Minimizers for the thin one-phase free boundary problem
Max Engelstein Aapo Kauranen Martí Prats Georgios Sakellaris Yannick Sire
math.AP

Collapse of layer dimerization in the photo-induced hidden state of 1T-TaS_2
    Tobias Ritschel
    December 30, 2023
============================================================================

alpha


We consider the “thin one-phase" free boundary problem, associated to minimizing a weighted Dirichlet energy of the function in ℝ^n+1_+ plus the area of the positivity set of that function in ℝ^n. We establish full regularity of the free boundary for dimensions n ≤ 2, prove almost everywhere regularity of the free boundary in arbitrary dimension and provide content and structure estimates on the singular set of the free boundary when it exists. All of these results hold for the full range of the relevant weight. 

While our results are typical for the calculus of variations, our approach does not follow the standard one first introduced in <cit.>. Instead, the nonlocal nature of the distributional measure associated to a minimizer necessitates arguments which are less reliant on the underlying PDE. 










 









§ INTRODUCTION


This article is devoted to the study of the regularity properties of a weighted version of the thin one-phase problem. More precisely we investigate even, nonnegative minimizers of the following functionals: denote x∈^n+1 by x=(x',y)∈^n×, and for β∈ (-1,1) we define


    𝒥(v,Ω):= ∫_Ω |y|^β |∇ v|^2 dx + m({v>0}∩^n∩Ω),

where m stands for the n-dimensional Lebesgue measure. 
Here, and throughout the paper, the integration is done with respect to the (n+1)-dimensional Lebesgue measure unless stated otherwise.

This functional is finite for open sets, Ω, and functions in the weighted Hilbert space,

    H^1(β,Ω): = { v∈ L^2(Ω;|y|^β) : ∇ v∈ L^2(Ω;|y|^β)},

equipped with the usual weighted norm.

Our main concern is to investigate fine regularity properties of the free boundary of minimizers v of (<ref>), that is the set,

    F(v):=∂_ℝ^n{ v(x,0)>0 }∩Ω.

Since the free boundary lies on a codimension 1 subspace of the ambient space ^n+1, such a problem is called a thin one-phase free boundary problem. This type of free boundary problem has been investigated for the first time by Caffarelli, Roquejoffre and the last author in <cit.> in relation with the theory of semi-permeable membranes (see, e.g., <cit.>). As we will describe later this is an analogue of the classical one-phase problem (also called the Bernoulli problem) but for the fractional Laplacian.

The Bernoulli problem was first treated in a rigorous mathematical way by Alt and Caffarelli in the seminal paper <cit.>: in the Bernoulli problem we consider minimizers of (<ref>) where β = 0 and the second term  is replaced by ℒ^n+1({v>0}∩Ω) (where ℒ^n+1 stands for the Lebesgue measure in ^n+1). In particular, for the Bernoulli problem, the free boundary fully sits in the ambient space, ^n+1. In <cit.>, the authors provided a general strategy to attack this type of problem.  Out of necessity we needed to modify this blueprint in several substantial ways (see below for a more detailed comparison).  For more information on the one-phase problem (and some of its variants) we refer to the book of Caffarelli and Salsa (and  references therein) <cit.>, and to the more recent survey of De Silva, Ferrari and Salsa <cit.>. 

As noticed in <cit.>, problem (<ref>) is related in a tight way to the standard  one-phase free boundary problem but with the Dirichlet energy replaced by the Gagliardo semi-norm [u]_Ḣ^α, for α=1-β/2∈ (0,1). This connection suggests that the thin one-phase problem is actually intrinsically a nonlocal  problem, though the energy in (<ref>) is clearly local. 


 §.§ Connection with the fractional one-phase problem

As previously mentioned, the functional 𝒥  introduced by Caffarelli, Roquejoffre and the last author in <cit.> is a local version of the following nonlocal free boundary problem: given a function f∈ L^1_(^n) with suitable decay at infinity, we can define its fractional Laplacian at x∈^n by

    (-Δ)^α f(x) = c_n,α  p.v. ∫_^nf(x)-f(ξ)/|x-ξ|^n+2α  dξ.

At the formal level, we are interested in solutions of the free boundary problem  

    (-Δ)^α f = 0     Ω∩{f>0},
    ∂_ν^α f = A     Ω∩ F(f),

where ∂_ν^α f (x):=lim_Ω∩{f>0}∋ξ→ xf(ξ)-f(x)/((ξ-x)·ν(x))^α and where f satisfies a given “Dirichlet boundary condition" on the complement of Ω. 



 
 


As in the case of the classical Laplacian (see <cit.>), we are interested in obtaining equation eqEulerLagrangeFrac as the Euler-Lagrange equation of a certain functional. Given a locally integrable function f, consider its fractional Sobolev energy

    [f]_Ḣ^α (^n):=∬_^2n|f(x)-f(ξ)|^2/|x-ξ|^n+2α  dξ  dx.

Since we want to study competitors which vary only in a certain domain Ω, it is natural to consider only the integration region which may suffer variations when changing candidates. Thus, we define the energy 

    J(f,Ω):= c_n,α∬_^2n∖ (Ω^c)^2|f(x)-f(ξ)|^2/|x-ξ|^n+2α  dξ  dx + m({f>0}∩Ω).

We say that f∈ L^1_ is a minimizer of J in Ω if J(f,Ω) is finite and J(f,Ω)≤ J(g,Ω) for every g satisfying that f-g∈Ḣ^α(^n) and such that f(x)=g(x) for almost every x∈Ω^c. We say that f is a global minimizer if it is a minimizer for every open set Ω⊂^n. Note that both terms in eqNonLocalFunctional are in competition, since a minimizer of the fractional Sobolev energy in Ω is α-harmonic and, thus, if it is non-negative outside of Ω it is strictly positive inside of Ω, maximizing the second term.











Consider now the Poisson kernel for fixed n∈ and 0<α<1

    P_y(ξ):=P_n,α(ξ,y)=c_n,α|y|^2α/|(ξ,y)|^n+2α    (ξ,y)∈^n×.

The Poisson extension of f∈ L^1_(^n) is given by

    u(x',y):= f * P_y(x')=∫_^n P_n,α(ξ,y) f(x'-ξ)  dξ    (x',y)∈^n×.

By <cit.>, with a convenient choice of the constant one gets

    lim_y ↘ 0y^1-2α u_y(x',y)=-(-Δ)^α f(x')
 in every point where f is regular enough. Moreover, the extension satisfies the localized equation ∇· (|y|^β∇ u)=0 weakly, away from ^n×{0}. The whole point is that local minimizers of (<ref>) can be extended via the previous Poisson kernel P_y to (even) minimizers of (<ref>) (see the Appendix for a precise statement). Therefore, the thin one-phase problem appears as a “localization" of the one-phase problem for the fractional Laplacian. Notice that, and this is of major importance for us, this localization technique does not carry over to other types of nonlocal operators besides pure powers of second-order elliptic operators. This is a major drawback of the theory, in the sense that, at the moment, it seems to be impossible to tackle one-phase problems involving more general operators than the fractional Laplacian. The main point is we do not know how to prove any kind of monotonicity for general integral operators.

This connection between the nonlocal analogue of the Bernoulli problem and our thin one-phase problem allows us to simplify several arguments by working in the purely nonlocal setting. However, this underlying nonlocality is also the reason why several results, which came more easily in the setting of <cit.>, are non-trivial or substantially harder for us. For example, perturbations of solutions need to take into account long range effects which makes classical, local, perturbation arguments much more difficult. 








In the paper <cit.>, the authors proved basic properties of the minimizers for the functional 𝒥 such as optimal regularity, non-degeneracy near the free boundary, and  positive densities of phases. Also they provided an argument for n=2 showing that Lipschitz free boundaries are C^1. A feature of the functional 𝒥 is that the weight |y|^β is either degenerate or singular at { y=0 } (except in the case β=0). Such weights belong to the Muckenhoupt class A_2 and the seminal paper of Fabes, Kenig and Serapioni <cit.> investigated regularity issues for elliptic PDEs involving such weights (among other things). After that, <cit.> proved an ε-regularity result and <cit.> showed the existence of a monotonicity formula for this setting.

In the case β=0, the problem is still degenerate in the sense that derivatives near the free boundary blow up. The case β=0 has been thoroughly investigated in the series of papers by De Silva, Savin  and Roquejoffre <cit.>. 

The main goal of our paper is to provide a full picture of the regularity of the free boundary for any power β∈ (-1,1), both in terms of measure-theoretic statements and partial (or full) regularity results. From this point of view our contribution is a complement of the paper by De Silva and Savin <cit.> for β=0. It has to be noticed that the standard approach to regularity of Lipschitz  free boundaries as developed by Caffarelli (see the monograph <cit.>) does not seem to work in our setting.







  §.§.§ Our approach to regularity


In <cit.> (and many subsequent works), the minimizing property of the solution is used to prove that the distributional Laplacian of that solution is an Ahlfors-regular measure supported on the free boundary. This implies (amongst other things) that the free boundary is a set of (locally) finite perimeter, and thus almost every point on the free boundary has a measure theoretic tangent. One can then work purely with the weak formula (i.e. the analogue of (<ref>)) to prove a “flat implies smooth" result which, together with the existence almost everywhere of a measure theoretic tangent, has as a consequence that the free boundary is almost everywhere a smooth graph and the free boundary condition in (<ref>) holds in a classical sense at the smooth points. 

A similar “flat implies smooth" result exists in our context (this is essentially due to De Silva, Savin and the last author, <cit.>, see Theorem <ref> below). However, showing that the free boundary is the boundary of a set of finite perimeter proves to be much more difficult. Due to the nonlocal nature of the problem, -div(|y|^β∇ u) (considered as a distribution) is not supported on the free boundary. Furthermore, the scaling of this measure does not allow us to conclude that the free boundary has the correct dimension (much less that it is Ahlfors regular).





To prove finite perimeter, we take the following approach inspired by the work of de Silva and Savin: after establishing some preliminaries 

we prove crucial compactness results. This, along with a monotonicity formula originally due to Allen <cit.>  allows us to run a dimension reduction argument in the vein of Federer or (in the context of free boundary problems) Weiss <cit.>. With this tool in hand, we show that the set of points at which no blow-up is flat is a set of lower dimension. Locally finite perimeter and regularity for the reduced boundary then follow from a covering argument and some standard techniques.

 Here and throughout  the paper, we will denote the ball of radius r in ^n+1 centered at the origin by  B_r, and B_r':=B_r∩^n×{0}. Moreover, for the definition of ^β, see Section <ref>. We may then summarize our regularity results in the following theorem.


[Main Regularity Theorem]
Let u∈^β(B_1) be a (non-negative, even) local minimizer of 𝒥 in B_1 ⊂ℝ^n+1. Let B_1,+'(u):={x=(x',0)∈ B_1: u(x)>0}, let F(u) be the boundary of B_1,+'(u) inside of ^n×{0}  and assume that 0∈ F(u). Then,


  * B_1,+'(u) (as a subset of ^n×{0}) is a set of locally finite perimeter in B_1'. 

  * We can write the free boundary as a disjoint union F(u) = ℛ(u) ∪Σ(u), where ℛ(u) is open inside F(u), and for x ∈ℛ(u) there exists an r_x > 0 such that B(x, r_x) ∩ F(u) can be written as the graph of a C^1,s-continuous function. 

  * Furthermore, the set Σ(u) is of Hausdorff dimension ≤ n-3 (and, therefore, of ℋ^n-1-measure zero). In particular, for n≤ 2, Σ(u) is empty, and moreover, if n=3 then Σ(u) is discrete.



The constants (implicit in the set of finite perimeter, and the Hölder continuity of the functions whose graph gives the free boundary) depend on n and β but not on u_^β(B_1). 



As usual Σ(u) ⊂ F(u) is called the singular set of the free boundary: the set of points around which F(u) cannot be parameterized as a smooth graph and all the blow-ups will be non-trivial minimal cones, see Theorem <ref>.

Our second contribution concerns the structure and size of the singular set. It builds on recent major works on quantitative stratification <cit.>, extended to free boundary problems (in particular the one-phase problem) by Edelen and the first author <cit.>. 


Let u∈^β(B_1) be a (non-negative, even) local minimizer of 𝒥 in B_1 and 0∈ F(u). Let B_1,+'(u):={x=(x',0)∈ B_1: u(x)>0} and F(u) be the boundary of B_1,+'(u) inside  B_1'. Then, there exists a k_α^*≥ 3  such that Σ(u) is (n-k_α^*)-rectifiable and 
    ℋ^n-k^*_α(Σ(u)∩ D)≤ C_n,α, (D,∂ B_1)    D⊂⊂ B_1.




In <cit.>, De Silva and Jerison constructed a singular minimizer for the Alt-Caffarelli one-phase problem in dimension 7, giving the dimension bound k^* ≤ 8 in the previous theorem in this case (see <cit.>). This result is not known for the thin one-phase problem. The reason is that the one-phase problem, seen from the nonlocal point of view involving the fractional Laplacian, is related to the so-called nonlocal minimal surfaces introduced by Caffarelli, Roquejoffre and Savin <cit.>. Indeed, in <cit.>, the authors proved that a fractional version of Allen-Cahn equation converges variationally to the standard perimeter functional for α≥ 1/2 and to the so-called nonlocal minimal surfaces for α <1/2. We can then conjecture the bound k^*_α≤ 8 for α≥ 1/2 by analogy with the result for the standard one-phase problem but the bound for α <1/2 is not clear at all. However, one knows that there is no singular cone in dimension 2 for nonlocal minimal surfaces <cit.> and that the Bernstein problem is known for those in dimensions 2 and 3 <cit.>. 



We would like also to make a last remark about a result which is of purely nonlocal nature. In the case of the one-phase problem, one can show that the distributional Laplacian is a Radon measure along the free boundary. In the case of the thin one-phase free boundary problem, due to the nonlocality of the problem, such a behavior does not happen in the sense that we will show that the fractional Laplacian  is an absolutely continuous  measure with respect to n-dimensional Lebesgue measure with a precise behavior. This phenomenon is of purely nonlocal nature and similar to the fact that the fractional harmonic measure is of trivial nature.  More precisely, every minimizer u satisfies ∇· (|y|^β∇ u)=0 weakly, away from ^n∩{u ≤ 0}. Thus, equation eqEulerLagrangeFrac above can be understood as an Euler-Lagrange equation for the functional 𝒥 in the sense that the restriction to ^n of a given minimizer u in Ω⊂^n+1 and with asymptotic behavior u(x,y)=𝒪(|(x,y)|^α) is always a solution to eqEulerLagrangeFrac for A=A(α) at “nice" points of the free boundary. 


A brief summary of this paper  follows. In Sections <ref> and  <ref> we discuss  compactness of minimizers and we recall Allen's monotonicity formula to derive some immediate consequences. In Section <ref> we show that the positive phase is a set of locally finite perimeter, establishing the first part of Theorem <ref> (modulo energy bounds), and we show that the singular set can be identified using the Allen-Weiss density. Section <ref> is devoted to deducing full regularity of minimizers in ^2+1 concluding the proof of Theorem <ref>.

