r_2,r_3)↦λ_1. Recall that if

    L>L=
        
        0    if r_1=r_3, 
    1/√(r_2-max(r_1,r_3))arccot(√(r_2-max(r_1,r_3)/|r_1-r_3|))    if r_1≠ r_3,

then λ_1 is characterized as the unique solution
in (-r_2,min(-max(r_1,r_3),π^2/L^2-r_2)) of the equation (<ref>),
whereas λ_1 = - max (r_1,r_3) if L ≤L.
Recall also that (L,r_1,r_2,r_3)↦λ_1 
is smooth in each of the parameter sets {L>L}, {L≤L, r_1<r_3} and 
{L≤L, r_1>r_3}.




 §.§ Monotonicity and symmetry


According to <cit.>, the function (L,r_1,r_2,r_3)↦λ_1 is nonincreasing and
concave with respect to each variable. It is also 1-Lipschitz-continuous with respect to r_1, r_2 and r_3. 
The following two propositions state the monotonicity properties of λ_1 with respect to L and r_2 in a more precise manner. 


The map L∈(0,+∞)↦λ_1(L) is continuous, constant in (0,L] (this interval might be empty), 
decreasing in (L,+∞), with the following asymptotic or particular values:


  * λ_1(L)→-max(r_1,r_3) as L→L^+;

  * λ_1(π/2√(2r_2-r_1-r_3/(r_2-r_1)(r_2-r_3)))=-r_2^2-r_1r_3/2r_2-r_1-r_3;

  * λ_1(L)→-r_2 as L→+∞.




The monotonicity of L∈(L,+∞)↦λ_1(L) follows directly from the chain rule applied 
to the equation satisfied by λ_1(L); in fact, ∂λ_1/∂ L<0.

The limit as L→+∞ is an immediate consequence of L√(r_2+λ_1)<π. The limit as L→L^+ follows similarly, separating the case r_1=r_3 and the case r_1≠ r_3. 
The continuity at L=L when L>0 follows.

The function 

    λ↦r_2+λ-√((r_1+λ)(r_3+λ))/√(r_2+λ)(√(-r_1-λ)+√(-r_3-λ))

is increasing, continuous and maps (-r_2,-max(r_1,r_3)) onto (-∞,+∞) if r_1=r_3 or
onto (-∞, √(r_2-max(r_1,r_3)/|r_1-r_3|)) if r_1≠ r_3. Denote in both
cases ζ∈ (0,+∞) ∪{ + ∞} the upper limit of this image interval.
Consequently, for any ζ∈(-∞,ζ), there exists
a unique preimage λ_1∈(-r_2,-max(r_1,r_3)) such that 

    r_2+λ_1-√((r_1+λ_1)(r_3+λ_1))/√(r_2+λ_1)(√(-r_1-λ_1)+√(-r_3-λ_1))=ζ,

and subsequently there exists a unique L_0∈(L,+∞) such that 
λ_1=λ_1(L_0), with L_0 given by the following formula:

    L_0=1/√(r_2+λ_1)arccot(ζ).


Now we are in a position to pick admissible values of ζ that correspond to remarkable values of the
 function. For instance, ζ=0 leads to:

    L_0√(r_2+λ_1(L_0))=π/2, 
    
    r_2+λ_1(L_0)=√((r_1+λ_1(L_0))(r_3+λ_1(L_0))).

After elementary manipulations, we deduce:

    λ_1(L_0)=-r_2^2-r_1r_3/2r_2-r_1-r_3, 
     L_0=π/2√(2r_2-r_1-r_3/(r_2-r_1)(r_2-r_3)),

which completes the proof.


Next, the function L is either 0 or a decreasing continuous function of r_2. In the latter case, it also satisfies that L→ 0 as r_2 → +∞. Thus, given a fixed value of L, there exists
a threshold r_2≥max(r_1,r_3) such that L > L for any r_2>r_2, and L < L for any r_2 ∈ (max (r_1,r_3), r_2) (this latter interval possibly being empty). 
Thus, in the latter case, λ_1 = - max (r_1,r_3) as mentioned above. 
Quite similarly to the previous proposition, we deduce the following result; for the sake of brevity, we omit the proof.


The map r_2∈(max(r_1,r_3),+∞)↦λ_1(r_2) is continuous, constant in (max(r_1,r_3),r_2]
(this interval might be empty), decreasing in (r_2,+∞), with the following asymptotic or particular values:


  * λ_1(r_2)→-max(r_1,r_3) as r_2→max (r_1,r_3); 

  * λ_1(r_2)→-∞ as r_2→+∞.




Finally, we point out that the eigenvalue λ_1 is symmetric with respect to the parameters r_1 and r_3.
 
λ_1(r_1,r_3)=λ_1(r_3,r_1).



By the change of the variable y↦ 1-y in (<ref>), the role of r_1 and r_3 are exchanged, therefore the above symmetry is obvious.





 §.§ More general heterogeneities and optimization issues


In view of our method of proof, it should be clear that the main result (the piece-by-piece formula 
for the spreading speed in Theorem <ref>) will extend to many equations of the form 

    ∂_t u -∂_xxu=f(t,x,u)

with ∂_u f(t,x,0)=m(x-c_At), m∈ L^∞(), inf m>0. More precisely, since
the existence of λ_1 and φ_1 is given by <cit.>, 
the following properties are the only true requirements to end the construction of super- and sub-solutions in the proof
of Theorem <ref>:

    
  * m=r_3 in a neighborhood of +∞ – this is used to construct the explicit super-solution in Step 1;
    
  * m=r_1 in a neighborhood of -∞ – this is used to construct the explicit super-solution in Step 2;
    
  * lim supφ_1'/φ_1≤√(-r_1-λ_1) at -∞ – this is used to validate the angle condition in Step 2;
    
  * ^-c_A/2xφ_1(x)→ 0 at +∞ – this is used to ensure that the super-solution of Step 2
    acts indeed as a barrier for the solution.

It turns out that the third and fourth ones are direct consequences of the first and second. Indeed, recall from Proposition <ref> (see also again <cit.>) that φ_1 is the limit of the eigenfunctions of a truncated Dirichlet problem. Using this together with the maximum principle and λ_1 ≤ -max (r_1,r_3), one may check that φ_1 is either affine or identical to

    e^√(-r_1 - λ_1)x

on a left-half line depending on whether λ_1 = -r_1 or λ_1 < -r_1. In the former case, φ_1 ' must be nonnegative and in the latter, φ_1 ' is precisely equal to √(-r_1 -λ_1)φ_1, that is, (3) holds true. The same argument shows that φ_1 grows at most linearly at +∞. 

Therefore the exact variations of m between the left half-line where m=r_1 and the right half-line where m=r_3
do not really matter, and we are able to handle arbitrary bounded variations[We actually conjecture
that m→ r_1 at -∞ and m→ r_3 at +∞ are sufficient for our purposes. This, however, requires more
work.].

Of course, such extensions are made at the expense of the formula (<ref>) that characterizes λ_1. 
But they make it possible to study, for instance, smooth growth rates or piecewise-constant growth rates with more than
one traveling patch. By doing so, we obtain a nice connection with classical shape optimization results. For instance,
if r_1=r_3, m≥ r_1, and L^∞ and L^1 constraints are imposed on m-r_1, then the function m which minimizes the eigenvalue λ_1 (and in turn maximizes the spreading speed) is bang-bang and contains precisely only one patch (i.e., m-r_1 is the indicator function of a bounded interval). In other words, the situation we studied in the present paper corresponds to this optimal situation.





§ ACKNOWLEDGMENT


This work was initiated during a visit of T.G. and L.G. at the Meiji Institute for Advanced Studies of Mathematical Sciences (MIMS).
H.M. is supported by KAKENHI 21H00995. T.G. and L.G. acknowledge support from the ANR via the project Indyana under grant agreement ANR-21-CE40-0008 and via the project Reach under grant agreement ANR-23-CE40-0023-01. The three authors also acknowledge support from the CNRS via IRN ReaDiNet.


plain




 Paper Title: Unique continuation for nonlinear variational problems.


Anti-correlation between Flux and Photon Index of Hard X-ray Emission from The Crab
    [
    August 5, 2024
===================================================================================




§ ABSTRACT 
This paper is dedicated to the unique continuation properties of the solutions to nonlinear variational problems. Our analysis covers the case of nonlinear autonomous functionals depending on the gradient, as well as more general double phase and multiphase functionals with (2,q)-growth in the gradient. We show that all these cases fall in a class of nonlinear functionals for which we are able to prove weak and strong unique continuation via the almost-monotonicity of Almgren's frequency formula. As a consequence, we obtain estimates on the dimension of the set of points at which both the solution and its gradient vanish. 
        

 
AMS-Subject Classification. 
35B40, 35J60, 


Keywords. 
Unique continuation, nonlinear variational problems, Whitney decomposition, Almgren's frequency function

















§ INTRODUCTION

Unique continuation type results for elliptic operator have been a central theme of investigation in PDEs for many years, we refer for instance to  <cit.> for the cases of linear and semilinear operators, the analysis of eigenfunctions and the fractional case. In this paper, on the other hand, we present a result on the (strong) unique continuation property for nonlinear elliptic equations. 

As an introductory example, consider the functional ℱ: W^1, q(B) → defined by


    ℱ(u) ∫_B L(∇ u) ,


where the lagrangian L: ^d → is 


    L(p) 1/2| p |^2 + 1/q| p |^q,   with   q >2,


and the associated variational problem is


    argmin{ℱ(φ): φ∈ W^1, q(B)  and φ = u_0  on ∂ B }


for some boundary datum u_0 ∈ W^1, q(B) and some ball B⊂^d.

Let the function u be the unique (by strict convexity) solution to (<ref>), then u solves the nonlinear elliptic equation


    ( (1+|∇ u |^q-2) ∇ u )=0.


Thanks to the pioneering work of Marcellini <cit.>, it is now known that if q satisfies some upper bounds depending on the dimension (see <ref> below), the solution u is at least C^2, α smooth (<cit.>). We notice that when 2<q<3, the function u is a solution to a problem of the form 


    (A(x)∇ u)=0,


where the matrix field


    A(x):=(1+|∇ u |^q-2) Id


is only (q-2)-Hölder continuous even when u∈ C^2(B), provided that ∇^2 u(x) ≠ 0.    

Hence, to the best of our knowledge, the well established theory of Garofalo and Lin <cit.>, which requires A∈ C^0,1, cannot be employed to study the property of unique continuation for the solution u. Thus, the main obstruction to the development of a unique continuation theory for this type of functionals is not the possible lack of regularity of u, but the nonlinear nature of the lagrangian. 



In this paper, using the strategy developed in <cit.>, which in turn was inspired by <cit.>, we are able to prove a unique continuation result for a class of nonlinear problems including the above case. In particular, even though the matrix field A(x) from (<ref>) is, in general, only Hölder continuous, the quasilinear structure of (<ref>) allows us to recover the strong unique continuation property. In fact, our strategy applies to more general functionals 


    ℱ: W^1, q(B) → ,    ℱ(φ) ∫_B L(x, φ,∇φ),


where the lagrangian L: ^d××^d → is 


    L(x, s,p) 1/2| p |^2 + F(x,s,p),


and where the function F satisfies the following conditions:



  * F is Lipschitz continuous in x and there are constants γ, C>0, and a neighborhood 𝒰⊂^d××^d of the origin such that 


    |F(x,s,p)|+|∇_xF(x,s,p)|≤ C(|p|^2+γ+|s|^2)   for every    (x,s,p)∈𝒰;




  * F is differentiable in p and there exist constants γ, C>0, and a neighborhood 𝒰⊂^d××^d of the origin such that 


    |∇_pF(x,s,p)|≤ C(|p|^1+γ+|s|^1+γ)   for every    (x,s,p)∈𝒰 ;




  * there exist constants γ, C>0 and a neighborhood 𝒰⊂^d××^d of the origin such that, 
for every (x,s,p)∈𝒰, the function 

    f_x,s,p(t):= F(p,s(1+t))

is differentiable at t=0 and we have the following estimate: 


    |f_x,s,p'(0)|≤ C(|p|^2+γ+|s|^2)  for every  (x,s,p)∈𝒰 .


We notice that F when differentiable at s, we have that 

    f_x,s,p'(0)=s ∂_sF(x,s,p).

From now on, for simplicity, we will often write s ∂_sF(x,s,p) in place of f_x,s,p'(0).



To state our results more clearly, it is convenient to work under the following


    
    There exist constants 0<α, δ_0<1 such that
    
    
    u _C^1, α( B )≤δ  for some δ∈(0,δ_0).

    
    Moreover, there exists a constant C = C(d, δ_0)>0 such that
    
    
    u _C^0, α( B_r/2(x) )≤ C ( _B_r(x) u^2 )   and  ∇ u _C^0, α( B_r/2(x) )≤C/r( _B_r(x) u^2 )

    
    for all balls B_r(x) with B_r(x) ⊂ B.



The regularity of the minimizers of ℱ is now known to be strongly related to the growth of ℱ in the p variable. The unique continuation, on the other hand, relies on the behavior of ℱ near singular points of u, where s=0 and p=0.



Since the unique continuation property is localized at singular points, i.e. where u=|∇ u | = 0, the linear estimates in <ref> are a consequence of the C^1, α regularity of u. For the validity of <ref> we refer to <ref> and <ref> below.    


Our main result is the following


    Let the function u ∈ W^1, q(B) be a local minimizer of (<ref>) under <ref> and suppose that for some point x_0 ∈ B
    
    
    lim_r→ 01/r^n_B_r(x_0) u^2 = 0     for all   n ∈ .

    
    Then,
    
    
    u ≡ 0  in   B.

    



    Let the function u ∈ W^1, q(B) ∩ C^1, α(B) be a local minimizer of (<ref>) and suppose that u= 0 in some open subset U⊂ B, then u≡ 0 in B.


Finally, as a consequence of the classical Federer's dimension reduction principle, we obtain the following 


 Let the function u ∈ W^1, q(B) ∩ C^1, α(B) be a local minimizer of (<ref>), then either u≡ 0 in B or

    ({u=0  and |∇ u|=0}∩ B)≤ d-2
    



The main idea to prove <ref> and <ref> is to show an (almost-)monotonicity of the frequency function for harmonic function by using ideas similar to <cit.>. We remark that the same strategy applies also to more general energies, which can also depend less regularly on the variable x. For instance, 

    L(x,s,p) 1/2 (p· A(x)p) + s b(x)· p+V(x) s^2+F(x, s,p),

with A,b Lipschitz continuous and  V∈ L^∞.

We chose to present this paper in the simpler setting above, but for the required modifications one can look for instance at <cit.> or <cit.>.































 §.§ Further examples



    
  * General autonomous functionals with (2,q) - growth (<cit.>)
    
    
    L(x,s,p)=ϕ(p),

    
    where ϕ:^d→ is a C^2,α function for some α>0. A (C^2,α-)regularity theorem for minimizers of the above functionals was proved in <cit.> and <cit.> for ϕ satisfying the following growth conditions:
   
    m|p|^2≤ϕ(p)≤ M(1+|p|^q) ,
    
       m|ξ|^2≤ξ·∇^2ϕ(p)ξ≤ M(1+|p|^2)^q-2/2|ξ|^2  for every  ξ∈^d ,

    for some constants 0<m≤ M<+∞, and under the following bounds on q:
   
    2≤ q≤ 2+min{2,4/d-1}.

   
Our unique continuation theorems (<ref>, <ref> and <ref>) apply to functionals with lagrangians of the form 


    L(x,s,p)=1/2|p|^2+ψ(p),


where ψ:^d→, d≥ 2, is a C^2,α-regular non-negative convex function satisfying 


    0≤ψ(p)≤ M|p|^q ,
    
      0≤ξ·∇^2ψ(p)ξ≤ M|p|^q-2|ξ|^2  for every  ξ∈^d ,
  
 where M>0 is a positive constant and where the exponent q satisfies the bounds
 
    2< q≤ 2+min{2,4/d-1} .

 Indeed, the conditions (<ref>) and (<ref>) on ψ assure that the lagrangian satisfies both the conditions from <cit.> (so the solutions are C^1,α regular) and the conditions (<ref>) and (<ref>). Finally, for this functional, the C^1,α regularity of u implies that for small δ the linear estimates from <ref> hold (see <ref>).
    
   
  * Double phase functionals. Consider the lagrangian 
    
    
    L(x,s,p)=|p|^2+a(x)|p|^q

    
    where the coefficient a is non-negative and Lipschitz continuous, and where q satisfies the condition 

    2<q≤2+2/d .

Then, the unique continuation theorems <ref>, <ref> and <ref> hold for any minimizer u of ℱ. Indeed, under the condition (<ref>), the minimizers u of ℱ are in C^1,α for some α>0 (see <cit.>). At the same time, it is immediate to check that the conditions (<ref>) and (<ref>) are fulfilled. Finally, as in the case of autonomous functionals, the linear estimates from <ref> hold as a consequence of <ref>.
    
  * Multiphase functionals. Consider the functional
    
    
    L(x,s,p)=|p|^2+a(x)|p|^q+b(x)|p|^s,

    where a and b are non-negative Lipschitz functions, and where the exponents q and s satisfy the condition
    
    2<q≤ s≤2+2/d .

Then, the conclusions of <ref>, <ref> and <ref> hold for any minimizer u of ℱ of this form. Indeed, in <cit.> it was shown that the condition (<ref>), together with the Lipschitz continuity and the positivity of a and b, implies the C^1,α regularity of the minimizers u to ℱ. Thus, in a neighborhood of a point x_0 such that u(x_0)=|∇ u(x_0)|=0 the C^1,α norm of u is small and, by Proposition <ref>, <ref> is fulfilled. Finally, it is immediate to check that (<ref>) and (<ref>) hold for this functional, so we can apply <ref>, <ref> and <ref>.




























§ LINEAR C^1, Α-L^2 ESTIMATES

In this section we show how a C^1, α regularity assumptions in B for minimizers of (<ref>) can be used to prove linear C^1, α-L^2 near singular points, i.e. where u=0 and ∇ u = 0. In particular, since in all the examples presented in <ref> the lagrangian depends only on x and ∇ u, for the sake of simplicity we only consider such dependence. 

More precisely, in this section we consider functionals ℱ: W^1, q(B) → of the form


    ℱ(φ) = ∫_B L(x,∇φ(x)) dx


where the lagrangian L: ^d×^d → is of the form 


    L(x,p) 1/2| p |^2 + F(x,p),


and the function F satisfies the hypotheses (<ref>) and (<ref>).

We are ready for to state the following


    Let the function u ∈ W^1, q(B) ∩ C^1, α(B) be a local minimizer of (<ref>) and suppose that
    
    
    u _C^1, α(B)≤δ.

    
    Moreover, assume also the following two conditions:
    
    
        
  (i) ∇_p F(·, p) ∈ C^0, α(B) for some α>0 and every p ∈ B, with the estimate
        
        
    [∇_p F(·, p) ]_C^0, α(B) = o(1)   as δ→ 0.

        
        where the quantity o(1) is uniform in p; 

        
  (ii) ∇_p F(x, ·) ∈ C^1(B) for all x ∈ B and
         
        
    ∇_p^2 F(x, 0) = 0.

        
    
    
    There exists a constant δ_0 = δ_0(d) > 0 such that, if δ≤δ_0 in (<ref>), then
    
    
    u _C^0, α( B_r/2(x) )≤ C ( _B_r(x) u^2 )   and  ∇ u _C^0, α( B_r/2(x) )≤C/r( _B_r(x) u^2 )

    
    for some constant C = C(d, δ_0)>0 and all balls B_r(x) with B_r(x) ⊂ B.



    Notice that, both conditions (i) and (ii) in <ref> are satisfied by all the examples in <ref>.


The proof of <ref> is divided in two steps:


    
  * in <ref> we show the linear C^0, α bound for u, using the classical De Giorgi-Nash-Moser iterations;

    
  * in <ref> we prove the linear C^0, α bound for ∇ u; this is carried out using step 1 and a linearization argument.


Let us proceed in order.


    Let u, δ, δ_0, C and B_r(x) be as in <ref>. Then, 


    u _C^0, α( B_r/2(x) )≤ C ( _B_r(x) u^2 ).





Testing the outer variation for (<ref>) with the competitors


    φ^±η^2 ( u - k )^±,    k ∈,


for some smooth cutoff function η:B_r(x_0) →, with computations analogous to <ref> it is possible to choose δ_0 = δ_0(d)>0 sufficiently small so that, thanks to (<ref>) and (<ref>), the Caccioppoli inequalities on super/sublevel sets are satisfied:


    ∫_{u > k}∩ B_ρ(x_0)|∇ u |^2 ≤C/(r- ρ)^2∫_{u > k}∩ B_r(x_0) (u-k)^2, 
    ∫_{u < k}∩ B_ρ(x_0)|∇ u |^2 ≤C/(r- ρ)^2∫_{u < k}∩ B_r(x_0) (u-k)^2.


for some constant C=C(d, δ_0)>0 and all balls B_ρ(x_0) ⋐ B_r(x_0) ⋐ B. The lemma now follows from the De Giorgi-Nash-Moser iterations.



