Paper Title: Rigidity for the non self-dual Chern--Simons--Schr{ö}dinger equation at the level of the soliton.


]Rigidity for the non self-dual Chern–Simons–Schrödinger equation at the level of the soliton





§ ABSTRACT 
In this paper we prove a rigidity result for a solution to the non self-dual Chern–Simons–Schrödinger equation at the level of the soliton.

[
    Benjamin Dodson
    August 5, 2024
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§ INTRODUCTION

In this paper we prove rigidity for blowup solutions for the equivariant Chern–Simons–Schrödinger equation,

    i u_t + Δ u = 2m/r^2 A_θ[u] u + A_0[u] u + 1/r^2 A_θ[u]^2 u - g |u|^2 u,      u : ℝ×ℝ^2→ℂ.

where

    A_θ[u](t, r)    = -1/2∫_0^r |u(t, s)|^2 s ds, 
    
    A_0[u](t, r)    = -∫_r^∞ (m + A_θ[u](t, s)) |u(t, s)|^2ds/s.

Here, we are in the equivariant case, which imposes the equivariant symmetry on the scalar field ϕ,

    ϕ(t, x) = u(t, r) e^im θ,      m ∈ℤ.


This problem shares a number of similarities with the nonlinear Schrödinger equation,

    i u_t + Δ u = α |u|^2 u.

Indeed, solutions to (<ref>) and (<ref>) conserve the quantities mass,

    M(u(t)) = ∫ |u(t, x)|^2 dx = M(u(0)),

and energy

    E(u(t)) = 1/2∫ |∂_r u|^2 + 1/2∫ (m + A_θ[u]/r)^2 |u|^2 - g/4∫ |u|^4.

Furthermore, (<ref>) is a mass-critical problem, as is (<ref>). Indeed, equation (<ref>) has the scaling symmetry

    u(t, x) ↦λ u(λ^2 t, λ x),      u_0(x) ↦λ u_0(λ x),     λ > 0.

The general Chern–Simons–Schrödinger equation is locally well-posed <cit.> for small data in H_x^s(ℝ^2), s > 0. The case when s = 0 is unknown. For the equivariant Chern–Simons–Schrödinger equation, let L_m^2 denote the space of functions in L^2(ℝ^2) that satisfy (<ref>). Similarly, we can define the equivariant Sobolev spaces.

Let m ∈ℤ. For each s ≥ 0, define the function space H_m^s to be the Sobolev space of functions f ∈ H_x^s that admit the decomposition f(x) = f(r, θ) = e^im θ u(r).

Crucially, like the mass-critical nonlinear Schrödinger equation, (<ref>) enjoys the virial identity

    d^2/dt^21/4∫ |x|^2 |u|^2 = d/dt∫ Im[u̅ r ∂_r u] = 4 E[u],

and the pseudoconformal transformation symmetry. If u solves (<ref>) then

    1/tu(t, x/t) e^i |x|^2/4t,

also solves (<ref>). Of course, by standard time translation and time reversal symmetry arguments, it is possible to replace the t in (<ref>) by T - t. In this case, the pseudoconformal transformation may be abbreviated PC_T.

Rewriting the energy,

    E[u(t)] = 1/2∫ |∂_r u - m + A_θ[u]/r u|^2 + 1 - g/4∫ |u|^4.

Thus, when g < 1, (<ref>) resembles a defocusing nonlinear Schrödinger equation (α > 0), when g > 1, (<ref>) resembles a focusing nonlinear Schrödinger equation (α < 0), and when g = 1, (<ref>) is called a self-dual Chern–Simons–Schrödinger equation.


Let g < 1 and m ∈ℤ. Then (<ref>) is globally well-posed in L_m^2 and solutions scatter both forward and backward in time.


See <cit.>. Compare to <cit.> for the mass-critical NLS.




Let g > 1 and m ∈ℤ_+. Then there exists a constant c_m, g > 0 such that if u_0∈ L_m^2 with u_0_L^2^2 < c_m, g, then (<ref>) is globally well-posed in L_m^2 and scatters forward and backward in time. Moreover, the minimum charge of a nontrivial standing wave solution in the class L_t^∞ L_m^2 is equal to c_m, g.


See <cit.>. Compare to <cit.> for the focusing, mass-critical NLS.




Let g = 1 and m ∈ℤ_+. Let u_0∈ L_m^2 with u_0_L^2^2 < 8 π (m + 1). Then (<ref>) is globally well-posed in L_m^2 and scatters both forward and backward in time.


See <cit.>.



In this paper we prove a rigidity result that is analogous to the rigidity result for the mass-critical nonlinear Schrödinger equation, see <cit.> and <cit.>.

For m ∈ℕ and g > 1, if u_0_L^2^2 = c_m, g then

    u = ψ^(m, α)(t, x),

 or u = e^i γ PC_T[λψ^(m, g)(λ^2·, λ·)](t, x) for some T > 0.


For a generic g > 1, (<ref>) has a standing wave solution of the form

    Δ u - 2m/r^2 A_θ[u] u - A_0[u] u - 1/r^2 A_θ[u]^2 u + g |u|^2 u -α u = 0,

for some α > 0. This gives a standing wave equation to (<ref>) that is in the form e^i α t Q, where Q is the positive solution to (<ref>). The existence of such a solution was proved by <cit.>. The unique positive, standing wave solution to (<ref>) with a given m and α > 0 may be abbreviated ψ^(m, α)(t, x).


Theorem <ref> improves an earlier result of <cit.>.

For m ∈ℕ and g > 1, if ϕ∈ H_m^1(ℝ^2), u_0_L^2^2 = c_m, g, and the solution blows up forward in finite time at T > 0, there exists γ∈ [0, 2 π), λ∈ℝ_+, and an m-equivariant standing wave solution

    ψ^(m, g)(t, x) = e^i α tϕ^(m, g)(x),

such that

    u(t,x) = e^i γ PC_T[λψ^(m, g)(λ^2·, λ·)](t,x).



The proof of Theorem <ref> is very similar to the argument in <cit.>. The main new difficulty is that equation (<ref>) is no longer a local equation.


The method proving Theorem <ref> does not extend to the self-dual, or g = 1 case. There are several reasons for this. The first is that the standing wave equation with g = 1 has α = 0 in (<ref>). In the non self–dual case, the fact that α > 0 is used extensively in the proof. Indeed, for the mass–critical problem, (<ref>), the soliton satisfies the elliptic equation

    Δ Q + |Q|^2 Q = Q.

Taking u close to the soliton, u = Q + ϵ,

    (∇ Q, ∇ϵ) - (|Q|^2 Q, ϵ) = -(Δ Q + |Q|^2 Q, ϵ) = -(Q, ϵ) = 1/2ϵ_L^2^2.

The last equality arises from the fact that Q + ϵ_L^2 =  Q _L^2. Since (<ref>) represents the ϵ^1 term in the expansion of E[Q + ϵ], we obtain

    E[Q + ϵ] = 1/2(ℒϵ, ϵ) + O(ϵ^3),

where (ℒϵ, ϵ) ≳ϵ_H^1^2 when ϵ is orthogonal to ∇ Q and the negative eigenvector of ℒ. Since Q is smooth and rapidly decreasing, (<ref>) is well–behaved under truncations in space and frequency. It is possible to obtain a similar estimate for (<ref>) when α > 0. However, when α = 0 we lose the ϵ_L^2^2 term, which adds additional technical difficulties for a mass–critical problem.

Secondly, the standing wave solution to (<ref>) is no longer rapidly decreasing. This is also a by-product of the fact that α = 0. Instead, the solution has the explicit form

    Q(r) = √(8) (m + 1) r^m/1 + r^2m + 2,      m ≥ 0.

This fact is also used heavily. It seems likely to the author that Theorem <ref> should be true in the self-dual case, since <cit.> proved that Theorem <ref> does hold under the additional assumption that u_0∈ H_m^1.

Additionally, it should be noted that when m < 0, <cit.> proved that global well-posedness and scattering hold for initial data in H_m^1,1, and when m ≥ 0, <cit.> proved that a blowup solution should resolve into a single soliton plus a radiative term.

In fact, using the arguments proving Theorem <ref> in <cit.>, when m < 0, a minimal mass blowup solution to (<ref>) can be reduced to one of three enemies:



  * N(t) = 1,


  * N(t) ≤ 1, t ∈ℝ, lim inf_t →±∞ N(t) = 0,


  * N(t) = t^-1/2, t ∈ (0, ∞).