Once we have established the finite perimeter, in Section <ref> we remove the dependence of the estimates on the energy of the minimizer in the previous theorems, using a rather subtle argument which combines results from all the previous sections. A crucial step is to analyze some basic properties of the distributional fractional Laplacian of our minimizer. As stated above this analysis will not be enough to establish that the positivity set of the minimizer is a set of locally finite perimeter.  We believe that many of these results may be of independent interest. For example, corresponding results for the classical Bernoulli problem have been used to understand free boundary problems for harmonic measure (see <cit.>).

Finally, Section <ref> is devoted to the proof of Theorem <ref>.



  §.§.§ Notation



We denote the constants that depend on the dimension n, α and perhaps some other fixed parameters which are clear from the context  by C. Their value may change from an occurrence to another. On the other hand, constants with subscripts as C_0 retain their values along the text. For a,b≥ 0, we write a≲ b if there is C>0 such that a≤ Cb. We write a≈ b to mean a≲ b≲ a.

Let u be a continuous function in ^n+1. Then we write Ω_+(u):=Ω∩{u>0}, and we denote the zero phase, the positive phase and the free boundary by

    Ω_0(u):={x∈^n×{0}: u(x)=0}^∘,
       Ω_+'(u):=Ω_+∩ (^n×{0})={x∈^n×{0}: u(x)>0},
       F(u):=F_Ω(u)=∂ (Ω_+(u)∩^n×{0}) ∩Ω,

respectively. Here both the boundary and the interior are taken with respect to the standard topology in ^n.
Note that ^n×{0} is the disjoint union of  Ω_0(u), Ω_+'(u) and F(u) whenever u is non-negative. We also call F_ red(u)=F_ red,Ω(u) the points of F_Ω(u) where the free boundary is expressed locally as a C^1 surface. Finally, let Σ(u)=Σ_Ω(u)=F_Ω(u)∖ F_ red,Ω(u). In general we will write Ω':=Ω∩(^n×{0}).

Throughout the paper we will often fix β∈ (-1, 1) but then refer to α∈ (0,1) or vice versa. These two numbers are always connected by the relationship α = 1-β/2.




























§ PRELIMINARIES


In this section, we provide the known results concerning the problem under consideration. We say that a function u is even if it is symmetric with respect to the hyperplane ^n×{0}, that is, u(x',y)=u(x',-y). The function spaces that we will consider are the following

    ^β (Ω):= {u ∈ H^1(β,Ω) : u}

and

    ^β_(Ω):= {u∈ L^2_(Ω): u ∈^β(B)  B⊂⊂Ω}.

We will omit Ω in the notation when it is clear from the context.

We say that a function u∈^β_(Ω) is a (local) minimizer of 𝒥 in a domain Ω if for every ball B⊂⊂Ω and for every function v∈^β(B) such that the traces v|_∂ B≡ u|_∂ B, the inequality

    𝒥(u,B)≤𝒥(v,B)

holds.




As usual for several free boundary problems, it is a natural question to exhibit a particular (global) solution so that one gets an idea of the qualitative properties of general solutions. Let us consider the following function: for every x∈^n let 
    f_n,α(x):= c_n,α (x_n)_+^α ,

where a_+=max{0,a}.
If n=1, f_1,α is a solution to eqEulerLagrangeFrac for a convenient choice of c_1,α (see <cit.>). In fact one can see that the same is true for n≥ 1 using Fubini's Theorem conveniently, with 

    -(-Δ)^α f_n,α(x) = c_n,α (x_n)_-^-α,

where a_-=max{0,-a}.

As a  toy question we wonder whether the trivial solutions are minimizers. Indeed, this is the case, as we will see later in Section <ref>.

Let n∈ and 0<α<1. Then the trivial solution u_n,α:=f_n,α*P_y is a minimizer of 𝒥 in every ball B⊂^n+1.



Next we collect the main properties of minimizers in the unit ball proven in <cit.>. 

If u∈^β(B_1) is a minimizer of 𝒥 in B_1  with u_^β(B_1):=∇ u_L^2(B_1,|y|^β)≤ E_0 and x_0∈ F(u)∩ B_1/2, then it satisfies


  * Optimal regularity (see <cit.>): u_Ċ^α(B_1/2)≤ C(1+E_0).

  * Nondegeneracy (see <cit.>): u(x)≥ C (x,F(u))^α for x∈ B_1/2'.

  * Interior corkscrew condition (see <cit.>):  there exists x_+∈ B_r'(x_0) so that  B'(x_+, C_0r)⊂Ω_+'(u).

  * Positive density (see <cit.>): |Ω_0∩ B_r'(x_0)| ≳ r^n. 

  * Blow-ups are minimizers (see <cit.>): The limit of a blow-up sequence u_k(x):=u(x_0+ρ_k x)/ρ_k^α converging weakly in H^1(β,B_1) and uniformly is a global minimizer.

  * Normal behavior at the free boundary (see <cit.>): the boundary condition in eqEulerLagrangeFrac is satisfied at every point on the free boundary with a measure theoretic normal (see <cit.>) for a prescribed value of A.

All the constants depend on n and α; and also on  E_0 except for the one in P1.




A major tool in the present paper is an ϵ-regularity result, i.e. in the language of free boundaries a statement of the type “flatness implies smoothness". In <cit.>, the authors proved such an ϵ-regularity result for viscosity solutions to the overdetermined system associated to minimizers of 𝒥. Here we establish that all local minimizers are in fact viscosity solutions. While this verification may be standard for experts in the field, we include it here for the sake of completeness. 



There exists ϵ>0 depending only on n, α and E_0 such that for every non-negative, even minimizer u of the energy eqLocalizedFunctional on a ball B⊂^n+1 with 
u_^β(B)≤  E_0 r(B)^n/2 and 

    {(x,0)∈ B: x_n≤ -ϵ}⊂ B_0(u) ⊂{(x,0)∈ B: x_n≤ϵ},

we have that F(u)∈ C^1,γ_ loc(1/2 B), with 0<γ<1.

Note that the dependence on E_0 will be removed in Section <ref>.

 We say that u is a viscosity solution to 

    ∇· (|y|^β∇ u) = 0    B_1^+(u),
    lim_t→ 0+u(x_0+tν (x_0),0)/t^α = 1,      (x_0,0)∈ F(u),

if 


  * u∈ C(B_1), u≥ 0,

  * u∈ C^1,1_ loc(B_1,+(u)), u is even and it solves ∇· (|y|^β∇ u) = 0 in the viscosity sense, and

  * any strict comparison subsolution (resp. supersolution) cannot touch from below (resp. from above) at a point (x_0,0)∈ F(u). 


We claim that 

    


Conditions (i) and (ii) have been verified  in <cit.>. To verify our claim it suffices to prove condition (iii) above: that any strict comparison subsolution cannot touch u from below at a point (x_0, 0) ∈ F(u). The analogous claim for strict comparison supersolutions will follow in the same way. 

Let us recall (see, e.g. Definition 2.2 in <cit.>),  that w∈ C(B_1) is a strict comparison subsolution (resp. supersolution) to (<ref>) if 


  * w ≥ 0,

  * w is even with respect to {y= 0}, 

  * w∈ C^2({w > 0}),

  * div(|z|^β∇ w) ≥ 0 in B_1 \{y = 0},

  * F(w) is locally given by the graph of a C^2 function and for any x_0 ∈ F(w) we may write 
    w(x,y) = aU((x-x_0)·ν(x_0), y) + o((x-x_0, y)^α),     (x,y) → (x_0,0).
 Here U is the extension of the trivial solution (see <cit.>), ν(x_0) is the unit normal to F(w) considered as a subset of ^n pointing into {w > 0} and a ≥ 1.

  * Furthermore, either the inequality is strict in d), or a >1 in e). 


So assume that w ≥ u where w is a strict comparison subsolution and u is some minimizer and that w = u at (x_0,0) ∈ F(u). Since u(x_0, 0) = 0 it follows that (x_0, 0) ∈ F(w) and with a harmless rotation we can guarantee that ν((x_0,0)) = e_n. We want to show that e_n is also the measure theoretic unit normal to F(u). Indeed, since F(w) is C^2 there must exist a ball B ⊂{w > 0} which is tangent to F(w) at (x_0,0). It must then be that case that B ⊂{u > 0} as well. Thus (x_0,0) ∈ F(u) has a tangent ball from the inside which, by <cit.> Proposition 4.5 implies that u has the asymptotic expansion 
    u(x,y) = U((x-x_0)·ν(x_0), y) + o((x-x_0, y)^α),     (x,y) → (x_0,0).
 

If u ≥ w this implies that w must satisfy the expansion in (<ref>) with a = 1 at the point x_0. This, in turn, implies that div(|z|^β∇ w) > 0 in B_1 \{y = 0} (by the definition of a strict subsolution). Furthermore, since w∈ C^2 where {w > 0} we can guarantee that div(|z|^β∇ w) ≥ 0 in all of B_1 ∩{w > 0}. 

Let us return to the ball B which is a subset of {u > 0} and {w > 0} and for which (x_0, 0) ∈B. We know that w-u ≠ 0 in B∖{y=0} (this is because w strictly satisfies the differential inequality in B away from {y = 0}) and we know that w-u is a subsolution in B. Furthermore (x_0, 0) ∈ B is a strict maximum, so by the Hopf lemma in <cit.> it must be that 
    lim_t↓ 0^+(w-u)(x_0+tν (x_0),0)/t^α > 0.
 This contradicts the fact u and w both satisfy (<ref>) at (x_0,0) with a = 1.  Therefore, (x_0,0) must not have been a touching point and u is indeed a viscosity solution. 

Since, u is a viscosity solution, <cit.> applies and we get the desired ε-regularity. 
















§ COMPACTNESS OF MINIMIZERS


In this section we prove important results on the compactness of minimizers. As we mentioned above, our contribution is that convergent sequences of minimizers also converge in the relevant weighted Sobolev spaces strongly rather than just weakly. This will prove essential to the compactness arguments used in the later sections of this paper. 





 §.§ Caccioppoli Inequality

First we want to show that  the distribution λ:=∇·(|y|^β∇ u) is in fact a Radon measure with support in the complement of the positive phase as long as u is a minimizer. In Section <ref> we will come back to this measure to understand its behavior around the free boundary.


Let Ω⊂^n+1 be an open set, and let u∈ W^1,2_(Ω, |y|^β) be such that ∇· (|y|^β∇ u)=0 weakly in Ω_+(u), i.e., for every η∈ C^∞_c(Ω_+(u)),

    ⟨∇·(|y|^β∇ u), η⟩ := - ∫ (|y|^β∇ u) ∇η=0.

Then λ:=∇·(|y|^β∇ u) is a positive Radon measure supported on {u=0}  and for every v∈ W^1,2(Ω, |y|^β) ∩ C_c(Ω) 

    ∫ v   dλ = -∫ |y|^β∇ u·∇ v.



Indeed, by eqBetaHarmonic the quantity

    -∫ |y|^β∇ u·∇ζ=-∫ |y|^β∇ u ·∇(ζmax{min{2-u/ε,1},0}) ≥ -∫_Ω∩{0<u<2ε} |y|^β |∇ u| |∇ζ| 0

defines a positive functional on positive  ζ∈ C^0,1_c(Ω). Moreover, for compact K⊂Ω, consider a Lipschitz function f_K such that χ_K≤ f_K≤χ_Ω. If ζ∈ C^0,1_c(K), by the positivity shown above we obtain

    -∫ |y|^β∇ u·∇ζ≤ - ζ_L^∞∫ |y|^β∇ u·∇ f_K≤ C_K,uζ_L^∞

and, by Hahn-Banach's theorem, we can extend the functional to a positive functional in C_c(Ω), that is given by integration against a positive Radon measure by the Riesz representation theorem.

The fact  that eqMeasureDef holds for all functions in W^1,2(Ω, |y|^β) ∩ C_c(Ω) follows by a standard density argument. 

 
The Caccioppoli inequality is the first step to proving convergence in a Sobolev sense. It will also be useful when we remove the a priori dependence of our results on the Sobolev norm of the minimizer. 



Let B ⊂^n+1 be a ball of radius r centered on ^n×{0}, and let u ∈ W^1,2(B, |y|^β) be such that ∇· (|y|^β∇ u)=0 weakly in B∩{u>0}. Then 

    ∫_1/2 B |y|^β |∇ u|^2 ≤4/r^2∫_B∖1/2 B |y|^β u^2.



Let η be a Lipschitz function such that χ_1/2 B≤η≤χ_B and with |∇η|≤1/r. By Lemma <ref>

    0= ∫_B u η^2 dλ = ∫_B |y|^β∇ u·∇(u η^2).

By the Leibniz rule 

    ∫_B |y|^βη^2 |∇ u|^2 =- ∫_B |y|^β  2 u η∇ u·∇η,
 
and using Hölder's inequality we get

    ∫_1/2 B |y|^β |∇ u|^2 ≤∫_B |y|^βη^2 |∇ u|^2  ≤∫_B |y|^β  4 u^2 |∇η|^2 ≤4/r^2∫_B∖1/2 B |y|^β u^2.














Let u ∈^β(B_r) be a minimizer of 𝒥 in B_2r and 0∈ F(u). Then 

    r^-n/2∇ u_L^2(1/2 B_r;|y|^Β)≤  r^-αu_L^∞ (B_r)≤u_Ċ^α(B_r)≤ C (1+r^-n/2∇ u_L^2( B_2r;|y|^Β)).





The first inequality is Caccioppoli, the middle estimate is trivial and the last follows from P1  in Theorem <ref>.









 §.§ Compactness


In the following lemma we prove the compactness of minimizers in the relevant Sobolev spaces. For convenience, we also detail several compactness results which were either already proven in <cit.> or are standard consequences of the non-degeneracy estimates in Theorem <ref>. Nevertheless, we include full proofs here for the sake of completeness. We note here (as we did above and will do again below) that while we currently need to assume the uniform bound on the Hölder norm of the functions u_k we can get rid of  this assumption in the light of the results of Section <ref>.


 Let {u_k}_k=1^∞⊂^β_(Ω) be a sequence of minimizers in a domain Ω⊂^n+1 with u_k_Ċ^α(Ω)≤ E_0 with non-empty free boundary. Then there exists a subsequence converging 
to some u_0∈^β_(Ω) such that for every bounded open set 
G⊂⊂Ω we have
 
  
  *  u_k→ u_0 in C^β(G) for 
every β<α,
  
  * u_k→ u_0 in L^p(G) for every p≤∞,
  
  * ∂{u_k>0}∩G̅→∂{u_0>0}∩G̅  in the
Hausdorff distance,
  
  * χ_{u_k>0}→χ_{u_0>0} in L^1(G), and
  
  * ∇ u_k →∇ u_0 in  L^p(G;|y|^β) for every p≤ 2.

  
  


 The first claim follows from uniform Hölder continuity and compact 
embeddings  of Hölder spaces. The claim (2) follows from (1) easily. 
 