    Let u, δ, δ_0, C and B_r(x) be as in <ref>. Moreover, suppose that for some linear function l:^d →
    
    
    u - u(x) - l _L^∞(B_r(x))≤ r ε

    
    and that
    
    
    ∇_p F( ·, ∇ l) - ∇_p F(x, ∇ l) _L^∞(B_r(x))≤ o(1) ε  as δ→ 0

    
    Then, there exist constants δ_0 = δ_0(d), ε_0 = ε_0(d, δ_0)>0, ρ=ρ(d, δ_0) ∈ (0, 1), C=C(d)>0 and a linear function l':^d → such that 
    
    
    u(y) -  u(x) - l' _L^∞( B_ρ r(x) )≤Cρ^1+α u(y) -  u(x) - l _L^∞( B_r(x) )

    
    and
    
    
    ∇ l' - ∇ l ≤Cε.

    



 We proceed by contradiction in ε and δ, via a linearizaion argument.    

 Suppose that there exist a sequence of local minimizers u_k to (<ref>), radii r_k → 0, points x_k ∈ B, constants δ_k, ε_k →  0 and linear functions l_k satisfying
 
 
    u_k - u_k(x_k) - l_k _L^∞(B_r_k(x_k)) = r_k ε_k

 
 but
    
    
    u_k(y) -  u_k(x_k) - l_k' _L^∞( B_ρ r_k(x_k) )≥ k ρ_k^1+α u_k(y) -  u_k(x_k) - l_k _L^∞( B_r_k(x_k) )

    
    for every ρ∈ (0, 1) and linear function l' with 
    
    
    ∇ l' - ∇ l_k ≤ c ε_k,

    
    for some large but fixed constand c=c(d)>0.

Without loss of generality we can assume that x_k=0 and, by translation invariance, also that u_k(x_k)=0. Let us introduce the linearized functions w_k:B → defined as
 
 
    w_k(x) u_k(r_k x ) - l_k(r_k x)/r_k ε_k.

 
 The functions w_k solve
 

    ∫_B∇ w_k ·∇ψ +  1/ε_k∫_B[ ∇_p F(r_k x , ∇ l_k + ε_k ∇ w_k) - ∇_p F(0, ∇ l_k)  ]·∇ψ = 0,


for all functions ψ∈ C_c^∞(B), and we have the estimate


    |1/ε_k[ ∇_p F(r_k x, ∇ l_k + ε_k ∇ w_k) - ∇_p F(0, ∇ l_k)  ] |≤
       ≤1/ε_k[ |∇_p F( r_k x, ∇ l_k) - ∇_p F(0, ∇ l_k) | + |∇_p F( r_k x, ∇ l +  ε_k ∇ w_k ) - ∇_p F(r_k x, ∇ l_k) |] 
       ≤ o(1) + ∇_p^2 F(x, ·) _L^∞(B_δ_0)|∇ w_k(x) |
       ≤ o(1) [ 1 + |∇ w_k(x) |]   as δ→ 0,


where we have used (<ref>) and (<ref>). In particular, the above estimate combined with (<ref>) implies the Caccioppoli inequality for the functions w_k. Hence, using also <ref>, the linearizations w_k converge weakly in H^1(K) and in C^0, α(K), for all K ⋐ B, to a limit function w_∞ which is harmonic in B, and satisfies w_∞(0) = 0 and w_∞_L^∞(B)≤ 1. Consequently, the function w_∞ satisfies


    w_∞ - ∇ w_∞· x _L^∞(B_ρ)≤ C ρ^1+α  for all ρ∈ (0, 1/2),


where the constant C=C(d)>0.

Now, as a consequence of (<ref>) and the C^0, α convergence of w_k to w_∞, for k sufficiently large there exist constants C=C(d)>0 and ρ=ρ(d) such that


    u_k(r_k x) - ( ∇ l_k - ε_k ∇ w_∞(0) ) · x _L^∞(B_ρ)≤ C ε_k ρ^1+α  in  B_ρ


Since (<ref>) is in contradiction with (<ref>) as long as |∇ w_∞(0) |≤ c (where c is the constant from (<ref>)), the proof is concluded.


We are ready for the


The linear bound for u in (<ref>) follows directly from <ref>.
In order to prove the linear bound on the gradient, we take any point y∈ B_r/2(x) and we choose δ in such a way that 

    u - u(y) _L^∞(B_r/4(y))≤r/4_0,

where ε_0 is the constant from <ref>. Thus, by iterating <ref> (which is possible thanks to (<ref>)) on a sequence of balls r_k:=ρ^kr/4, where we choose ρ such that C̃ρ^α/2≤ 1, we obtain a sequence of linear functions l_k such that l_0≡ 0 and 


    1/r_k u - u(y) - l_k _L^∞(B_r_k(y))   ≤ρ^kα/21/r/4 u - u(y)_L^∞(B_r/4(y))
       ≤ρ^kα/2C/r^(d+2)2 u - u(y) _L^2(B_r/2(y)).

Moreover, 

    |∇ℓ_k+1-∇ℓ_k|≤Cρ^kα/2C/r^(d+2)2 u - u(y) _L^2(B_r/2(y)),

which implies that 

    |∇ u(y)|≤∑_k=0^∞|∇ℓ_k+1-∇ℓ_k|≤C/1-ρ^α/2C/r^(d+2)2 u - u(y) _L^2(B_r/2(y)).

Finally, we notice that 

    1/r^d2 u - u(y) _L^2(B_r/2(y))   ≤ C_d|u(y)|+1/r^d2 u_L^2(B_r/2(y))
       ≤ C_du_L^∞(B_r/2(x))+1/r^d2 u_L^2(B_r/2(y))≤C/r^d2 u_L^2(B_r(x)),

where in the last inequality we used the linear bound for u from (<ref>). Thus, we get 

    |∇ u(y)|≤C/r^(d+2)2 u _L^2(B_r(x)),

which implies the second bound in (<ref>).
































§ OUTER AND INNER VARIATIONS


In this section we compute the outer and inner variations of the functional (<ref>), centered at the local minimizer u. We use a strategy analogous to  <cit.>. Let the function φ:^+ →^+ be defined as


    φ(x) 
        1    if  x ∈ (0, 1-υ], 
    1-x/1-υ   if  x ∈ (1-υ, 1], 
    
        0    if  x ∈ (1, +∞),


for some υ∈ (12, 1) and define the rescaled cutoff functions


    ψ_r:^d→^+ ,    ψ_r(x) φ( | x |/r).


We introduce the height function,


    H(r)  - ∫φ'( | x |/r) u(x)^2/| x |


the energy terms D_0(r), D_l(r) defined as


    D_0(r)    ∫φ( | x |/r) |∇ u |^2, 
    
        D_l(r)    ∫φ( | x |/r) u ∂_sF(x,u,∇ u) ,


and the energy function


    D(r)  D_0(r)+D_l(r).


For notational convenience, let us also define the quantities


    G(r)    ∫φ( | x |/r) u^2, 
    
        A(r)     - ∫φ'( | x |/r) | x |( ∇ u ·x/| x |)^2, 
    
        B(r)     - ∫φ'( | x |/r) u  (∇ u ·x/|x|) .


The main content of this section is the following


For all r ∈ (0, 1), 

we have

    H'(r) - d-1/r H(r) - 2/r B(r) = 0 ,


    D(r) - 1/r B(r)+ e_O(r)=0 ,


    (d-2) D(r) - r D'(r) + 2/r A(r)
    + e_I(r)  = 0 ,

where the errors e_O, e_I (produced by the outer and the inner variations) are defined as


    e_O(r)   ∑_k=1^4 E^o,k(r),
    
    e_I(r)    2∑_k=1^4 E^i, k(r).


and for some constant C=C(d, δ_0) the following estimates hold, with κ = γ/2:


    |E^o,1(r)|   ≤ C∫φ( | x |/r) |u|^2+κ, 
    
    |E^o,2(r)|   ≤ C∫φ( | x |/r)|∇ u|^2+κ, 
    
    |E^o,3(r)|   ≤ - C∫φ'( | x |/r) |u|^2+κ, 
    
    |E^o,4(r)|   ≤ - C∫φ'( | x |/r)|u| |∇ u|^1+κ.



    |E^i,1(r)|   ≤ C∫φ( | x |/r) |u|^2, 
    
    |E^i,2(r)|   ≤ C∫φ( | x |/r)|∇ u|^2+κ, 
    
    |E^i,3(r)|   ≤ - C∫φ'( | x |/r) |u|^2, 
    
    |E^i,4(r)|   ≤ - C∫φ'( | x |/r)|∇ u|^2+κ.







 §.§ Height derivative: proof of  (<ref>)

 We can rewrite
    
    
    H(r) =∫_^d rφ( | x |/r) ( x/|x|u^2(x)/| x |),

    
    so that a direct computation gives
    
    
    H'(r)
           =H(r)/r-2/r∫φ'( | x |/r) u∇ u·x/|x|-1/r∫ |x|φ'( | x |/r) u^2(x) (x/|x|^2) = 
       = d-1/r H(r) - 2/r∫φ'( | x |/r) u(∇ u·x/|x|)
       = d-1/r H(r) + 2/r B(r),

    
    where in the last identity we have used that
    
    
    (x/|x|^2)=d-2/|x|^2.

    
This concludes the proof of (<ref>).



    



 §.§ Outer variation: proof of (<ref>)


By a direct computation


    0   =d/dt|_t=0ℱ(u+t ψ_r u;r)
       =d/dt|_t=0∫(1/2|∇(u+t ψ_r u)|^2+F(x,u+t ψ_r u,∇(u+t ψ_r u)))
       =∫(∇ u·∇ (ψ_r u)+ψ_ru ∂_sF(x,u,∇ u)+∇(ψ_r u)·∇_pF(x,u,∇ u))
       =∫(∇ u·∇ (ψ_r u)+ψ_ru ∂_sF(x,u,∇ u)
            +ψ_r∇ u·∇_pF(x,u,∇ u)+u/rϕ'(|x|/r)x/|x|·∇_pF(x,u,∇ u))
       =∫φ( | x |/r) |∇ u |^2 + ∫φ( | x |/r) u ∂_sF(x,u,∇ u) + 1/r∫φ'( | x |/r) u ∇ u ·x/| x | + ∑_k=1^2 e^o, k(r)
       =D(r)-1/rB(r)+ ∑_k=1^2e^o, k(r) .


where we have introduced the errors


    e^o, 1(r)    ∫φ(|x|/r) ∇  u·∇_pF(x,u,∇ u), 
    
        e^o, 2(r)    1/r∫φ'(|x|/r) u  ∇_p F(x,u,∇ u) ·x/|x|.


Now, using the bounds (<ref>), (<ref>) and (<ref>) we have the estimates


    | ∇  u·∇_pF(x,u,∇ u) |
       ≤ C(|∇ u| |u|^1+γ+|∇ u|^2+γ) 
       ≤ C(|u|^2+γ/2+|∇ u|^2+γ/2),



    | u ∇_pF(x,u,∇ u) |
       ≤ C(|u| |∇ u|^1+γ+|u|^2+γ).


Hence, there exist error terms E^o, k(r) with k=1, ..., 4 with estimates


    |E^o,1(r)|   ≤ C∫φ( | x |/r) |u|^2+γ/2, 
    
    |E^o,2(r)|   ≤ C∫φ( | x |/r)|∇ u|^2+γ/2, 
    
    |E^o,3(r)|   ≤ - C∫φ'( | x |/r) |u|^2+γ, 
    
    |E^o,4(r)|   ≤ - C∫φ'( | x |/r)|u| |∇ u|^1+γ.


such that


    D(r)-1/rB(r)+ ∑_k=1^4 E^o, k(r) = 0.


The above computations give the proof of (<ref>), (<ref>), (<ref>), (<ref>) and (<ref>).


























 §.§ Inner variation: proof of (<ref>)

Let T_ε:B_r → B_r be the family of diffeomorphisms defined as


    T_ε(x)  x + εψ_r(x) x,

with ψ_r as in (<ref>). Let u_ε be defined as


    u_ε u ∘ T_ε^-1.


By changing coordinates and differentiating in , we get:

    ∫(1/2|∇ u_|^2+F( x, u_ε, ∇ u_ε))
               = ∫(1/2|D(T_ε^-1)[∇ u ∘ T_ε^-1]|^2+F( x, u ∘ T_ε^-1, D(T_ε^-1)[∇ u ∘ T_ε^-1]) )
       = ∫(1/2|(DT_ε)^-1∇ u|^2+F( T_ε, u, (DT_ε)^-1∇ u )) |(DT_ε)| 
       = ∫(1/2|∇ u|^2+F(x,u,∇ u)) 
           +∫(1/2|∇ u|^2 div(xψ_r)-∇ u· D(ψ_r x)∇ u)
                +∫ F(x,u,∇ u)  div(xψ_r)
                    + ∫ψ_r x·∇_x F( x, u, ∇ u )  
                       -∫∇_p F( x, u, ∇ u ) · D(ψ_r x) ∇ u  +o().


Now, a standard computation gives


    ∫(1/2|∇ u|^2 div(xψ_r)-∇ u· D(ψ_r x)∇ u)   =d-2/2∫φ( | x |/r) |∇ u|^2 + 1/2r∫φ'( | x |/r) | x | |∇ u|^2  
             - 1/r∫φ'( | x |/r) | x |( ∇ u ·x/| x |)^2
       =d-2/2 D_0(r) -r/2D_0'(r) +1/rA(r)
       =d-2/2 D(r) -r/2D'(r) +1/rA(r)-d-2/2 D_l(r) +r/2D_l'(r).


The above computations imply 


    0=d/dε|_ε=0ℱ(u_;r) = d-2/2 D(r) -r/2D'(r) +1/rA(r) + e^i,1(r)+e^i,2(r)+r/2D_l'(r),


where we have defined the errors e^i,1(r) and e^i,2(r) as


    e^i,1(r)    ∫ψ_r [ d   F(x,u,∇ u) +   x·∇_x F( x, u, ∇ u )-∇ u·∇_p F( x, u, ∇ u )] + d-2/2 D_l(r), 
    
        e^i,2(r)    1/r∫|x|φ'(| x |/r) [  F(x,u,∇ u)-(x/|x|·∇ u) (x/|x|·∇_pF(x,u,∇ u))].


Using the bounds (<ref>), (<ref>) and (<ref>), we can deduce the following estimates:


    |d   F(x,u,∇ u)    +   x·∇_x F( x, u, ∇ u )-∇ u·∇_p F( x, u, ∇ u )+d-2/2u∂_sF(x,u,∇ u)|
       ≤ dC(|u|^2+|∇ u|^2+γ)+|x|C(|u|^2+|∇ u|^2+γ)
            +|∇ u|C(|u|^1+γ+|∇ u|^1+γ)+C(|∇ u|^2+γ+|u|^2)
       ≤ C(|u|^2+|∇ u|^2+γ/2),



    |F(x,u,∇ u)   -(x/|x|·∇ u) (x/|x|·∇_pF(x,u,∇ u))|
       ≤|F(x,u,∇ u)|+|∇ u| |∇_pF(x,u,∇ u)|
       ≤ C(|u|^2+|∇ u|^2+γ)+|∇ u|C(|u|^1+γ+|∇ u|^1+γ)
       ≤ C(|u|^2+|∇ u|^2+γ/2).

Finally, we compute the derivative of D_l as

    D_l'(r) = - 1/2r∫|x|φ'(| x |/r) s  ∂_s F(x, u, ∇ u),

and we notice that 

    |D_l'(r)| ≤  C∫|φ'|(| x |/r) |u|^2.

Combining the above estimates, we get that there exist error terms E^i, k(r) with k=1, ..., 4 satisfying

    |E^i,1(r)|   ≤ C∫φ( | x |/r) |u|^2, 
    
    |E^i,2(r)|   ≤ C∫φ( | x |/r)|∇ u|^2+γ/2, 
    
    |E^i,3(r)|   ≤ - C∫φ'( | x |/r) |u|^2, 
    
    |E^i,4(r)|   ≤ - C∫φ'( | x |/r)|∇ u|^2+γ/2.


such that


    d-2/2 D(r) -r/2D'(r) +1/rA(r) + ∑_k=1^4 E^i,k(r) = 0.


This concludes the proof of (<ref>) and (<ref>), (<ref>), (<ref>) and (<ref>).





























§ SCALE-PRESERVING L^∞-L^2 ESTIMATES


The main content of this section is to prove L^∞-L^2 estimates for minimizers u of (<ref>) at the same scale (see <ref> below). To this aim we use the strategy of <cit.>, thus we proceed via a Whitney decomposition argument in the spirit of Almgren-De Lellis-Spadaro <cit.> (see <ref> below).

In the following of this section, it will be useful to work under the following


    The origin is a singular point in the sense that
    
    
    u(0) = 0   and  ∇ u(0) = 0.

    































 §.§ Some weighted inequalities

In this section we mainly recall some weighted inequalities that were derived in <cit.>, which will be useful for the subsequent analysis.






















  §.§.§ Height inequality


To begin with, we recall a bound from <cit.> for the weighted L^2 norm G(r) from (<ref>) in terms of the height function H(r) from (<ref>).


For every function u∈ H^1(B_r)


    G(r)= ∫_B_rφ(|x|/r) u^2≤∫_0^r H(ρ) dρ,


where H is the height function from (<ref>).


















 






  §.§.§ Poincaré inequality


Now recall a weighted Poincaré-type inequality from <cit.>.


    There exist constants c=c(d, δ_0)>0 and r_0=r_0(d, δ_0)>0 with the following property. Suppose that the function u is a local minimizer of (<ref>) under <ref> and <ref>, then
    
    
    r D_0(r)≥ c  H(r)  for every  r ∈ (0, r_0).

    


As a consequence of the L^2-norm bound <ref>, the Poincaré inequality from <ref> and the bound (<ref>), we have the following


There exist constants c=c(d, δ_0)>0 and r_0=r_0(d, δ_0)>0 with the following property. Suppose that the function u is a local minimizer of (<ref>) under <ref> and <ref>, then


    
  (i)
    
    
    G(r) ≤ c  r^2   D_0(r)   for every    r ∈ (0, r_0),

    

    
  (ii)
    
    
    0 < (1-cr^αγ) D_0(r)≤ D(r)≤ (1+cr^αγ) D_0(r)  for every  r ∈ (0, r_0).

    





Point (i) follows as in <cit.>. Concerning point (ii), from (<ref>), (<ref>) and point (i) we see that


    | D_l(r) |   ≤ C ∫φ( | x |/r) (| u |^2 + |∇ u |^2+γ) 
       ≤ C [ G(r) + r^αγ D_0(r) ] 
       ≤ C [ r D_0(r) + r^αγ D_0(r) ],


so that the conclusion follows.
















 §.§ Whitney decomposition

In the proof of the monotoncity of the frequency function (<ref>), the estimates of the error terms produced from the nonlinearity F rely on a Whitney decomposition type argument (see <ref>). The construction of this Whitney decomposition is exactly the same as in <cit.>, but since it is a key step in the proof of the main theorems (<ref> and <ref>), we explain the detailed construction in this subsection, keeping the notations from <cit.>.


Consider a function u:B_R→, u∈ H^1(B_R), defined in some sufficiently large ball B_R⊂^d with R chosen in such a way that the cube [-4,4]^d is contained in B_R. We define the Whitney decomposition of the cube [-1,1]^d as follows.

Basic notations. Given a=(a_1,…,a_d)∈^d and ℓ>0, we denote by L=L_ℓ(a) the closed cube of center a and side 2ℓ as follows 

    L:=[a_1-l,   a_1+l]×…× [a_d-l,   a_d+l] .

Vice versa, for a cube L of the form (<ref>), we will use the notation 

    a(L):=(a_1,…,a_d)    and    ℓ(L):=ℓ.

Moreover, we will denote by B_L the ball 

    B_L:=B_3 l(L)( a(L) ).
 
Collections of cubes. We define the collections of cubes 𝒞_j, j≥ 1, as follows: the only element of the set 𝒞_1 is the cube [-1,1]^d; the collection of cubes 𝒞_2 is obtained by dividing [-1,1]^d into 3^d cubes with disjoint interiors and with the same side-length; similarly, for every j≥ 1, the collection 𝒞_j+1 is obtained by dividing each of the cubes from 𝒞_j into 3^d cubes with disjoint interiors and with the same side-length. In particular, if L∈𝒞_j, for some j, then 

    l(L)=3^1-j    and     a(L)∈ (3^1-jℤ)^d.