Also following the arguments in <cit.>, it is possible to show that for an almost periodic solution to (<ref>), u(t) ∈ H^s for s ≤ 2, which furthermore implies that if u is one of the three enemies, E(u(t)) = 0. However, using the estimate in <cit.>, which shows that E(u(t)) ∼_M[u] u _Ḣ_m^1^2 when m < 0, gives a contradiction.





§ SEQUENTIAL CONVERGENCE

We begin with a sequential convergence result, comparable to the sequential convergence result for the mass-critical NLS in <cit.>, <cit.>, and <cit.>. The argument here follows the argument in <cit.> for the self-dual Chern–Simons–Schrödinger equation.

Let u be a solution to (<ref>) that blows up forward in time and satisfies u _L^2^2 = c_m, g. That is,

    lim_T ↗sup(I) u _L_t,x^4([0, T] ×ℝ^2) = +∞.

Then there exists t_n↗sup(I) and sequences λ(t_n) > 0, γ(t_n) ∈ [0, 2 π), such that

    e^i γ(t_n)λ(t_n) u(t_n, λ(t_n) ·) →ψ^(m, g)(·),     in     L^2,

where ψ^(m, g) is the real, positive standing wave solution to (<ref>).


The proof uses the fact that if u _L^2^2 = c_m, g, then u is a minimal mass blowup solution. Thus, it is possible to make use of much of the analysis in <cit.>.



Let ψ_n, n = 1, 2, ... be a bounded sequence in L_m^2. Then, after passing to a subsequence if necessary, there exists a sequence of functions ϕ^j∈ L_m^2, group elements g_n^j, and times t_n^j∈ℝ such that we have the decomposition

    ψ_n = ∑_j = 1^J g_n^j e^i t_n^jΔϕ^j + w_n^J,     ∀ J = 1, 2, ...

where g_n^j belongs to the group of transformations of L^2(ℝ^2) generated by the scaling symmetry (<ref>) and multiplying by e^i γ for some γ∈ℝ,

    g_n^j u(x) = e^i γ(t_n^j)λ(t_n^j) u(λ(t_n^j) x).


Moreover, w_n^J∈ L_m^2 is such that its linear evolution has asymptotically vanishing scattering size

    lim_J →∞lim sup_n →∞ e^it Δ w_n^J_L_t,x^4 = 0.

Moreover, for any j ≠ j',

    λ_n^j/λ_n^j' + λ_n^j'/λ_n^j + |t_n^j (λ_n^j)^2 - t_n^j' (λ_n^j')^2|/λ_n^jλ_n^j'→∞.

Furthermore, for any J ≥ 1, we have the mass decoupling property

    lim_n →∞ [M(u(t_n)) - ∑_j = 1^J M(ϕ^j) - M(w_n^J)] = 0,



This is Proposition 3.1 of <cit.>.



Specifically, let t_n↗sup(I) be a sequence and let ψ_n = u(t_n) and apply Proposition <ref>. Then possibly after passing to a subsequence,


If u is a blowup solution to (<ref>), there exists some j such that ϕ^j≠ 0.


Otherwise, by a perturbative argument, (<ref>) implies that u is a scattering solution. Relabeling, suppose ϕ^1≠ 0.



ϕ^j_L^2 = 0 for j > 1. 


Otherwise, by (<ref>), if ϕ^2_L^2 > 0 for j > 2, then ϕ^j_L^2^2 < c_m, g for all j. By <cit.> and standard perturbative arguments, u scatters forward in time.




For any J > 1,

    lim_n →∞ w_n^J_L^2 = 0.



Otherwise by (<ref>), if lim sup_n →∞ w_n^J_L^2 > 0, and therefore ϕ^1_L^2^2 < c_m, g. Then by standard perturbative arguments, u scatters forward in time.




After possibly passing to a subsequence, the sequence t_n^1 converges as n →∞. 


If t_n^1→ +∞ then we have scattering forward in time. If t_n^1→ -∞, we have scattering backward in time, which contradicts

    u _L_t,x^4(inf(I), t_n] ×ℝ^2)→ +∞,

as n →∞.


Therefore, possibly after passing to a subsequence,

    (g_n^1)^-1 u(t_n, x) →ϕ^1,     ϕ^1∈ L^2,     ϕ^1_L^2 = c_m, g.

Now then, by construction, ϕ^1 is the initial value of a blowup solution to (<ref>) that blows up both forward and backward in time. Let ϕ be the solution to (<ref>) with initial data ϕ^1. Since ϕ is a minimal mass blowup solution, then after making the concentration compactness argument in <cit.>, there exists λ(t), γ(t) such that

    e^i γ(t)λ(t) ϕ(t, λ(t) ·) ∈ K ⊂ L^2,

where K ⊂ L^2 is a precompact set. Furthermore, following the reduction to three enemies in  <cit.>, see also <cit.>. there exist t_n such that

    λ(t_n) e^i γ(t_n)ϕ(t_n, λ(t_n) ·) → v_0∈ L^2.

Furthermore, v_0 is the initial data for a solution to (<ref>) that satisfies (<ref>) and λ(t) satisfies one of three cases:



  * λ(t) = 1 for all t ∈ℝ,


  * λ(t) ≥ 1 for all t ∈ℝ and lim sup_t →±∞λ(t) = ∞,


  * λ(t) = t^1/2 for t ∈ (0, ∞).


Using the additional regularity argument in <cit.>, E(v) = 0, so since v _L^2^2 = c_m, g, v_0 is a soliton. See Proposition 3.7 of <cit.> for the proof that the solitons are the only zero energy m-equivariant functions satisfying v _L^2^2 = c_m, g. Therefore,

    e^i γ(τ_n')λ(τ_n') ϕ(τ_n', λ(τ_n') ·) →ψ^(m, g),     in     L^2.

Therefore, choosing n(n') sufficiently large,

    e^i γ(τ_n')λ(τ_n') (g_n^1)^-1 u(t_n + λ(t_n)^-2τ_n', λ(τ_n') ·) →ψ^(m, g),     in     L^2.






§ REDUCTION OF A BLOWUP SOLUTION




Fix some 0 < η_∗≪ 1 sufficiently small. To prove Theorem <ref>, it suffices to prove the following. To simplify notation let Q = ψ^(m, g) be the positive solution to the standing wave equation (<ref>). Theorem <ref> can be reduced to Theorem <ref>.


If u is a blowup solution to (<ref>) that satisfies u _L^2^2 = c_m, g and for all t ≥ 0,

    inf_λ > 0, γ∈ℝ e^i γλ u(t, λ x) - Q _L^2≤η_∗,

then Theorem <ref> is true.



Suppose that u is a solution to (<ref>) that blows up forward in time and satisfies u _L^2^2 = c_m, g. Consider two cases separately.

Case 1: There exists some t_0 > 0 such that, for all t ∈ [t_0, sup(I)),

    inf_λ > 0, γ∈ℝ e^i γλ u(t, λ·) - Q _L^2≤η_∗.

In this case, Theorem <ref> reduces to Theorem <ref>.


Case 2: There exist a sequences t_n, t_n^-↗sup(I) such that

    sup_t ∈ [t_n^-, t_n]inf_λ > 0, γ∈ℝ e^i γλ u(t_n, λ·) - Q _L^2≤η_∗,


    inf_λ > 0, γ∈ℝ e^i γλ u(t_n^-, λ·) - Q _L^2 = η_∗,

and

    lim_n →∞ u _L_t,x^4((inf(I), t_n^-] ×ℝ^2) = lim_n →∞ u _L_t,x^4([t_n^-, t_n] ×ℝ^2) = ∞.


To see why (<ref>)–(<ref>) must hold for a blowup solution to (<ref>) that does not satisfy Case 1, observe that by Theorem <ref>, there exists a sequence t_n↗sup(I) such that (<ref>) holds. Since (<ref>) does not hold, we also have t_n^-↗sup(I).

Next, recall the Strichartz estimates of <cit.>, <cit.>, and <cit.>.