 We now prove the third claim. Let ϵ>0.  We will first  show that 
for x ∈^n we have 

    d(x, 
    F(u_0))>ϵ⇒  d(x,F(u_k))>ϵ/2

for large k. 
This implies that F(u_k)⊂{x  d(F(u_0),x)<2ϵ} for k 
large enough.

Let B(x,ϵ)⊂ F(u_0)^c. If u_0 is positive in B(x,ϵ) then 
it is bounded from below by a positive number in  B(x,ϵ/2). In this 
case u_k are also positive in  B(x,ϵ/2) for large k due to uniform 
convergence in G. Thus B(x,ϵ/2)⊂ F(u_k)^c for large k.  If 
u≡0 in B'(x,ϵ) then due to the uniform convergence we know that  
for k large enough u_k<Cϵ^α in B'(x,ϵ), where C is a constant given by P2 in Theorem <ref> 
so that   u_k has no free boundary points in  B(x,ϵ/2) for all large k.  This proves (<ref>).

Next we will show that for all large k

    F(u_0)⊂{x  d(F(u_k),x)<ϵ}.

 If this was not true we could find a point x∈ F(u_0) and a subsequence  of 
u_k such that B'(x,ϵ)⊂ F(u_k)^c for every k in the subsequence.
If the subsequence contains infinitely many u_k such that u_k≡0 in 
B(x,ϵ) then also u_0≡ 0 due to uniform convergence.
Otherwise, the sequence contains infinitely many u_k for which B(x,ϵ)
is contained in the positive phase. In this case the non-degeneracy implies 
that 
in B(x,ϵ/2) we have u_k>Cϵ^α, with C independent of k. Again 
uniform 
convergence implies the same lower bound for u_0, which contradicts our 
choice 
 x∈ F(u_0).  
 
 To show the fourth claim we notice that F(u_0) has zero Lebesgue measure by the Lebesgue differentiation Theorem and the positive density of the zero phase. Take 
an 
open set V⊃ F(u_0) with m(V∩  G )<ϵ. For large k 
we have F(u_k)∪ F(u_0)⊂  V∩  G, so  χ_{u_k>0}- χ_{u_0>0}_L^1(G)<ϵ.

Also the sequence is uniformly bounded in H^1,p(G;|y|^β) by the Caccioppoli inequality. This implies 
by compactness <cit.> the  
weak convergence of ∇ u_k in L^p(G;|y|^β). To obtain strong convergence, use Lemma <ref> below.
 
 


It remains to show that weak convergence implies strong convergence. 


Any sequence of minimizers  {u_k}_k=0^∞ in Ω⊂^n+1 with u_k→ u_0 uniformly and 
∇ u_k⇀∇ u_0 weakly in  L^2_(Ω,|y|^β) satisfies that ∇ u_k→∇ u_0 in  L^2_(Ω,|y|^β).


Let η∈ C^0,1_c(Ω) be a non-negative function. We claim that for every ε>0 there exists j_0 so that 

    ∫ |y|^βη |∇ u-∇ u_j|^2≤ε

for j≥ j_0.

First we isolate the main difficulty

    ∫ |y|^βη |∇ u_0-∇ u_j|^2=∫ |y|^βη (∇ u_0-∇ u_j)·∇ u_0  - ∫ |y|^βη (∇ u_0-∇ u_j)·∇ u_j.

By weak convergence, 

    |∫ |y|^βη (∇ u_0-∇ u_j)·∇ u_0|≤ε/4

for j big enough. Note that this is true even if the u_j are not minimizers. The bound on the second term, however, needs the minimization property. 

We observe that 
    ∫ |y|^βη (∇ u_0-∇ u_j)·∇ u_j = ∫ |y|^β  (∇ u_0-∇ u_j)·∇ (η u_j)_=:I - ∫ |y|^β  u_j (∇ u_0-∇ u_j)·∇η_=:II.


To estimate I in (<ref>), let λ_j be the measures corresponding to u_j from Lemma <ref>. By eqMeasureDef we get that

    ∫ |y|^β  (∇ u_0-∇ u_j)·∇ (η u_j) = ∫η u_j   dλ_0 - ∫η u_j  dλ_j.

Since λ_j is supported on {u_j = 0} we have that 

    ∫η u_j   dλ_j=0

for every j (including j = 0 as u_0 is also a minimizer to 𝒥, see Corollary 3.4 in <cit.>). 

 To finish the estimate on I in (<ref>) we observe that

    ∫η u_j   dλ_0=∫η (u_j-u_0)   dλ_0≤sup_ η |u_j-u_0| ∫η  dλ_0 .

By uniform convergence on compact subsets, for j big enough, sup_ η |u_j-u_0|≤ϵ/4η_L^1(λ_0). 

We turn towards estimating II in (<ref>):

    |II|= | ∫ |y|^β  u_j (∇ u_0-∇ u_j)·∇η|
       ≤| ∫ |y|^β   (∇ u_0-∇ u_j)· (u_0 ∇η)| 
          + sup_ η |u_j-u_0|  ∇ u_0-∇ u_j_L^2(Ω,|y|^β)∇η_L^2(Ω,|y|^β).

The first term goes to zero by weak convergence of ∇ u_j to ∇ u_0. The second term satisfies 

    sup_η |u_j-u_0|  ∇ u_0-∇ u_j_L^2(η,|y|^β)∇η_L^2(Ω, |y|^β)≤ε/4

for j big enough, by uniform convergence and the uniform bound of ∇ u_j_L^2( η,|y|^β) derived from the Caccioppoli inequality in Lemma <ref> together with uniform convergence. 


























Lemma <ref> implies that minimizers converge to minimizers (which was observed in Corollary 3.4 in <cit.>), but also implies the stronger fact that the energy is continuous under this convergence:


Let u_k be a sequence of minimizers in Ω⊂ℝ^n+1 with u_k → u_0 locally uniformly and sup_k u_k_^β < ∞. Then u_0 is also a minimizer to 𝒥 in Ω and for any B ⊂⊂Ω we have 𝒥(u_k, B) →𝒥(u_0, B). 
 






§ MONOTONICITY FORMULA AND SOME IMMEDIATE CONSEQUENCES


From <cit.> we have the following monotonicity formula:

Let u∈^β(B_δ(x_0)) be a minimizer in B_δ(x_0) for the functional 𝒥 with x_0∈ F(u).
Then the function

    r↦Ψ^u_r(x):=Ψ(r)=𝒥(u,B_r(x_0))/r^n  - α/ r^n+1∫_∂ B_r(x_0) |y|^Β u^2   dℋ^n

is defined and nondecreasing in (0,δ), and for 0<ρ<σ<δ, it satisfies

    Ψ(σ)-Ψ(ρ) = ∫_B_σ(x_0)∖ B_ρ(x_0) |y|^Β2|α u (x) -(x-x_0)·∇ u(x)|^2 /|x_0-x|^n+2dx ≥ 0.



As a consequence, the blow-up limits are cones, in the sense of the following corollary.

Let u∈^β(B_δ(x_0)) be a  minimizer in B_δ(x_0) with x_0=(x_0',0). Consider a decreasing sequence 0<ρ_k0 and the associated rescalings u_k(x):=u(x_0+ρ_k x)/r^α.
Then  the Allen-Weiss density

    Ψ^u_0(x_0):=lim_r↘ 0Ψ^u_r(x_0)

is well defined. Furthermore, for every bounded open set D⊂^n+1 and k≥ k(D) this subsequence u_k is bounded in H^1,2(D;|y|^β) and, passing to a subsequence u_k_j, converges (in the sense of Lemma <ref>) to u_0 which is a globally defined minimizer of 𝒥 that is homogeneous of degree α.

The proof is the same as  in <cit.>



We call the function u_0 appearing in Corollary <ref> a blow-up of u at x_0. A priori, the function u_0 may depend on the subsequence u_k_j. However, a simple scaling argument shows that for all radii r ≥ 0 and all blow-ups u_0 to u at x_0 we have 
    Ψ^u_0_r(0) ≡Ψ^u_0(x_0).
 





 §.§ Dimension reduction

We use the homogeneity of the blow-ups to obtain dimension estimates on the points in the free boundary for which there exists a non-flat blow-up. This process is known as “dimension reduction" and has been applied to a variety of situations (see <cit.> for its application to the Bernoulli problem).

The first lemma shows that blow-up limits of blow-up limits have additional symmetry:


Let u∈^β_(^n+1) be an α-homogeneous  minimizer of 𝒥 and let x_0∈ F(u)∖{0}. Then any blow-up limit u_0 at x_0 is invariant in the direction of x_0, i.e., for every x∈^n+1 and every λ∈,

    u_0(x+λ x_0)=u_0(x) .





Let x ∈^n+1, and consider its decomposition x=x+λ x_0 with x∈⟨ x_0 ⟩^. We only need to check that 
    u_0(x)=u_0(x).





Consider a ball B=B(0,r) ⊂^n+1 so that x, x∈ B.
Let {ρ_k} be a sequence of radii converging to zero and such that u_k(x):=u(x_0+ρ_k x)/ρ_k^α converges  to u_0

uniformly on B_r. For k big enough, u_k-u_0_L^∞(B_r)<ε. Then,

    |u_0(x)-u_0(x)|≤ 2ε + |u_k(x)-u_k(x)|.


To control the last term above, we use the homogeneity of u.
Writing P_1:=x_0+ρ_k x and P_2:=x_0+ρ_k x we have ρ_k^α u_k(x)=u(P_1) and ρ_k^α u_k(x)=u(P_2). Let P_3 be the intersection between the line through P_1 and x_0 and the line through the origin and P_2 (see Figure <ref>). By homogeneity of u

    u(P_2)=u(P_3)(|P_2|/|P_3|)^α=u(P_3)(1 ±|P_2-P_3|/|P_3|)^α.

Thus,

    ρ_k^α|u_k(x)-u_k(x)|
           ≤|u(P_1) - u(P_3)(1 + 𝒪(ρ_k))^α|
               ≤|u(P_1) - u(P_3)| + |u(P_3)|𝒪(ρ_k).

By Thales' Theorem, |P_1- P_3|=|P_1- P_2||P_3-x_0|/|x_0|=𝒪(ρ_k^2) and using the Ċ^α character of u and the fact that u(x_0)=0, we get

    ρ_k^α|u_k(x)-u_k(x)|
            ≤u_Ċ^α( |P_1- P_3|^α  + |P_3|^α𝒪(ρ_k))
             =  𝒪(ρ_k^2α )+ 𝒪(ρ_k),

and eqTargetIdentity follows by eqPartialStep since ρ_k→ 0.


We then recall that a minimizer with a translational symmetry is actually a minimizer without that symmetry in one dimension less. This is known as “cone splitting":


Let u∈^β_(^n+1) be an α-homogeneous minimizer of 𝒥 in ^n+1 which is invariant in the direction e_n. Then  u(x',y):=u(x',0,y) is a minimizer of 𝒥 in one dimension less.
 
 
 The proof is a slight variation of <cit.>.
  


Next we provide a non-standard proof of Proposition <ref>, that is, to show that the trivial solution is a minimizer. We use P<ref> in a sequence of conveniently chosen blow-ups and a dimension reduction argument, based on the following lemma. Note that the proposition could also be proven via a classical dimension reduction argument.


Consider a non-zero minimizer u with non-empty free boundary (see <cit.> for its existence), choose a free boundary point x_0∈ F(u) and consider u_0 to be a blow-up weak limit at this point, which exists and is α-homogeneous by Lemma <ref>. Then u_0  is also  a global minimizer by P<ref> and not null by the nondegeneracy condition.

Next we argue by induction: given 0≤ j≤ n-2 let u_j be an α-homogeneous global minimizer different from 0 such that it is invariant in a j-dimensional linear subspace H_j⊂^n, i.e., for every v∈ H_j and every x'∈^n,

    u_j(x',y)=u(x'+v,y).



Consider a point x_j ∈ F(u_j)∖ (H_j×{0}) which exists as long as j<n-1 by the interior corkscrew condition and positive density, and let u_j+1 be a blow-up limit at this point, which is again an α-homogeneous global minimizer. We claim that u_j+1 is invariant in fact in the (j+1)-dimensional subspace H_j+⟨ x_j' ⟩.

Indeed u_j+1 is invariant in ⟨ x_j'⟩ by Lemma <ref>. On the other hand, since u_j is invariant in H_j, so are the functions in the blow-up sequence and, thus, u_j+1 is invariant in H_j. Thus, for v∈ H_j, v_0∈⟨ x_j'⟩ and x∈^n+1 we get

    u(x+v+x_j')=u(x+v)=u(x),

and the claim follows.


Thus, after n-1 steps, we obtain u_n-1 which is an α-homogeneous global minimizer invariant in an (n-1)-dimensional space H_n-1, with non-empty free boundary. Thus,

    u_n-1(x',0)=C_n,α(x'_n)_+^α,

where the constant is given by P<ref>.
The proposition follows by  Proposition <ref>.
























 §.§ Upper semicontinuity

Next we show that Allen-Weiss' energy at a fixed radius is continuous both with respect to the minimizer and with respect to the point:

Let u_j∈^β_(Ω) be minimizers of 𝒥 in Ω and u_j → u_0 in the sense of Lemma <ref>. Then, for x_j→ x_0 and r<(x_0,∂Ω), 

    Ψ_r^u_j(x_j) Ψ_r^u_0(x_0).



Let ε>0. We want to check that  for j big enough, 

    |  Ψ_r^u_j(x_j) -Ψ_r^u_0(x_0)|≤ε.


We will consider the three terms of the energy separately. 
For the first term, 

    ∫_B_r(x_j)|y|^Β |∇ u_j|^2- ∫_B_r(x_0)|y|^Β |∇ u_0|^2 ≤ r^n ε/3

follows from the L^2 convergence of the gradients. Indeed, if δ_j:=|x_j-x_0|≤δ for j big enough and B_r+δ⊂Ω, then

    ∫_B_r(x_j)|y|^Β |∇ u_j|^2- ∫_B_r(x_0)|y|^Β |∇ u_0|^2 
     	   ≤∫_B_r(x_j)|y|^Β(|∇ u_j|^2- |∇ u_0|^2)+ ∫_B_r(x_j)Δ B_r(x_0)|y|^Β |∇ u_0|^2  
       ≤∫_B_r+δ(x_j)|y|^Β(|∇ u_j|^2- |∇ u_0|^2)+ ∫_(B_r+δ_j∖ B_r-δ_j)(x_0)|y|^Β |∇ u_0|^2  
       ≤ r^n ε/3.


For the measure, we estimate

    |∫_B_r(x_j)'χ_Ω_+(u_j) dm - ∫_B_r(x_0)'χ_Ω_+(u_0)dm | ≤ r^n ε/3

for j big enough as a consequence of χ_Ω_+(u_j)→χ_Ω_+(u_0) in L^1_ as before.
The fact that

    α| ∫_∂ B_r(x_j) u_j^2- ∫_∂ B_r(x_0) u_0^2  | ≤ r^n+1ε/3

for j big enough is a straight consequence of the uniform convergence and the continuity of u_0.