By construction, if L∈𝒞_j and H∈𝒞_k for some k>j, then we have only two possibilities:


  * H⊂ L;

  * L and H have disjoint interiors. 

If case (1) occurs, then we say that H is a descendant of L and that L is an ancestor of H.
Moreover, if two cubes H and L are such that L∈𝒞_j, H∈𝒞_j+1 and H⊂ L, we will say that L is the father of H and that H is a son of L. 

Whitney decomposition. 

From now on we fix two constants 

    C_0 > 0    and    α∈ (0, 1/2).

We define the family of cubes (with disjoint interiors) 𝒲 as: 

    𝒲= 𝒲^e∪𝒲^h,
 
where of the family of excess cubes 𝒲^e and the family of height cubes 𝒲^h, are the unions

    𝒲^e=⋃_j𝒲^e_j    and    𝒲^h=⋃_j𝒲^h_j.

We construct the families of cubes 𝒲^h_j and 𝒲^e_j inductively. When j=0, we set 𝒲_0=∅. For j≥ 1, the families 𝒲^h_j and 𝒲^e_j are disjoint subsets of the collection 𝒞_j and are obtained as follows. Consider a cube L∈𝒞_j such that 


no ancestor of L is in ⋃_i=1^j-1𝒲^e_i or in ⋃_i=1^j-1𝒲^h_i.



    
  * We say that L∈𝒲^e_j if
        

    ∫_B_L|∇ u |^2 ≥ C_0  l(L)^d+2α ,
  
where C_0 and α are the constants from (<ref>).

  * We say that L∈𝒲^h_j, if L∉𝒲^e_j and 
        

    ∫_B_L  u^2 ≥ C_0   l(L)^d+2+2α ,

where again C_0 and α are the constants from (<ref>).

  * If none of the above occurs we say that L∈𝒮_j.

   
It is immediate to check that the decomposition 𝒲 has the following properties:


  * for every j, 𝒲^h_j, 𝒲^e_j and 𝒮_j are disjoint subsets of 𝒞_j;

  * 𝒲 is a countable union of cubes with disjoint interior; 

  * the residual set of points [-1,1]^d∖⋃_L∈𝒲L is contained in the compact set

    Γ:=⋂_j≥ 1⋃_L∈𝒮_jL ;


  * for every x_0∈Γ it holds


    u(x_0)=0    and    ∇ u(x_0)=0 ;



  * if L∈𝒲^e and H is the father of L, then H∉𝒲^e, and H∉𝒲^h, so we have


    ∫_B_H u^2 ≤ C   l(L)^2 ∫_B_L|∇ u |^2   and  ∫_B_H|∇ u |^2 ≤ C ∫_B_L|∇ u |^2,


where C depends only on the dimension d  and the constants C_0 and α from (<ref>);


  * finally, if L∈𝒲^h and H is the father of L, then L∉𝒲^e, H∉𝒲^e, H∉𝒲^h, and


    ∫_B_H u^2 ≤ C ∫_B_L u^2   and  ∫_B_H|∇ u |^2 ≤C/l(L)^2∫_B_L u^2 ,

where as above C depends on C_0, α, and d.


We conclude this section with the following lemma, which contains two properties of the Whitney decomposition for solutions u satisfying <ref> and <ref>. For the proof we refer to <cit.>. 


There exist constants R=R(d, δ_0)>0, λ=λ(d, δ_0)>0 and C = C(d, δ_0)>0 with the following property. Suppose that the function u is a minimizer of (<ref>) under <ref> and <ref>. Then, for all cubes L ∈𝒲 with


    L ∩ B_r ≠∅,


the following estimate holds:

    
    u _L^∞( L )+∇ u _L^∞( L )≤ C D_0(r)^λ  for all  r ∈ (0, R).

    





















































§ FREQUENCY (ALMOST-)MONOTONICITY

Let us introduce the Almgren-type frequency function N(r) defined as


    N(r) r D(r)/H(r).


The main content of this section is the following


   There exist constants R=R(d, δ_0)>0, λ=λ(d, δ_0)>0 and C = C(d, δ_0)>0 with the following property. Suppose that the function u in B_R is a local minimizer of (<ref>) under <ref> and <ref>, and that
   
   
    H(r_0)>0   for some  r_0 ∈ (0, R).

   
   Then,
   
   
    e^ g(r) N(r)  is non-decreasing in a neighborhood of  r_0,

    
    where the function g(r): ^+ → is defined as
    
    
    g(r) C/β[ r^β + D(r)^β]

    
    and satisfies
    
    g(r) → 0  as  r → 0^+.

    





























 §.§ Frequency derivative

To begin with, we compute the derivative of the frequency function intoduced in (<ref>). To this aim, we first introduce the auxiliary quantity


    F(r) 1/r B(r) - E^o,4(r),


and then we prove the following


 There exists a constant R=R(d, δ_0)>0 with the following property. Suppose that


    H(r_0) > 0   for some   r_0 ∈ (0, R).


Then, for all r in a neighborhood of r_0, the following identity holds


    d/drlnN(r) = 1/r + D'/D - H'/H = 2/r^21/F(r) H(r)[A(r)H(r) - B(r)^2 ]+ ∑_k=1^3 e_k(r),


where we have defined the error terms in the following way:


    e_1(r) 1/r∑_k=1^4 E^i, k(r)/D(r), 
        e_2(r)  - 1/r^2A(r)/D(r)F(r)∑_k=1^3 E^o, k(r), 
        e_3(r) 1/rB(r) E^o,4(r)/F(r) H(r).





To begin with, we have that


    d/drlnN(r) = 1/r + D'(r)/D(r) - H'(r)/H(r).


Now, from (<ref>) we have


    D'(r)/D(r) = d-2/r +2/r^2A(r)/D(r) + e_3(r).


On the other hand, (<ref>) implies that


    H'(r)/H(r) = d-1/r + 2/rB(r)/H(r).


Combining (<ref>), (<ref>) and (<ref>) we deduce


    d/drlnN(r) = 2 [1/r^2A(r)/D(r) - 1/rB(r)/H(r)] + e_3(r).


Since we wish to avoid comparing most error terms with the height H(r), we can split the terms in square brackets as


    2 [1/r^2A(r)/D(r) - 1/rB(r)/H(r)]     = 2 [1/r^2A(r)/F(r) - 1/rB(r)/H(r)] + 2 A(r)/r^2[1/D(r) - 1/F(r)] 
        = 2 [1/r^2A(r)/F(r) - 1/rB(r)/H(r)] + e_2(r),


where the quantity F(r) is the one from (<ref>). Hence, we can rewrite (<ref>) as


    d/drlnN(r)    = 2/r^21/F(r) H(r)[A(r)H(r) - rF(r)B(r) ] + e_1(r) + e_2(r) 
        = 2/r^21/F(r) H(r)[A(r)H(r) - B(r)^2 ] + 2/r^2B(r)/F(r) H(r)( B(r) - rF(r) ) + e_1(r) + e_2(r) 
        = 2/r^21/F(r) H(r)[A(r)H(r) - B(r)^2 ] + e_1(r) + e_2(r) + e_3(r) ,


which is exactly (<ref>).





























 §.§ Error estimates

The (almost-)monotonicity of the frequency function will follow as a consequence of the following proposition, which deals with the estimate for the error terms.


    There exists constants R=R(d, δ_0)>0, C=C(d, δ_0), β=β(d, δ_0) with the following properties. Suppose that the function u is a local minimizer of (<ref>) under <ref> and <ref> and that
    
    
    H(r_0 > 0)   for some  r_0 ∈ (0, R).

    
    Then, for all r in a neighborhood of r_0 the following estimates hold:
    
    
    | e_1(r) |≤ C [ r^β-1 + D_0(r)^β-1 D_0'(r) ], 
       | e_2(r) |≤ C D_0(r)^β-1 D_0'(r), 
       | e_3(r) |≤ C D_0(r)^β-1 D_0'(r).

    


We proceed to prove the estimates (<ref>), (<ref>) and (<ref>) in order.
















 §.§ Proof of (<ref>)

To begin with, combining the estimates (<ref>), (<ref>), (<ref>) and (<ref>), together with the Whitney decomposition from <ref>, the bounds from <ref>, <ref> and <ref>, we have


    | E^i, 1(r) |   ≤ G(r) ≤ C r^2 D_0(r),



    | E^i, 2(r) |   ≤ C ∑_L ∈𝒲∫_Lφ( | x |/r)|∇ u|^2+κ
       ≤ C ∇ u _L^∞( L )^κ∫_B_rφ( | x |/r)|∇ u|^2  ≤ C D_0(r)^1+λκ,



    | E^i, 3(r) |   ≤ C r H(r) ≤ C  r^2 D_0(r),



    | E^i, 4(r) |   ≤ -C ∑_L ∈𝒲∫_L ∩ B_rφ'( | x |/r) |∇ u|^2+κ
       ≤ -C ∇ u _L^∞( L )^κ∫_B_rφ'( | x |/r)|∇ u|^2 ≤ C r D_0(r)^λκ D_0'(r).


Combining (<ref>), (<ref>), (<ref>), and (<ref>) with the definition of the error e_1(r) from (<ref>) gives


    | e_1(r) |≤ C [ r^αλκ - 1 + D_0^1-λκ D_0'(r) ],


where we have used <ref> and <ref>. This concludes the proof of (<ref>).





























 §.§ Proof of (<ref>)

From the bounds (<ref>), (<ref>) and (<ref>), using the Whitney decomposition from <ref> together with <ref> and <ref>, we have the estimates


    | E^o, 1(r) |   ≤ C ∑_L ∈𝒲∫_Lφ( | x |/r) |u|^2+κ
       ≤ C  u _L^∞( L )^κ∫_B_rφ( | x |/r)|u|^2  ≤ C D_0(r)^λκ G(r)  ≤ C r^2 D_0(r)^1+λκ,



    | E^o, 2(r) |   ≤ C ∑_L ∈𝒲∫_Lφ( | x |/r)|∇ u|^2+κ
       ≤ C ∇ u _L^∞( L )^κ∫_B_rφ( | x |/r)|∇ u|^2  ≤ C D_0(r)^1+λκ,




    | E^o, 3(r) |   ≤ -C ∑_L ∈𝒲∫_L ∩ B_rφ'( | x |/r) |u|^2+κ
       ≤ -C  u _L^∞( L )^κ∫_B_rφ'( | x |/r)|u|^2  ≤ C r D_0(r)^λκ H(r)  ≤ C r^2 D_0(r)^1+λκ,


where in the last inequality we have also used <ref>. In particular, combining (<ref>), (<ref>) and (<ref>) we get that


    ∑_k=1^3 | E^o, 3(r) |   ≤ C D_0(r)^1+λκ≤ C r^αλκ D_0(r),


where in the last passage we have used <ref> and <ref>. Combining the bound (<ref>) with the definition of the quantity F(r) from (<ref>) and the outer variation (<ref>) gives the equivalence


    ( 1- C r^αλκ) D_0(r) ≤ F(r) ≤( 1 + C r^αλκ) D_0(r).


Now, the definition of A(r) from (<ref>) implies that


    | A(r) |≤ C r^2 D_0'(r),


so that, combining the definition of e_2(r) from (<ref>) with the estimates (<ref>) and (<ref>) we have


    | e_2(r) |≤ C D_0^1-λκ D_0'(r),


which is exactly (<ref>).































 §.§ Proof of (<ref>)

From the estimate (<ref>), using the Whitney decomposition from <ref> together with <ref>, we have that


    | E^o, 4(r) |   ≤ -C ∑_L ∈𝒲∫_L ∩ B_rφ'( | x |/r)|u| |∇ u|^1+κ
       ≤ -C ∇ u _L^∞( L )^κ∫_B_rφ'( | x |/r)|u| |∇ u| 
       ≤ C D_0(r)^λκ( r^2 H(r) D_0'(r) )^1/2,


where in the last inequality we have also used the definitions of H(r) and D_0(r) from (<ref>) and (<ref>) respectively. Moreover, from the definition of B(r) in (<ref>), we also see that


    | B(r) |≤ C ( r^2 H(r) D_0'(r) )^1/2.


Combining the estimates (<ref>) and (<ref>) with the definition of the error e_1(r) from (<ref>), <ref> (ii) and the equivalence (<ref>), we have


    | e_3(r) |≤ C D_0(r)^1-λκ  D_0'(r),


which concludes the proof of (<ref>).






























 §.§ Proof of <ref>

To begin with, we observe that thanks to the hypothesis H(r)>0 the frequency function is well defined. Moreover, combining <ref> and <ref> we have


    d/drlnN(r)≥2/r^21/F(r) H(r)[A(r)H(r) - B(r)^2 ] - C [ r^β-1 + D_0(r)^β-1 D_0'(r) ]


in a neighborhood of r_0 and for some constants C=C(d, δ_0)>0 and β=β(d, δ_0)>0. By a standard Cauchy-Schwarz inequality 


    A(r)H(r) - B(r)^2 ≥ 0,


so that, from (<ref>) and (<ref>)


    d/drlnN(r)≥ - C [ r^β-1 + D_0(r)^β-1 D_0'(r) ].


Now let the function g(r) as in (<ref>). Using the estimate (<ref>) we have


    d/dr e^g(r) N(r) = e^g(r)[ N'(r) + g'(r) N(r) ] ≥ 0,


in a neighborhood of r_0, which gives the monotonicity (<ref>). To conclude, the condition (<ref>) follows at once combining the definition of D_0(r) in (<ref>) together with <ref> and <ref>.



























§ PROOF OF <REF>

Without loss of generality we can assume x_0 = 0, and let R=R(d, δ_0)>0 be the radius from <ref>. In order to prove <ref> it is sufficient to show that


    u≡ 0


Indeed, if (<ref>) holds true, we can can iterate <ref> for any point in B_R and so on, thus covering all B.

To begin with, thanks to (<ref>) we can suppose that there exists a radius r_0 ∈ (0, R) such that


    H(r_0)>0.



Let also


    r_1 sup{ r ∈ [0, r_0]   :   H(r) = 0 }.


Combining the height derivative (<ref>) and the outer variation (<ref>) we have that


    H'(r) = d-1/r H(r) + 2 D(r) + 2 ∑_k=1^4 E^o,k,


and estimating the errors E^o, k using (<ref>), (<ref>), (<ref>) and (<ref>) 


    H'(r)    ≤d-1/r H(r) + C D_0(r) + D_0(r)^β( r^2 H(r) D_0'(r) )^1/2
       ≤C/r H(r) + r^2 D_0(r)^β D_0'(r),


for some constant C=C(d, δ_0) and all r ∈ (r_1, r_0), where we have also used <ref> (ii). Now, from the (almost-)monotonicity of the frequency function (<ref>), we have that


    H(r) ≥C r D_0(r)   for all  r ∈ (r_1, r_0),


and a constant C C N(r_0) where C=C(d, δ_0)>0, thanks to (<ref>). In particular, from (<ref>) and (<ref>) we see that


    H'(r)/H(r)≤C/r + C D_0(r)^β-1 D_0'(r)   for all  r ∈ (r_1, r_0).


Integrating the estimate (<ref>) in the interval (s, t), with r_1 ≤ s≤ t ≤ r_0 we get


    H(t)/H(s)≤C( t/s)^C   for all  r_1 ≤ s≤ t ≤ r_0,


and some constants C = C(d, δ_0)>0 and C >0 depending only on d, δ_0 and N(r_0). In particular, (<ref>) implies that r_1 = 0 and so H(r)>0 and  <ref> applies for all r∈(0,r_0].

Finally, integrating (<ref>) on (0, r) for any r ∈ (0, r_0) we obtain the doubling inequality



    _B_rφ(| x |/r) u^2 ≤C_B_r/2φ(| x |/r) u^2,


for a constant C >0 depending only on d, δ_0 and N(r_0), and passing to the limit as υ→ 1^- in (<ref>) (where υ is the parameter introduced in (<ref>)), we reach a contradiction with (<ref>). Consequently, we have that


    H(r) = 0   for all  r ∈ (0, R),


which concludes the proof of (<ref>) and thus of <ref>.


































§ ACKNOWLEDGEMENTS
 LS acknowledges the support of the NSF Career Grant DMS 2044954. LF and BV are supported by the European Research Council (ERC), under the European Union's Horizon 2020 research and innovation program, through the project ERC VAREG - Variational approach to the regularity of the free boundaries (No. 853404). LF and BV acknowledge the MIUR Excellence Department Project awarded to the Department of Mathematics, University of Pisa, CUP I57G22000700001. LF is a member of INdAM-GNAMPA. BV acknowledges support from the projects PRA 2022 14 GeoDom (PRA 2022 - Università di Pisa) and MUR-PRIN “NO3” (No. 2022R537CS). We warmly thank Roberto Ognibene for the useful discussions on the existing literature about the unique continuation property for elliptic equations. 


























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Classe di Scienze, Scuola Normale Superiore
 
piazza dei Cavalieri 7, 56126 Pisa (Italy)







Department of Mathematics, UC San Diego,

 AP&M, La Jolla, California, 92093, USA







Dipartimento di Matematica, Università di Pisa
 
largo Bruno Pontecorvo 5, 56127 Pisa (Italy)






 Paper Title: Precomputing approach for a two-scale phase transition model.





























































definitionDefinition
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acknowledgements § ABSTRACT  

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†,1]Michael Eden
⋆]Tom Freudenberg
†]Adrian Muntean

[†]Department of Mathematics and Computer Science, Karlstad University, Sweden
[⋆]Center for Industrial Mathematics, University of Bremen, Germany



Precomputing approach for a two-scale phase transition model
    [
    August 5, 2024
============================================================

§ ABSTRACT 
In this study, we employ analytical and numerical techniques to examine a phase transition model with moving boundaries.
The model displays two relevant spatial scales pointing out to a macroscopic phase and a microscopic phase, interacting on disjoint  inclusions.
The shrinkage or the growth of the inclusions is governed by a modified Gibbs-Thomson law depending on the macroscopic temperature, but without accessing curvature information.
We use the Hanzawa transformation to transform the problem onto a fixed reference domain.
Then a fixed-point argument is employed to demonstrate the well-posedness of the system for a finite time interval. 
Due to the model's nonlinearities and the macroscopic parameters, which are given by differential equations that depend on the size of the inclusions, the problem is computationally expensive to solve numerically.
We introduce a precomputing approach that solves multiple cell problems in an offline phase and uses an interpolation scheme afterward to determine the needed parameters.
Additionally, we propose a semi-implicit time-stepping method to resolve the nonlinearity of the problem.
We investigate the errors of both the precomputing and time-stepping procedures and verify the theoretical results via numerical simulations.


Keywords:  Phase transitions, two-scale model, moving boundary problem, numerical analysis, simulation

MSC2020: 35R37, 65M60, 80A22, 35K55


[1]<michael.eden@kau.se>
footnote







§ INTRODUCTION

Phase transition models describe the heat dynamics in a system where phase transitions can occur, such as water-ice or austenite-pearlite microstructures in steel.
One of the main difficulties is the need to account for the changing geometries inherent to the transition process.
In some cases, there is an additional scale separation: the phase transition processes might happen at the microscale while our interest might be in the effective macroscopic behaviors.
Examples include microstructural changes in steel <cit.>, liquid–solid phase transitions in metals <cit.>, or thawing processes in permafrost <cit.>. In these cases, the focus is not on the geometric changes at the microscale but on their impact on macroscopic properties.
However, via homogenization techniques (e.g., two-scale convergence or asymptotic expansion) it is easy to see that the microstructural changes can not be neglected: In the resulting models, the phase transitions are still present on the microscale and directly influence the effective material parameters, see, e.g., <cit.>.
Such two-scale models (sometimes also called micro-macro model or distributed microstructures model) are computationally extremely costly, mainly due to the following two factors: 

    
  (a) Strong non-linear coupling between the scales: The macroscopic temperature field is driving the microstructural changes and the microstructural changes are in turn non-linearly influencing the macroscopic material parameters.
    
  (b) Family of moving boundary problems: Every macroscopic point is associated with corresponding moving boundary problems at the microscale both via the interaction with the microscopic temperature but also via the geometric changes influencing the heat conductivity via the associated cell problems.
 
Since one of the main goals of homogenization is the derivation of simpler models that are computationally less heavy, these issues might at first glance call into question the usefulness of such limit models.
There is, however, one important advantage these two-scale models have, namely the explicit separation of the scales.
This can be exploited in many different ways: via iteration procedures <cit.>, parallel computing <cit.>, machine learning <cit.>, and/or interpolation techniques <cit.>.

In this work, we propose, analyze, and implement a precomputing strategy in which the calculations of the macroscopic coefficients are pushed into an offline phase.
Instead of having to calculate the effective conductivity via geometry-dependent cell problems at every macroscopic point and at each time step, we precompute the conductivity instead for a range of potential parameter values and interpolate in between.
Please note, that this precomputing is perfectly parallizable.
We show that this interpolation is stable and that the introduced error in the two-scale system can be controlled via the step size in the parameter space and the order of the interpolation. 