Let (i ∂_t + Δ) u = f on a time interval I with t_0∈ I and u(t_0) = u_0. A pair (p, q) of exponents is called admissible if 2 ≤ p, q ≤∞, 1/p + 1/q = 1/2, and (p, q) ≠ (2, ∞). Let (p, q) and (p̃, q̃) be admissible pairs of exponents. Then,

    u _L_t^∞ L_x^2(I ×ℝ^2) +  u _L_t^p L_x^q(I ×ℝ^2)≲ u_0_L^2 +  f _L_t^p̃' L_x^q̃'(I ×ℝ^2).






Let (i ∂_t + Δ) u = f on a time interval I with t_0∈ I and u(t_0) = u_0, and suppose that m ∈ℤ and u_0∈ L_m^2, f ∈ L_t^1 L_m^2(I ×ℝ^2). Let (p, q) be an admissible pair of exponents. Then,

    u _L_t^2 L_x^∞(I ×ℝ^2)≲ u_0_L^2 +  f _L_t^p' L_x^q'(I ×ℝ^2).



Finally,

Let

    Λ(u) =  2m/r^2 A_θ[u] u + A_0[u] u + 1/r^2 A_θ[u]^2 u - g |u|^2 u.

We have

    Λ(u) _L_t,x^4/3(I ×ℝ^2)≲ u _L_t,x^4(I ×ℝ^2)^3,

 and
 
    Λ(u) - Λ(ũ) _L_t,x^4/3(I ×ℝ^2)≲ u - ũ_L_t,x^4(I ×ℝ^2) ( u _L_t,x^4(I ×ℝ^2)^2 + ũ_L_t,x^4(I ×ℝ^2)^2).



This is proved in <cit.>.


It follows from Lemma <ref> that

    inf_λ > 0, γ∈ℝ e^i γλ u(t, λ x) - Q _L^2,

is continuous as a function in t. Therefore, for each t_n∈ I, there exists some t_n^-∈ I, t_n^- < t_n, such that

    inf_λ > 0, γ∈ℝ e^i γλ u(t_n^-, λ x) - Q _L^2 = η_∗,

and

    sup_t ∈ [t_n^-, t_n]inf_λ > 0, γ∈ℝ e^i γλ u(t_n^-, λ x) - Q _L^2 = η_∗.

Thus, (<ref>) and (<ref>) hold. Finally, using the perturbation result in Lemma <ref>, (<ref>),

    inf_λ > 0, γ∈ℝ e^i γλ u(t', λ x) - Q _L^2≲inf_λ > 0, γ∈ℝ e^i γλ u(t, λ x) - Q _L^2,

with implicit constant depending only on u, for any pair of times t, t' such that

    u _L_t,x^4([t, t'] ×ℝ^2)≤ 1.

Since

    inf_λ > 0, γ∈ℝ e^i γλ u(t_n, λ x) - Q _L^2→ 0,

(<ref>) and (<ref>) imply (<ref>).

Now, using Proposition <ref>, there exists a sequence g_n∈ G and u_0∈ L^2, u_0_L^2^2 = c_m, g, such that

    g_n^-1 u(t_n^-1, x) → u_0,     in     L^2.

Furthermore, by (<ref>)–(<ref>), u_0 is the initial data to a solution to (<ref>) that blows up both forward and backward in time, satisfies (<ref>), and satisfies

    inf_λ > 0, γ∈ℝ e^i γλ u_0(λ x) - Q _L^2 = η_∗.

But then by Theorem <ref>, u must be a pseudoconformal transformation of a soliton. However, this gives a contradiction, since pseudoconformal transformations of a soliton blow up in one time direction and scatter in the other. Therefore, Case 2 cannot happen.




§ DECOMPOSITION OF THE ENERGY

Now decompose the energy. Recall that

    E[u] = 1/2∫ |(∂_r - m + A_θ[u]/r) u|^2 dx + 1 - g/4∫ |u|^4.

Now then, let u = Q + ϵ for ϵ_L^2≪ 1, ϵ is real valued.

    (∂_r - m + A_θ[u]/r) u _L^2 =  (∂_r - m + A_θ[Q]/r) Q + (Re ∫_0^r Q ϵ̅ s ds/r) Q + (∂_r - m + A_θ[Q]/r) ϵ_L^2
     + O(ϵ_L^2ϵ_Ḣ_m^1 + ϵ_L^2^2).

Indeed, decompose

    (∂_r - m + A_θ[u]/r)(Q + ϵ) = (∂_r - m + A_θ[Q]/r)Q - (A_θ[u] - A_θ[Q]/r) Q + (∂_r - m + A_θ[Q]/r) ϵ - (A_θ[u] - A_θ[Q]/r) ϵ.

By direct computation,

    (A_θ[u] - A_θ[Q]/r) ϵ_L^2 = 1/2 (∫_0^r [|u|^2 - Q^2] s ds) ϵ/r_L^2≲ϵ_L^2ϵ_Ḣ_m^1.

Meanwhile,

    -(A_θ[u] - A_θ[Q]/r) Q = 1/r Re (∫_0^r Q ϵ̅ s ds)Q + 1/2r (∫_0^r |ϵ|^2 s ds) Q.

Again by direct computation,

    Q/2r (∫_0^r |ϵ|^2 s ds) _L^2≲ϵ_L^2^2.

Expanding

    1/2 (∂_r - m + A_θ[Q]/r) Q + (Re ∫_0^r Q ϵ̅ s ds/r) Q + (∂_r - m + A_θ[Q]/r) ϵ_L^2^2 + 1 - g/4 u _L^4^4
    
    = 1/2 (∂_r - m + A_θ[Q]/r) Q _L^2^2 + 1 - g/4 Q _L^4^4 + ⟨ (∂_r - m + A_θ[Q]/r) Q, (Re ∫_0^r Q ϵ̅ s ds/r) Q + (∂_r - m + A_θ[Q]/r) ϵ⟩
     + (1 - g) Re ∫ Q^3ϵ̅ + 1/2  (Re ∫_0^r Q ϵ̅ s ds/r) Q + (∂_r - m + A_θ[Q]/r) ϵ_L^2^2 + 3(1 - g) ∫ Q^2ϵ^2
     + O(ϵ_L^2^5/2ϵ_Ḣ_m^1^1/2 + ϵ_Ḣ_m^1^2ϵ_L^2^2).

Since E[Q] = 0,

    1/2 (∂_r - m + A_θ[Q]/r) Q _L^2^2 + 1 - g/4 Q _L^4^4 = 0.

Next, integrating by parts,

    ⟨ (∂_r - m + A_θ[Q]/r) Q, (Re ∫_0^r Q ϵ̅ s ds/r) Q + (∂_r - m + A_θ[Q]/r) ϵ⟩
     = ⟨ -∂_rr Q - 1/r∂_r Q + ∂_r(A_θ[Q]) Q/r + (m + A_θ[Q]/r)^2 Q - 1/2 Q^3, ϵ⟩ - ∫m + A_θ[Q]/r^2 Q^2 Re(∫_0^r Q ϵ̅ s ds) r dr 
    
    = ⟨ -∂_rr Q - 1/r∂_r Q - Q^3 + (m + A_θ[Q]/r)^2 Q + A_0[Q] Q, ϵ⟩.

Since

    (∂_r^2 + 1/r∂_r - α - (m + A_θ[Q]/r)^2 - A_0[Q] - g Q^2)Q = 0,


    ⟨  -∂_rr Q - 1/r∂_r Q - Q^3 + (m + A_θ[Q]/r)^2 Q + A_0[Q] Q, ϵ⟩ + (1 - g) ⟨ Q^3, ϵ⟩ = -α⟨ Q, ϵ⟩ = α/2ϵ_L^2^2.

The last equality uses the fact that Q + ϵ_L^2 =  Q _L^2.


Now let

    L_Q f = (Re ∫_0^r Q f̅ s ds/r) Q + (∂_r - m + A_θ[Q]/r) f.

We have proved

    E[Q + ϵ] = α/2ϵ_L^2^2 + 1/2 L_Qϵ_L^2^2 + 3/2(1 - g) ∫ Q^2ϵ^2 + O(ϵ_Ḣ_m^1^2ϵ_L^2^2 + ϵ_L^2^4 + ϵ_Ḣ_m^1^1/2ϵ_L^2^5/2).