It is well known that the limit of a decreasing sequence of continuous functions is upper semicontinuous (see <cit.>). The monotonicity formula also implies the following result.

Let u_j∈^β_(Ω)  be minimizers of 𝒥 in Ω and u_j u_0 in the sense of Lemma <ref>, with x_j∈ F(u_j) for j∈. Then, if x_j→ x_0 and r_j→ 0,

    lim sup_jΨ_0^u_j(x_j)≤lim sup_jΨ_r_j^u_j(x_j) ≤Ψ_0^u(x_0).



The first inequality comes from monotonicity.

To see that

    lim sup_jΨ_r_j^u_j(x_j) ≤Ψ_0^u_0(x_0),

it is enough to check that for every r>0

    lim sup_jΨ_r_j^u_j(x_j) ≤Ψ_r^u_0(x_0),

or using monotonicity it suffices to show that for every ε>0 and j big enough, 

    Ψ_r^u_j(x_j) - Ψ_r^u_0(x_0) ≤ε.

But this is true for j big enough because the left-hand side converges to 0 by the continuity of the energy from Lemma <ref>.






























































§ MEASURE-THEORETIC PROPERTIES




 §.§ Finite perimeter

We will show that Ω'_+(u) is a set of locally finite perimeter. Then  F_ red(u)  will coincide with the measure-theoretic reduced boundary by the ϵ-regularity theorem, see <cit.>.


For every 0<α<1 we can define
 
    k^*_α:=inf{k∈: ∃}.


Note that, to the best of our knowledge, there is no result showing that k^*_α needs to be finite.








Let u be an α-homogeneous minimizer of 𝒥 in ^n+1 with n < k^*_α. Then u is a rotation of the trivial solution.

See <cit.> for the proof.








From the positive density properties, we know that k^*_α≥ 2.  From the homogeneity of the blow-ups we find out that the free boundary in ^1+1 is in fact  a collection of isolated points. Later in Theorem <ref> we will show that in fact k^*_α≥ 3.


Let u∈^β_(Ω) for Ω⊂^1+1 be a minimizer of 𝒥 in Ω. Then F(u) has no accumulation points in Ω.


Arguing by contradiction, we assume that F(u) has an interior accumulation point which, without loss of generality, we assume to be the origin. 

Let (x_k,0)  be a sequence of singular points converging to 0 with x_k >0. Consider the blow-up rescaling u_k(x):=u(x_k x)/x_k^α. Note that u_k(0,0)=u_k(1,0)=0. Moreover, by the interior corkscrew condition, there exist z_k∈ (1/2, 3/2) such that u_k|_B'_c(z_k, 0)>0, so u(z_k,0)≳ C by the non-degeneracy condition. 

Choosing a subsequence, we may assume that z_k→ z_0≥ 1/2, and u_k→ u_0 in the sense of Lemma <ref>. In particular u_0 is homogeneous by Corollary <ref>, reaching a contradiction with the fact that u_0(1,0)=0 and u_0(z_0,0)≳ C.



We will prove the local finiteness of the perimeter of the free boundary adapting a proof of De Silva and Savin in <cit.>. Our proof is essentially the same, but we repeat it for the sake of completeness.

As in <cit.> we say that a set A⊂^n satisfies the property (P) if following holds: for every x∈ A there exists an r_x>0 such that for every 0<r<r_x, every subset S of B(x,r)∩ A can be covered with a finite number of balls B(x_i,r_i) with x_i∈ S such that 

    ∑_i r_i ≤ r^α/2.




 If  ℋ^t (Σ(U))=0 for some α>0 and for every minimal cone U in ^n+1 then ℋ^t (Σ(u))=0 for every minimizer u of 𝒥 defined on Ω⊂^n+1 



 We first show that Σ(u) satisfies the property (P). If (P) does not hold we find a point y∈Σ(u) for (P) is violated for a sequence r_k→ 0. We consider the blow-up sequence 
 
    u_r_k(x)=r_k^-αu(y+r_k x).

By Corollary <ref> we may assume, by taking  a subsequence,  that  u_r_k converges to a minimal cone U. By our assumptions we may cover Σ(U)∩ B(0,1) with a finite collection of balls {B(x_i,ρ_i/10)}_i=1^k  with 

    ∑_i ρ_i^t≤1/2.


By Lemma <ref> we know that free boundaries converge in Hausdorff sense and thus the set
F(u_r_k)∩ B(0,1)∖⋃_i B(x_i,ρ_i/5) is flat for all large k. From Theorem <ref> we infer that all singularities must be covered by the same balls, that is, for all k≥ k_0

    Σ(u_r_k)∩ B(0,1)⊂⋃_i B(x_i,ρ_i/5).

After rescaling we see that u satisfies the condition for property (P) in the ball B(y,r_k), which is a contradiction. Therefore the property (P) holds as claimed.

Consider the set D_k:={y∈Σ(u): r_y≥1/k }. Fix a point y_0∈ D_k.  By property (P) applied to r_0=1/k we find  a finite cover of D_k∩ B(y_0,r_0) with balls B(y_i,r_i), y_i∈ D_k, satisfying

    ∑_i r_i^t≤ r_0^t/2.

Similarly, for each ball B(y_i,r_i) in the cover we use the property (P) to find a finite number of balls B(y_ij, r_ij), y_ij∈ D_k, which cover D_k∩ B(y_i,r_i) and satisfy

    ∑_j r_ij^t≤ r_i^t/2,

and thus ∑_i,j r_ij≤ r_0^t/4.
By repeating the argument N times we obtain a cover of D_k∩ B(y_0,r_0) by balls B(z_l,r_l) which satisfies

    ∑_l r_l^t ≤ 2^-Nr_0^t.

This implies that ℋ^t(B(y_0,r_0)∩ D_k)=0 and thus ℋ^t(D_k)=0. By countable additivity we obtain the claim.








 If  ℋ^t (Σ(U))=0 for some t>0 and for every minimal cone in ^n+1, we then have that ℋ^t+1 (Σ(V))=0 for every minimal cone V in ^(n+1)+1. 


Without loss of generality we may assume Σ(V)≠{0}. 
Let x∈Σ(V)∖{0}.
By Corollary <ref> the blow-ups at any point of Σ(V)∩∂ B converge to a minimal cone  in dimension (n+1)+1 up to a subsequence. Let V_x be a blow-up at x. By Lemma <ref> V_x is a minimal cone which is invariant in at least one direction. By Lemma <ref>, using our assumption this implies that ℋ^t+1(Σ(V_x))=0,  and thus the singular set of every possible blow-up cone of any minimizer V  has zero ℋ^t+1-measure. 

Arguing as in Lemma <ref> we obtain  ℋ^t+1(Σ(V))=0.















Combining Lemmas <ref>, <ref> and  <ref> we obtain the following corollary. Notice that we will be able to replace n-1 by n-2 by Theorem <ref>.


Every minimizer satisfies 

    ℋ^n-1(Σ(u))=0.







 Let u∈^β (2B) be a minimizer of 𝒥 in 2B with u_Ċ^α(2B)<E_0. Then there exists a constant C depending on n, α and E_0  and a finite collection of balls {B(X_i,r_i)} s.t.
 
    ℋ^n-1 ( (F(u)∩ B)∖⋂_i= 1^m B(X_i,r_i) )≤ C

and
 
    ∑_i= 1^m r_i^n-1≤1/2.



 Proof is by contradiction. For k∈ assume we have u_k_Ċ^α(2B)< E_0 and the left-hand side of (<ref>) is bounded below by k>0 for every collection of balls satisfying (<ref>). By Lemma <ref> we know the sequence u_k is bounded in ^β(B).  
  Taking  a subsequence we may assume that u_k converges locally uniformly to a minimizer  u (see Corollary <ref>).
 
 By Corollary <ref> the set of singularities Σ(u) has ℋ^n-1 -measure zero and thus they can be covered with finitely many balls B_i  satisfying (<ref>). 
 
 Since F(u)∖Σ(u) is a C^1,γ-surface by Theorem <ref>, using the Hausdorff convergence of the free boundaries we apply again Theorem <ref> to see that F(u_k)∩ B_1∖⋃_i=1^M B_i are also  C^1,γ-surfaces converging to F(u)∩ B_1∖⋃_i=1^M B_i uniformly in the  C^1-norm. This is  a contradiction with the assumption that the Hausdorff measure blows up as k goes to ∞. 


The fact that the free boundary has finite perimeter follows now from the same iteration argument as <cit.>. 




 Let u be as in Lemma <ref>.
 Then for some constant C depending only on E_0,
 
    ℋ^n-1( F(u)∩ B)≤ C.




 By Lemma <ref> we find a finite collection of balls B_r_i such that
 
    F(u)∩ B ⊂Γ∪⋃ B_r_i,

with ℋ^n-1(Γ)≤ C and ∑ r_i^n-1≤1/2.

Applying Lemma <ref> again for each ball B_r_i we have

    F(u)∩ B_r_i⊂Γ_i ∪⋃ B_r_ij,

with ℋ^n-1(Γ_i)≤ Cr_i^n-1 and ∑ r_ij^n-1≤1/2r_i^n-1.
Moreover, we have 

    ℋ^n-1((F(u)∩ B_1)∩⋃_i,j B_r_ij)≤ℋ^n-1(Γ)+∑ℋ^n-1(Γ_i)≤ C (1+∑_i,j r_ij^n-1)≤ C(1+1/2).




Continuing inductively, after k steps we have  that

    F(u)∩ B_1 ⊂Γ' ∪⋃_q=1^N B_r_q,

with 

    ℋ^n-1(Γ')≤ C (∑_i=0^k 2^-i)≤ 2C,

and ∑ r_q^n-1≤ 2^-k.
This gives the claim.



Finally the fact that {u>0}∪Ω has locally finite perimeter in Ω follows from the previous lemma and well-known results of Federer, see for example <cit.> or <cit.>. 
























































































































































 §.§ Energy gap

Next we will check that the Allen-Weiss density can also be used to identify singular points. First let us state a useful identity for minimizers (which is also valid in the context of variational solutions in the sense of <cit.>).

Let u∈^β_(Ω) be a minimizer to eqLocalizedFunctional in Ω. For every B⊂⊂Ω we have

    ∫_B  |y|^Β |∇ u|^2=∫_∂ B |y|^Β  u ∇ u ·ν  dℋ^n.




Let u be a minimizer and x_0∈ F(u). If we consider a blow-up u_0 at x_0, then 

    Ψ^u_0(x_0)=Ψ^u_0_1(0)=∫_B_1 |y|^Β |∇ u_0|^2 + m({u_0>0}∩^n∩ B_1)  - α∫_∂ B_1 |y|^Β u_0^2   dℋ^n .

By Lemma <ref> we get

    Ψ^u_0_1(0) =∫_∂ B_1(x_0) |y|^Β  u_0 ∇ u_0 ·ν  dℋ^n+ m({u_0>0}∩^n∩ B_1)  - α∫_∂ B_1 |y|^Β u_0^2   dℋ^n .

Since ∇ u_0(x)  ·ν (x) = α/|x| u_0(x) almost everywhere on the sphere, the first and the third terms cancel out and we obtain

    Ψ^u_0_1(0)= m({u_0>0}∩ B_1').

Thus, the density Ψ^u_0 at a free boundary point is given by the area of the positive phase of any blow-up at the same point.

We write ω_n:=m( B_1') for the volume of the n-dimensional ball. 

Every homogeneous minimizer u∈^β_(^n+1) has density 

    Ψ^u_1(0)= m({u>0}∩ B_1') ≥ω_n/2  ,
 and equality is only attained when u is the trivial minimizer. 






Let u be a minimizer such that Ψ^u_1(0)≤ω_n/2. 

Let x_1∈ F_red(u). Being a regular point, Ψ^u_0(x_1)=ω_n/2. On the other hand, by the homogeneity and the continuity in Lemma <ref>,

    lim _r→∞Ψ^u_r(x_1)=lim _r→∞Ψ^u_1(x_1/r)=Ψ^u_1(0)≤ω_n/2.

Combining both assertions with the monotonicity of Ψ we get that Ψ^u_r(x_1)≡ω_n/2. But using the second formula in Theorem <ref>, one can see that this is  true only whenever Ψ is α-homogeneous with respect to x_1. Thus, u is 1-symmetric and invariant in the direction of ⟨ x_1⟩.

Since Ω_0 is a finite perimeter set (see Section <ref>), F_red(u) has full ℋ^n-1 measure on F(u).  Thus, we can find x_1,… x_n-1∈ F_red(u) linearly independent. By the previous discussion u is invariant on an (n-1)-dimensional affine manifold and, thus, it is the trivial solution.



There exists ϵ>0 depending only on n and α such that every minimizer u∈^β_(Ω) and every singular point  x_0∈Σ(u) satisfy  

    Ψ^u_1(x_0)-ω_n/2≥ϵ.



Assume the conclusion to be false. Then there exist u_j minimizers in B_1 with 

    Ψ^u_j_1(0)≤ω_n/2 + 1/j.
 
Passing to a subsequence, u_j→ u_0 as in Lemma <ref>. Using Lemma <ref> we get that

    Ψ^u_0_1 (0)= lim_j Ψ^u_j_1(0) ≤ω_n/2 .

But then u_0 is the trivial cone by Proposition <ref>. Since F(u_j)→ F(u) in the Hausdorff distance, using ϵ-regularity (see Theorem <ref>) we get that u_j is the trivial cone for j big enough. 

The value ϵ above depends on the constants and on u_Ċ^α in a neighborhood of x_0. In the next section we will show that ϵ does not depend on u at all.





















§ FULL REGULARITY IN ^2+1


In the case of n=2, we prove full regularity of the free boundary for minimizers of our functional. Note that this result does not depend on the previous sections except that we use dimension reduction and blow-ups to deduce regularity of the free boundary.



Let n=2. Then there is no singular minimal cone. In particular, the free boundary F(u) of every minimizer u is C^1,α everywhere.


We follow closely the arguments in <cit.>, building on <cit.>. The case β=0 has been considered in <cit.>. The idea is to construct a competitor by a perturbation argument. We note at this point that the argument is two dimensional in nature and does not generalize to higher dimensions. Recall the functional under consideration:

    𝒥 (u,Ω) = ∫_Ω |y|^β |∇ u|^2 +m({ u>0}∩ℝ^n ∩Ω).

Let V be a non trivial minimal cone. Define, as in <cit.>, the Lipschitz continuous function 

    ψ_R(t)=
    {[               1,      0 ≤ t ≤ R,; 2-ln(t)/ln(R),      R ≤ t ≤ R^2,;                 0,       t ≥ R^2 ] .

Define now the bi-Lipschitz change of coordinates

    Z(x',y)=(x',y)+ψ_R(|(x',y)|)e_1

and set V^+_R(Z)=V(x',y). Clearly, one has 

    D_(x',y) Z=Id+A

where A≤ |ψ'_R(|(x',y)|) | <<1. Defining now V^-_R exactly as V^+_R changing ψ_R into -ψ_R, the very same computation as in <cit.>  gives 


    𝒥(V^+_R, B_R^2) +𝒥(V^-_R, B_R^2) ≤ 2 𝒥(V, B_R^2)+∫_B_R^2 |y|^β  |∇ V|^2 A^2.