This idea is not completely new and similar approaches have been outlined before, see <ref> for a short literature overview.
The non-linearities in the model are resolved with a semi-implicit time-stepping method which we show to converge linearly.
We confirm these convergence results via simulations and the corresponding FEniCS code is made freely available <cit.>.



 §.§ Two-scale phase transition models

We want to elaborate a bit on the origin and structure of the two-scale phase transition problem that we are looking at in this work.
The model in question is the homogenization limit of a highly heterogeneous two-phase system consisting of a connected phase Ω_^a(t) with disconnected inclusions of a different phase Ω_^b(t), see <ref>. 
Both domains depend on time due to the potential of phase transitions. 
Here, >0 is a small parameter denoting the heterogeneity of the system and the homogenization limit can be achieved via a (formal or rigorous) limit process →0.




A two-phase problem for such a phase transition problem could look like

    ∂_tθ_^a-k^aΔθ_^a   =f_^a        in Ω_^a(t), t>0,

    ∂_tθ_^b-^2k^bΔθ_^b   =f_^b        in Ω_^b(t), t>0,

    θ_^a   =θ_^b        on Γ_(t), t>0,

    -(k^a∇θ_^a+^2k^b∇θ_^b)·n_   =Lv_        on Γ_(t), t>0,

where L∈ denotes the Latent heat and v_ the normal velocity of the moving phase interface Γ_(t).
The particular -scalings are natural for this particular setup with inclusions, cf. <cit.>.
The normal velocity has to be implicitly or explicitly described via an additional equation.
The usual Stefan condition is θ_^a=θ_ref, where θ_ref is the reference temperature of the phase transition in question.[In a water/ice system under normal conditions, we have θ_ref=0^∘ C.]
Another option is Gibbs-Thomson law without curvature which postulates


    v_= (θ_^a-θ_ref)  on Γ_(t), t>0.



The model in question in this work is the limit of System (<ref>) for →0.
Such a limit procedure was conducted via two-scale convergence techniques for a slightly simplified one-phase equivalent of this system in <cit.>, and the results can easily, at least formally, be extended to this two-phase system via asymptotic expansion.
Via such a limit process, we can identify limits such that, in some sense, θ_^a→Θ and θ_^b→θ.
Here, Θ is the temperature field of the macroscopic phase which is defined on a non-changing macroscopic domain Ω and θ is the temperature field of the microscopic phase given on a changing microscopic domain Y(t).
Of course, Θ and θ are intricately coupled via energy conservation and the continuity condition <ref>.
Moreover, all material coefficients for the macroscopic heat problem are now dependent on the microstructural changes as well, rendering the problem highly nonlinear.
The exact setup of the two-scale problem is given in <ref>.

 



 §.§ Discussion on related literature

An early work on two-scale models in phase transitions is due to A. Visintin <cit.>, where the transition between the two scales is achieved via convolution with a Gaussian kernel but without explicitly looking at evolving microstructures.
More closely related to our work, in <cit.>, two-scale models for dendritic growth with prescribed geometry changes are formally derived via asymptotic expansion.
In <cit.> two-phase models for phase transitions are derived, albeit with a priori known geometry evolution.
Very recently, in <cit.>, an effective one-phase model for phase transitions was rigorously derived via two-scale convergence.

Very similar models and challenges can be found in the context of reactive transport or coagulation/fragmentation dynamics in evolving porous media which has garnered quite a bit of interest in recent years.
We point to the upscaling works in <cit.>, which are concerned with the derivation of two-scale models in that context, and note that the resulting models are structurally quite similar to the phase transition problem considered in this work.
Naturally, there are quite a few works looking more closely at the resulting two-scale models: In <cit.>, the well-posedness for such a two-scale model was established for a simplified setup where the geometry changes do not impact the diffusivity except via the porosity.
In <cit.>, strong local-in-time solutions for a two-scale model, even allowing for nonuniform geometry changes, are established via maximal parabolic regularity and a level set approach.

Such models often come with huge computational challenges due to their nonlinear nature and complex coupling.
Our main approach of dealing with this is a precomputing strategy in which the calculations of the effective macroscopic coefficients are pushed into an offline phase; introducing an interpolation error but significantly speeding up the calculations.
Similar ideas are often explored in the context of shape or topology optimization, see e.g.,  <cit.> and references therein.
In <cit.>, the convergence of a precomputing scheme for a two-scale dispersion model for porous media is shown, however, there are no geometry changes present.

Naturally, there are other approaches to tackle the computational complexity of two-scale models:
Instead of solving the entire coupled system in a classical FE approach, an alternative idea is to explicitly decouple the macro and micro temperatures by solving between them iteratively.
This is closely related to the FE^2 approach <cit.> and the Heterogeneous Multiscale Method <cit.>.
This approach has two main advantages: the calculations on the microscale can be run in parallel and an adaptive interpolation strategy can be used to only solve the microscale problems for a small number of macroscopic points.
However, for each time step, there may be a need for several iteration steps, which again can be costly, especially when the coupling is complex.
In addition, the iteration process introduces a new error.
The approach has been studied and investigated for many different two-scale problems, see <cit.>.
As a direct extension of the iterative scheme, an adaptive clustering approach has been developed in <cit.>.
Here, micro problems with comparable macroscopic data are paired together and solved via one representative system.
This allows for a speed-up and even enables fast simulations in higher-dimensional setups.
Of course, this approach produces further error sources, for which, to our knowledge, no estimates are available.
But the numerical results in <cit.> show good approximation accuracy. 

Another approach is the hierarchical FEM as they are used for multiscale problems in <cit.>.
Here, special FE spaces are constructed to reduce the DOFs of those functions which depend on multiple scales.
This method is comparable to a wavelet approach and allows for analytical error estimates, but is more complicated to set up than the previous methods.
Finally, we want to mention a model reduction approach which is often encountered in mechanics, see <cit.> and the references therein.
A model reduction for microscopic crystal growth was considered in <cit.>. 
This approach can lead to significant speed-ups, but requires, especially for time-dependent problems, a large amount of data to obtain reasonable models.



 §.§ Outline of the paper

The paper is structured as follows: In <ref>, we present the two-scale system, collect the mathematical assumptions, and introduce our notion of weak solutions.
In <ref>, we establish the existence of local-in-time weak solutions via a fixed-point argument by utilizing Hanzawa transformations (<ref>).
Moreover, we show that uniqueness and long-time existence are given when some additional estimates are met.
This is followed up by <ref>, where we introduce and discuss our precomputing strategy.
Here, we first show the stability of the resulting scheme with respect to the interpolation error (<ref>) before then analyzing our semi-implicit time-stepping method with which we resolve the non-linearities (<ref>).
We present some simulation scenarios in <ref>: in particular, we are showing convergence curves for our precomputing strategy.
Finally, we close with a short conclusion and discussion (<ref>).








§ SETTING OF THE GEOMETRY AND MODEL EQUATIONS

In this section, we introduce the geometric setup and the two-scale system modeling a two-phase composite medium with phase transformations taking place at the microscale.
We also collect the mathematical assumptions and present our notion of weak convergence.

To this end, let S = (0, T ), for T∈∪{∞}, denote the time interval of interest.
Let a Lipschitz domain Ω⊂^d (d=2,3, outside normal vector ν) represent the macroscopic domain, set Y = (0, 1)^d and let a C^3-domain Y_0⊂⊂ Y (outer normal vector n) represent the initial micro-domain.
We set Γ_0=∂ Y_0.
The micro-domain Y_0 is allowed to grow or shrink based on the dynamics of the problem and this change is characterized via a height parameter h inside a tubular neighborhood U_Γ⊂⊂ Y of Γ_0 with width 2a for some a>0.
Here, h=0 indicates no changes, h<0 shrinkage, and h>0 growth.
We denote the dynamic domain by Y(h) and interface by Γ(h),[Of course, Y(0)=Y_0 and Γ(0)=Γ_0.] respectively, via


    Γ(h):={γ+h n(γ) :  γ∈Γ_0},   Y(h):=(Γ(h)).

Please note that the growth of the micro-domain is cell uniform, i.e., the height h=h(t,x) only depends on time t∈ S and the macro variable x∈Ω.
As a consequence, shape changes are not possible (e.g., if Y_0 is a ball of radius r, Y(h) will be a ball of radius r+h).
For the exact setup of the transformation, we refer to <ref>, and, for a sketch of this setup, we refer to <ref>.




Now, let Θ=Θ(t,x) and θ=θ(t,x,y) denote the macroscopic and microscopic temperature, respectively, and h=h(t,x) the height function characterizing the micro-domain changes.

On the macroscale, we have a standard heat equation where all parameters non-linearly depend on h:


∂_t(C(h)Θ)-(K(h)∇Θ)   =F- L(h)∂_th-q(h)[θ]        in S×Ω,

-K(h)∇Θ·n   =0        on S×∂Ω,

Θ(0)   =Θ_0        in Ω.



Here, C denotes the heat capacity, K the effective heat conductivity, and L is the latent heat.
Moreover, q(h)[θ] S×Ω→ accounts for the energy flow from the micro into the macro region and F is a generic volume source.
The h-dependent coefficients are given via



    C   (-a,a)→(0,∞),     C(h)   =1-|Y(h)|,

    K   (-a,a)→^d×d,     K(h)_i,j   =∫_Y∖Y(h)𝒦(∇_yξ_j(h)+e_j)·(∇_yξ_i(h)+e_i)y,

     L   (-a,a)→,     L(h)   =|Γ(h)|,

    q(·)[θ]   (-a,a)→,     q(h)[θ]   =∫_Γ(h)κ∇_yθ·nσ_y.

Here, 𝒦, κ>0 are the heat conductivity of the macro- and micro-phase and ξ_j(h)∈ H^1_#(Y∖ Y(h)), j=1,...,d, solutions to the cell problems


∫_Y∖Y(h)𝒦(∇_yξ_j(h)+e_j)·∇_yϕy=0


for all ϕ∈ H^1_#(Y∖ Y(h)).


Please note that the coefficients <ref> represent what we typically expect in a two-scale limit system; see <cit.> for similar setups.
For example, C(h) represents the effective volumetric heat capacity of the macroscopic phase which is given by C(h)=|Y∖ Y(h)|=1-|Y(h)|.
This accounts for the competition with the micro-phase.


The heat problem for the macroscopic phase (<ref>) is strongly coupled with a Stefan-type problem for the microscopic phase in the form of



∂_tθ-_y(κ∇_yθ)   =f        in Y(h),  in  S×Ω,

θ   =Θ        on Γ(h),  in  S×Ω,

∂_th   =θ-Θ^ref        on Γ(h),  in S×Ω,

θ(0)   =θ_0        in Ω×Y(0).



The moving boundary condition (<ref>) is a modified Gibbs-Thomson law without curvature, where the normal velocity of the moving boundary, i.e., ∂_th, is determined by the difference of the temperature to a reference temperature.
As an example, for the water-ice system under normal conditions, we have Θ^ref=0^∘ C.
Here, we just assume Θ^ref∈.
The overall problem is both highly nonlinear due to the changing domains and is fully coupled via the following mechanism:


    
  * The flux term q in <ref> tracks the heat exchange between the phases. As a consequence, the microphase can act as a heat storage or heat sink.
    
  * <Ref> enforces continuity of the micro and macro temperatures at the micro-interface Γ(h). This also implies cell-uniform growth: Since Θ is y-independent, h must be as well.
    
  * The growing or shrinking domains directly influence the macroscopic coefficients (<ref>).

We point out that the Neumann flux appearing via q(h)[θ] in the strong form <ref> disappears in the weak form as it balances with the corresponding term in the micro part after integration by parts.
The main difficulties in the analysis of this system are the changing micro-geometries and the inherent non-linearity associated with those changes.
We do this by transforming the geometries into the fixed, initial geometries via the so-called Hanzawa transformation.
Details are given in <ref>.


    In our setup, it holds h(0)=0 which means that the initial micro-geometry is uniform in x∈Ω.
    However, we could easily consider an initial condition h∈ L^∞(Ω) as long as it is small enough (see also <ref>).
    As this would introduce some additional technical estimates in our analysis, we stick to a uniform initial setup. 




 §.§ Assumptions and weak forms

A few words, regarding our notation: In general, we use lowercase and capitalized symbols for the microscopic and macroscopic counterparts, e.g., f and F for the volume source densities.
For functions depending on both the macroscopic variable x∈Ω and y∈ Y, we take _y,∇_y to denote the spatial differential operators with respect to y.
Moreover, we take c,C>0 as generic constants (whose value can change even from line to line) and where we explicitly point out when they are independent of a specific parameter.
We start by formulating the assumptions we place on our geometry and data:

    
  (A1) The heat conductivities κ,𝒦∈^d× d are symmetric and positive definite.
    
  (A2) The volume source densities satisfy f∈ L^∞(S×Ω;C^0,1(Y_0)) and F∈ L^∞(S×Ω).
    
  (A3) The initial geometry Y(0)=Y_0⊂⊂(0,1)^d is a C^3 domain and Ω⊂^d is a Lipschitz domain. 
    The tubular neighborhood U_Γ⊂(0,1)^d of the interface Γ=∂ Y_0 has tubular radius a>0.
    
  (A4) The initial conditions satisfy θ_0∈ L^∞(Ω× Y_0) and Θ_0∈ L^∞(Ω).


Please note that the regularity presented in Assumptions (A2) and (A4) is higher than the standard parabolic regularity for two reasons: To be able to work with coordinate transforms, the micro heat production source needs to be continuous in the y-Variable.
We chose Lipschitz continuous to guarantee the uniqueness of solutions.
The boundedness of the data is needed to ensure boundedness of the solutions; without this, it is impossible to control the height functions (see <ref>).

Now, for any 0<T^*≤ T and h∈ L^∞((0,T^*)×Ω) satisfying h_∞≤ a^*=a/8, we introduce the corresponding non-cylindrical micro-domain


    Q[h](T^*)=⋃_(t,x)∈ (0,T^*)×Ω[{(t,x)}× Y(h(t,x)) ]⊂ (0,T')×Ω× Y.

and the Banach space


    𝒳[h](T^*)={(Θ,θ)∈ L^2(0,T^*;H^1(Ω))× L^2(Q[h](T^*)) : θ(t,x)∈ H^1(Y(h)),
    θ(t,x,y)=Θ(t,x) on Γ(h) for a.a. (τ,x)∈ (0,T')×Ω, y∈Γ(h) }.


Assuming that the height function h∈ L^∞((0,T^*)×Ω) describes the moving boundaries and taking any function (Φ,ϕ)∈𝒳[h](T^*), we introduce the weak form for the PDE system (<ref>):


    ∫_Ω∂_t(C(h)Θ)Φx+∫_Ω∫_Y(h)∂_tθϕyx+∫_ΩK(h)∇Θ·∇Φx
    +∫_Ω∫_Y(h)κ∇_yθ·∇_yϕyx
    
    =∫_Ω(F- L(h)∂_th)Φx
    +∫_Ω∫_Y(h)fϕyx


Please note that the q(h)[θ] term in <ref> cancels with the micro boundary integral due to ϕ=Φ on Γ(h).
With that in mind, we can now define a solution of the two-scale heat problem in moving coordinates:


We say that (h,Θ,θ) is a local-in-time weak solution in moving coordinates, when there is a time horizon S^*=(0,T^*), such that


    
  (i) The domain changes are characterized by the height function h∈ W^1,∞(S^*;L^∞(Ω)) which satisfy
    
    h_L^∞(S^*×Ω)≤ a^*.

    
  (ii) The velocity of the domain evolution is driven by the temperature:
    
    ∂_th=Θ-Θ^ref in L^2(S^*×Ω).

    
  (iii) The temperatures (Θ,θ)∈𝒳[h](T^*) with ∂_t(Θ,θ)∈ L^2(S^*×Ω)× L^2(Q[h](T^*)) satisfy
    
    
    ∫_S∫_Ω∂_t(C(h)Θ)Φx+∫_Ω∫_Y(h)∂_tθϕyx+∫_ΩK(h)∇Θ·∇Φx
        +∫_Ω∫_Y(h)κ∇_yθ·∇_yϕyx
    
        =∫_Ω(F- L(h)∂_th)Φx
        +∫_Ω∫_Y(h)fϕyx

    for all (Φ,ϕ)∈𝒳[h].
    
  (iv) The initial conditions are satisfied: (Θ(0),θ(0), h(0))=(Θ_0,θ_0,0).



This solution concept, however, is inconvenient for the mathematical analyses due to the changing domains.
We use the concept of the Hanzawa transformation to reformulate this problem in the fixed domain Y_0=Y(0).
The precise definition and properties of this transformation are introduced in <ref> and specifically <ref>.

Now, for any 0<T^*<T and height function h∈ W^1,∞(0,T^*;L^∞(Ω)) with |h|≤ a^*, let s_h∈ W^1,∞(0,T^*;L^∞(Ω;C^2(Y))) be the corresponding Hanwaza transformation.

















We introduce the Jacobian matrix F_h=D_ys_h, the Jacobian determinant J_h= F_h, and the velocity ∂_ts_h. 
Also, we denote the push-forward S_h and the pull-back S_h^-1 for any sufficiently regular function fY→ defined via S_hf=f∘ s_h and S_h^-1f=f∘ s_h^-1.
Finally, we introduce the transformed quantities 


    c(h)   =J_h,       κ(h)   =J_hF_h^-1κF_h^-T,

    f(h)   =J_hS_hf,       v(h)   =J_hF_h^-1∂_ts_h.  

With this transformation, we can alternatively formulate the problem in fixed coordinates where we use 𝒳_0(T^*):=𝒳[0](T^*):




We say that (h,Θ,ϑ) is a local-in-time weak solution in moving coordinates, when there is a time horizon S^*=(0,T^*), such that:


    
  (i) The domain changes are characterized via the height function h∈ W^1,∞(S^*;L^∞(Ω)) which satisfy
    
    h_L^∞(S^*×Ω)≤ a^*.

    
  (ii) The velocity is driven by the temperature:
    
    ∂_th=Θ-Θ^ref in L^2(S^*×Ω).

    
  (iii) The temperatures (Θ,ϑ)∈𝒳_0(T^*) with ∂_t(Θ,ϑ)∈ L^2(S^*×Ω)× L^2(S^*×Ω× Y_0) satisfy
    
    
    ∫_S∫_Ω∂_t(C(h)Θ)Φx+∫_Ω∫_Y_0∂_t(c(h)ϑ)ϕyx+∫_ΩK(h)∇Θ·∇Φx
        +∫_Ω∫_Y_0κ(h)∇_yϑ·∇_yϕyx
    
        +∫_Ω∫_Y_0ϑ v(h)·∇_yϕyx
        =∫_Ω(F- L(h)∂_th)Φx
        +∫_Ω∫_Y_0f(h)ϕyx

    for all (Φ,ϕ)∈𝒳_0.
    
  (iv) The initial conditions are satisfied: (Θ(0),ϑ(0),h)=(Θ_0,ϑ_0,0).



It is relatively straightforward to show that (h,Θ,θ) is a solution in the sense of <ref> if and only if (h,Θ,S_h^-1ϑ) is a solution in the sense of <ref> (see, e.g., <cit.>).
In <ref>, we will work with <ref> to show the existence of local-in-time weak solutions.







§ ANALYSIS OF THE MOVING BOUNDARY PROBLEM

In this section, we investigate the two-scale problem with changing micro-domains and show that there are weak solutions (h,Θ,ϑ) in the sense of <ref> over some possibly small time interval (0,T^*)⊂ S.
We also establish uniqueness in case of higher regularity and long-time existence under certain smallness conditions.
Our strategy and outline of this section is the following:


    
  (i) In <ref>, we introduce the Hanzawa transformation (<ref>) and establish some uniform positivity and Lipschitz estimates (<ref>).
    
  (ii) We go on, in <ref>, to look at the linearized problem by solving the heat problem for a prescribed height function and establishing a-priori estimates (<ref>).
    
  (iii) Finally, in <ref>, we employ a Schauder fixed-point argument to prove the existence of weak solutions (<ref>).
    We go on to show uniqueness (<ref>) and long-time existence (<ref>) when certain conditions are met.    




 §.§ Hanzawa transformation and analysis of the cell problems

In the following, we present some auxiliary results regarding the Hanzawa transformation which are necessary to estimate and control the impact of the changing geometry on the overall system.
For more details regarding this type of setup, including the case of nonuniform growth, we refer to <cit.>.

As a C^3-interface, Γ_0=∂ Y_0 has a tubular neighborhood U_Γ⊂⊂ Y with tubular radius a>0.
In U_Γ, every point is uniquely characterized by its projection onto Γ_0 (via P U_Γ→Γ_0), and its signed distance (via d U_Γ→(-a,a))[Here, we follow the convention that negative distances correspond to points inside Y_0.].
Moreover, the function


    ΛΓ_0×(-a,a)→ U_Γ,  Λ(γ,h)=γ+hn(γ)


is a C^2-isomorphism with inverse


    Λ^-1 U_Γ→Γ_0×(-a,a),    Λ^-1(y)=(P(y),d(y)).