Therefore, if

    L_Qϵ_L^2^2 + 3(1 - g) ∫ Q^2ϵ^2≳ϵ_Ḣ_m^1^2,

then

    E[Q + ϵ] ≳ϵ_L^2^2 + ϵ_Ḣ_m^1^2.




We claim that there exists a rapidly decreasing negative eigenfunction ψ of (<ref>) that satisfies

    ⟨ Q + x ·∇ Q, ψ⟩≠ 0.

Furthermore, if ϵ⊥ψ, then (<ref>) holds.



Using Lemma 2.1 of <cit.>,

    L_Q f _L^2∼ f _ℋ̇_m^1,

for f orthogonal to 𝒵_1 and 𝒵_2 satisfying the transversality condition. Then for g sufficiently close to 1, (<ref>) holds. The transversality condition from <cit.> is that

    det [ (Λ Q, 𝒵_1)  (iQ, 𝒵_1); (Λ Q, 𝒵_2)  (iQ, 𝒵_2) ]≠ 0,     Λ = 1 + x ·∇,

so (<ref>) certainly holds. We conjecture that (<ref>) holds for any g > 1.


If the above claim is true, then by the implicit function theorem we can prove the following.

Suppose there exists some λ_0(t) > 0, γ_0(t) ∈ℝ such that

    e^i γ_0(t)λ_0(t) u(t, λ_0(t) x) - Q(x) _L^2≤η_0.

Suppose without loss of generality that λ_0(t) = 1 and γ_0(t) = 0. Then there exists γ(t) ∈ℝ, λ(t) > 0 such that

    e^i γ(t)λ(t) u(t, λ(t) x) - Q(x) _L^2≤ 2η_0,


    ⟨ e^i γ(t)λ(t) u(t, λ(t) x) - Q(x), ψ⟩ = ⟨ e^i γ(t)λ(t) u(t, λ(t) x) - Q(x), i ψ⟩ = 0,

and

    |γ(t)| + |λ(t) - 1| ≲ϵ_L^2 + ϵ_L^2ϵ_L^4^2.








§ LONG TIME STRICHARTZ ESTIMATES



 §.§ Estimates for the mass-critical NLS

As a warm-up, we prove an estimate for a mass-critical NLS.

Suppose u is a solution to the mass-critical nonlinear Schrödinger equation,

    i u_t + Δ u = -|u|^2 u,      u(0, x) = u_0,      u_0_L^2 =  Q _L^2.

Furthermore, suppose that for some interval [a, b] with a > 0,

    ∫_a^bλ(t)^-2 dt = T,      T > η_∗^-1,


    sup_t > 0 dist(u(t), ℳ) ≤η_∗,

and that u is equivariant of order m. Finally, suppose that for all t ∈ [a, b],

    1 ≤λ(t) ≤ T^1/100.

Then,

    ∫_a^bϵ(t) _L^2^2λ(t)^-2 dt ≤ 3(ϵ_2(a), Q + x ·∇ Q)_L^2 - (3 ϵ_2(b), Q + x ·∇ Q)_L^2 + O(T^-8).
 


It suffices to prove the proposition under the bootstrap assumption,

    ∫_a^bϵ(t) _L^2^2λ(t)^-2 dt ≤η_∗^1/2.

Indeed, since ϵ(t) _L^2^2≲η_∗^2 for all t ∈ [a, b],

    ∫_a^b'ϵ(t) _L^2^2 dt ≤η_∗^1/2,

where

    ∫_a^b'λ(t)^-2 dt ∼ϵ_∗^-3/2≫ϵ_∗^-1.

If b' ≥ b, then the proof is complete. If b' < b, then since (<ref>)–(<ref>) hold on [a, b'], (<ref>) implies

    ∫_a^b'ϵ(t) _L^2^2λ(t)^-2 dt ≲η_∗≪η_∗^1/2.

By standard bootstrap arguments, the proof of Proposition <ref> would then be complete.

The proof of Proposition <ref> depends on two propositions: the long time Strichartz estimate and  the almost conservation of energy.

Under the conditions of Proposition <ref> and (<ref>), if N = T^1/3,

    P_> N u _U_Δ^2([a, b] ×ℝ^2)^2≲1/T^10 + 1/T∫_a^bϵ(t) _L^2^2λ(t)^-2 dt.




Under the conditions of Proposition <ref> and (<ref>), if N = T^1/3,

    sup_t ∈ [a, b] E(P_≤ N u)(t) ≲1/T^10 + N^2/T∫_a^bϵ(t) _L^2^2λ(t)^-2 dt.


Let χ∈ C_0^∞(ℝ^2) be a radially symmetric function, χ = 1 for r ≤ 1, χ supported on r ≤ 2, χ(r) is decreasing as a function of r, and let

    ϕ(r) = ∫_0^rχ^2(s/R) ds,

and let

    M(t) = ∫ϕ(x) x/|x|· Im[P_≤ N u∇ P_≤ N u](t, x) dx.

By direct computation,

    d/dt∫ϕ(x) x/|x|· Im[u̅∇ u](t,x) dx = 2 ∫χ^2(x/R) |∇ u(t,x)|^2 dx - ∫χ^2(x/R) |u(t,x)|^4 dx 
    
    - 1/2∫ [1/|x|ϕ(x) - χ^2(x/R)] |u(t,x)|^4 dx + 2 ∫ [1/|x|ϕ(x) - χ^2(x/R)] x_j x_k/|x|^2 Re(∂_j u∂_k u)(t,x) dx 
    
    + O(1/R^2∫_|x| > R |u(t,x)|^2 dx).


The last estimate follows from the fact that 1/|x|ϕ(x) = 1 for |x| ≤ R.

For R ≫λ(t), say R = T^1/25, since Q is rapidly decreasing,

    ∫_|x| > R1/R^2 |u(t,x)|^2 dx ≲1/R^2ϵ(t) _L^2^2 + 1/R^2 T^10≪1/λ(t)^2ϵ(t) _L^2^2 + 1/λ(t)^2 T^10.

Next,

    2 ∫ [1/|x|ϕ(x) - χ^2(x/R)] x_j x_k/|x|^2 Re(∂_j u∂_k u)(t,x) dx ≥ 0.

Next, since Q is rapidly decreasing,

    - 1/2∫ [1/|x|ϕ(x) - χ^2(x/R)] |u(t,x)|^4 dx ≲1/λ(t)^2 T^10 + ϵ(t) _L^4^4.

Finally,

    ∫χ^2(x/R) |u(t,x)|^4 dx = ∫χ^4(x/R) |u(t,x)|^4 dx + ∫χ^2(x/R) |u(t,x)|^4 - ∫χ^4(x/R) |u(t,x)|^4
     = ∫χ^4(x/R) |u(t,x)|^4 dx + O(1/λ(t)^21/T^10) + O(ϵ(t) _L^4^4).

Therefore,

    ∫_a^b E[χ(x/R) u(t, x)] dt ≤∫ϕ(x) x/|x|· Im[u̅∇ u](t,x)|_a^b + o(∫_a^b1/λ(t)^2ϵ(t) _L^2^2 dt) + ∫_a^b1/λ(t)^21/T^10 dt.

The last estimate uses the Strichartz estimate

    ∫_a^bϵ(t) _L^4^4 dt ≪∫_a^b1/λ(t)^2ϵ(t) _L^2^2 dt.

Using the energy lower bound, E[χ(x/R) u] ≥1/2 λ(t)^2ϵ(t) _L^2^2 - O(1/λ(t)^2 T^10), then

    1/4∫_a^b1/λ(t)^2ϵ(t) _L^2^2 dt ≤∫ϕ(x) x/|x|· Im[u̅∇ u](t,x)|_a^b + O(1/T^9).

Replacing u by P_≤ N u and ignoring the error terms arising from the frequency truncation, (which are handled by Proposition <ref>),

    ∫ϕ(x) x/|x|· Im[P_≤ N u∇ P_≤ N u](t,x) dx = 2 ∫ Im[ϵ(t,x) (Q(x) + x ·∇ Q(x))] dx 
     + ∫ϕ(x) x/|x|· Im[P_≤ N1/λ(t)ϵ(t, x/λ(t))∇ P_≤ N1/λ(t)ϵ(t, x/λ(t))] dx.