Now, we have 

    ∫_B_R^2 |y|^β  |∇ V|^2 A^2= ∫_R^R^2∫_∂ B_r |y|^β  |∇ V|^2 A^2   dℋ^n  dr.

Now since V is homogeneous of degree α by assumption, the function g(x,y)=|y|^β  |∇ V|^2 is homogeneous of degree β+2α-2=-1. Therefore by a trivial change of variables on the sphere of radius r and using the fact that n=2, we get the very same estimate 

    ∫_B_R^2 |y|^β  |∇ U|^2 A^2 ≤C/ln (R) 0.

The rest of the proof follows verbatim <cit.>, page 1318 since this is only based on energy considerations and we refer the reader to it. 














§ UNIFORM BOUNDS AROUND THE FREE BOUNDARY

The optimal regularity bound and the non-degeneracy described in Theorem <ref> were obtained in <cit.> with bounds that depend on the seminorm u_^β(B_1). As a consequence, this dependence propagates to many of our estimates above. 
In this chapter we use the semi-norm dependent estimates (e.g. Lemma <ref>) to prove semi-norm independent non-degeneracy estimates. Re-running the arguments above yields the  semi-norm independent results presented in our main Theorem <ref>. 

The question of semi-norm independence may seem purely technical; however, independence allows the compactness arguments of the next section to work without additional assumptions on the minimizers involved. 









 §.§ Uniform non-degeneracy

We will begin by showing uniform non-degeneracy  from scratch to deduce uniform Hölder character from this fact, reversing the usual arguments in the literature. 

The following lemma was shown in <cit.> in a more general setting. Here we give a more basic approach based on <cit.>. The main difference is that where Alt and Caffarelli could use the energy to directly control the H^1 norm of the minimizer, in our case we need to find an alternative because the measure term of the functional is computed on the thin phase (as opposed to the H^1 norm which is computed on the whole space). To bypass this difficulty we will use Allen's monotonicity formula.

The drawback of our approach is that we need the ball to be centered on the free boundary, while in the original lemma, Alt and Caffarelli could center the ball in the zero phase, allowing for a slightly better result.

Let u be a minimizer in B_r with 0∈ F(u). Then sup_∂ B_r u  ≥ Cr^α with C depending only on n and α.  


By rescaling we can assume that r=1. 

Let ℒu := -∇· (|y|^β∇ u), consider Γ(x)=1/|x|^n-2α which is a solution of ℒΓ=0 away from the origin (or Γ(x)=log |x| if n=1 and α=1/2), and let 

    v(x):=ℓmax{1-Γ(2x),0}/1-Γ(2),

where 

    ℓ:=sup_∂ B_1 u.


It follows that u≤v on ∂ B_1 and thus

    𝒥(u, B_1 )≤𝒥(min{u,v},B_1),


 and observing that ṽ = 0 on B_1/2 and ṽ > 0 on the annulus A:=B_1∖ B_1/2, we get

    ∫_B_1/2|y|^Β |∇ u|^2 + m(B_1/2,+(u))
       ≤∫_A |y|^Β (|∇ (min{u,v})|^2-|∇ u|^2) + m(A_+'(min{u,v})) - m(A_+'(u)) 
       ≤ -2∫_A |y|^β∇max{u-v,0}·∇v.

By Green's theorem,  writing dσ=|y|^β dℋ^n we get

    ∫_B_1/2|y|^Β |∇ u|^2 + m(B_1/2,+(u))
     ≤ -2∫_∂ B_1/2  u ∂_νv  dσ 
    	 = C_n, αℓ∫_∂B_1/2 u  dσ,

with C_n,α > 0. 




Using the monotonicity formula and  Proposition <ref>, we get that ψ^u(r) ≥ψ^u(0)≥ω(B_1)/2 and, therefore

    α/r∫_∂B_r u^2  dσ + ω(B_1) r^d/2≤𝒥_r(u),

so using Hölder's inequality and the AM-GM inequality we obtain 

    ∫_∂B_1/2 u  dσ≤(∫_∂B_1/2 u^2  dσ)^1/2 C_n,α^1/2≤1/2∫_∂B_1/2 u^2  dσ + 1/2 C_n,α≤ C_n,α𝒥_1/2(u).


Combining  eqBoundByEll, eqControlJBelow and eqControlByMoreThanHolder we obtain

    0 < 𝒥_1/2(u)≤ C_n, αℓ𝒥_1/2(u),

and therefore ℓ≥ C_n,α^-1.




To show averaged non-degeneracy we need a mean value principle which is well-known, but we include its proof for the sake of completeness.


Let u∈ H^1(β,Ω) be a  weak solution to ℒu:=∇· (|y|^β∇ u)=0 in Ω, and let x_0∈^n ×{0} with B_r(x_0) ⊂Ω. Then 

    u(x_0)=_B_r  u   dω


where the mean is taken with respect to the measure dω:= |y|^β  dx.



Changing variables, we have that

    A(ρ):=1/ρ^β+n+1∫_B_ρ(x_0) |y|^β u(x) dx=∫_B_1 |y|^β u(ρ x+x_0) dx.


On the other hand, set

    A(ρ)	
    		    := ∫_B_1 |y|^β∇ u(ρ x + x_0)· x  dx
        = ∫_B_ρ(x_0)(|y|/ρ)^β∇ u(x)· (x-x_0)/ρdx/ρ^n+1
    		= 1/2 ρ^β+n+2∫_B_ρ(x_0) |y|^β∇ u(x)·∇ |x-x_0|^2   dx.

Since u is a weak solution to ∇· (|y|^β∇ u)=0 in Ω, we can apply Green's formula twice to obtain

    A(ρ)
    	   = 1/2 ρ^β+n+2∫_∂ B_ρ(x_0)  |x-x_0|^2 |y|^β∇ u(x)·ν  dx = 1/2 ρ^β+n∫_∂ B_ρ(x_0)  |y|^β∇ u(x) ·ν  dx=0.

Since u is absolutely continuous on lines (see <cit.>), for almost every x we have ∫_ρ^r ∇ u(t x + x_0)· x   dt=u(rx+x_0)-u(ρ x+x_0). Applying Fubini's Theorem we get

    ∫_ρ^r A(t)   dt = ∫_B_1|y|^β∫_ρ^r ∇ u(t x + x_0)· x   dt  dx = ∫_B_1|y|^β  (u(rx+x_0)-u(ρ x+x_0)) dx = A(r)-A(ρ) .

So
A(r)-A(ρ)= 0 for all ρ<r.

On the other hand, taking the mean with respect to the measure dω:= |y|^β  dx and using the continuity of u (see <cit.>)
we obtain 

    |u(x_0) -1/ω(B_1)lim_ρ→ 0 A(ρ) | =  lim_ρ→ 01/ω(B_ρ(x_0))|∫_B_ρ(x_0) (u(x_0) -u(x))   dω(x)| ≤lim_ρ→ 0 o_ρ→ 0 (1)=0.






Let u be a minimizer in B_r with 0∈ F(u) and let dσ=|y|^β dℋ^n. Then _∂ B_r u   dσ≥ Cr^α with C depending only on n and α. 


Let v be the ℒ-harmonic replacement of u in B_r, that is, the solution to

    ℒv=0     B_r, 
    
    v≡ u    ∂ B_r,

see <cit.>. After differentiating with respect to the radius, by the mean value principle we get that v(0)=_∂ B_r u   dσ. By the comparison principle and the Harnack inequality we get that 

    C r^α≤sup_B_r/2 u ≤sup_B_r/2 v ≤ C _∂ B_r u   dσ.


















 §.§ Behavior of the distributional fractional Laplacian








Next we use an idea of <cit.> and investigate the behavior of the distributional α-Laplacian of the minimizer introduced in Section <ref>. As mentioned in the introduction, in <cit.> this investigation immediately yields that the positivity set is a set of locally finite perimeter, and more precisely, that it is Ahlfors regular of the correct dimension. However, the nonlocal nature of this problem indicates that the distributional fractional Laplacian may not be supported on the free boundary and thus we cannot expect to immediately gain such strong geometric information. 



























First we can bound the growth of the fractional Laplacian measure around a free boundary point. Note that this growth is the natural counterpart to the upper Ahlfors regularity in the case of Alt-Caffarelli minimizers.

Let u ∈^β(B_2r(x_0)) be a minimizer of 𝒥 in B_2r(x_0), and let x_0∈ F(u). 
Then, we have

    λ(B_r(x_0)) ≤ C r^n-α.

 In particular λ(F(u))=0. 

A glance at (<ref>) will convince the reader that these estimates are sharp, for they cannot be  improved even in the case of the trivial solution. 


Without loss of generality we may assume that x_0=0.  Let ℒu := -∇· (|y|^β∇ u) and let v be the ℒ-harmonic replacement of u in B_2r, see eqHarmonicReplacement. Write   dσ=|y|^β dℋ^n and M:=_∂ B_2r u   dσ. By Harnack's inequality (see <cit.>, for instance) and the mean value principle in Lemma <ref>,

    inf_B_r v≥ C v(0)=CM.

We have that

    λ(B_r)=∫_B_rdλ≤1/CM∫_B_rv dλ.

Since u≡ 0 in the support of  λ and u is ℒ-subharmonic  (see <cit.>) we get

    ∫_B_rv dλ=∫_B_r(v-u)dλ≤∫_B_2r(v-u)dλ .

By the properties of the measure λ, we obtain

    ∫_B_2r(v-u)dλ= -∫_B_2r|y|^β∇ (v-u)·∇ u =  ∫_B_2r|y|^β(|∇ u|^2-|∇ v|^2),

and using the definition of the functional and the fact that u is a minimizer, we get

    ∫_B_2r|y|^β(|∇ u|^2-|∇ v|^2) = 𝒥(u,B_2r)-m(B_2r^+(u)) -𝒥(v,B_2r)+m(B_2r') ≤ C r^n.

All together, we have that

    λ(B_r)≤1/CMCr^n,

and, since uniform non-degeneracy (see Lemma <ref>) implies that M≥ Cr^α we can conclude the proof of the first statement.

To show the second one, note that since the free boundary has locally finite (n-1)-dimensional Hausdorff measure, given a set E⊂ F(u) and k∈ we can find a collection of balls I_k={B^k_i}_i such that 

    E⊂⋃_B∈ I_k B,     sup_B∈ I_kr(B)≤ 1/k         ∑_B∈ I_k r(B)^n-1≤ 2ℋ^n-1(E).

Thus, 

    λ(E)≤∑_B∈ I_kλ(B)≲∑_B∈ I_k r(B)^n-α≤sup_B∈ I_kr(B)^1-α∑_B∈ I_k r(B)^n-1 0.











 Next we study the measure away from the free boundary. We should emphasize here that even though the estimates in Lemma <ref> and Theorem <ref> depend on E_0, they will be used to remove the dependence of our other estimates on E_0. More precisely, Theorem <ref> will play a role in establishing the continuity of the Green function in Lemma <ref>.  This qualitative fact is used to prove  the quantitative uniform Hölder character in Theorem <ref>. 

After proving Theorem <ref>, we may drop the hypothesis u_^β(B_2)≤ E_0  from both Lemma <ref> and Theorem <ref>.



If u ∈^β_(B_2) is a minimizer of 𝒥 in the ball B_2 with u_^β(B_2)≤ E_0 and 0∈ F(u), then for every x_0=(x',0)∈ B_1,0(u) we get

    lim_y→ 0 |y|^β |u_y(x',y)| ≈ C (x_0,F(u))^-α.

Moreover, for every ball B centered at ^n×{0} with B'⊂⊂ B_1,0(u), we have that

    |y|^β |u_y(x',y)| ≤ C (x,F(u))^-α,

for |y|<C_B(x,F(u)), where the constant C_B may depend on B.


Let u be a minimizer, and let B:=B_r(x_0) with B'⊂ B_1,0(u). 
 By <cit.>, we can write u(x',y)=|y|^1-βg(x') +𝒪(y^2), where g is a C^1+β(1/2B') function, with a uniform control on the error term in terms of  u_L^2(B,|y|^β). In particular, lim_y→ 0 |y|^β-1 u(x',y)=g(x').

Let us define 

    u(x',y):=
    u(x',y)    y≥ 0
    
    -u(x',-y)    y< 0.

It is clear that ℒu≡ 0 in B. According to  <cit.> v(x',y)=|y|^β y^-1u(x',y) is an even C^∞(1/2B) function in ^2-β(B) (note that 1<2-β<3 is out of the usual range of β) and satisfying ∇·(|y|^2-β∇ v)=0. The mean value principle (see Lemma <ref>) applies also to this case, so

    g(x_0')=v(x_0)= 1/∫_1/2B|y|^2-β∫_1/2B|y|^2-β v(x)  = C 1/r^2-β+n+1∫_1/2B|y| u(x),

and using P1-P3 , if r=(x_0,F(u)) we get

    g(x_0')=v(x_0)≈ C r^β-2+1+α=Cr^-α.


On the other hand, on the upper half plane we have u_y=(y^1-βv)_y=(1-β) y^-β v + y^1-βv_y, so

    y^β u_y(x',y)=(1-β) v(x',y) + y v_y(x',y),

and 

    lim_y→ 0^+ y^β u_y(x',y)  = (1-β) g(x')≈ r^-α,

the limit being uniform on compact subsets of B.





If u ∈^β_(B_2) is a minimizer of 𝒥 in the ball B_2 with u_^β(B_2)≤ E_0,

then the measure λ is absolutely continuous with respect to the Lebesgue measure, and for m-almost every x∈ B_1'(u) we have that

    dλ/dm(x)=2lim_y→0|y|^β u_y(x',y)≈χ_B_1,0(u)(x) (x,F(u))^-α,

with constants depending on n, α and E_0.








By Theorem <ref> we only need to show absolute continuity in  B_1,0(u)∪ B_1,+'(u).
For x=(x',0)∈ B_1,+'(u) by <cit.> we have that

    lim_y→ 0|y|^β u_y(x',y)=0,

and, for x∈ B_1,0(u) we have seen in Lemma <ref> that

    lim_y→ 0|y|^β u_y(x',y)≈(x,F(u))^-α,

showing the second part of the statement.


















Consider a ball B_r(x_0) with x_0∈^n×{0} and a collection of even smooth functions χ_B_r≤ψ_k≤χ_B_r+1/k. Then

    λ(B_r)≤ -∫ |y|^β∇ u·∇ψ_k≤λ(B_r+1/k),

and for every ε>0 we use the Green's theorem to get

    -∫ |y|^β∇ u·∇ψ_k= -∫_|y|≤ε |y|^β∇ u·∇ψ_k - ∫_|y|=ε |y|^βψ_k ∇ u ·ν  dm.

Using the symmetry properties and taking limits,

    -∫ |y|^β∇ u·∇ψ_k =2 lim_ε→ 0∫ε^βψ_k(x',ε)  u_y (x',ε)  dm(x').