We set a^*=a/8. 
For h∈[-a^*,a^*], we introduce


    Γ(h)={γ+h n(γ) :  γ∈Γ_0}, Y(h)=(Γ(h)).


Clearly, Γ(0)=Γ_0, Y(0)=Y_0, as well as ∂ Y(h)=Γ(h).
In the following, let χ→[0,1] be a smooth cut-off function that satisfies


(i)    χ(r)=1  for r∈(-1/2,1/2),        (ii)    χ(r)=0  for r∉(-1,1),

(iii)    sign(r)χ'(r)≤0,     (iv)    |χ'(r)|≤5/2

and introduce ψU_Γ→^d given by ψ(y)=n(P(y))χ(d(y)/a).
Note that ψ∈ C^2(U) with |D_yψ(y)|<4/a.[This follows as the mean curvature is bounded by a^-1.]
Then, the so-called Hanzawa transform s_hY→Y defined via


    s_h(y)=
         y+hψ(y)     y∈ U_Γ,
    
         y     y∈Y∖ U_Γ


is a C^2-transformation satisfying s_h(Γ)=Γ(h) and |D_ys_h|, |D_ys_h^-1|≤ 2 for all h∈[-a^*,a^*].
Please note that due to h being constant this is all a special case of the more general results for the Hanzawa transformation as outlined in <cit.>.
In the following, we denote the Jacobian and the Jacobian derivative by F_h=D_ys_h and J_h= F_h, respectively.


    
        
  (i) There are 0<c<C independent of h∈[-a^*,a^*] and y∈Y such that
        
        
    c≤ J_h(y)≤ C,   c|ξ|^2≤ F^-1_h(y)F^-T_h(y)ξ·ξ≤ C|ξ|^2   (ξ∈^d).

        
  (ii) Let h_1,h_2∈ [-a^*,a^*] and let s_i=s_h_i, F_i=F_h_i, J_i=J_h_i denote the corresponding transformation functions.
        Then,
        
    F_1-F_2_L^∞(Y)^d× d+F_1^-1-F_2^-1_L^∞(Y)^d× d
            +J_1-J_2_L^∞(Y)≤ C|h_1-h_2|

        
        where C>0 independent of h.
    


  (i). These bounds exist as |F_h(y)|, |F_h^-1(y)|≤ 2 uniformly and independently of h∈[-a^*,a^*].
  
  (ii). Outside the tubular neighborhood U_Γ, the transformation is trivial, i.e., F_i= and J_i=1, so we only have to look at y∈ U.
  We immediately get
  
  
    |F_1(y)-F_2(y)|=|(h_1-h_2)D_yψ(y)|≤ C|h_1-h_2|.

  
  We already know that the F_i are invertible with
  
  
    F_i^-1(y)=(+h_iD_yψ(y))^-1

  
  from where
  
  
    F_i^-1(y)=-h_iD_yψ(y)F_i^-1(y)

  follows.
  As a consequence,
  
  
    F_1^-1(y)-F_2^-1(y)=D_yψ(y)(h_2F_2^-1(y)-h_1F_1^-1(y))

  
  or,
  
    |F_1^-1(y)-F_2^-1(y)|≤ C|h_1-h_2|+|h_2D_yψ(y)||F_1^-1(y)-F_2^-1(y)|.

  
  Since |h_2D_yψ(y)|≤1/2, the Lipschitz estimate follows.
  Finally, for the determinant a second order Taylor approximation yields
  
  
    J_i(y)=(+h_i D_yψ(y))=1+h_i((D_y ψ(y))+𝒪(h_i))

  
  showing the Lipschitz estimate.


In the following, for any h∈[-a^*,a^*], we note that


    S_h H^1(Y∖ Y_0)→ H^1(Y∖ Y(h)),      S_h^-1 H^1(Y∖ Y(h))→ H^1(Y∖ Y_0)).


Based on the estimates in <ref>, there are constants c,C>0 such that


    cS_hϕ_H^1(Y∖ Y(h))≤ϕ_H^1(Y∖ Y_0)≤ CS_hϕ_H^1(Y∖ Y(h))     (ϕ∈ H^1(Y∖ Y_0)).




In the next step, we establish how the different coefficients depend on the height parameter and the corresponding transformation.
To that end, let h∈[-a^*,a^*] be given.
It is clear, that h↦|Y(h)| and h↦|Γ(h)| are C^1 and therefore also Lipschitz. 
We now look at the cell problem given via <ref>:



    For every h∈[-a^*,a^*], there is a unique zero-average solution ξ_j,h∈ H_#(Y∖ Y(h)), j=1,…,d, satisfying    
    
    
    ∫_Y∖Y(h)𝒦(∇_yξ_j,h+e_j)·∇_yϕy=0
    
    
    for all ϕ∈ H^1_#(Y∖ Y(h)) as well as ∇ξ_j,h_L^2(Y∖ Y(h))≤ C where C>0 is independent of h.
    
    Moreover, for h_1,h_2∈[-a^*,a^*], we have
    
    
    |∫_Y∖ Y(h_1)𝒦∇ξ_j,h_1· e_iy-∫_Y∖ Y(h_2)𝒦∇ξ_j,h_2· e_iy|≤ C |h_1-h_2|.



    The existence of a unique zero-average solution is standard and follows via Lax-Milgram <cit.>.
    Let h∈[-a^*,a^*].
    By testing with φ = ξ_j,h and by assumption (A1) we obtain the desired estimate
    
    c ∇ξ_j,h_L^2(Y∖ Y(h))^2 ≤ C e_j_L^2(Y∖ Y(h))∇ξ_j,h_L^2(Y∖ Y(h)) = C √(|Y∖ Y(h)|)_≤ 1∇ξ_j,h_L^2(Y∖ Y(h)).

    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    

    Now, let h_1,h_2∈[-a^*,a^*], take the difference of their respective weak forms, and test with ξ̅_j:=S_1^-1ξ_j,h_1-S_2^-1ξ_j,h_2:
    
    
    ∫_Y∖ Y_0J_1F_1^-1𝒦F^-T_1∇_yξ̅_j·∇_yξ̅_j
    
          =∫_Γ_0(J_1-J_2)e_j· nξ̅_jy
          +∫_Y∖ Y_0(J_2F_2^-1𝒦F^-T_2-J_1F_1^-1𝒦F^-T_1)∇_y(S_2^-1ξ_j,h_2)·∇_yξ̅_jy.

    
    With the uniform positivity of J_1F^-T_1 and the Lipschitz continuity of J_h and F_h^-T (<ref>), we estimate
    
    
    c∇_yξ̅_j^2_L^2(Y∖ Y_0)
    ≤ C|h_1-h_2|(ξ̅_j_L^1(Γ_0)+∇_y(S_2^-1ξ_j,h_2)_L^2(Y∖ Y_0)∇_yξ̅_j_L^2(Y∖ Y_0)).

    
    Via Poincare's inequality and (<ref>), we therefore get
    
    
    ξ̅_j_H^1(Y∖ Y_0)≤ C|h_1-h_2|.

    
    From here, we can conclude the desired Lipschitz estimate via
    
    
    ∫_Y∖ Y(h_1)𝒦∇ξ_j,h_1· e_iy
        -∫_Y∖ Y(h_2)𝒦∇ξ_j,h_2· e_iy
    
        =
        ∫_Y∖ Y_0(J_1F_1^-1𝒦 F^-T_1∇_y(S_1ξ_j,h_1)-J_2F_2^-1𝒦 F^-T_2∇_y(S_2ξ_j,h_2))· e_iy

    
    and <ref>.






    
  (i)  Let h_1,h_2∈[-a^*,a^*].
    There is a C>0 independent of h_1,h_2 such that
    
    
    |C(h_1)-C(h_2)|+|K(h_1)-K(h_2)|+| L(h_1)- L(h_2)|≤ C|h_1-h_2|.

    
  (ii)
    There is c>0 such that, for all h∈[-a^*,a^*] and all ρ∈^d,
    
    |C(h)|≥ c,   K(h)ρ·ρ≥ c|ρ|^2.




    (i). This is easy to check for C and L by directly utilizing the transformation <ref> and the Lipschitz estimates in <ref>.

    For the heat conductivity, we rely on additional estimates for the cell problem given in <ref>:
    Their difference can be estimated by
    
    
    K(h_1)_i,j-K(h_2)_i,j=∫_Y∖ Y(h_1)𝒦∇_yξ_j,h_1· e_iy
        -∫_Y∖ Y(h_2)𝒦∇_yξ_j,h_2· e_iy
    
        +∫_Y∖ Y(h_1)𝒦_ijy
        -∫_Y∖ Y(h_2)𝒦_ijy

    
    where we can apply <ref> to get the Lipschitz estimate.

    (ii). Trivially true for C(h)=1-|Y(h)| as (Γ_0,∂ Y)≥ 2a^*.
    For the macroscopic conductivity matrix, positivity can be shown easily for a fixed h with standard techniques (e.g., <cit.>, i.e., K(h)ρ·ρ≥ c_h|ρ|^2.
    Since K is continuous in h (shown in (i)) and [-a^*,a^*] is compact, uniform positivity follows with c=min_|h|≤ a^*c_h>0.    






 §.§ Analysis for prescribed height function

We take a look at the weak form for the heat system, i.e., <ref>, for a prescribed height function h and establish well-posedness and some further auxiliary results for this linearized problem.
For some arbitrary and later to be fixed M>0, we introduce the space of admissible height functions whose time derivatives are bounded by M:


    ℋ_M(t)={h∈ W^1,∞((0,t); L^∞(Ω)) :  h(0,x)=0, |∂_th(τ,x)|≤ M  for a.a. τ∈(0,t) and x∈Ω}.


Please note that every h∈ℋ_M(t) satisfies |h|≤ a^* as long as t≤a^*/M=:T_M; therefore the results from <ref> apply pointwise almost everywhere.
We set S_M=(0,T_M).


Let M>0 and h∈ℋ_M(T_M).
There exists a unique weak solution (Θ,ϑ)∈𝒳_0 with ∂_t(Θ,ϑ)∈ L^2(S_M×Ω)× L^2(S_M×Ω× Y_0) satisfying


    ∫_Ω∂_t(C(h)Θ)Φx+∫_Ω∫_Y_0∂_t(c(h)ϑ)ϕyx+∫_ΩK(h)∇Θ·∇Φx
        +∫_Ω∫_Y_0κ(h)∇_yϑ·∇_yϕyx
    
        +∫_Ω∫_Y_0ϑ v(h)·∇ϕyx
        =∫_Ω(F- L(h)∂_th)Φx
        +∫_Ω∫_Y_0f(h)ϕyx


for all (Φ,ϕ)∈𝒳_0 and almost all t∈ S_M.
In addition, there is C>0, which is independent of h and M, such that


    Θ_L^∞(S_M;L^2(Ω))+Θ_L^2(S_M;H^1(Ω))+ϑ_L^∞(S_M;L^2(Ω× Y_0))+ϑ_L^2(S_M×Ω;H^1(Y_0))≤ Cexp(CM)


Moreover, there is C>0 such that for all h∈ℋ_M(T_M):


    Θ_L^∞((0,t)×Ω) + ϑ_L^∞((0,t)×Ω× Y_0)≤ C(1+M^2√(t))




    For any such height function, all coefficients are well-defined and essentially bounded.
    Moreover, c(h) and C(h) are positive and κ(h) and K(h) are positive definite (see <ref>).
    As a consequence, this is a linear implicit evolution equation in the form of
    
    
    (ℬ(t)u(t))+𝒜(t)u(t)=f(t)

    
    which under the given regularity admits a unique solution.
    We refer to <cit.>.

    Regarding the estimates: First of all, the advective velocity v(h)=J_hF_h^-1∂_ts_h and the time derivatives of the coefficients c(h) and C(h) can be estimated by
    
    |v(h)|+|∂_tc(h)|+|∂_tC(h)|≤ C|∂_th|≤ CM.

    
    
    Now, testing with (Φ,ϕ)=(Θ,ϑ) shows:
    
    
    ∫_Ω1/2C(h)|Θ|^2x
        +∫_Ω∫_Y_01/2c(h)|ϑ|^2x
        +c∇Θ^2_L^2(Ω)
        +c∇_yϑ^2_L^2(Ω× Y_0)
    ≤ C∫_Ω M(|Θ|+|Θ|^2+∫_Y_0|ϑ|^2+|ϑ∇_yϑ|y)x
        +F_L^2(Ω)Θ_L^2(Ω)
        +f(h)_L^2(Ω× Y_0)ϑ_L^2(Ω× Y_0)

    
    for almost all t∈ S_M.
    Integrating over (0,t) for t∈ S_M and using standard energy techniques, we get 
    
    
    Θ(t)_L^2(Ω)^2
        +ϑ(t)_L^2(Ω× Y_0)^2
        +∫_0^t∇Θ^2_L^2(Ω)
        +∇_yϑ^2_L^2(Ω× Y_0)τ
    ≤ C∫_0^t∫_Ω M(|Θ|+|Θ|^2+(1+M)∫_Y_0|ϑ|^2y)xτ
    
        +∫_0^tF_L^2(Ω)Θ_L^2(Ω)
        +f(h)_L^2(Ω× Y_0)ϑ_L^2(Ω× Y_0)τ.

    
    On the first term on the right hand side, we can further estimate:
    
    
    ∫_0^t∫_Ω|∂_th|(|Θ|+|Θ|^2)xτ≤
        C∫_0^tM(Θ^2_L^2(Ω)+ 1 )τ

    
    As a result, we arrive at
    
    
    Θ(t)_L^2(Ω)^2
        +ϑ(t)_L^2(Ω× Y_0)^2
        +∫_0^t∇Θ^2_L^2(Ω)
        +∇_yϑ^2_L^2(Ω× Y_0)τ
    ≤ C(1+ ∫_0^t(1+M+M^2)(Θ^2_L^2(Ω)+ϑ_L^2(Ω× Y_0)^2)τ).

    
    With Grönwalls inequality, we are led to 
    
    
    Θ_L^∞(0,t;L^2(Ω))^2
        +ϑ_L^∞(0,t;L^2(Ω× Y_0))^2
        +∇Θ^2_L^2(0,t;L^2(Ω))
        +∇_yϑ^2_L^2(0,t;L^2(Ω× Y_0))
    ≤ Cexp(CtM^2).

    

    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    Regarding the boundedness of the solution, for k∈ with k≥max{Θ_0_L^∞(Ω),θ_0_L^∞(Ω× Y_0)}, we introduce the functions Θ^k∈ L^2(S_M;H^1(Ω)) and ϑ^k∈ L^2(S_M×Ω;H^1(Y_0)) via
    
    
    Θ^k=Θ-k    for Θ> k
    
                0   for |Θ|≤ k
    Θ+k    for Θ< -k
                ,    ϑ^k=ϑ-k    for ϑ> k
    
                0   for |ϑ|≤ k
    ϑ+k    for ϑ< -k
                ,

    
    as well as the sets
    
    
    A_Θ(t,k)   ={x∈Ω :  |Θ(τ,x)|≥ k},
     A_ϑ(t,x,k)   ={y∈  Y_0 :  |ϑ(τ,x,y)|≥ k}.

    
    Please note that ∂_tΘ^k=χ_A_Θ∂_tΘ, ∇Θ^k=χ_A_Θ∇Θ, and ΘΘ^k≥|Θ^k|^2 (similar for ϑ).
    With (Θ^k,ϑ^k) as a test function and the <ref>, standard energy estimates lead to (t∈ S_M)
    
    
    Θ^k(t)_L^2(Ω)^2+ϑ^k(t)^2_L^2(Ω× Y_0)
        +∇Θ^k^2_L^2((0,t)×Ω)
        +∇_yϑ^k^2_L^2((0,t)×Ω× Y_0)
    ≤ C∫_0^t∫_Ω(|∂_th||Θ||Θ^k|+|F- L(h)∂_th||Θ^k|)xτ
    
        +C∫_0^t∫_Ω∫_Y_0|∂_th||ϑ||ϑ^k|+|ϑ| |v(h)||∇_yϑ^k|
        +|f(h)||ϑ^k|yxτ

    
    where C>0 does not depend on h, k, or M.
    Using |Θ|≤ k+|Θ^k|, |ϑ|≤ k+|ϑ^k|, and |∂_th|≤ M, we estimate
    
    
    ∫_Ω|∂_th||Θ||Θ^k|x   ≤ 2M(Θ^k_L^2(Ω)^2+k^2|A_Θ(t,k)|),
    ∫_Y_0|∂_th||ϑ||ϑ^k|y   ≤ 2M(ϑ^k_L^2(Y_0)^2+k^2|A_ϑ(t,x,k)|).

    
    The latent heat term, we estimate via (with k≥max_h∈[-a^*,a^*]| L(h)|)
    
    
    ∫_Ω| L(h)∂_th||Θ^k|x≤M/2(Θ^k_L^2(Ω)^2+k^2|A_Θ(t,k)|).

    
    Looking at the convection term (v(h)≤ CM):
    
    
    ∫_Y_0|ϑ| |v(h)||∇_yϑ^k|y≤δ∇_yϑ^k_L^2(Y_0)^2+M^2/4δ(ϑ^k_L^2(Y_0)^2+k^2|A_ϑ(τ,x,k)|).

    
    for any δ>0.
    
    We set
    
    
    k^*=max{Θ_0_L^∞(Ω),θ_0_L^∞(Ω× Y_0),F_L^∞((0,T_M)×Ω), f_L^∞((0,T_M)×Ω× Y_0),max_h∈[-a,a]| L(h)|}

    
    and conclude, for all k≥ k^* and assuming M≥1,
    
    
    Θ^k(t)^2_L^2(Ω)+ϑ^k(t)^2_L^2(Ω× Y_0)
        +∇Θ^k^2_L^2((0,t)×Ω)
        +∇_yϑ^k^2_L^2((0,t)×Ω× Y_0)
    ≤ CM^2(θ^k_L^2((0,t)×Ω)^2+    
        ϑ^k_L^2((0,t)×Ω× Y_0)^2
        +k^2∫_0^t(|A_θ(τ,k)|+∫_Ω|A_ϑ(τ,x,k)|x)τ).

    

    
    Thus, Gronwall's inequality allows us to conclude for 0<t<T_M
    
    
    Θ^k_L^∞(0,t;L^2(Ω))^2+ϑ^k^2_L^∞(0,t;L^2(Ω× Y_0))
        +∇Θ^k_L^2(0,t×Ω)^2
        +∇_yϑ^k^2_L^2(0,t×Ω× Y_0)
    ≤ Cexp(tM^2)M^2k^2∫_0^t(|A_θ(τ,k)|+∫_Ω|A_ϑ(τ,x,k)|x)τ.

    
    where the constant C>0 is still independent of k, h, and M.
    From here, we apply <cit.> to conclude the existence of k_M, which is independent of h but does depend on M, such that |A_θ(τ,k_M)|,|A_ϑ(τ,x,k_M)|=0 almost everywhere.
    
    More precisely
    
    
    Θ_L^∞((0,t)×Ω)+ϑ_L^∞((0,t)×Ω)≤ 2k^*(1+Cexp(5/4M^2t)M^2 √(t))   (t∈(0,T_M))

    
    Moreover, tM^2≤ a^* for 0≤ t≤√(T_M).
    Over this time interval, we have
    
    
    Θ_L^∞((0,t)×Ω)+ϑ_L^∞((0,t)×Ω)≤ 2k^*(1+CM^2√(t))    (t∈ (0,√(T_M)).

    This implies uniformity in M of the L^∞ estimates for t≤a^*/M^4.
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    


























































 §.§ Fixed-point argument and long time behavior

We first note that <ref> induces a solution map ℒ:=(ℒ_1(h),ℒ_2(h))ℋ_M(T_M)→𝒳_0(T_M) for every M>0.
Here, ℒ_1(h)∈ L^2(S_M×Ω) denotes the macroscopic temperature and ℒ_2(h)∈ L^2(S_M×Ω× Y_0) the microscopic temperature, respectively.
Also, with <ref>, it holds


    C_L:=sup_h∈ℋ_M(t)ℒ_1(h)_L^∞((0,t)×Ω)<∞


where, importantly, C_L does not depend on M as long as t∈(0,a^*/M^4).
Setting


    𝒴(t):={(Θ,ϑ)∈𝒳_0(t) :  Θ_L^∞((0,t)×Ω)≤ C_L}⊂𝒳(t),

and T^*_M= a^*/M^-4, it is clear that ℒ[ℋ_M(T_M^*)]⊂𝒴(T_M^*).
Let (Θ,ϑ)∈𝒴(T_M^*).
To get to a fixed-point argument, we must be able to construct a new height function h̃ based on the macroscopic temperature Θ.
We introduce h̃(t,x)=∫_0^tΘ(τ,x)-Θ^refτ and note that h̃∈ W^1,∞((0,T_M^*);L^∞(Ω)) via the regularity of Θ.
Moreover, 

    |∂_th̃(t,x)|=|Θ(t,x)-Θ^ref|≤ C_L+|Θ^ref|

for almost all (t,x)∈ (0,T_M^*)×Ω.
As a consequence, h̃∈ℋ_M(T_M) as long as M≥ C_L+|Θ^ref|.
We fix M^*=C_l+|Θ^ref|, set T^*:=T_M^* and S^*=(0,T_M^*), and introduce


    𝒯𝒴(T^*)→𝒴(T^*),  𝒯(Θ,ϑ)=ℒ(∫_0^tΘ(τ,x)-Θ^refτ).