Now then, using (<ref>) and Proposition <ref>,

    ∫ϕ(x) x/|x|· Im[P_≤ N1/λ(t)ϵ(t, x/λ(t))∇ P_≤ N1/λ(t)ϵ(t, x/λ(t))] dx 
    ≲ R N ϵ(t) _L^2∇ P_≤ Nϵ(t) _L^2≲ R T^1/100 E[P_≤ N u(t)] ≲1/T^9 + N^2 R T^1/100/T∫_a^b1/λ(t)^2ϵ(t) _L^2^2 dt.

Absorbing the second term on the right hand side of (<ref>) into the left hand side of (<ref>) proves Proposition <ref>, if Propositions <ref> and <ref> hold.




By the intermediate value theorem, there exists t_0∈ [a, b] such that

    ϵ(t_0) _L^2^2≤1/T∫_a^b1/λ(t)^2ϵ(t) _L^2^2 dt.

By the Sobolev embedding theorem,

    E(P_≤ N u(t_0)) ≲N^2/T∫_a^b1/λ(t)^2ϵ(t) _L^2^2 dt.

Computing the change of energy,

    d/dt E(P_≤ N u(t)) = (-P_> N(|u|^2 u) + [|u|^2 u - |P_≤ N u|^2 (P_≤ N u)], P_≤ N u_t)_L^2.

By Proposition <ref> and the properties of the Littlewood–Paley projection operator,

    ∫_a^b |(-P_> N (|u|^2 u), P_≤ N u_t)_L^2| dt ≲ N^2 P_> N (|u|^2 u) _L_t^2 L_x^1 P_N/2 < ·≤ N u _L_t^2 L_x^∞
     +  P_> N (|u|^2 u) _L_t^2 L_x^1 P_> N/8 u _L_t^2 L_x^∞ P_≤ N u _L_t,x^∞^2 +  P_N < · < 2N (|u|^2 u) _L_t,x^2 P_N/2 < · N (|P_> N/8 u|^2 (P_> N/8 u)) _L_t,x^2
    ≲N^2/T + N^2/T∫_a^bϵ(t) _L^2^21/λ(t)^2 dt.

Now decompose

    |u|^2 u - |P_≤ N u|^2 (P_≤ N u) = (P_> N u)^3 + 3 (P_> N u)^2 (P_≤ N u) + 3 (P_> N u)(P_≤ N u)^2.

Now then,

    ∫ |((P_> N u)^2 u, Δ P_≤ N u)_L^2| dt ≲ N^2 P_> N u _L_t^2 L_x^∞^2 u _L_t^∞ L_x^2^2≲N^2/T (∫_a^bϵ(t) _L^2^2λ(t)^-2 dt) + N^2/T^10.


    ∫ |((P_> N u)^2 u, P_≤ N (|u|^2 u))_L^2| dt ≲ P_N < · < 2N ((P_> N u)^2 u) _L_t^1 L_x^∞ u _L_t^∞ L_x^2^2 P_≤ N u _L_t,x^∞
    
    +  P_≤ 2N((P_> N u)^2 u) _L_t,x^2 P_≤ N((P_> N u)^2 u) _L_t,x^2≲N^2/T (∫_a^bϵ(t) _L^2^2λ(t)^-2 dt) + N^2/T^10.



    ∫ |((P_> N u)(P_≤ N u)^2, Δ P_≤ N u)_L^2| dt ≲ N^2 P_> N u _L_t^2 L_x^∞ P_> N/8 u _L_t^2 L_x^∞ u _L_t^∞ L_x^2^2≲N^2/T (∫_a^bϵ(t) _L^2^2λ(t)^-2 dt) + N^2/T^10.


    ∫ |((P_> N u)(P_≤ N u)^2, P_≤ N((P_> N u) u^2))_L^2| dt ≲ P_≤ N u _L_t,x^∞^2 P_> N u _L_t^2 L_x^∞^2 u _L_t^∞ L_x^2^2
    ≲N^2/T (∫_a^bϵ(t) _L^2^2λ(t)^-2 dt) + N^2/T^10.


    ∫ |((P_> N u)(P_≤ N u)^2, P_≤ N((P_≤ N u)^3))_L^2| dt ≲ P_≤ N u _L_t,x^∞^2 P_> N/8 u _L_t^2 L_x^∞^2 u _L_t^∞ L_x^2^2
    ≲N^2/T (∫_a^bϵ(t) _L^2^2λ(t)^-2 dt) + N^2/T^10.

This completes the proof of Proposition <ref>.





This proposition is proved using induction on frequency. Fix T^1/6≤ M ≤ T^1/3.

    P_> M u _U_Δ^2([a, b] ×ℝ^2)≲1/T^5 + inf_t ∈ [a, b]ϵ(t) _L^2 +  P_> M (|u|^2 u) _DU_Δ^2([a, b] ×ℝ^2).

Since u = 1/λ Q(x/λ) + 1/λϵ(t, x/λ), for any δ > 0,

    P_> M(|u|^2 u) _DU_Δ^2([a, b] ×ℝ^2)≲_δ (P_> M/8 u) _L_t^∞ L_x^21/λϵ(t, x/λ) _L_t^2 L_x^∞^2
    
    +  (P_> M/8 u) (1/λ(t) Q(x/λ(t))^2) _L_t^1 L_x^2([a, b] ×ℝ^2)^δ· (M^-1/2∑_j (P_> M/8 u) (1/λ(t) Q(x/λ(t))^2) _L_t, x^2 ([a, b] ×{ |x| ∼ 2^j}))^1 - δ.

Using standard Strichartz estimates,

    (P_> M/8 u) _L_t^∞ L_x^21/λϵ(t, x/λ) _L_t^2 L_x^∞^2≲ P_> M/8 u _U_Δ^2([a, b] ×ℝ^2) (∫_a^b1/λ(t)^2ϵ(t) _L^2^2 dt) ≲η_∗^1/2 P_> M/8 u _U_Δ^2([a, b] ×ℝ^2).

Since λ(t) ≥ 1,

    (P_> M/8 u)(1/λ(t) Q(x/λ(t))) _L_t,x^2≲sup_R > 0 R^-1/2 P_> M/8 u _L_t,x^2(|x| ≤ R)·∑_j ≥ 0 2^-j∑_k 2^k/2 Q(2^-j 2^k) 
    ≲1/M^1/2 P_> M/8 u _U_Δ^2([a, b] ×ℝ^2).

Also, since λ(t) ≥ 1,

    ∑_k 2^k/21/λ(t) Q(x/λ(t)) _L_t,x^∞(|x| ∼ 2^k)≲∑_j ≥ 0 2^-j∑_k 2^k/2 Q(2^-j 2^k) ≲ 1.

Finally, make the trivial estimate

    1/λ(t) Q(x/λ(t)) _L_t^2 L_x^∞≲ (∫_a^b1/λ(t)^2 dt)^1/2≲ T^1/2.

Plugging (<ref>)–(<ref>) into (<ref>),

    (<ref>) ≲ (η_∗^1/2 + T^δ/2/M^1 - δ/2)  P_> M/8 u _U_Δ^2([a, b] ×ℝ^2).

Arguing by induction on frequency, starting from (<ref>), which implies

    u _U_Δ^2([a, b] ×ℝ^2)^2≲ T,

for N = T^1/3, there exists some c > 0 such that

    P_> N u _U_Δ^2([a, b] ×ℝ^2)^2≲1/T^10 + 1/T∫_a^bϵ(t) _L^2^21/λ(t)^2 dt + T(η_∗^c/2·ln(T) + T^-c ln(T)) ≲1/T^10 + 1/T∫_a^bϵ(t) _L^2^21/λ(t)^2 dt.







 §.§ Estimates for the Chern–Simons–Schrödinger equation

Now we can prove a similar estimate for a solution to the Chern–Simons–Schrödinger equation.

Suppose u is a solution to (<ref>),

    i u_t + Δ u = 2m/r^2 A_θ[u] u + A_0[u] u + 1/r^2 A_θ[u]^2 u - g |u|^2 u,      u _L^2 =  Q _L^2.

Furthermore, suppose that for some interval [a, b] with a > 0,

    ∫_a^bλ(t)^-2 dt = T,      T > η_∗^-1,


    sup_t > 0 dist(u(t), ℳ) ≤η_∗,

and that u is equivariant of order m. Finally, suppose that for all t ∈ [a, b],

    1 ≤λ(t) ≤ T^1/100.