Next we want to apply the dominated convergence theorem. 








Let us begin by considering a ball B_r(x_0)⊂ B_1 centered in the zero phase, with (B'_r(x_0),F(u))≥ 2r. In this case, by Lemma <ref> we have

    ε^β   u_y(x',ε)≲  r^-α,

with constants depending perhaps on u and B_r as well.

If instead B'_r(x_0)⊂⊂ B_1,+'(u), by <cit.> u is an even C^∞ function on B'_r(x_0), so |y|^β u_y=𝒪(|y|^1+β). Thus

    ε^β   u_y(x',ε)≲  r^2-2α .


In both cases, the dominated convergence theorem applies and

    lim_ε→ 0∫_B_r+1/k∩{y=ε}ε^βψ_k  u_y   dm= ∫_B_r+1/k'ψ_k lim_ε→ 0 (ε^β  u_y)  dm,

and by eqSandwich and eqControlByUy we obtain

    λ(B_r)≤ 2 ∫_B_r+1/k'ψ_k lim_ε→ 0 (ε^β  u_y)   dm≤λ(B_r+1/k).

In particular lim_ε→ 0 (ε^β  u_y)∈ L^1_(B_1,0(u)∪ B_1,+'(u))  and taking limits in k we get

    λ(B_r)= 2 ∫_B_r'lim_ε→ 0 (ε^β  u_y)   dm.







A consequence of our control of the behavior of λ is that we can establish the existence of exterior corkscrews. We should note that exterior corkscrews can be also obtained by a purely geometric argument given the non-degeneracy and positive density of Theorem <ref> (see, e.g. the proof of Proposition 10.3 in <cit.>). 

If u∈ is a minimizer in B_2 with u_^β(B_2)≤ E_0, then B_1,+'(u) satisfies the exterior corkscrew condition, i.e. there exists a constant C_1 such that for every x∈ F(u) and every 0<r<(x,∂ B_1) one can find x_0∈ B_r(x) so that

    B(x_0,C_1r)∩ B_1,+' (u)= ∅.





This is a consequence of Theorems <ref> and <ref>, and the positive density condition for the zero phase. Indeed, given a ball B_r⊂^n+1, combining both theorems we get

    r^n-α   ≳λ(B_1,0(u)∩ B_r) ≥ C_E_0∫_B_1,0(u)∩ B_r(x,∂ B_1)^-α
       ≥ C (sup_B_1,0(u)∩ B_r(x,∂ B_1))^-α |B_1,0(u)∩ B_r |,

and the positive density condition implies that

    |B_1,0(u)∩ B_r | ≥ C_E_0 r^n  .

Thus, 
    sup_B_1,0(u)∩ B_r(x,∂ B_1)≥ C_E_0r,

which is equivalent to the exterior corkscrew condition.














 §.§ Uniform Hölder character


The uniform non-degeneracy of Section <ref> lets us conclude uniform control on the Hölder norm of u. 


Let u be a minimizer of 𝒥 in B_r with 0∈ F(u). Then |u(x)|≤ C|x|^α for every x∈∂ B_r/2 with C depending only on n and α.


Again we set v to be the ℒ-harmonic replacement of u inside of B_r as in eqHarmonicReplacement. Let u:=v- u, so that

    ℒu = ℒv - ℒu = -λ = - ∇· (|y|^β∇ u),

and u∈ H^1,2_0(B_r; |y|^β). 

Consider the Green function G: B_r× B_r→ such that ℒ G(· , z) = δ_z, and G(· , z)∈ H^1,2_(B_r∖{z}) with null trace on ∂ B_r

(see <cit.>).

By <cit.> there exists p_0>1 so that u is the unique function in H^1,p_0_0(B_r;|y|^β) such that ℒu=λ, and moreover

    u(z)=∫_B_r G(z,x)  dλ(x),

for almost every z∈ B_r. 



Below, in Lemma <ref>, we will see that the equality eqGrennRules is in fact valid for every z∈ B_r/4, that is, 
u=∫_B_r G(·,x)  dλ(x). In particular

    v(0)=u(0)=∫_B_r G(0,x)  dλ(x).






































Next we use the following estimate (see <cit.>): let z,x∈ B_r/4. Then

    G(z,x)≈∫_|x-z|^r s   ds/w(B(x,s)),

where w is the A_2 weight w(x)=|y|^β. Computing, for x=(x',y) we obtain

    w(B(x,s))≈ s^n ∫_y-s^y+s |t|^β  dt ≈ s^n max{|y|,s}^β+1.

First we assume that n-2α>0.
Thus, if x∈ B_r/4' then

    G(z,x) ≈∫_|x-z|^r s^-n-β  ds ≈ |x-z|^-n-β+1 = |x-z|^2α-n.




 Note that λ(B_r) ≤ Cr^n-α by Theorem <ref>. Thus, writing A_t,s:=B_s∖ B_t, we have that
 
    v(0)= ∫_B_r G(0,x)  dλ(x) ≤∫_cr^2α-n^∞λ({x∈ B_r/4: G(0,x) > t}) dt + ∫_A_r/4,r G(0,x)  dλ(x).

By the strong maximum principle, the Green function in the annulus is bounded by C r^n-2α. This fact, together with Theorem <ref>, implies that
 
    v(0) ≤∫_cr^2α-n^∞λ(B_Ct^-1/(n-2α)) dt +Cr^α≤ C ∫_cr^2α-n^∞ t^-n-α/n-2α dt +Cr^α = Cr^α.

 By the mean value theorem we conclude that

    _∂ B_r v   dσ≤ C r^α,
 where dσ=|y|^β dℋ^n. 
The theorem follows by observing that, as in eqComparisonAndHarnack, the mean of v dominates u by sup_∂ B_r/2 u ≤sup_∂ B_r/2 v ≤ C _∂ B_r v  dσ.

In case n-2α=0, which could only happen for n=1 and α=1/2, estimate eqGreenEstimate reads as

    G(z,x) ≈log(r/|x-z|),

and the proof follows the same steps.

In case n-2α<0, then  estimate eqGreenEstimate reads as

    G(z,x) ≈  r^n-2α,

and the estimate is even better compared to the above.







∫_B_r G(z,x)  dλ(x) is continuous in z∈ B_r/4.


Let ε<r/2 and let z_1, z_2∈ B_r/4,  with |z_1-z_2|≤ε/2. Then 

    ∫_B_r|G(z_1,x)-G(z_2,x)|  dλ(x)
       ≤∫_B_r∖ B_ε(z_1)|G(z_1,x)-G(z_2,x)|  dλ(x) 
          + ∫_B_ε(z_1)G(z_1,x)dλ(x) + ∫_B_ε(z_1)G(z_2,x)dλ(x).


Next we use eqGreenEstimate and Theorems <ref> and <ref>. By decomposing the domain on dyadic annuli, in case n-2α>0 we get

    ∫_B_ε(z_1)G(z_1,x)dλ(x)
       ≤∑_j≤0∫_A_2^j-1ε,2^jε(z_1)G(z_1,x)dλ(x)
    		≲∑_j≤0λ(B_2^jε(z_1))(2^j-1ε)^2α-n≲ε^α∑_j≤0 2^jα.

In case n-2α=0 we obtain ε^α∑_j≤0 2^jαlog(r/2^jε) on the right-hand side instead, and in case n-2α<0 we obtain ε^n-αr^2α-n∑_j≤0 2^j(n-α). In every case, fixing ε small enough this term can be as small as wanted. The same will happen with the last term on the right-hand side of eqBreakGMinusG.

On the other hand, by <cit.> Green's function is uniformly continuous on the set {(z,x)∈ B_r× B_r: |z-x|>ε} so |G(z_1,x_1)-G(z_2,x_2)|≤δ_ε(|z_1-z_2|+|x_1-x_2|) with δ_ε(t) 0. Thus, 

    ∫_B_r∖ B_ε(z_1)|G(z_1,x)-G(z_2,x)|  dλ(x) ≤δ_ε(|z_1-z_2|)λ(B_r)→ 0 .

Assuming that |z_1-z_2| is small enough, we obtain that ∫_B_r|G(z_1,x)-G(z_2,x)|  dλ(x) is as small as wanted and the claim follows.






































 In light of Theorem <ref>, arguing as in <cit.> we obtain that every minimizer u in a ball B_r with 0∈ F(u) has uniform C^α character in B_r/2. By the Caccioppoli inequality (see Section <ref>) we also obtain the same for the ^β norm. Moreover, using <cit.> we can find interior corkscrew points with constants not depending on these norms. This allows us to remove the a priori dependence on u_^β from all of our results above. 









 §.§ Lower estimates for the distributional fractional Laplacian

Next we bound the growth of the measure around a free boundary point from below. None of these results will be used in the present paper, but we include them to give a complete picture of the tools under consideration.

Let u ∈^β(B_2r) be a minimizer of 𝒥 in B_2r such that 0∈ F(u). Then we have

    λ(B_r) ≥ C r^n-α.




Let ℒu := -∇· (|y|^β∇ u) and let v be the ℒ-harmonic replacement of u in B_r (see eqHarmonicReplacement). Let u:=v- u and
consider the Green function G: B_r× B_r→  as in the proof of Theorem <ref>.







Let 0<κ<1 to be fixed later. By P1-P3 in Theorem <ref> there exists a point z_0∈ B_κ r with 

    u(z_0)≈ (κ r)^α,

with constants depending only on n and α by Remark <ref>.
By P1 there is a constant c such that for every z∈ B(z_0,cκ r)  we have that u(z)≈ (κ r)^α. Since λ is supported on the zero phase of u, the ball B(z_0,cκ r) is away from its support, and

    u(z)=∫_B_r∖ B(z_0,cκ r)  G(z,x)  dλ(x).


Using  the strong maximum principle (see <cit.>) and eqGreenEstimate, for almost every z∈ B(z_0,cκ r/2) we get

    u(z)
    	   ≤λ(B_r) sup_x∈ B_r∖ B(z_0,cκ r)  G(z,x) = λ(B_r) sup_x∈ B_r/4∖ B(z_0,cκ r)  G(z,x) 
       ≈λ(B_r)sup_x∉ B(z_0,cκ r)   |x-z|^2α-n =λ(B_r) (c κ r)^2α-n.













That is,

    u(z)
    	 ≲λ(B_r)  (cκ r)^2α-n.



On the other hand, note that u is continuous. By the Riesz representation theorem, there exists a probability measure ω^z_ℒ such that

    v(z)= ∫_∂ B_r u(x) dω^z_ℒ(x).


We can choose r so that ∂ B_r intersects a big part of a corkscrew ball, i.e., assume that there exists a point ξ_0∈∂ B_r' which is the center of a ball B'(ξ_0, cr) where u has positive values. This can be done by the interior corkscrew condition, with all the constants involved depending only  on n and α. Then, changing the constant if necessary, all points  ξ∈ B(ξ_0, cr) satisfy that u(ξ)≥ C r^α by the non-degeneracy condition and the optimal regularity. Call U:=∂ B_r∩ B(ξ_0, cr). Then

    v(z)≳ r^αω^z_ℒ(U).

But ω^z_ℒ(U) is bounded below by a constant by <cit.> and the Harnack inequality (use a convenient Harnack chain). All in all, we have that

    v(z)≳ r^α.











 Combining eqTildeU, eqU and eqV and choosing κ small enough, depending in n and α, we get

    λ(B_r) ≳u(z_0)/(c κ r)^2α-n≥ Cr^α - C' (κ r)^α/(c κ r)^2α-n≥ C_n,α  r^n-α ,

for κ small enough.

In case n-2α=0, that is for n=1 and α=1/2, using similar changes as in the proof of Theorem <ref>  we get  u(z)≲λ(B_r)sup_x∉ B(z_0,cκ r)log(r/|x-z|) ≈λ(B_r)|logκ| instead of eqTildeU. In case n-2α<0, the proof is even easier than before.




Theorem <ref> implies that the (n-α)-Hausdorff measure of the free boundary is locally finite. This does not suffice to show finite perimeter of the positive phase and, therefore, we had to use the approach in Section <ref>.
















The following theorem summarizes the information that we have gathered so far about the measure λ.


If u ∈^β_(Ω) is a  minimizer of 𝒥 in Ω, then the measure λ is absolutely continuous with respect to the Lebesgue measure in Ω'(u). Moreover, given x_0∈ F(u) and r>0 such that B_2r(x_0)⊂Ω, then

    λ(B_r(x_0)) ≈ r^n-α,




and for almost every x∈ B_r'(x_0) we have that

    dλ/dm(x)=2lim_y→0|y|^β u_y(x',y)≈χ_Ω_0(u)(x) (x,F(u))^-α,

with constants depending only on n and α.














































































§ RECTIFIABILITY OF THE SINGULAR SET


In this section we use the Rectifiable-Reifenberg and quantitative stratification framework of Naber-Valtorta <cit.> to prove Hausdorff measure and structure results for the singular set. Recall that k^*_α is the first dimension in which there exists non-trivial α-homogeneous global minimizers to (<ref>) defined in Section <ref>. 


Let u∈^β_(Ω) be a minimizer of eqLocalizedFunctional in a domain Ω. Then Σ(u) is (n-k^*_α)-rectifiable and for every D⊂⊂Ω, we have

    ℋ^n-k^*_α(Σ(u)∩ D)≤ C_n,α, (D,∂Ω).



Part of the power of this framework is that it is very general. One needs certain compactness properties on the minimizers and a connection between the drop in the monotonicity formula and the local flatness of the singular set (see Theorem <ref> below). To avoid redundancy and highlight the original contributions of this article, we omit many details here and try to focus on the estimates needed to apply this framework to minimizers of (<ref>). Whenever we omit details we will refer the interested reader to the relevant parts of <cit.>. 

The key first step is to introduce the appropriate formulation of quantitative stratification. First introduced by Cheeger and Naber <cit.> in the context of manifolds with Ricci curvature bounded from below, this is a way to quantify the intuitive fact that F(u) should “look" (n-k^*_α)-dimensional near a point x_0 ∈ F(u) at which the blow-ups have (n-k^*_α)-linearly independent translational symmetries. 



 §.§ Quantitative stratification for minimizers to 𝒥


We have seen in Section <ref> that homogeneous functions have linear spaces of translational symmetry. Here we want to quantify (both in terms of size and stability) how far a function is from having no more than k directions of translational symmetry. 


We write V^k for the collection of linear k-dimensional subspaces of ^n. A function u is said to be k-symmetric if it is α-homogeneous with respect to some point, and there exists a  L∈ V^k so that 

    u(x+v)=u(x),      v∈ L.

A function u is said to be (k,ϵ)-symmetric in a ball B if for some k-symmetric u we have

    r(B)^-2-n∫_B |y|^Β |u-u|^2 dy <ϵ.




Next we define the k-stratum S^k(u), the (k,ϵ)-stratum S^k_ϵ(u) and the (k,ϵ,r)-stratum S^k_ϵ,r(u). A key insight here is to define these strata by the blow-ups having k or fewer symmetries as opposed to exactly k symmetries. 