    There is at least one solution (h,Θ,ϑ)∈ℋ_M^*(T^*)×𝒳_0(T^*) for the moving boundary problem in fixed coordinates in the sense of <ref> over the time interval S^*=(0,T^*).


We wirst note, that (h,Θ,ϑ)∈ℋ_M^*(T^*)×𝒳_0(T^*) is a solution if, and only if, 𝒯(Θ,ϑ)=(Θ,ϑ) and h(t,x)=∫_0^tΘ(τ,x)-Θ^refτ.

(i) Continuity of fixed-point operator over L^2. Let (Θ_n,ϑ_n),(Θ_∞,ϑ_∞)∈𝒴(T^*) such that (Θ_n,ϑ_n)→(Θ_∞,ϑ_∞) in L^2(S^*×Ω)× L^2(S^*×Ω× Y_0).
Let h_n,h_∞∈ℋ_M^*(T^*) denote the corresponding height functions.
Due to ∂_th_k=Θ_k (k∈_∞), we immediately have ∂_th_n→∂_t h_∞ and h_n→ h_∞ in L^2(S^*×Ω).
Now, we set (Θ_k,ϑ_k):=𝒯(Θ_k,ϑ_k)∈𝒴(T^*) for k∈_∞.
Utilizing the a priori estimates established with <ref>, there exist (Θ,ϑ)∈𝒴(T^*) such that, at least along a subsequence, (Θ_n,ϑ_n)→ (Θ,ϑ) weakly in 𝒳_0(T^*) and, with Lions-Aubin, strongly in L^2.
If we can show that (Θ,ϑ)=(Θ_∞,ϑ_∞) and that the full sequence converges, continuity holds.


Looking at the most critical terms in the weak forms, namely the heat conductivities, we have the limits

    lim_j→∞∫_ΩK(h_n_j)∇Θ_n_j·∇Φx   =∫_ΩK(h_∞)∇Θ·∇Φx,
    lim_j→∞∫_Ω∫_Y_0κ(h_n_j)∇_yϑ_n_j·∇_yϕyx   =∫_Ω∫_Y_0κ(h)∇_yϑ·∇_yϕy


for all (Φ,φ)∈𝒳_0(T^*).
Here, we used the weak convergences of (∇Θ_n,∇_yϑ_n) along a subsequence n_j and the strong convergence of h_k together with the uniform bounds for K,κ.
After identifying all terms in the weak form in the same way, we can see that Θ=Θ_∞, which implies Θ_n_j→Θ_∞ strongly in L^2(S^*×Ω), similarly for ϑ̂_n_j.
Finally, since every subsequence (Θ_n,ϑ_n) has a converging subsequence which all converge to the same limit, the whole sequence converges implying L^2-continuity of 𝒯.

(ii) Existence of solutions. We showed that

    𝒯𝒴(T^*)⊂ L^2(S^*×Ω)× L^2(S^*×Ω× Y_0)→ L^2(S^*×Ω)× L^2(S^*×Ω× Y_0)


is continuous.
Moreover 𝒯[𝒴(T^*)] is compact in  L^2(S^*×Ω× Y_0) via Lions-Aubin and 𝒴(T^*) is closed and convex.
With Schauders fixed point theorem, there must therefore be at least one fixed point (Θ,ϑ)∈𝒴(T^*) satisfying 𝒯(Θ,ϑ)=(Θ,ϑ).



    Please note that all the arguments hold true when starting with a non-trivial height function, i.e., when h(0)=h_0 for some h_0∈ L^∞(Ω) with c_0:=h_0_L^∞(Ω)<a^*.
    It does however lead to a shorter time horizon, as now M^*=c_0+C_L+|Θ^ref| and T^*=(a^*-c_0)/(M^*)^4.





This argument does not guarantee uniqueness, however, this follows under slightly higher regularity via Lipschitz arguments (see <ref> below).
<Ref> ensures the existence of solutions for the moving boundary problem in fixed-coordinates, but using the push forward S_h the corresponding solutions in moving coordinates are given via (Θ,S_hϑ, h).



If 𝒯 has a fixed point (Θ,ϑ)∈𝒴(T^*) with the additional regularity ∂_tΘ, ∇Θ∈ L^∞((0,T^*)×Ω) and ∂_tϑ, ∇_yϑ∈ L^∞((0,T^*)×Ω;L^2(Y_0)).
Then, (Θ,ϑ) is the only fixed point.


    Let (Θ_i,ϑ_i)∈𝒴(T^*) (i=1,2) be fixed-points of 𝒯 with the higher regularity for i=2 with corresponding height functions h_i∈ℋ_M^*(T^*).
    Taking the difference of their weak forms and testing with (Θ̅,ϑ̅):=(Θ_1-Θ_2,ϑ_1-ϑ_2):
    
    
    ∫_Ω∂_t(C(h_1)Θ_1-C(h_2)Θ_2)Θ̅x_I_1+∫_Ω∫_Y_0∂_t(c(h_1)ϑ_1-c(h_2)ϑ_2)ϑ̅yx_I_2
    
        +∫_Ω(K(h_1)∇Θ_1-K(h_2)∇Θ_2)·∇Θ̅x_I_3
        +∫_Ω∫_Y_0(κ(h_1)∇_yϑ_1-κ(h_2)∇_yϑ_2)·∇_yϑ̅yx_I_4
    
        +∫_Ω∫_Y_0(ϑ_1 v(h_1)-ϑ_2 v(h_2))·∇_yϑ̅yx_I_5
    
        =∫_Ω( L(h_2)∂_th_2- L(h_1)∂_th_1)Θ̅x_I_6
        +∫_Ω∫_Y_0(f(h_1)-f(h_2))ϑ̅yx_I_7.

    
    Using the estimates from <ref>, we can now estimate the individual terms.
    Starting with the macroscopic time derivative, we get
    
    
    I_1   =
        ∫_Ω∂_t(C(h_1)Θ̅)Θ̅x+∫_Ω∂_t[(C(h_1)-C(h_2))Θ_2]Θ̅x
       =1/2∫_Ω C(h_1)Θ̅^2x
        -1/2∫_Ω∂_tC(h_1)Θ̅^2x
        +∫_Ω∂_t[(C(h_1)-C(h_2))Θ_2]Θ̅x.

    
    The second and third term on the right hand side satisfy
    
    
    |1/2∫_Ω∂_tC(h_1)Θ̅^2x|
        +|∫_Ω∂_t[(C(h_1)-C(h_2))Θ_2]Θ̅x|
        ≤ CΘ̅_L^2(Ω)(Θ̅^2_L^2(Ω)+∂_th̅_L^2(Ω)
        +h̅_L^2(Ω)).

    
    Similarly, we get for the microscopic time derivative:
    
    
    I_2≥1/2∫_Ω∫_Y_0c(h_1)ϑ̅^2yx
        -Cϑ̅_L^2(Ω× Y_0)(ϑ̅_L^2(Ω× Y_0)+∂_th̅_L^2(Ω)+h̅_L^2(Ω))

    
    Looking at the macroscopic heat conductivity term:
    
    
    I_3   =∫_ΩK(h_1)∇Θ̅·∇Θ̅x+∫_Ω(K(h_1)-K(h_2))∇Θ_2·∇Θ̅x,
       ≥ c∇Θ̅_L^2(Ω)^2-Ch̅_L^2(Ω)∇Θ̅_L^2(Ω)

    
    The same type of estimate also holds for the microscopic heat conductivity term I_4:
    
    
    I_4   =∫_Ω∫_Y_0κ(h_1)∇_yϑ̅·∇_yϑ̅yx
        +∫_Ω∫_Y_0(κ(h_1)-κ(h_2))∇_yϑ_2·∇_yϑ̅yx
       ≥ c∇_yϑ̅_L^2(Ω× Y_0)^2
        -Ch̅_L^2(Ω)∇_yϑ̅_L^2(Ω× Y_0)

    
    The convective part on the microscale can be estimated as
    
    
    |I_5|   ≤ C(ϑ̅_L^2(Ω× Y_0)∇_yϑ̅_L^2(Ω× Y_0)+h̅_L^2(Ω)∇_yϑ̅_L^2(Ω× Y_0)),

    
    and the macroscopic heat production via
    
    |I_6|   ≤ CΘ̅_L^2(Ω)(h̅_L^2(Ω)
        +∂_th̅_L^2(Ω))

    For I_7, we use the uniform Lipschitz continuity of f to estimate
    
    |I_7|≤ Ch̅_L^2(Ω)ϑ̅_L^2(Ω;L^1(Y)).

    
    Summarizing and sorting these individual estimates while subsuming the gradient differences via Young’s inequality, we get
    
    
    ∫_Ω C(h_1)Θ̅^2x+∫_Ω∫_Y_0c(h_1)ϑ̅^2yx
            +∇θ̅_L^2(Ω^2+∇_yϑ̅_L^2(Ω× Y_0)^2
    ≤ C(Θ̅^2_L^2(Ω)+ϑ̅^2_L^2(Ω× Y_0)+h̅^2_L^2(Ω)+∂_th̅^2_L^2(Ω)).

    
    Integrating over any time interval (0,t) and keeping in mind the uniform positivity of C(h_1) and c(h_1), we are led to
    
    
    Θ̅(t)^2_L^2(Ω)+ϑ̅(t)_L^2(Ω× Y_0)
            +∇θ̅_L^2((0,t)×Ω^2+∇_yϑ̅_L^2((0,t)×Ω× Y_0)^2
    ≤ C∫_0^t(Θ̅^2_L^2(Ω)+ϑ̅^2_L^2(Ω× Y_0)+h̅^2_L^2(Ω)+∂_th̅^2_L^2(Ω))τ.

    
    Now, looking at the terms with the height function, we first note that ∂_th̅=Θ.
    Similarly, we estimate 
    
    
    ∫_0^th̅(τ)^2_L^2(Ω)τ≤∫_0^t(∫_Ω∫_0^τ|Θ̅(s,x)|^2sx)^2τ
        =t∫_0^tΘ̅^2_L^2((0,τ)×Ω)τ≤ t^2Θ̅^2_L^2((0,t)×Ω).

    
    Applying Grönwall's inequality to <ref>, we conclude Θ̅=0, ϑ̅=0 which also implies h̅=0.    


The solution given by <ref> is only local-in-time, which is expected as the Hanzawa transformation can only ever track changes inside the tubular neighborhoods.
In the following lemma, we show that there are no other factors limiting the time horizon. 
This also means that in some very specific scenarios, i.e., whenever the data is very well prepared to ensure that the growth/shrinkage of the cells is controlled, global existence holds.
To that end, we introduce the supremal time horizon


    T_sup:=sup{t∈ :  there exists a solution (Θ,ϑ,h) in the sense of <ref>}.



    Let T∈, let (Θ,ϑ,h)∈ X_0(T)×ℋ_M^*(T) be a solution in the sense of <ref>, and let h_L^∞((0,T)×Ω)<a^*.
    Then, the solution can be extended, i.e., T_sup>T.

    Moreover, if h_L^∞((0,T')×Ω)<a^* for all 0<T'<T_sup, then the solution can be extended globally, i.e, T_sup=∞.


Let T∈ and let (Θ,ϑ,h)∈𝒳_0(T)×ℋ_M^*(T) be a solution with h_L^∞((0,T)×Ω)=a^*- for some >0.
Now, taking Y(a^*-) and Γ(a^*-) as the initial geometry and (Θ(T),ϑ(T)) as initial conditions, all results from this section can be applied.
In this case, we do not have a uniform initial geometry, i.e., the height function is not uniformly 0, but this does not pose problems (see <ref>).
Therefore, there is a time horizon T_>0 and a solution (Θ',ϑ',h') on (0,T_). 
Gluing (Θ',ϑ',h') to (Θ,ϑ,h) at time T results in a solution on (0,T+T_).

For the second proposition, assume that h_L^∞((0,T')×Ω)<a^* for all T'<T_sup.
There are two cases:[The limit must exist as the norm is monotonically increasing and bounded.]


    (i) lim_T'→ T_suph_L^∞((0,T')×Ω)=a-<a^*,   (ii) lim_T'→ T_suph_L^∞((0,T')×Ω)=a^*.


In the first case, T_sup=∞ because otherwise we could extend via the first part (choose T'<T_sup such that T_sup<T'+T_).
In the second case, if T_sup is finite the solution can not be extended to T_sup (since we assumed h_L^∞((0,T')×Ω)<a^*).

To show, on the contrary, that it can be extended, we first glue all extensions together to get a height function h∈ L^∞((0,T_sup)×Ω) which automatically satisfies h_L^∞((0,T_sup)×Ω)=a^*.
Similarly, we get the temperatures (Θ,ϑ).
Now, (Θ,ϑ,h) is a solution on (0,T') for all 0<T'<T_sup.
More specifically, it satisfies all conditions in <ref> for all smaller time intervals.
Since none of the estimates for (Θ,ϑ,h) blow-up for T'→ T_sup (note that h_L^∞((0,T_sup)×Ω)≤ a^* and T_sup<∞), if follows that (Θ,ϑ,h)∈𝒳(T_sup)×ℋ(T_sup) is a solution over (0,T_sup).
As a consequence, T_sup=∞.










§ NUMERICAL ANALYSIS

From a numerical perspective, the present model is quite challenging and computationally expensive to solve due to the different scales, changing domains, and non-linear coupling.
In this section, we introduce and investigate a precomputing strategy together with a semi-implicit time-stepping scheme to resolve these problems.

First note, that independent of the underlying numerical algorithm, the effective conductivity K has to be recomputed multiple times throughout the simulation process: after each discrete time step and at every macroscopic node since K depends on the local value of the height function h.
This is quite expensive since differential equations (cell problems) have to be solved to compute K.
Seeing that in the underlying setup, the general cell topology does not change for different heights and only depends on h, it would be advantageous if one could precompute the conductivity K for a discrete number of height values in an offline phase and utilize these values in the simulation by interpolation.
We start by analyzing this idea and the influence of the solution in more detail.



 §.§ Precomputing strategy

Given N ∈, we evenly split the interval [-a^*, a^*] into N sub-intervals [h_i,h_i+1] with h_i = -a^* + 2a^*iN, for i=0,…,N.
We then compute for each h_i the corresponding conductivity K_i = K(h_i) given by <ref>.
We also set Δ h = 1/N in the following.[
    Depending on the expected behavior of K(h) with respect to h, it might be advantageous to utilize a nonuniform grid (see e.g., <cit.>).
    In our case, this is unnecessary as the general behavior is quite smooth and uniform (for comparison, see <ref>).
]

For any h ∈ [-a^*, a^*] the precomputed conductivity K_pre is then defined via an interpolation scheme. This raises the question of what interpolation scheme is suitable given the properties of K.
In line with the Lipschitz continuity of K (<ref>), a piecewise linear interpolation is a natural candidate, e.g.,


    K_lin(h) = (1 - h - h_i/Δ h)K_i +  h - h_i/Δ h K_i+1 ,   if  h ∈ [h_i, h_i+1),


This Lipschitz continuity immediately guarantees at least a first-order interpolation error:

    |K(h) - K_lin(h)| ≤ C Δ h/2.













We need higher regularity results for the effect conductivity to ensure higher order errors, which we investigate in the following lemma.


    For the effective conductivity it holds K ∈ C^1,1([-a^*, a^*], ^d× d).



    Recall, that the effective conductivity is given by
    
    
    K_ij(h) = ∫_Y∖ Y(h)𝒦(∇ξ_h,j + e_j) · e_iy
            = ∫_Y∖ Y_0𝒦 (F_h^-1∇(S_h^-1ξ_h,j) + e_j) · e_i J_hy.

    
    Therefore, by the product rule, we obtain the following expression assuming that all parts are well-defined:
    
    
    ∂_h K_ij(h) 
                   = 
                ∫_Y∖ Y_0 K (∂_h F_h^-1∇(S_h^-1ξ_h,j)) · e_i J_h y
                + 
                ∫_Y∖ Y_0 K (F_h^-1∂_h ∇(S_h^-1ξ_h,j)) · e_i J_hy
       + 
                ∫_Y∖ Y_0 K (F_h^-1∇(S_h^-1ξ_h,j) + e_j) · e_i ∂_h J_hy.

    
    Now, we investigate if all the derivatives on the right-hand side are well-defined and Lipschitz with respect to h.
    For the Matrix F_h^-1 = (+h D_yψ)^-1, it holds ∂_h F_h^-1 = -F_h^-1 D_yψ F_h^-1. 
    
    
    
    
    
    
    
    
    
    
    
    Therefore, ∂_h F_h^-1 is well-defined and also Lipschitz in h.
    For the determinate J_h = (+hD_y ψ), it holds
    
    
    ∂_h J_h    = ∂_h∑_σ∈ S_dsgn(σ) ∏_i=1^d (δ_i,σ(i) + h D_yψ_i, σ(i)) 
            
            = ∑_k=1^d  ∑_σ∈ S_dsgn(σ) ∏_i=1^d p_k, i, σ(i),

    
    where S_d denotes the set of all permutations in dimension d, sgn(σ)=±1 depending on if the permutation is even or odd, and
    
    
    p_k, i, σ(i) = 
            
                (δ_i,σ(i) + h D_yψ_i, σ(i)),     if  i≠ k, 
    
                D_yψ_i, σ(i),     if  i = k.

    
    This is a sum of multiple determinants (each time removing the identity and h from a different column of the matrix +hD_y ψ), hence ∂_h J_h is also well-defined and by the same arguments as in <ref> also Lipschitz with respect to h.

    Next, we investigate the derivative of the cell solution S_h^-1ξ_h, j with respect to h. To this end, we define the operator E_j:[-a^*, a^*] × V → V^*, where 
    
    
    V {v ∈ H^1_#(Y∖ Y_0) : ∫_Y ∖ Y_0 v y  = 0 },

    
    and the operator is given by
    
    
    E_j(h, v) = ∫_Y ∖ Y_0 J_h F^-1_h 𝒦 F^-T_h ∇_y v ·∇_y ∗y - ∫_Γ_0 J_h e_j · n ∗y.

    
    Here ∗ denotes a placeholder, for the argument of the functional.
    Note, that the cell solutions S_h^-1ξ_h, j are implicitly defined by E_j(h, S_h^-1ξ_h, j) = 0.
    We want to utilize the implicit function theorem <cit.> to show the existence of the Fréchet-derivative of the cell solution and find a characterization of it.
    Given that E_j is linear in v∈ V and F_h, J_h are differentiable in h∈[-a^*,a^*], it is straightforward to check that E_j is Fréchet-differentiable. The derivative ∂_u E_j(h, S_h^-1ξ_h, j) is a bounded linear operator from V → V^* and defined by 
    
    
    ∂_u E_j(h, S_h^-1ξ_h, j)[v] = ∫_Y ∖ Y_0 J_h F^-1_h 𝒦 F^-T_h ∇_y v ·∇_y ∗y.

    
    For v = 0 ∈ V it holds ∂_u E_j(h, S_h^-1ξ_h, j)[v]_V^* = 0 and for v ≠ 0
    
    
    ∂_u E_j(h, S_h^-1ξ_h, j)[v]_V^* 
            = sup_φ∈ V, φ≠ 0|∂_u E_j(h, S_h^-1ξ_h, j)[v] (φ) |/φ_H^1(Y∖ Y_0)≥|∂_u E_j(h, S_h^-1ξ_h, j)[v] (v) |/v _H^1(Y∖ Y_0)≥ C v _H^1(Y∖ Y_0),

    
    where in the last inequality we applied assumption (A1), <ref> (i) and the Poincaré inequality. Therefore, ∂_u E_j(h, S_h^-1ξ_h, j)[v]_V^*≥ C v _H^1(Y∖ Y_0) for all v ∈ V. With <cit.> we can conclude that ∂_u E_j(h, S_h^-1ξ_h, j) is injective and the image is closed, hence the inverse exists. Additionally, the inverse is also bounded. By the implicit function theorem, we obtain that the solution operator h ↦ S_h^-1ξ_h, j is well-defined and Fréchet-differentiable, where the derivative is given by
    
    
    ∂_u E_j(h, S_h^-1ξ_h, j) [∂_h (S_h^-1ξ_h, j)]    = -∂_h E_j(h, S_h^-1ξ_h, j) 
                
       ⇔   ∫_Y∖ Y_0 J_h F^-1_h 𝒦 F^-T_h ∇_y (∂_h (S_h^-1ξ_h, j)) ·∇_y∗y   = - ∫_Y∖ Y_0∂_h(J_h F^-1_h 𝒦 F^-T_h) ∇_y (S_h^-1ξ_h, j) ·∇_y∗y
             + ∫_Γ_0∂_h J_h e_j· n ∗y,

    
    understood as an element in V^*. With the same arguments as for the estimate in <ref> one can show that ∂_h (S_h^-1ξ_h, j) is Lipschitz in h as well. 
    Finally, we conclude that ∂_h K_ij is well-defined and Lipschitz in h since all parts of the right-hand side of <ref> are bounded and Lipschitz. 