Then,

    ∫_a^bϵ(t) _L^2^2λ(t)^-2 dt ≤ 3(ϵ_2(a), Q + x ·∇ Q)_L^2 - (3 ϵ_2(b), Q + x ·∇ Q)_L^2 + O(T^-8).
 


Proof of Proposition <ref> is the same as the proof of Proposition <ref>, making use of a long time Strichartz estimate that is analogous to Proposition <ref> and an almost conservation of energy result analogous to Proposition <ref>. The main difficulty is the fact that the Chern–Simons–Schrödinger equation is nonlocal. Once again, we make a bootstrap assumption analogous to (<ref>).


If u satisfies the conditions of Proposition <ref>,

    P_> N u _U_Δ^2([a, b] ×ℝ^2)^2≲1/T^10 + 1/T∫_a^bϵ(t) _L^2^2λ(t)^-2 dt.



As in the proof of Proposition <ref>, for any T^1/6≤ M ≤ T^1/3,

    P_> M (g |u|^2 u) _DU_Δ^2([a, b] ×ℝ^2)≲ g (η_∗^1/2 + T^δ/2/M^1 - δ/2)  P_> M/8 u _U_Δ^2([a, b] ×ℝ^2).

Expanding

    A_θ[u] = -1/2∫_0^r |u(t, s)|^2 s ds = -1/2∫_0^r1/λ(t)^2 Q(s/λ(t))^2 s ds 
     - Re ∫_0^r1/λ(t)^2 Q(s/λ(t)) ϵ(t, s/λ(t)) s ds
    - 1/2∫_0^r1/λ(t)^2 |ϵ(t, s/λ(t))|^2 s ds.

Since

    -1/2 r^2∫_0^r1/λ(t)^2 Q(s/λ(t))^2 s ds ≲inf{1/λ(t)^2, 1/r^2λ(t)}, 
    
    -1/2 r^2∫_0^r1/λ(t)^2 |ϵ(t, s/λ(t))|^2 s ds ≲1/λ(t)^2ϵ(t) _L^∞^2,

then using an argument similar to (<ref>)–(<ref>),

    (1/r^2 A_θ[u]) (P_> M/8 u) _DU_Δ^2([a, b] ×ℝ^2)≲ (η_∗^1/2 + T^δ/2/M^1 - δ/2)  P_> M/8 u _U_Δ^2([a, b] ×ℝ^2).

Meanwhile,

    P_> M(1/r^2 A_θ[u] (P_≤M/8 u)) = P_> M(P_> M/2(1/r^2 A_θ[u]) · (P_≤M/8 u)).


Making a change of variables,

    m/r^2 A_θ[u] = -m/2∫_0^1 |u(t, s r)|^2 s ds.

Therefore,

    P_> M/2 (2m/r^2 A_θ[u]) _L_t,x^2≲ P_> M/8 u _L_t^2 L_x^∞.

Decomposing u = 1/λ(t) Q(x/λ) + 1/λ(t)ϵ(t, x/λ(t)) and using (<ref>) for the Q term and (<ref>) for the ϵ term,

    P_> M/2 (2m/r^2 A_θ[u]) (P_≤M/8 u) _DU_Δ^2≲ (η_∗^1/2 + T^δ/2/M^1 - δ/2)  P_> M/8 u _U_Δ^2([a, b] ×ℝ^2).

By a similar argument,

    P_> M/2 (1/r^2 A_θ[u]^2) (P_≤M/8 u) _DU_Δ^2≲ (η_∗^1/2 + T^δ/2/M^1 - δ/2)  P_> M/8 u _U_Δ^2([a, b] ×ℝ^2).

Indeed, let χ∈ C_0^∞(ℝ^2), χ(x) = 1 for |x| ≤ 1. Let

    Ã_θ[u] = ∫_0^r |P_≤M/8 u(t, s)|^2 s ds.

Then,

    Ã_θ[u]^2/r^2 = r^2∫_0^1 |P_≤M/8 u(t, s r)|^2 s ds ·∫_0^1 |P_≤M/8 u(t, s' r)|^2 s' ds'.

Then by Fourier support arguments, for any N,

    P_> M(Ã_0[u]^2/r^2) _L_t,x^2≲ P_> M/2(χ(r) r^2) _L_t^∞ L_x^2 P_≤M/8 u _L_t^8 L_x^∞^4≲1/M^N T^1/2 M^3.

Now let ψ_j(r) = χ(2^-j r) - χ(2^-j + 1 r). Then,

    ∑_j P_> M/2(ψ_j(r) r^2) _L_x^2 P_≤M/8 u _L_t^8 L_x^∞^4≲∑_j 2^-j1/M^N T^1/2 M^3≲T^1/2/M^N.

Next, for any R,

    1/r^2∫_0^r |P_> M/8 u(t, s)| |u(t, s)| s ds _L^2(R ≤·≤ 2R)≲∑_j ≤ 0 2^j P_> M/8 u _L^∞ u _L^2(2^j R ≤·≤ 2^j + 1 R).

Therefore, by Young's inequality, A_0[u] _L^∞ + Ã_0[u] _L^∞≲ u _L^2^2, (<ref>), (<ref>), M ≥ T^1/6, and again using (<ref>) for the Q term and (<ref>) for the ϵ term,

    P_> M(A_0[u]^2/r^2) u _DU_Δ^2([a, b] ×ℝ^2) (η_∗^1/2 + T^δ/2/M^1 - δ/2)  P_> M/8 u _U_Δ^2([a, b] ×ℝ^2) + 1/T^10.


Finally turn to the A_0[u] u term. Since Q ∈ℋ_m^1,

    m/r Q _L^2∼∫_0^∞m^2/r Q(r)^2 dr < ∞.

Therefore,

    ∫_r^∞1/λ(t)^2 Q(s/λ(t))^21/s ds ≲inf{1/λ(t) r^2, 1/λ(t)}.

As in (<ref>),

    ∫_r^∞1/λ(t)^2 |ϵ(t, s/λ(t))|^21/s ds _L^2(R ≤·≤ 2R)≲1/λ(t)∑_j ≥ 0 2^-j/21/λ(t)ϵ(t, s/λ(t)) _L^2(2^j R ≤·≤ 2^j + 1 R)ϵ_L^∞.

Again, as in (<ref>)–(<ref>),

    A_0[u] (P_> M/8 u) _DU_Δ^2([a, b] ×ℝ^2)≲ (η_∗^1/2 + T^δ/2/M^1 - δ/2)  P_> M/8 u _U_Δ^2([a, b] ×ℝ^2).


It remains to estimate

    P_> M(A_0[u] (P_≤M/8 u)).

Let

    Ã_0[u] = -∫_r^∞ (m + Ã_θ[u])(t, s) |P_≤M/8 u(t, s)|^2ds/s.

By direct computation, following (<ref>) and using the fact that A_θ[u] _L^∞ + Ã_θ[u] _L^∞≲ u _L^2^2 and |A_θ[u] - Ã_θ[u]|(r) ≲ |∫_0^r |u| |P_≥M/8 u| s ds| ≲ r  u _L^2 P_≥M/8 u _L^∞,

    A_0[u] - Ã_0[u] _L_t,x^2≲∫_r^∞(m + Ã_θ[u](t, s)/s |u(t, s)| |P_> M/8 u(t, s)| ds _L_t,x^2 + ∫_r^∞|A_θ[u] - Ã_θ[u]/s |u(t, s)|^2 ds _L_t,x^2
    ≲ u _L^2^2∫_r^∞|u(t, s)| |P_≥M/8 u|/s ds _L_t,x^2 + ∫_r^∞ u _L^2 P_≥M/8 u _L^∞ |u(t, s)|^2 ds _L_t,x^2≲ u _L_t^∞ L_x^2^3 P_≥M/8 u _L_t^2 L_x^∞.

By (<ref>), again using (<ref>) for the Q term and (<ref>) for the ϵ term,

    (A_0[u] - Ã_0[u]) (P_≤M/8 u) _DU_Δ^2([a, b] ×ℝ^2)≲ (η_∗^1/2 + T^δ/2/M^1 - δ/2)  P_> M/8 u _U_Δ^2([a, b] ×ℝ^2).