Let 0≤ k≤ n,  0<ε<∞ and 0<r<(x,∂Ω), let u be a continuous function in Ω and let x∈ F(u). We say that:


  * x∈ S^k(u) if u has no  (k+1)-symmetric blow-ups at x.

  * x∈ S^k_ϵ(u) if u is not  (k+1,ϵ)-symmetric in B_s(x) for every 0<s≤min{1,(x,∂Ω)}.

  * x∈ S^k_ϵ,r(u) if u is not  (k+1,ϵ)-symmetric in B_s(x) for every r≤ s≤min{1,(x,∂Ω)}.

If it is clear from the context we will omit u from the notation.


We now detail some standard properties of the strata defined above and how they interact with the free boundary F(u). While the proofs are mostly standard, we give the details as the scaling associated to the problem (<ref>) adds some technical difficulties. This proof also provides a blueprint for fleshing out the details in Sections <ref> and <ref>.


Let 0≤ j ≤  k≤ n,  0< ε≤τ<∞, 0<r≤ s<(x,∂Ω), and let u∈^β_(Ω) be a minimizer in Ω. Then:


  * S^0⊂ S^1⊂⋯⊂ S^n-1 = S^n= F(u). Moreover, for the reduced boundary, we have that
F_red(u) ⊂ S^n-1∖ S^n-2 and Σ (u) ⊂ S^n-k^*_α.

  * We have S^j_τ⊂ S^k_ϵ⊂ S^k  and, moreover,  S^k=⋃_ϵ>0 S^k_ϵ. 

  * Also S^j_τ⊂ S^j_τ, r⊂ S^k_ϵ, s and, moreover,  S^k_ϵ= ⋂_r>0 S^k_ϵ, r.

  * The sets S^k_ϵ are closed, in both x and u: if u_i u and x_i→ x with x_i∈  S^k_ϵ(u_i), then x∈ S^k_ϵ(u).

  * If u_i  u,  ϵ_i→ 0, and u_i are (k,ϵ_i)-symmetric in B_1, then u is k-symmetric in B_1.



1. The inclusions S^k⊂ S^k+1 of the first property are trivial. The last equalities are consequences of the non-degeneracy. The fact that F_red(u) ∩ S^n-2 =∅ can be deduced from the Hausdorff convergence of the free boundaries described in Lemma <ref> and Theorem <ref>. Finally, Σ (u) ⊂ S^n-k^*_α is a consequence of Lemmas <ref> and <ref>.

2. The inclusions S^j_τ⊂ S^k_ϵ of the second property come from the definitions: if x∉ S^k_ϵ then there exist a ball B⊂Ω centered at x and a (k+1)-symmetric u so that r(B)^-2-n∫_B |y|^Β |u-u|^2 dy <ϵ≤τ. But  u is also (j+1)-symmetric. Thus, x∉ S^j_τ. 

The fact that S^k_ϵ⊂ S^k is a consequence of the uniform convergence on Lemma <ref>: if x∉ S^k, then u has a  (k+1)-symmetric blow-up sequence u_i→ u_0 at x converging uniformly. Thus,

    ∫_B_ρ_i(x) |y|^Β|u(x)- ρ_i^α u_0(x-x_0/ρ_i)|^2 dx 
     	    =  ρ_i^Β+2α+n+1∫_B_1(x) |y|^Β|u(x_0+ ρ_i x)/ρ_i^α-u_0(x)|^2 dx
       ≤ρ_i^n+2ω(B_1) u_i-u_0_L^∞.

That is, 

    ρ_i^-n-2∫_B_ρ_i(x) |y|^Β|u(x)- r_i^α u_0(x-x_0/ρ_i)|^2 dx  0,

and therefore, for every ε there exists a ball small enough so that u is (k+1,ε)-symmetric in it. In particular S^k ⊃⋃_ϵ>0 S^k_ϵ. 

To see the converse, assume that x∉⋃_ϵ S^k_ϵ. Then for every i∈ there exist a (k+1)-symmetric function u_i, invariant with respect to  L_i∈ V^k+1  and r_i<min{1,(x,∂Ω)} such that 

    1/r_i^n+2∫_B_r_i |y|^Β| u(x)-u_i(x)|^2   dx < 1/i.

In the case when  r_i stays away from zero, since r_i<1, we can take a subsequence converging to r_0∈(0,1), and one can see that u is (k+1)-symmetric in the ball B_r_0(x_0). Otherwise, 
consider u_i:=u(x_0+ r_i x)/r_i^α and u_i,i=u_i(x_0+ r_i x)/r_i^α. Taking subsequences, we can assume that L_i→ L_0 locally in the Hausdorff distance, and that u_i → u_0 locally uniformly. One can check also using the Hölder character of u that {u_i,i} is uniformly bounded in L^2( B; |y|^Β), so taking subsequences again, we can assume the existence of u_0 so that
 u_i,i→u_0 in L^2( B; |y|^Β). This function will be (k+1)-symmetric, being invariant in the directions of L_0. By the triangle inequality we get

    ∫_B_1 |y|^Β |u_0- u_0|^2   dx ≲∫_B_1 |y|^Β |u_0- u_i|^2   dx +∫_B_1 |y|^Β |u_i- u_i,i|^2   dx +∫_B_1 |y|^Β |u_i,i- u_0|^2   dx .

The first and the last integrals converge to zero by our choice of the subsequence. For the middle term just change variables as before:

    ∫_B_1 |y|^Β |u_i- u_i,i|^2   dx = 1/r_i^n+2∫_B_r_i |y|^Β| u(x)-u_i(x)|^2   dx→ 0.

Thus we have that u_0=u_0 and, therefore, x∉ S_k. 


3. The inclusions S^j_τ⊂ S^j_τ, r⊂ S^k_ϵ, s of the third property come from the definitions and thus, S^k_ϵ⊂⋂_r>0 S^k_ϵ, r. The converse implication is also trivial.

4. The closedness is obtained by a contradiction argument again. It is straightforward but we write it here for the sake of completeness.
 
Assume by contradiction that x∉ S^k_ϵ(u). Then there exist a (k+1)-symmetric function u and a radius r such that

    ϵ_0:=1/r^n+2∫_B_r(x) |y|^Β| u(x)-u(x)|^2   dx <ϵ.

Let τ<1 to be fixed and consider  i_0∈ so that B_τ r(x_i)⊂ B_r(x) for every i≥ i_0. By the triangle inequality

    1/(τ r)^n+2∫_B_τ r(x_i) |y|^Β| u_i(x)-u(x)|^2   dx ≤1/(τ r)^n+2u_i-u _L^2(B_τ r(x_i);|y|^Β)^2 + ϵ_0/τ^n+2.

We define τ so that ϵ_0/τ^n+2=ϵ+ϵ_0/2. Choose  i_0 big enough so that every i≥ i_0 satisfies that u_i-u _L^2(B_τ r(x_i);|y|^Β)^2 < (τ r)^n+2ϵ-ϵ_0/2. Then x_i∉ S^k_ϵ(u_i), contradicting the hypothesis. 

5. Assume that  u_i is invariant with respect to  L_i∈ V^k+1 and

    ∫|y|^Β |u_i-u_i|^2≤ϵ_i.
 Consider a subsequence {u_i} so that the varieties L_i→ L locally in the Hausdorff distance. Using the triangle inequality as in 4 it follows that u is (k,δ_i)-symmetric with δ_i→ 0. 




There exists ϵ(n,α) > 0 such that if u∈^β_(Ω) is a minimizer of 𝒥 in a domain Ω⊂^n+1, then Σ(u)⊂ S^n-k^*_α_ϵ(u). 


It is enough to show that if u is a minimizer of 𝒥 in B_2(0), then Σ(u)∩ B_1(0)⊂ S^n-k^*_α_ϵ(u). 

By contradiction, let us assume that there is a sequence of positive numbers ϵ_i 0, functions u_i minimizing 𝒥 in B_2(0) and x_i∈Σ (u_i)∩ B_1(0), r_i ∈ (0,1], with u_i being (n-k^*_α+1, ϵ_i)-symmetric  in B_r_i(x_i), and let L_i be an (n-k^*_α+1)-dimensional subspace that leaves invariant one of the admissible (n-k^*_α+1)-symmetric approximants. By rescaling we can assume that r_i=1. 

Passing to a subsequence we can assume that L_i → L_0∈ V^n-k^*_α+1 locally in the Hausdorff distance and x_i→ x_0. By the compactness results in Lemma <ref> we have a uniform limit u_0 which is a minimizer as well, and it is (n-k^*_α+1)-symmetric with invariant manifold L_0. By Lemma <ref> any blow-up u_0,0 at x_0 will be (n-k^*_α+1)-symmetric as well. Applying Lemma <ref> (n-k^*_α+1) times we find that the restriction of u_0,0 to the orthogonal manifold L_0^⊥ is a (k^*_α-1)-dimensional minimal cone which, by Lemma <ref> is the trivial solution, and so is u_0,0. Thus, x_0 is a regular point for u_0.

On the other hand,  the Hausdorff convergence of Lemma <ref> together with the improvement of flatness of Theorem <ref> imply that for i big enough x_i∈ F_ red(u_i), reaching a contradiction.





 §.§ The Refined Covering Theorem


Our estimates on the size and structure of the singular set Σ(u)  come from similar results concerning the S^k_ϵ(u). In particular, we prove the following covering result:


Let u∈^β(B_5) be a  minimizer to eqLocalizedFunctional in B_5  with   0∈ F(u). For given real numbers ϵ>0, 0<r≤ 1 and every natural number 1≤ k≤ n-1, we can find a collection of balls {B_r(x_i)}_i=1^N with N≤ C_n,α,ϵ r^-k such that 

    S^k_ϵ,r(u)∩ B_1⊂⋃_i B_r(x_i).

In particular, |B'_r(S^k_ϵ,r∩ B_1)|≤ C_n,α,ϵ r^n-k for every 0<r≤ 1 and 

    ℋ^k(S^k_ϵ(u) ∩ B_1)≤ C_n,α,ϵ.



From Proposition <ref> and Theorem <ref> we can conclude the following corollary which comprises the second part of Theorem <ref> above. 


If u∈^β(B_5) is a minimizer to eqLocalizedFunctional  in B_5 with  0∈ F(u), then Σ(u) is (n-k^*_α)-rectifiable and for every 0<r≤ 1 we have 

    |B_r (Σ(u)∩ B_1)|≤ C_n,α r^k^*_α.

In particular, 

    ℋ^n-k^*_α(Σ(u)∩ B_1)≤ C_n,α.



Rectifiability is encoded in the following result. We omit the details of proof here but it is a consequence of the packing result above, the Rectifiable-Reifenberg theorem of <cit.> and Theorem <ref> below. For more details see Sections 2 and 8 of <cit.> (particularly Theorem 2.2 in the former and the proof of Theorem 1.12 in the latter). 


Let u be a non-negative, even minimizer to eqLocalizedFunctional in a domain Ω. Then S^k_ϵ(u) is k-rectifiable for every ϵ and, hence, each stratum S^k(u) is k-rectifiable as well.


The proof of Theorem <ref> follows from inductively applying the following, slightly more technical, packing result (for details see Section 4 of <cit.>).


Let ϵ>0. There exists η (n,α, ϵ) such that, for every minimizer u∈^β(B_5) of 𝒥 in B_5 with 0∈ F(u) and 0<R<1/10, there is a finite collection 𝒰 of balls B with center x_B ∈ S^k_ϵ, η R and  radius R ≤ r_B ≤ 1/10 which satisfy the following properties:


  * Covering control:

    S^k_ϵ, η R∩ B_1  ⊂⋃_B∈𝒰 B.


  * Energy drop:
For every B∈𝒰, 

    r_B=R,       sup_2BΨ^u_2r_B≤sup_B_2Ψ^u_2 -η.


  * Packing:

    ∑_B∈𝒰 r_B^k ≤ c(n, α, ϵ).




We construct the balls of Theorem <ref> using a “stopping time" or “good ball/bad ball" argument. Much of this argument uses harmonic analysis and geometric measure theory and is completely independent of the original problem (<ref>). However, there are a few places in which we need to connect the behavior of minimizers to the geometric structure of the singular set. Here we will sketch the “good ball/bad ball" argument, taking for granted the estimates needed to apply this argument to our functional. In the next few subsection we will provide these estimates. For more details on the construction itself we refer the reader to Section 7 in <cit.>.



Outline of the Construction in Theorem <ref> To find this covering we define good and bad balls as follows: imagine our ball, B, has radius 1. We say that B is a good ball, if at every point in x∈ S^k_ε(u)∩ B the monotone quantity centered at that point at some small scale, ρ, is not much smaller than the monotone quantity on ball B (we say these points have “small density drop"). A ball B is a bad ball if all the points in S^k_ε(u)∩ B with small density drop are contained in a small neighborhood of a (k-1)-plane. This good/bad is a dichotomy follows from Theorem <ref> in Section <ref>. 

In a good ball of radius r we cover S^k_ε(u) with balls of radius ρ r iterating the construction until we find a bad ball or until the radius of the ball becomes very small. In a bad ball, we cover S^k_ε(u) away from the (k-1)-plane without much care. Close to the (k-1)-plane we cover S^k_ε(u) with balls of radius ρ r iterating the construction until we reach a good ball or until the radius of the ball becomes very small. 

Inside long strings of good balls, the packing estimates follow from powerful tools in geometric measure theory (see Theorem <ref> below) and the connection between the drop in monotonicity and the local flatness of the singular strata (see Theorem <ref> below). We give more details in Section <ref>. 

Inside  long strings of bad balls each of which is near the (k-1)-plane of the previous bad ball, we have even better packing estimates than expected (as we are effectively well approximated by planes which are lower dimensional). This leaves only points which are in many bad balls and in most of those balls they are far away from the (k-1)-plane. However, at these points the monotone quantity drops a definite amount many times, which contradicts either finiteness or monotonicity. This implies that the points and scales inside the bad balls which are not close to the (k-1)-plane form a negligible set (the technical term is a Carleson set).  We give more information about the bad balls in Section <ref>. 




































































































 §.§ Tools for bad balls: key dichotomy




Let  ϵ, ρ, γ, η'>0 be fixed numbers with ργ<2. There exists an
η_0 (n, α, ϵ, ρ, γ, η')<ρ/100 such that for every η≤η_0, every r>0, every E>0 and every  minimizer u∈^β(B_4r) of 𝒥 in B_4r with 0∈ F(u) and sup_B_rΨ^u_2r≤ E,
then either


  * Ψ_γρ r^u≥ E-η' on S^k_ϵ,η r∩ B_r, or

  * there exists ℓ∈ L^k-1 so that {x∈ B_r: Ψ^u_2η r(x)≥ E-η}⊂ B_ρ r(ℓ).



The key dichotomy is a direct consequence of the Lemma <ref> below. The core idea is to make effective the following assertion: if u is k-symmetric, then along the invariant manifold the Allen-Weiss density is constant, and every point away from the manifold will have (k+1)-symmetric blow-ups by Lemma <ref>. 