The above regularity ensures a quadratic error order if we utilize linear interpolation:


    |K(h) - K_lin(h)| ≤ C (Δ h)^2  for all  h ∈ [-a^*, a^*].



    Since F_h^-1 and J_h are smooth regarding h, the implicit function theorem can also be used to show the existence of higher derivatives of the solution operator. Therefore, higher-order interpolation schemes can be applied as well. But we have to note, that the Lipschitz constant grows exponentially with 1/a^*. E.g., for the first derivative we see in the above proof, that ∂_h F_h^-1≤ C D_y ψ≤5C/a^*, which leads to
    
    
    |∂_h K(h + Δ h) - ∂_h K(h)| ≤ C (1 + 1/(a^*)^2) Δ h.

    
    This means that high-order interpolations may need a small step size Δ h to ensure the desired convergence order.



    By assumption (A1) we have that the underlying conduction matrix 𝒦 is not dependent on the space variable. For some applications this may not hold and 𝒦 may be a function of y. In this case the results of <ref> are still valid under the condition that 𝒦 is Lipschitz in regard to y. The property K ∈ C^1,1([-a^*, a^*], ^d× d) requires the additional assumption that 𝒦∈ C^1,1(Y∖ Y_0). 


In the following, K_pre stands for the precomputed conductivity, while ℰ_pre:= sup_|h|≤ a^*|K(h)-K_pre(h)| denotes the corresponding interpolation error.
In the case that K_pre = K_lin, or a higher order interpolation, K_pre is a Lipschitz continuous function in h and the results of <ref> can be transferred to ensure the existence of solutions for the precomputed system.
The important question is how the interpolation error propagates over time, this is analyzed in the following theorem.


    Let (h, Θ, ϑ) denote the solution with the exact conductivity and (h_pre, Θ_pre, ϑ_pre) the solution with the precomputed values K_pre. Set T^* = min(T^*, T_pre^*) such that both solutions are defined on (0, T^*). Assume that 
    ∂_t Θ_pre, ∇Θ_pre∈ L^∞((0, T^*) ×Ω), 
    ∂_t ϑ_pre, ∇_y ϑ_pre∈ L^∞((0, T^*) ×Ω× Y_0). Then, there exists a C > 0 (depending on T^*) such that
    
    
    Θ_L^∞((0, T^*); L^2(Ω)) + ϑ_L^∞((0, T^*); L^2(Ω× Y_0)) 
            + ∇Θ_L^2((0, T^*) ×Ω)
            + ∇ϑ_L^2((0, T^*) ×Ω× Y_0)≤ C ℰ_pre,

    
    where (h, Θ, ϑ)  (h-h_pre, Θ - Θ_pre, ϑ - ϑ_pre)_|(0, T^*) denotes the difference of both solutions.



    The proof follows the same idea as the uniqueness proof in <ref>. Here, we only have to consider the macroscopic conductivity term in more detail, all other terms can be estimated as previously shown. For the integral I_3 in the proof of <ref>, we have
    
    
    I_3 = 
            ∫_ΩK(h)∇Θ̅·∇Θ̅x
            + 
            ∫_Ω[K(h) - K_pre(h_pre)] ∇Θ_pre∇Θ̅x.

    
    The integral on the right can be further estimated by
    
    
    ∫_Ω   | [K(h) - K_pre(h_pre)] ∇Θ_pre·∇Θ̅|x
       ≤∫_Ω|(K(h)  - K(h_pre )) ∇Θ_pre·∇Θ̅|x
            + ∫_Ω|(K(h_pre ) - K_pre(h_pre)) ∇Θ_pre·∇Θ̅|x
       ≤ C ∇Θ_pre_L^∞(Ω)∇Θ_L^2(Ω)(h_L^2(Ω) + ℰ_pre).

    
    Now plugging this estimate into the proof steps of <ref>, we obtain, for any time interval (0, t) ⊂ (0, T̅^*), the following estimate
    
    
    Θ̅(t, ·)_L^2(Ω)^2
            +ϑ̅(t, ·)_L^2(Ω× Y_0)^2
            +∇Θ̅^2_L^2((0,t) ×Ω)
            +∇_yϑ̅^2_L^2((0,t) ×Ω× Y_0)
    ≤ C ℰ_pre^2 + C (1 + t^2) (Θ̅_L^2((0, t) ×Ω)^2 + ϑ̅_L^2((0, t) ×Ω× Y_0)^2).

    
    Applying the Grönwall inequality leads to the desired result.    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    




 §.§ Resolving the nonlinearity.
 Even with the precomputing strategy at hand, we are still facing a nonlinear two-scale problem. We propose a semi-implicit time-stepping method to resolve the nonlinearity, which is often applied to nonlinear equations in numerical simulations. To this end, given a time step Δ t we define the discrete time point t_i = i Δ t. Then we are looking at each i > 0 for the time-discrete solution (Θ_i, ϑ_i) ∈X of


    ∫_Ω C(h_i) Θ_i - Θ_i-1/Δ tΦx
            + ∫_Ω∫_Y_0 c(h_i)  ϑ_i - ϑ_i-1/Δ tϕyx
            + ∫_Ω K_pre(h_i) ∇(Θ_i + Θ_i-1/2) ·∇Φx
       + ∫_Ω∫_Y_0κ(h_i) ∇_y ϑ_i + ϑ_i-1/2·∇_y ϕyx
            +  ∫_Ω∫_Y_0ϑ_i v(h_i)·∇_yϕyx
            = ∫_Ω(F_i- L(h_i)h_i - h_i-1/Δ t)Φx
       +∫_Ω∫_Y_0f(h_i)ϕyx 
            - ∫_Ω∂_h C(h_i) h_i - h_i-1/Δ tΘ_i-1Φx
            - ∫_Ω∂_h c(h_i) h_i - h_i-1/Δ tϑ_i-1ϕx


for all (Φ,ϕ) ∈X where 


    X{
            (Φ,ϕ) ∈ H^1(Ω) × L^2(Ω, H^1(Y_0)) : Φ(x) = ϕ(x, y)  a.e. on Ω×Γ_0 
        }.


To find (Θ_i, ϑ_i) a differential equation has to be solved at each time step. But we now linearize the problem by using an explicit scheme to compute the height function h, to be precise


    h_i = h_i-1 + Δ t (Θ_i-1 - Θ^ref)    for  i > 0   and   h_0 = 0.


This also leads to the simplification of (h_i - h_i-1)/Δ t = Θ_i-1 - Θ^ref in the above scheme. In (<ref>) we used the product rule to split the time derivative into the derivative of Θ and of C, respectively for ϑ and c. This simplifies the proof for the time stepping error. Note, that also the more natural discretization 

    ∂_t(C(h) Θ) ≈C(h_i) Θ_i - C(h_i-1) Θ_i-1/Δ t

could be used and only deviates from the above scheme by a term of order Δ t. For the gradients, we utilize an average between the current and previous time step since this fits more naturally in the following proof idea. Note, that this scheme requires ∇Θ_0 ∈ L^2(Ω) and ∇_y ϑ_0 ∈ L^2(Ω× Y_0). But again, just ∇Θ_i, and ∇_y ϑ_i, could be used.

If |h_i| ≤ a^*, the solution of the system (<ref>) exists and is unique since it is a linear time-discretized parabolic two-scale equation. Given that Θ_0 ∈ L^∞(Ω) we can always carry out one time step if Δ t is small enough. To show that the discrete solution exists on a larger time interval (0, T^*_dis), we need to show that the discrete solution Θ_i can be bounded in L^∞(Ω) independent of the height function. For this, we can proceed as in <ref>, e.g., we would need to carry out the following steps:


    
  * Choose a M ≥ 1 and a N ∈, then define the time horizon T^*_dis=a^*/M^4 and step Δ t = T^*_dis/N.
    
  * Start with a fixed height function {h_i}_i=0^N defined by 
    
    
    h_i = h_i-1 + Δ t b_i,    h_0 = 0    and   b_i_L^∞≤ M.

    
    
  * Derive a L^∞(Ω) bound for the solution of <ref> for the prescribed height function {h_i}_i=0^N independent of M. This follows with similar computations as in <ref> and by testing with Φ = Θ_i + Θ_i-1, ϕ = ϑ_i + ϑ_i-1 and summing over all time steps. As the proof is similar to the previous one, it will not be presented in detail here.
    
  * Choose the M corresponding to the solution bound and obtain that the discrete method is reasonable and the solution exists on a nonempty interval (0, T^*_dis).


Note, that the above procedure also ensures that for a smaller time step Δ t the discrete solution exists, at least, on the same time interval. The error we introduce through this scheme is analyzed in the next lemma.


    Let (ĥ, Θ̂, ϑ̂) denote the solution that is continuous in time, exists on the time interval (0, T^*) and uses the precomputed values K_lin. Denote with {h_i, Θ_i, ϑ_i}_i=0^N the time discrete solution of <ref> that exists on [0, T^*_dis], with K_pre = K_lin. Let the assumptions
    
    
    

    [t]0.49
      
        
  * Θ̂∈ W^2, ∞((0, T^*); W^1, ∞(Ω)),
        
  * F ∈ C(S ×Ω),
        
  * Δ t ≤C(a^*)/8K(a^*),
      
    
    
    [t]0.49
      
        
  * ϑ̂∈ W^2, ∞((0, T^*); W^1, ∞(Ω× Y_0)),
        
  * f ∈ C(S ×Ω; C^0, 1(Y_0)),
        
      
    
    
    

    
    be fulfilled. Let N_t ∈ be such that Δ t N_t < min(T^*, T^*_dis) and Δ t(1 + N_t) ≥min(T^*, T^*_dis). Then, there exists a C > 0 (depending on (Θ̂, ϑ̂)) such that
    
    
    sup_i=1,…, N_tΘ̂(t_i) - Θ_i_H^1(Ω) + ϑ̂(t_i) - ϑ_i_H^1(Ω) + ĥ(t_i) - h_i_L^2(Ω)≤ C (Δ t + Δ h).

    


 
    We follow the proof ideas of <cit.> where numerical approaches for similar nonlinear problems are analyzed. The main difference in our case is, that we consider a two-scale system, the nonlinearity is coupled to the height function h, a less regular conductivity, and a different right-hand side because of C(h). But the general idea from <cit.> can be transferred to our case. We denote by ·_i the difference between the continuous and discrete solution at time point t_i, e.g., Θ_i = Θ̂(t_i) - Θ_i. We start by investigating the difference between the height functions of both solutions. By definition of both functions, we obtain
    
    
    h_i_L^2(Ω)   = ĥ(t_i) - h_i_L^2(Ω)
       ≤ĥ(t_i-1) - h_i-1_L^2(Ω) 
            +  ∫_t_i-1^t_i (Θ̂(s) - Θ^ref) s - Δ t (Θ_i-1 - Θ^ref) _L^2(Ω)
       ≤ĥ(t_i-1) - h_i-1_L^2(Ω) 
            + Δ t Θ_i-1_L^2(Ω)
            + ∫_t_i-1^t_i∫_t_i-1^s ∂_t Θ̂(τ) τs_L^2(Ω)
       ≤ĥ(t_i-1) - h_i-1_L^2(Ω) 
            + Δ t Θ_i-1_L^2(Ω) + C (Δ t )^2

    
    in the estimates we used the fundamental theorem of calculus and that ∂_t Θ̂ is bounded. Iteratively applying the above argument leads to
    
    
    h_i_L^2(Ω)≤∑_n = 1^i-1[ Δ t Θ_n_L^2(Ω) + C (Δ t )^2 ].

    
    Next, we derive the estimates for the temperature. To this end, we only show the error estimate for the macroscopic function Θ. The reason for this is, that the estimates for ϑ follow with the same steps since the microscopic equation has a similar structure. This allows us to reduce the length of the proof considerably. Therefore, let the continuous solution Θ fulfill
    
    
    ∫_Ω C(h) ∂_t ΘΦx 
            + ∫_Ω∂_h C(h) ∂_t hΘΦx
            + ∫_Ω K_lin(h) ∇Θ·∇Φx 
            = ∫_Ω  L(h) ∂_t hΦx

    
    and the time-discrete solution 
    
    
    ∫_Ω C(h_i) Θ_i - Θ_i-1/Δ t   Φx
                + ∫_Ω∂_h C(h_i) (Θ_i-1 - Θ^ref)  Θ_i-1Φx
       + ∫_Ω K_lin(h_i) ∇(Θ_i + Θ_i-1/2) ·∇Φx  
                = ∫_Ω  L(h_i) (Θ_i-1 - Θ^ref) Φx.

    
    To derive an error estimate at  t_i, we consider (<ref>) localized at t_i, subtract  the <ref> and (<ref>) from each other and test with Φ = Θ_i - Θ_i-1. We obtain four terms which we estimate individually. Let us introduce the discrete time derivative d_t which is defined by d_t Θ_i = Θ_i - Θ_i-1/Δ t. Additionally, we abuse the notation and write d_t Θ̂_i = Θ̂(t_i) - Θ̂(t_i-1)/Δ t. We start with the time derivative term, which yields
    
    
    ∫_Ω   C(h_i) d_tΘ_i Φx - ∫_Ω C(h) ∂_t Θ(t_i) Φx 
            = ∫_Ω C(h_i) d_tΘ_i Φx 
            - ∫_Ω C(h_i) d_tΘ̂_i Φx_=I_1
       + ∫_Ω C(h_i) d_tΘ̂_i Φx 
            - ∫_Ω C(h(t_i)) d_tΘ̂_i Φx_=I_2
            + ∫_Ω C(h(t_i)) d_tΘ̂_i Φx
            - ∫_Ω C(h(t_i)) ∂_t Θ(t_i) Φx_=I_3.

    
    For terms on the right-hand side, it holds
    
    
    I_1    = Δ t ∫_Ω C(h_i) (d_t Θ_i)^2 x≥Δ t C(a^*)  d_t Θ_i_L^2(Ω)^2,
            
    
            |I_2|    ≤ C h_i_L^2(Ω)Θ_i - Θ_i-1_L^2(Ω),
            
    
            |I_3|    ≤ C Δ t ∂_ttΘ̂(s) s_L^∞((0, T^*) ×Ω)Θ_i - Θ_i-1_L^2(Ω)≤ C (Δ t)^3 + Δ t d_t Θ_i_L^2(Ω)^2.

    
    Here, we used for I_2 and I_3 that C is Lipschitz, the Hölder inequality, and the Young inequality with a small > 0. I_2 can be estimated further by utilizing <ref>,
    
    
    |I_2|
               ≤  C ∑_n = 1^i-1[ Δ t Θ_n_L^2(Ω) + C (Δ t )^2 ] Θ_i - Θ_i-1_L^2(Ω)
       = C ∑_n = 1^i-1[ Δ t Θ_n_L^2(Ω)Δ t d_t Θ_i_L^2(Ω) + C (Δ t )^2 Δ t d_t Θ_i_L^2(Ω)] 
            
       ≤ C ∑_n = 1^i-1[(Δ t)^2 Θ_n_L^2(Ω)^2 +  (Δ t)^2 d_t Θ_i_L^2(Ω)^2 + C (Δ t )^4 +  (Δ t)^2 d_t Θ_i_L^2(Ω)^2 ] 
            
       ≤  C (Δ t )^3 + C Δ t d_t Θ_i_L^2(Ω)^2 + C Δ t ∑_n = 1^i-1Δ t Θ_n_L^2(Ω)^2.

    
    Next, the term with ∂_h C, where we obtain
    
    
    ∫_Ω∂_h C(h_i)    (Θ_i-1 - Θ^ref)  Θ_i-1Φx - ∫_Ω∂_h C(h(t_i)) (Θ(t_i) - Θ^ref) Θ(t_i) Φx
       = ∫_Ω∂_h C(h_i) (Θ_i-1 - Θ^ref)  Θ_i-1Φx 
            - ∫_Ω∂_h C(h_i) (Θ_i-1 - Θ^ref)  Θ̂(t_i-1) Φx_=I_4
       + ∫_Ω∂_h C(h_i) (Θ_i-1 - Θ^ref)  Θ̂(t_i-1) Φx -
            ∫_Ω∂_h C(h_i) (Θ̂(t_i-1) - Θ^ref)  Θ̂(t_i-1) Φx_=I_5
       + ∫_Ω∂_h C(h_i) (Θ̂(t_i-1) - Θ^ref)  Θ̂(t_i-1) Φx - ∫_Ω∂_h C(h(t_i)) (Θ(t_i-1) - Θ^ref) Θ̂(t_i-1)Φx_=I_6
       + ∫_Ω∂_h C(h(t_i)) (Θ̂(t_i-1) - Θ^ref)  Θ̂(t_i-1) Φx - ∫_Ω∂_h C(h(t_i)) (Θ(t_i) - Θ^ref) Θ(t_i) Φx_=I_7

    Using that ∂_h C is well-defined and bounded (since ∂_h J_h is well-defined and bounded; see the proof of <ref>) and that the solutions are bounded, we can obtain with similar arguments as before
    
    
    |I_4| + |I_5| 
               ≤ C Θ_i-1_L^2(Ω)Θ_i - Θ_i-1_L^2(Ω)≤ C Δ t Θ_i-1_L^2(Ω)^2 + Δ t  d_t Θ_i_L^2(Ω)^2,
            
    
            |I_6|    ≤ C h_i_L^2(Ω)Θ_i - Θ_i-1_L^2(Ω),
            
    
            |I_7|    ≤ C Δ t ∂_t Θ̂_L^∞((0, T^*) ×Ω)Θ_i - Θ_i-1_L^2(Ω)≤ C (Δ t)^3 + Δ t d_t Θ_i_L^2(Ω)^2.

    
    Since the term containing L is comparable to the previous ones, we first handle it and afterward estimate the gradients. By introducing the mixed terms, we have the following equality 
    
    
    ∫_Ω  L(h_i) (Θ_i-1 - Θ^ref) Φx   - ∫_Ω  L(h(t_i)) (Θ̂(t_i) - Θ^ref) Φx
       =  ∫_Ω  L(h_i) (Θ_i-1 - Θ^ref) Φx - ∫_Ω  L(h_i) (Θ(t_i-1) - Θ^ref) Φx_=I_8
       + ∫_Ω  L(h_i) (Θ(t_i-1) - Θ^ref) Φx - ∫_Ω  L(h_i) (Θ(t_i) - Θ^ref) Φx_=I_9
       +
            ∫_Ω  L(h_i) (Θ(t_i) - Θ^ref) Φx - ∫_Ω  L(h(t_i)) (Θ̂(t_i) - Θ^ref) Φx_=I_10

    All of I_8, I_9 and I_10 are of the same structure as before and we can find the following estimates
    
    
    |I_8|    ≤  C Δ t Θ_i-1_L^2(Ω)^2 + Δ t  d_t Θ_i_L^2(Ω)^2,
            
    
            |I_9|    ≤ C (Δ t)^3 + Δ t d_t Θ_i_L^2(Ω)^2,
            
    
            |I_10|    ≤ C h_i_L^2(Ω)Θ_i - Θ_i-1_L^2(Ω).

    
    Let us next consider the gradient terms, here the estimates are more involved but follow the general idea outlined in <cit.>. Introducing mixed terms into the difference first leads to
    
    
    ∫_Ω   K_lin(h_i) ∇(Θ_i + Θ_i-1/2) ·∇Φx 
            - ∫_Ω K_lin(h(t_i)) ∇Θ(t_i) ·∇Φx
       = ∫_Ω K_lin(h_i) ∇(Θ_i + Θ_i-1/2) ·∇Φx 
            - ∫_Ω K_lin(h_i) ∇(Θ̂(t_i) + Θ̂(t_i-1)/2) ·∇Φx_=I_11, i
       + ∫_Ω K_lin(h_i) ∇(Θ̂(t_i) + Θ̂(t_i-1)/2) ·∇Φx 
            -  ∫_Ω K_lin(ĥ(t_i)) ∇(Θ̂(t_i) + Θ̂(t_i-1)/2) ·∇Φx_= I_12, i
       + ∫_Ω K_lin(ĥ(t_i)) ∇(Θ̂(t_i) + Θ̂(t_i-1)/2) ·∇Φx - ∫_Ω K_lin(h(t_i)) ∇Θ(t_i) ·∇Φx_=I_13, i.