Letting

    c = -∫_0^∞(m + Ã_θ[u](t, s)/s |P_≤M/8 u(t, s)|^2 ds,


    P_> M (Ã_0[u] (P_≤M/8 u)) = P_> M ((Ã_0[u] - c) · (P_≤M/8 u)) = P_> M (-∫_0^rm + Ã_θ[u](t, s)/s |P_|≤M/8 u(t, s)|^2 ds · (P_≤M/8 u(t,r))).

Expanding out (<ref>), using (<ref>),

    (<ref>) = -∫_0^1 m |P_≤M/8 u(t, sr)|^2ds/s· (P_≤M/8 u(t, r)) - ∫_0^1Ã_θ(t, sr)/s^2 r^2 |u(t, sr)|^2 s ds · (P_≤M/8 u(t, r)) 
    
    = -∫_0^1 m |P_≤M/8 u(t, sr)|^2ds/s· (P_≤M/8 u(t, r)) 
     - ∫_0^1∫_0^1 |P_≤M/8 u(t, s s' r)|^2 |P_≤M/8 u(t, sr)|^2 s s' ds ds' · (P_≤M/8 u(t, r)).

Therefore, P_> M (<ref>) = 0. Arguing using induction on frequency proves Proposition <ref>.




In fact, the same argument implies

    P_> N u _U_Δ^2([a, b] ×ℝ^2)^2≲1/T^10 + inf_t ∈ [a, b]ϵ(t) _L^2^2.

Of course, by the intermediate value theorem,

    inf_t ∈ [a, b]ϵ(t) _L^2^2≲1/T∫_a^bϵ(t) _L^2^2λ(t)^-2 dt,

but to prove rigidity we will use a better bound.








If u satisfies the conditions of Proposition <ref>, and N ≥ T^1/3,

    sup_t ∈ [a, b] E(P_≤ N u)(t) ≲ N^2inf_t ∈ [a, b]ϵ(t) _L^2^2 + N^2/T^10.

In particular,

    sup_t ∈ [a, b] E(P_≤ N u)(t) ≲N^2/T∫_a^bϵ(t) _L^2^2λ(t)^-2 dt + N^2/T^10.



Recall that

    E[u] = 1/2∫ |∂_r u|^2 + 1/2∫ (m + A_θ[u]/r)^2 |u|^2 - g/4∫ |u|^4.

Now compute

    d/dt1/2∫ |∂_r P_≤ N u|^2 = Re ∫ (∂_r P_≤ N u_t)(∂_r P_≤ N u) r dr = -Re ∫ (P_≤ N u_t)(∂_rr + 1/r∂_r)(P_≤ N u) r dr.


    d/dt(-g/4∫ |P_≤ N u|^4) = -g Re ∫ (P_≤ N u_t)(|P_≤ N u|^2 (P_≤ N u)) dx.


    d/dt1/2∫ (m + A_θ[P_≤ N u]/r)^2 |P_≤ N u|^2 = Re ∫ (m + A_θ[P_≤ N u]/r)^2 (P_≤ N u_t)(P_≤ N u) 
     + Re ∫Ȧ_θ[P_≤ N u]/r (m + A_θ[P_≤ N u]/r) |P_≤ N u|^2.


Split

    |P_≤ N u|^2 (P_≤ N u) = P_≤ N(|u|^2 u) + P_> N(|u|^2 u) + |P_≤ N u|^2 (P_≤ N u) - |u|^2 u.


    2m A_θ[P_≤ N u]/r^2 (P_≤ N u) = P_≤ N (2m A_θ[u]/r^2 u) + P_> N (2m/r^2 A_θ[u] u) + 2m A_θ[P_≤ N u]/r^2 (P_≤ N u) - 2m A_θ[u]/r^2 u.


    A_θ[P_≤ N u]^2/r^2 (P_≤ N u) = P_≤ N (A_θ[u]^2/r^2 u) + P_> N(A_θ[u]^2/r^2 u) + A_θ[P_≤ N u]^2/r^2 (P_≤ N u) - A_θ[u]^2/r^2 u.

Finally,

    Re ∫Ȧ_θ[P_≤ N u]/r (m + A_θ[P_≤ N u]/r) |P_≤ N u|^2 = ∫ (P_≤ N u_t) A_0[P_≤ N u](P_≤ N u).

Expanding,

    A_0[P_≤ N u](P_≤ N u) = P_≤ N(A_0[u] u) + P_> N(A_0[u] u) + A_0[P_≤ N u](P_≤ N u) - A_0[u] u.

Since u is m-equivariant,

    (∂_rr + 1/r∂_r) (P_≤ N u) - m^2/r^2 (P_≤ N u) = Δ P_≤ N u = P_≤ NΔ u = P_≤ N (∂_rr + 1/r∂_r - m^2/r^2) u.

Taking the first term in (<ref>)–(<ref>),

    Re ∫ (P_≤ N u_t) · P_≤ N(-∂_rr u - 1/r∂_r u + m^2/r^2 u + 2m A_θ[u]/r^2 u + A_θ[u]^2/r^2 u + A_0[u] u - g |u|^2 u) = Re ∫ (P_≤ N u_t) (P_≤ N i u_t) = 0.

Next, taking the second terms in (<ref>)–(<ref>),

    Re ∫ (P_≤ N u_t) · P_> N (-g |u|^2 u + 2m/r^2 A_θ[u] u + A_θ[u]^2/r^2 u + A_0[u] u) 
    
    = Re ∫ (P_N/2≤·≤ N u_t) · P_N < · < 2N (-g |u|^2 u + 2m/r^2 A_θ[u] u + A_θ[u]^2/r^2 u + A_0[u] u).

Following the computations in the proof of Proposition <ref>,

    (<ref>) ≲ N^2 P_> N/2 u _L_t^2 L_x^∞ P_> N (-g |u|^2 u + 2m/r^2 A_θ[u] u + A_θ[u]^2/r^2 u + A_0[u] u) _L_t^2 L_x^1
    
    +  P_N/2 < · < 2N (-g |u|^2 u + 2m/r^2 A_θ[u] u + A_θ[u]^2/r^2 u + A_0[u] u) _L_t,x^2^2≲ N^2inf_t ∈ [a, b]ϵ(t) _L^2^2 + N^2/T^10.


Now take the third term in (<ref>)–(<ref>). To simplify notation let

    𝒩 = -g (|u|^2 u - |P_≤ N u|^2 (P_≤ N u)) + 2m/r^2 (A_θ[P_≤ N u](P_≤ N u) - A_θ[u] u) 
     + 1/r^2(A_θ[P_≤ N u]^2 (P_≤ N u) - A_θ[u]^2 u) + (A_0[P_≤ N u](P_≤ N u) - A_0[u] u).

Since the computations proving Proposition <ref> still hold,

    𝒩_L_t^2 L_x^1^2≲inf_t ∈ [a, b]ϵ(t) _L^2^2 + 1/T^10.




Now split,

    ∫ (P_≤ N u_t) 𝒩 = ∫ (P_N/8 < · < N u_t)𝒩 
    + ∫ (P_≤N/8 u_t)𝒩.

Again following the computations in (<ref>),

    ∫ (P_N/8 < · < N u_t)𝒩≲N^2/T^10 + N^2inf_t ∈ [a, b]ϵ(t) _L^2^2.


By the Sobolev embedding theorem,

    2m/r^2 A_θ[P_≤ N u] (P_≤ N u) + 1/r^2 A_θ[P_≤ N u]^2 (P_≤ N u) + A_0[P_≤ N u](P_≤ N u) + g |P_≤ N u|^2 (P_≤ N u) _L_t^∞ L_x^2≲ N^2 u _L^2^3.

Also, by (<ref>) and the Sobolev embedding theorem,

    P_≤N/8𝒩_L_t^2 L_x^∞≲ N^2𝒩_L_t^2 L_x^1≲ N^2 (inf_t ∈ [a, b]ϵ(t) _L^2^2 + 1/T^10)^1/2.

Combining (<ref>) and (<ref>),

    P_≤N/8 u_t_L_t^∞ L_x^2 + L_t^2 L_x^∞≲ 1 + (inf_t ∈ [a, b]ϵ(t) _L^2^2 + 1/T^10)^1/2.