Let ϵ, ρ, γ, η'>0 be fixed numbers with γρ <2. There exist η_0, θ>0 such that for every η<η_0, every E>0 and every minimizer u of 𝒥 in B_4 with  0∈ F(u) and sup_B_1Ψ^u_2≤ E, if  there exist w_0,…, w_k∈ B_1 and affine manifolds L^i:=⟨ w_0,…, w_i⟩∈ V^i with 

    w_i∉ B_ρ(L^i-1),        Ψ^u_2η(w_i) ≥ E-η    i∈{0,⋯,k},

then, 

    Ψ^u_γρ(x)≥ E-η'     B_θ(L^k)∩ B_1

and

    S^k_ϵ,η∩ B_1⊂ B_θ(L^k)



The proof follows (with only minor modifications) the proof in <cit.>. We end this subsection by formally defining the good/bad balls alluded to above: 


Let x∈ B_2, 0< R<r<2 and u be a minimizer to 𝒥 in B_5. We say that the ball B_r(x) is good if 

    Ψ^u_γρ r≥ E-η'      S^k_ϵ, η R∩ B_r(x),

and  otherwise we say that B_r(x) is bad. 

By Theorem <ref> in any bad ball B there exists an affine (k-1)-manifold ℓ_B with

    {w∈ B: Ψ^u_2η r(w)≥ E-η}⊂ B_ρ r(ℓ_B^k-1).










 §.§ Tools for good balls: packing estimates and GMT



In this section we control the local flatness of the singular strata by the drop in monotonicity. To do this we introduce a key tool from geometric measure theory which estimates the flatness of a set. Given a Borel measure μ, a point x and a radius r, the beta coefficient is defined as follows:

    β^k_μ,2(B_r(x))^2:=β^k_μ,2(x,r)^2 =inf_L ∈ V^k_a1/r^k∫_B_r(x)(z,L)^2/r^2  dμ (z)

where V^k_a stands for the collection of k-dimensional affine sets of ^n. The beta coefficients are meant to measure in a scale invariant way how far is a measure from being flat, in this case in the L^2 distance, although other L^p versions have been used in the literature for 1≤ p≤∞ quite often, dating back to <cit.> (for the L^∞ version) and David-Semmes <cit.> (for the L^p version). 

If we control the size of the β^k's we can conclude size and structure estimates on the measure μ. The following theorem says exactly this and represents a major technical achievement. It differs (importantly) from prior work in this area by the lack of a priori assumptions on the upper or lower densities of the measure involved.



Let {B_r_q(q)}_q be a collection of disjoint balls, with q∈ B_1(0) and 0<r_q≤1, and let μ be the packing measure μ:=∑_q r_q^kδ_q, where δ_q stands for the Dirac delta at q.  There exist constants τ_DR, C_DR>0 depending only on the dimension such that if

    ∫_0^2r∫_B_r(x)β^k_μ,2(z,s)^2   dμ (z) ds/s≤τ_DR r^k      x∈ B_1(0),   0<r≤ 1,

then

    μ(B_1(0))=∑_q r_q^k≤ C_DR.




















To obtain the packing estimates required for the Discrete-Reifenberg Theorem, we need to control the beta coefficients. The key estimate of this entire framework lies in the following theorem, which shows the drop in monotonicity at a given point and a given scale controls the beta coefficient at a comparable scale.

Let ϵ>0 be given. There exist δ(n,α, ϵ) and c(n, α, ϵ) such that  for every u∈^β(B_5r) minimizing 𝒥 in B_5r(x) with  x∈ F(u) and

    u (0,δ) B_4r(x)
    
    u (k+1,ϵ) B_4r(x),

and every Borel measure μ, we have that

    β^k_μ,2(B_r(x))^2 
    	≤ c(n,α, ϵ)/r^k∫_B_r(x)( Ψ_4r^u (w)-Ψ_r^u (w))   dμ (w).




We follow the proof of <cit.> closely. First the authors give an explicit formula for the beta coefficients. 



Let X be the center of mass of a  Borel measure μ on B=B_r(x). Let {λ_i}_i=1^n be the decreasing sequence of eigenvalues of the non-negative bilinear form

    Q(v,w):=  _B(v· (z-X)) (w· (z-X))  dμ (z),

and let {v_i}_i=1^n be a corresponding orthonormal sequence of eigenvectors, that is v_i· v_j=δ^ij and Q(v_i , v)=λ_i v_i· v. Then

    β_μ,2^k(B)^2= 1/r^k∫_B(z,L^k)^2/r^2  dμ (z) = μ(B)/r^k(λ_k+1 + … + λ_n)/r^2,

where L^k := X+ span⟨ v_1,…, v_k ⟩.






































Next we find a relation between the eigenvalues of Q and Allen-Weiss' energy.

 
Under the hypothesis of Lemma <ref>, for every u∈^β(B_5r) minimizing 𝒥 in B_5r(x) and every i≤ n, we have that

    λ_i 2/r^n+2∫_A_2r,3r(x) |y|^Β (v_i· Du(z))^2  dz≤ C _B_r(x)( Ψ_4r^u (w)-Ψ_r^u (w))   dμ (w) .



The argument follows as in <cit.>. In formula (18) one needs to change u(z) by α u(z), which can be done with exactly the same argument.



Finally, using compactness, we bound the left-hand side of lemLambda2 from below.

Let ϵ>0 be given. There exists a δ(n,α, ϵ) and c(n, α, ϵ) such that,  for  every orthonormal basis {v_i}_i=1^n and every u∈^β(B_5r) minimizing 𝒥 in B_5 r(x) with x∈ F(u) and satisfying eq0DeltaNonK1Epsilon, we have that

    1/c(n, α, ϵ)≤ r^-n∫_A_2r,3r(x) |y|^Β∑_i=1^k+1 (v_i· Du(z))^2  dz.



The proof follows that of <cit.> and we omit it.





By Lemmas <ref>,  <ref> and   <ref> we get that

    β^k_μ,2(B)^2 
    	   ≤μ(B)/r^k+2(n-k) λ_k+1
       ≤μ(B)/r^k (n-k) c(n,α, ϵ)  ∑_i=1^k+1λ_i/r^n+2∫_A_2r, 3r(x) |y|^Β (v_i· Du(z))^2   dz
       ≤ c(n,α, ϵ)/r^k∫_B_r(x)( Ψ_4r^u (w)-Ψ_r^u (w))   dμ (w).

















































































































































































































M.E. was partially supported by an NSF postdoctoral fellowship, NSF DMS 1703306 and by David Jerison's grant DMS 1500771. A.K. acknowledges Financial support from the Spanish Ministry of
Economy and Competitiveness, through the María de Maeztu Programme for Units of
Excellence in R&D (MDM-2014-0445). M.P. was funded by the European Research Council under the grant agreement 307179-GFTIPFD. G.S. has received funding from the European Union's Horizon 2020 research and innovation programme under Marie Skłodowska-Curie grant agreement No 665919.  
A.K., M.P. and G.S. were also partially funded by 2017-SGR-0395 (Catalonia) and MTM-2016-77635-P (MINECO, Spain). 
Y.S. is partially supported by the Simons foundation.
	
M.E. would also like to thank Nick Edelen for many fruitful conversations regarding the quantitative stratification and rectifiable Reifenberg framework. 

A.K., M.P., G.S. would like to thank Xavier Cabré, Tomás Sanz, Matteo Cozzi, Albert Mas, Maria del Mar González, Luis Silvestre and Stefano Vita for some conversations around <cit.>. They would also like to thank Mihalis Mourgoglou for some conversations regarding the degenerate elliptic measure. 





§ APPENDIX



§ RELATION WITH THE NONLOCAL BERNOULLI PROBLEM



As in <cit.>, we see that the study of minimizers of 𝒥 includes the study of minimizers of J.


If f is a minimizer of J in the unit ball of ^n then f*P_y is a minimizer of 𝒥 in every ball B such that B'⊂⊂ B_1'.

If u=f*P_y is a minimizer of 𝒥, then f is a minimizer for J. In particular, if u is a minimizer of 𝒥 in every ball, positive outside the hyperplane {y=0}, and u(x,y)=𝒪(|(x,y)|^α),  then u|_^n×{0} is a minimizer for J in every ball.



We follow <cit.>, that is, we use the following result from <cit.>.

Let f,g satisfy that J_0(f,B_1), J_0(g,B_1) < ∞, and suppose that f-g is compactly supported in B_1⊂^n.
Then we have that

    J_0(g,B_1)  - J_0(f,B_1)
    = c_n,αinf∫_Ω |y|^β (|∇ v(x,u)|^2 - |∇ (f*P_y)(x)|^2),

where the infimum is taken among all the symmetric bounded Lipschitz domains Ω with the property that Ω∩ (^n×{0})⊂ B_1 and among all symmetric functions v with trace g satisfying that v-f*P_y is compactly supported on Ω.



Let f be a minimizer of J in the unit ball of ^n and let B_r be a ball such that B_r'⊂⊂ B_1'. We want to show that u:=f*P_y is a minimizer of 𝒥 in B_r.

Let v:^n+1→ so that v≡ u in ^n+1∖ B_r and v ∈ H^1(β, B_r). Let g be the trace of v in ^n×{0}.
By Lemma <ref> we have that

    J_0(g,B_1)  - J_0(f,B_1)≤ c_n,α∫_B_r+ε |y|^β (|∇ v|^2 - |∇ u|^2)

for every ε>0.

In particular, since g|_(B')^c ≡ 0,  g is an admissible competitor for f
and
J(f,B_1)≤ J(g,B_1), i.e.,

    J_0(g,B_1)  - J_0(f,B_1)       ≥
    -m({g>0}∩ B_1) + m({f>0}∩ B_1)
        =  m({u>0}∩ B_r')-m({v>0}∩ B_r').

The proposition follows combining eqQuality and eqFinestQuality and letting ε→ 0.

The converse follows the same sketch: every global minimizer can be expressed as the Poisson extension of its restriction to the hyperplane by Proposition <ref> and it is left to the reader.


As a consequence of the previous proposition, all the results that we have proven for minimizers of 𝒥 also apply to minimizers of J:

If u:^n→ is a minimizer to J in B_2⊂^n and 0∈ F(u), then u_C^α(B_1)≤ C, it satisfies the nondegeneracy condition u(x)≥ C (x,F(u))^α for x∈ B_1, the positive phase satisfies the corkscrew condition, every blow-up limit is α-homogeneous, and the boundary condition in eqEulerLagrangeFrac is satisfied at F_(u). 

Moreover, the positive phase {u>0}∩ B_1 is a set of  finite perimeter, the singular set is an (n-3)-rectifiable set, it is discrete whenever n=3 and it is empty if n≤ 2. 

All the constants depend only on n and α.





§ UNIQUENESS OF EXTENSIONS


In Proposition <ref> we have used the following result, included in <cit.>. Here we provide a proof which is different than the one appearing in <cit.>.


Let α∈(0,1), β=1-2α, and set ℒu=-(|y|^β∇ u) in ℝ^n+1. Suppose that v:ℝ^n+1_+→ℝ is nonnegative outside ℝ^n, it is a solution to ℒv=0 in ℝ^n+1_+ with v(x',0)=0 for all x'∈ℝ^n and |v(x)|≤ C|x|^α. Then v≡ 0.


First, since |y|^β is C^∞ away from the hyperplane ℝ^n, v∈ C^∞_(ℝ^n+1_+). Let now i∈{1,… n}, and set

    f_m(x)=v(x+1/me_i)-v(x)/1/m.

Let B_r=B_r(x',0) be a ball centered at (x',0)∈ℝ^n ×{0} with radius r, and let B_2r be its double ball. Set also w(x)=w(x',y)=y^β for y>0. Since f_m is a solution of ℒf_m=0 in B_r^+=B∩ℝ^n+1_+, <cit.> shows that

    max_B_r^+|f_m(x)|≤ C(1/w(B_2r^+)∫_B_2r^+|f_m|^2w)^1/2.

From convergence of difference quotients (similarly to <cit.>), if v∈ H^1(β,B_2r^+), the last estimate will imply that f_m is uniformly bounded in B_r^+ by a constant C_r. Therefore, from the boundary Caccioppoli estimate  (<cit.>) we have that

    ∫_ B_r/2^+|∇ f_m|^2w≤C/r^2∫_ B_r^+|f_m|^2w≤C/r^2∫_ B_r^+C_r^2w≤ C_n,r,w<∞,

hence {f_m} is bounded in H^1(β,B_r/2^+). From weak compactness, a subsequence of {f_m} converges to a solution of ℒu=0 in B_r/2^+, and since f_m→∂_i v pointwise, we obtain that ∂_iv is  an H^1(β,B_r/2^+) solution in B_r/2^+. Hence ∂_iv is a solution to ℒu=0 in ℝ^n+1_+.

Now, for x=(x',y)∈ℝ^n+1_+, let R=|x|. We distinguish between two cases: y>R/16, and y<R/16. 

In the first case, set B_R to be the ball of radius R, centered at x. Note then that B_R/16⊆ℝ^n+1_+. Then, from <cit.>, Caccioppoli's estimate and the assumption |v(x)|≤ C|x|^α,

    |∂_iv(x)|^2   ≤C/w(B_R/32)∫_B_R/32|∂_i v|^2w≤C/w(B_R/32)C/R^2∫_B_R/16|v|^2w≤C/R^2w(B_R/16)/w(B_R/32)sup_B_R/16|v|≤ CR^2α-2.

In the second case, let B_R be the ball centered at (x',0) with radius R, and denote B_R^+=B_R∩ℝ^N+1_+. Then x∈ B_R/8^+, therefore from <cit.> and the boundary Caccioppoli estimate,

    |∂_iv(x)|^2   ≤C/w(B_R/8^+)∫_B_R/8^+|∂_i v|^2w≤C/w(B_R/8^+)C/R^2∫_B_R/4^+|v|^2w≤C/R^2w(B_R/4^+)/w(B_R/8^+)sup_B_R/4^+|v|≤ CR^2α-2.

So, in all cases, |∂_i v(x)|≤ C|x|^α-1. Letting R→∞ and using the maximum principle, we find that ∂_iv=0 for any i=1,… n. Therefore v does not depend on the first n variables,  so v(x',y)=v(y). Hence, in ℝ^n+1_+,

    0=-(y^β∇ v(y))=-∂_y(y^βv'(y))  ⇒   y^βv'(y)=c̃,

for some constant c̃. From <cit.>, v is Hölder continuous up to the boundary, therefore for any y>0,

    v(y)=v(y)-v(0)=∫_0^yv'=∫_0^yc̃s^-β ds=c̃/1-βy^1-β,

which implies that

    |c̃|=(1-β)y^β-1|v(y)|=(1-β)y^β-1|v(0,y)|≤ (1-β)y^β-1y^α=(1-β)y^-α,

for any y>0. Letting y→∞ we obtain that c̃=0, hence v'(y)=0 as well, which implies that v is a constant. Since v vanishes on ℝ^n, this implies that v≡ 0.