    Starting with I_11, i, where we obtain
    
    
    2 I_11, i   = ∫_Ω K_lin(h_i) ∇ (Θ_i + Θ_i-1) ·∇ (Θ_i - Θ_i-1) x =  ∫_Ω K_lin(h_i) (∇Θ_i ·∇Θ_i - ∇Θ_i-1·∇Θ_i-1) x.

    
    To further estimate the error, we now sum over m=1, …, i, and using summation by parts we obtain
    
    
    ∑_m=1^i ∫_Ω K_lin(h_i) (∇Θ_i ·∇Θ_i - ∇Θ_i-1·∇Θ_i-1) x   = ∑_m=1^i-1∫_Ω∇Θ_m ·∇Θ_m (K_lin(h_m)  - K_lin(h_m+1)) x
        + ∫_Ω∇Θ_i ·∇Θ_i K_lin(h_i) x.

    
    The sum on the right can be further estimated by 
    
    
    | ∑_m=1^i-1∫_Ω∇Θ_m ·∇Θ_m (K_lin(h_m)  - K_lin(h_m+1)) x | ≤  C ∑_m=1^i-1Δ t  ∇Θ_m_L^2(Ω)^2,

    
    where we used that K_lin is Lipschitz. For I_12 we start with a similar strategy,
    
    
    ∑_m=1^i I_12, i = ∫_Ω (K_lin(h_i) - K_lin(h(t_i)))  ∇Θ_i ·∇(Θ̂(t_i) + Θ̂(t_i-1)/2) x_= I_12, a
       + ∑_m=1^i-1∫_Ω (K_lin(h_m)  - K_lin(ĥ(t_m))) ∇Θ_m  ·∇(Θ(t_m) + Θ(t_m-1)/2 - Θ(t_m+1) + Θ(t_m)/2) x_= I_12, b
       + ∑_m=1^i-1∫_Ω[K_lin(h_m)  - K_lin(ĥ(t_m)) - K_lin(h_m+1) + K_lin(ĥ(t_m+1))] ∇Θ_m  ·∇(Θ(t_m) + Θ(t_m-1)/2) x_= I_12, c.

    
    The first term I_12, a can be estimated with the same ideas as before, leading to
    
    
    |I_12, a| ≤∇Θ_i_L^2(Ω) + C (Δ t )^2 + C ∑_n=1^i-1Δ t Θ_n_L^2(Ω)^2.

    
    For I_12, b we need <ref> and that the time derivative of ∇Θ̂ exists, to get the estimate
    
    
    |I_12, b|    ≤ C ∑_m=1^i-1Δ t ∇Θ_m_L^2(Ω)[Δ t  + ∑_n=1^m-1Δ t Θ_n_L^2(Ω)] 
            
       = C ∑_m=1^i-1Δ t ∇Θ_m_L^2(Ω)[Δ t + (i-1 - m) Δ t Θ_m_L^2(Ω)] 
            
       ≤ C (Δ t)^2 + C ∑_m=1^i-1Δ t Θ_m_H^1(Ω)^2.

    
    To bound I_12, c we use that K_lin∈ H^1((-a^*, a^*)) and since the exact K is in C^1,1 we also have |∂_h K_lin(h_1) - ∂_h K_lin(h_2)| ≤ C |h_1 - h_2| + Δ h, for a.a. h_1, h_2 ∈ (-a^*, a^*). This yields
    
    
    |I_12, c|    ≤ C ∑_m=1^i-1Δ t ∫_Ω∇Θ_m∫_0^1 |∂_h K(h_m + τ (h_m+1 - h_m)) 
            
        - ∂_h K(ĥ(t_m) + τ (ĥ(t_m+1) - ĥ(t_m)))| τx
       ≤ C ∑_m=1^i-1Δ t ∫_Ω∇Θ_m∫_0^1  |Δ h + h(t_m) - h_m + τ[h_m+1 - h_m - ĥ(t_m+1) + ĥ(t_m)]  | τx
       ≤ C ∑_m=1^i-1Δ t ∇Θ_m_L^2(Ω) [ Δ h + Δ t + ∑_n=1^m-1Δ t Θ_n_L^2(Ω)].

    
    Next, we apply the Young-inequality and can proceed in the same way as for the estimate of I_12, a. Lastly, I_13 can again be estimated by summation, leading to
    
    
    | ∑_m=1^i  I_13, i | 
               ≤∫_Ω | K_lin(h(t_i)) ∇(Θ(t_i-1) - Θ(t_i)/2) ∇Θ_i x | 
            
       + ∑_m=1^i-1 | ∫_Ω (K_lin(h(t_m))  - K_lin(h(t_m+1))) ∇Θ_m  ·∇(Θ(t_m-1) - Θ(t_m)/2) x | 
            
       + ∑_m=1^i-1 | ∫_Ω K_lin(h(t_m+1)) ∇Θ_m  ·(Θ(t_m-1) - Θ(t_m)/2 - Θ(t_m) - Θ(t_m+1)/2) x | 
            
       ≤ C Δ t ∇Θ_i_L^2(Ω) + ∑_m=1^i-1 C ∇Θ_m_L^2(Ω) (Δ t )^2
            
       ≤ C (Δ t)^2 + ∇Θ_i_L^2(Ω)^2 + ∑_m=1^i-1Δ t ∇Θ_m_L^2(Ω)^2.

    In the first inequality above the Lipschitz continuity of K_lin, the Hölder inequality, and the time-regularity of ∇Θ̂ were utilized. In the second inequality, we used Young again.

    We can now combine all of the above estimates. To this end, we also have to sum I_1, … I_10 over 1,…,i to combine them with the gradient estimates, to be precise we would first sum and then carry out the above computations inside the sum. This leads to the estimate
    
    
    ∑_m=1^i    Δ t C(a^*) d_t Θ_m^2_L^2(Ω) + K(a^*) ∇Θ_i_L^2(Ω)^2 
                ≤ C (Δ t)^2 +  
                C ∇Θ_i_L^2(Ω)^2
                + ∑_m=1^i Δ t C d_t Θ_m^2_L^2(Ω)
       + C ∑_m=1^i-1Δ t [ Θ_m^2_H^1(Ω) + Θ_m^2_L^2(Ω) + ∇Θ_m^2_L^2(Ω)]
                + C ∑_m=1^iΔ t ∑_n=1^m-1Δ t Θ_n^2_L^2(Ω) +  C (Δ h)^2.

    
    We are missing the L^2-norm of Θ_i on the left for the final estimate. Therefore, we use the same computation as <cit.> to obtain
    
    
    K(a^*)(Θ̅_i^2_L^2(Ω) + Θ̅_i-1^2_L^2(Ω)) 
                   ≤ 2 K(a^*) Δ t ∫_Ω d_t Θ_i Θ_i-1x + K(a^*) (Δ t)^2 d_t Θ̅_i^2_L^2(Ω)
       ≤C(a^*)Δ t/4d_t Θ̅_i^2_L^2(Ω) + C Δ t Θ̅_i-1^2_L^2(Ω).

    
    For the last inequality, we used the assumption Δ t ≤C(a^*)/8K(a^*). Next, we sum <ref> from 1,…,i and add the estimate to <ref>. Then the sum over d_t Θ̅ which originates when summing  (<ref>) can be collected into the already present term in (<ref>). 
    Additionally, the  on the right side originates from the Young inequality and can be chosen such, that all terms connected to  can be incorporated into the left-hand side. All other sums on the right can be further bounded by the H^1-norm, which yields
    
    
    ∑_m=1^i Δ t C d_t Θ_m^2_L^2(Ω) + C Θ_i_H^1(Ω)^2 
            ≤ C (Δ t)^2 + C (Δ h)^2 + C ∑_m=1^i-1Δ t Θ_m^2_H^1(Ω).

    
    Finally, applying the discrete Grönwall lemma gives the desired result. As mentioned at the start the estimates for ϑ follow analogously. Plugging the estimate for Θ back into <ref>, we find the estimate for h. 



    The convergence of the time-stepping scheme requires that the precomputing also gets refined with smaller time steps. This condition appears because the linear interpolation has a piecewise constant derivative. This is not necessary for a smoother interpolation, for example, quadratic splines.




 §.§ Further error sources
 In addition to the discretization in time, we also have to discretize the function space.
To this end, we consider a finite element (FE) approach. 
Define the triangulation 𝒯_H_M,Ω of Ω and 𝒯_H_m,Y_0 of Y_0, where H_M and H_m denote the largest edge length of the simplices in 𝒯_H_M,Ω and 𝒯_H_m,Y_0 respectively.
Denote with ℙ_1 the space of linear polynomials. On the macro scale, we define the finite element space X_H_M,Ω⊂ H^1(Ω) by


    X_H_M,Ω = {Φ∈ C(Ω) : Φ_|T∈ℙ_1(T) 
             for all  T ∈𝒯_H_M,Ω}.


Similarly, on the microscale the space X_H_m, Y_0⊂ H^1(Y_0) is given by


    X_H_m, Y_0 = {ϕ∈ C(Y_0) : ϕ_|T∈ℙ_1(T) 
             for all  T ∈𝒯_H_m,Y_0}.


Let N_M denote the degrees of freedom (DOFs) on the macroscale and N_m DOFs on the microscale. Define the basis functions {Φ_n}_n=1^N_M such that span(Φ_n) = X_H_M,Ω and equally the micro basis functions {ϕ_k}_k=1^N_m. The time  discretized finite element approximation of the solutions (Θ, ϑ, h) are then given by (H=(H_M,H_m))


    Θ_i, H = ∑_n=1^N_Mα_i, nΦ_n(x),
          ϑ_i, H = ∑_n=1^N_M∑_k=1^N_mβ_i, n, kΦ_n(x) ϕ_k(y) 
           and    
        h_i, H = ∑_n=1^N_M h_i, nΦ_n(x),


with the additional condition Θ_i, H = ϑ_i, H   |Γ_0 for all i=1,…,N. For the coefficients it holds, α_i, n, β_i, n, k, h_i, n∈.

The above FE approach introduces another error in the numerical approximation. The error estimate from <ref> can also be extended to include the projection error in regards to the FE space. This was carried out for a comparable problem in <cit.> and could be transferred to our setup under regularity assumptions of the solutions. Here, we do not carry out the complete proof, instead, we demonstrate the error in the following section through numerical simulations. 

Note that the above FE approach is a natural candidate for discretizing the problem, but can become quite expensive due to the high dimensionality of ϑ, especially in three dimensions. We still use the above method since we only want to demonstrate the previous analytical results on a two-dimensional setup. Parallelization is used to speed up the calculations; see <ref> for details. However, in other applications, this procedure may not be feasible. In <ref> we gave an overview of possible extensions and different approaches. Note, that the presented precomputing method and theory could also be combined with the alternatives to reduce the computational effort.













Next to the discretization of (Θ, ϑ) we also need to solve the cell problems <ref> with a FE approach. To this end introduce the mesh size H_cell of the mesh for Y∖ Y(h). Let K_H denote the effective conductivity computed with the FE solution. Under the assumption that we use piecewise linear FEs for solving the cell problems and the cell solutions are sufficiently smooth, we can obtain the error estimate


    |K(h) - K_H(h)| ≤ C H_cell,


for a C < ∞ independent of H_cell.







§ SIMULATIONS

In this last section, we want to investigate and back up the previous analysis results with numerical simulations. To this end, we apply the time-discretized, finite element approach with precomputation as explained in <ref>. Before we study the convergence behavior, we introduce the general problem setup and explain some implementation details.
We use the FEM library FEniCS <cit.> and implement a custom parallelization for the micro problems. Our code is freely available online <cit.>. 

As mentioned in the previous section, we want to apply an approach where the time step for Θ and ϑ is carried out simultaneously. Therefore, at time step i, we have to solve the linear system


    [     A   C_1   C_2     … C_N_M;   D_1   B_1     0     …     0;   D_2     0     ⋱           ⋮;     ⋮     ⋮           ⋱     0; D_N_M     0     …     0 B_N_M ][      α_i;   β_i, 1;        ⋮;        ⋮; β_i, N_M ]
        = 
        [      F_i;   f_i, 1;        ⋮;        ⋮; f_i, N_M ]
        .


Here, we do not want to go too deep into the implementation details of each matrix block, for this, we point to the available source code. Instead, we only explain the role of each part:


    
  * The matrix A ∈^N_M × N_M encodes the macroscopic bilinear form.
    
  * The matrices B_k ∈^N_m × N_m represent the microscopic bilinear form, for k=1,… N_M.
    
  * C_k ∈^N_M × N_m describes the heat flow across the micro boundary Γ_0, and can be created by computing κ(h_i) ∇ϑ_i · n over Γ_0.
    
  * D_k ∈^N_m × N_m enforces the coupling that Θ = ϑ on Γ_0. Therefore this matrix only contains 0 and 1 on the corresponding entries. In particular, the row structure of all D_k is the same, only one column contains ones and for each k this column is shifted to a different position. In the corresponding rows of matrix B_k only a -1 is present and on the right-hand side a 0.
    
  * The right-hand side F_i ∈^N_M and f_i, k∈^N_m include the heat sources and information from the previous time step.


The role of C_k and D_k is the coupling between macro and micro scale. Note that there is no direct coupling between the different cells and the block containing the B_k matrices is diagonal. Additionally, all matrices are sparse.
The matrices A, B_k, C_k need to be recomputed in each time step, since they depend on the current height h_i. This is easily parallelizable once h_i is known.

Before we start presenting the simulation results, we collect here all information regarding the data and geometry. For the macro domain we use Ω = [0, 1]^2, the time interval is set to S=[0, 10], and the microdomain Y_0 is defined as a disk with radius r = 0.25. Note, that for a disk the Hanzawa transformation only is a scaling in the direction of the radius. The conductivity values are set to κ = 𝒦 = 0.1 and the initial temperature to Θ_0 = ϑ_0 = Θ^ref = 0. We only consider a macroscopic heat source, meaning f = 0, given by


    F(x, t) = 0.75 max{0,  min[1, g(t, x)]}max{0, min[1, 5 - t]}


with


    g(t, x) = 2 - 10 x - [ 0.2 + 0.6 t/5;           0.7 ]_∞.


F describes a smoothed step function that moves to the right and becomes 0 for t ≥ 5.
Additionally, we introduce a speed variable v=0.1 to better control the change of the microdomain, e.g., ∂_t h = v(Θ - Θ^ref). This does not influence the previous analysis. We remark, that we do not have a concrete application in mind for this setup. Instead, the values are chosen to observe interesting aspects of the model.

If not mentioned otherwise, we use Δ t = 0.1, H_M = 0.05, H_m = 0.06, H_cell = 5e-4 and Δ h = 1/320 in the following simulations. 

Since our microdomains are symmetric around the center of the domain, the effective conductivity K(h) reduces to a scalar value. The dependence of K of different radii r, is shown in <ref>. Note, that we characterize the change in radius by the height function h, a negative h equals shrinkage and a positive h growth of the disk radius. 





 §.§ Precomputing strategy


We start with the numerical investigation of the precomputing method. Both a piecewise linear interpolation as well as a quadratic spline interpolation will be studied. For the precomputation, we solve (Δ h)^-1 = 320 cell problems for different height values h that are taken from a uniform grid in [-0.245, 0.245]. Outside of this interval, we extend the interpolation by a constant value. Since no analytic solution is at hand, we compute a reference solution that uses all cell problems and the quadratic spline interpolation. The error is then computed between the reference solution and simulation results that utilize fewer cell problems for the interpolation.

The results are presented in <ref>. For the linear interpolation, we can observe the expected convergence order of 𝒪((Δ h)^2). In the case of quadratic interpolation, we note for a large Δ h, meaning the use of fewer precomputed values, we detect that the error decreases with the expected order of 𝒪((Δ h)^3). However, as soon as finer interpolation is used, the error stops decreasing and reaches a plateau. This is where the discretization error of the cell problems comes into play, affecting the accuracy of the interpolation. This effect is further described for different H_cell in <ref>. There we focus only on the function Θ to simplify the visualization. It can be seen that for larger H_cell, meaning a coarser cell mesh, the plateau occurs earlier. Therefore, to obtain the expected convergence in the precomputing scheme, an accurate solution of the cell problems is mandatory.



Another aspect we want to highlight is the interpolation property when the disk gets close to the outer boundary of the unit cell, meaning h → 0.25 or equally r → 0.5. Corresponding to the previous theory, for a radius close to 0.5 we would obtain a small a^* since the disk is not allowed to grow much further. Therefore, by <ref> the slope of K(h) may become large. This is also observable in <ref>. To better demonstrate the effect on the interpolation, we construct linear and quadratic interpolations for different Δ h and compare them on the interval I_h = [-0.25, h_max]. Again as a reference, we use the finest interpolation with Δ h = 1/320 and compare by computing the L^∞-difference on a uniform grid in I_h with 1000 points. This comparison is visualized in <ref>. We notice, that when h_max gets closer 0.25 the convergence order gets reduced. To circumvent this effect one could employ a nonuniform sampling of the height values used for the interpolation, e.g., solve more cell problems when h is close to 0.25. Note, that in the quadratic spline interpolation, we again observe the influence of H_cell once Δ h is small. From a theoretical point of view, this effect could also appear when h → 0, but there we observe that the slope of K(h) is rather flat. 

Finally, we want to briefly mention the computation time. On our setup, an AMD EPYC 7282 16-core CPU, it takes about 10 minutes to solve a cell problem for H_cell=5e-4, including the mesh generation. Since the precomputation is easily parallelizable, it took about 5 hours to compute the full dataset of 320 cell problems, as we were able to spread the work over several CPUs. Then, solving the entire problem in parallel on 8 cores, using the discretization mentioned above, takes about 13 seconds. If we had not used the precomputation scheme, the cell problems would need to be solved at each macro DOF and possibly at each time step, this would result in computation times of several days or weeks. This is also the reason why we do not use such a solution as a reference for the convergence study. 





 §.§ Time and space discretization error

In this second simulation section, we briefly present the convergence behavior with respect to the spatial discretization of the macro and micro scales and the time discretization. We study the influence of each discretization independently and fix the other parameters in each simulation. As before, no analytical solutions are available and a comparison is made with a finer numerical solution. For the fine reference solution, we use H_M=0.01, H_m=0.006, and Δ t = 0.003 respectively for each convergence study and apply quadratic spline interpolation for the precomputing.

The results are presented in <ref>. The order of error for the time-stepping method is consistent with the theoretical results of <ref>. Given the FE spaces of piecewise linear polynomials, both on the macro and micro scale, we also obtain the expected convergence behavior in regard to the space discretization.





§ CONCLUDING REMARKS

We have suggested, analyzed, and implemented a precomputing strategy for non-linearly coupled two-scale systems with evolving microstructures.
Utilizing the Hanzawa transformation, the existence of local-in-time, weak solutions was established.
Under more restrictive conditions on the data, it is expected that the solutions are unique and strong (see also <cit.>).
We showed that the additional interpolation error introduced via the precomputing can be controlled by the step size and the interpolation order given that the underlying relationship in the material parameters is smooth.
We also introduced a semi-implicit time-stepping method to resolve the nonlinearities inherent to the moving microsctructures and showed linear convergence.
We note that it is possible to improve convergence by using, e.g., a Crank-Nicolson-Galerkin
approximation (<cit.>).
The simulation results presented in <ref> confirm the analytical convergence.
The implementation was done in FEniCS and our code is freely available online <cit.>.

While the model in question in this paper is concerned with phase transitions on the microscale, we want to point out that the same kind of model with very similar analytical and numerical challenges appears in many different applications: precipitation in porous media, swelling of biological tissues, and degradation of concrete to name just a few.
The main advantage of our approach is the fact that a significant part of the computational effort is pushed into an offline phase in a way that is additionally perfectly parallelizable.
There are of course potential drawbacks as well: When the underlying functions are discontinuous or extremely volatile, the interpolation error might be difficult to control.
Moreover, in our case we only have a one-dimensional bounded parameter space h∈[-a^*,a^*], which is relatively easy to exhaust.
In cases where shape changes are possible (e.g., <cit.>) the parameter space becomes much larger.






§ ACKNOWLEDGEMENTS

The research activity of ME is funded by the European Union’s Horizon 2022 research and innovation program under the Marie Skłodowska-Curie fellowship project MATT (funding number 101061956, <https://doi.org/10.3030/101061956>).

TF acknowledges funding by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) – project nr.  
281474342/GRK2224/2 and by project nr. 439916647.























































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