Now turn to the third terms in (<ref>)–(<ref>) with (P_≤ N u_t) replaced by (P_≤N/8 u_t). First take the third term in (<ref>). By Fourier support arguments,

    ∫ (P_≤N/8 u_t) (|P_≤ N u|^2(P_≤ N u) - |u|^2 u) ≲ P_> N u _L_t^2 L_x^∞ P_> N/8 u _L_t^2 L_x^∞∩ L_t^∞ L_x^2 u _L_t^∞ L_x^2 P_≤N/8 u_t_L_t^2 L_x^∞ + L_t^∞ L_x^2
    ≲ N^2 (inf_t ∈ [a, b]ϵ(t) _L^2^2 + 1/T^10).

Next take (<ref>),

    A_θ[u] u - A_θ[P_≤ N u](P_≤ N u) = A_θ[u] (P_> N u) + (A_θ[u] - A_θ[P_≤ N u])(P_≤ N u).

By standard Fourier support arguments,

    ∫ (P_≤N/8 u_t)(P_> N u) 2m/r^2 A_θ[u] = ∫  (P_≤N/8 u_t)(P_> N u) P_> N/2 (2m/r^2 A_θ[u]) ≲ N^2 (inf_t ∈ [a, b]ϵ(t) _L^2^2 + 1/T^10).

Next,

    ∫2m/r^2 (P_≤ N u)(P_≤N/8 u_t) { A_0[u] - A_0[P_≤ N u] } = ∫2m/r^2 (P_≤ N u)(P_≤N/8 u_t) { Re ∫_0^r 2 Re((P_≤ N u)(P_> N u)) s ds }
    
    + ∫2m/r^2 (P_≤ N u)(P_≤N/8 u_t) { Re ∫_0^r |P_> N u)|^2 s ds }.

Changing the order of integration,

    ∫2m/r^2 (P_≤ N u)(P_≤N/8 u_t) { Re ∫_0^r 2 Re((P_≤ N u)(P_> N u)) s ds }
     = Re ∫ (P_> N u)(P_≤ N u) { Re ∫_r^∞2m/s (P_≤ N u)(P_≤N/8 u_t) ds }.

Following the computations in (<ref>)–(<ref>),

    (<ref>) ≲N^2/T^10 + N^2inf_t ∈ [a, b]ϵ(t) _L^2^2.

Also,

    2m/r^2∫_0^r s |P_> N u|^2 ds _L_t^1 L_x^∞≲ P_> N u _L_t^2 L_x^∞^2,     2m/r^2∫_0^r s |P_> N u|^2 ds _L_t,x^2≲ u _L^2 P_> N u _L_t^2 L_x^∞.

Combining (<ref>) with (<ref>),

    ∫2m/r^2 (P_≤N/8 u_t)(P_≤ N u) (A_θ[u] - A_θ[P_≤ N u) ≲ N^2inf_t ∈ [a, b]ϵ(t) _L^2^2 + N^2/T^10.


Now turn to (<ref>). Expanding,

    A_θ[P_≤ N u]^2 (P_≤ N u) - A_θ[u]^2 u = A_θ[u]^2 (P_> N u) + (A_θ[P_≤ N u] - A_θ[u])(A_θ[P_≤ N u] + A_θ[u])(P_≤ N u).

Following (<ref>),

    ∫1/r^2 (P_≤N/8 u_t)(P_> N u) A_θ[u]^2≲ N^2inf_t ∈ [a, b]ϵ(t) _L^2^2 + N^2/T^10.

Next, as in (<ref>), after changing the order of integration and using (<ref>)–(<ref>),

    ∫1/r^2 (P_≤N/8 u_t)(P_> N u) (A_θ[P_≤ N u] - A_θ[u])(A_θ[P_≤ N u] + A_θ[u]) ≲ N^2inf_t ∈ [a, b]ϵ(t) _L^2^2 + N^2/T^10.

Finally, decompose

    A_0[P_≤ N u] (P_≤ N u) - A_0[u] u = -A_0[u](P_> N u) + (A_0[P_≤ N u] - A_0[u]) (P_≤ N u).

Again following (<ref>)–(<ref>),

    ∫ (P_≤N/8 u_t) A_0[u] (P_> N u) = ∫ P_> N/2 ((P_≤N/8 u_t) A_0[u]) (P_> N u) ≲ N^2inf_t ∈ [a, b]ϵ(t) _L^2^2 + N^2/T^10.

Finally, changing the order of integration,

    ∫ (P_≤N/8 u_t)(P_≤ N u)(A_0[P_≤ N u] - A_0[u]) 
     = ∫{ (m + A_θ[u]/r^2) |u|^2 - (m + A_θ[P_≤ N u]/r^2) |P_≤ N u|^2}·{∫_0^r (P_≤N/8 u_t)(P_≤ N u) s ds } r dr dt.

Then, applying the arguments in (<ref>)–(<ref>) to (<ref>) proves that

    (<ref>) ≲ N^2inf_t ∈ [a, b]ϵ(t) _L^2^2 + N^2/T^10.

This finally proves Proposition (<ref>).





Now we are finally ready to prove Proposition <ref>. The proof uses the Morawetz estimate.

Let u be a solution to (<ref>) and let ψ∈ C_0^∞(ℝ^2) be a smooth, radially symmetric function such that ψ(r) = r for r ≤ 1, ψ(r) = 3/2 for r > 2, and ∂_r(ψ(r)) = ϕ(r)^2 ϕ(r) ∈ C_0^∞(ℝ^2), and ϕ(r) ≥ 0. Then if

    M(t) = R ∫ψ(r/R) Im[u̅∂_r u](t, r) r dr,

then

    d/dt M(t) = 2 ∫ϕ^2(r/R) r |u_r|^2 + 2 ∫ϕ^2(r/R) (m + A_θ[u]/r)^2 |u|^2 - g ∫ϕ^2(r/R) |u|^4 r 
      + O(∫_r ≥ R1/r^2 |u|^2) + O(∫_r ≥ R |u|^4).



This follows by direct computation and integrating by parts.


Taking R = T^1/25, since Q is rapidly decreasing,

    ∫_r ≥ R |u(t, x)|^21/r^2 dx ≲1/R λ(t)^21/T^10 + 1/R λ(t)^2ϵ(t) _L^2^2.

Therefore,

    ∫_a^b∫_r ≥ R |u(t, x)|^21/r^2 dx dt ≪1/T^9 + ∫_a^b1/λ(t)^2ϵ(t) _L^2^2 dt.

Also using the fact that Q is rapidly decreasing combined with the Strichartz estimates, for any m ≥ 0,

    ∫_m^m + 1ϵ(s) _L^4^4 ds ≲ (∫_m^m + 1ϵ(s) _L^2^2)^2,


    ∫_a^b∫_r ≥ R |u(t, x)|^4 dx dt ≲∫_a^b1/R λ(t)^21/T^10 dt + ∫_a^b1/λ(t)^4∫ |ϵ(t, x/λ(t))|^4 dx dt ≤1/R1/T^9 + η_∗^2∫_a^b1/λ(t)^2ϵ(t) _L^2^2 dt.

Replacing u by P_≤ N u with N = T^1/3,

    2 ∫ϕ^2(r/R) r |∂_r P_≤ N u|^2 + 2 ∫ϕ^2(r/R) (m + A_θ[P_≤ N u]/r)^2 |P_≤ N u|^2 - g ∫ϕ^2(r/R) |P_≤ N u|^4 r 
    
    = 4 E[ϕ(r/R) P_≤ N u] + O(∫_r ≥ R1/r^2 |u|^2) + O(∫_r ≥ R |u|^4).

Therefore,

    ∫_a^bϵ(t) _L^2^21/λ(t)^2 dt ≲ R ∫ψ(r/R) Im[P_≤ N u∂_r P_≤ N u]|_a^b + ℰ,

where ℰ are the error terms arising from frequency truncation, see (<ref>)–(<ref>). Now then, using (<ref>) and the fact that R ψ(r/R) is smooth,

    ∫ Re[P_> N (-g |u|^2 u + 2m/r^2 A_θ[u] u + A_θ[u]^2/r^2 u + A_0[u] u) ·∂_rP_≤ N u] R ψ(r/R) 
    ≲ RN  P_> N (-g |u|^2 u + 2m/r^2 A_θ[u] u + A_θ[u]^2/r^2 u + A_0[u] u) _L_t^2 L_x^1 P_N/2 < · < N u _L_t^2 L_x^∞
    
     + RN  P_> N (-g |u|^2 u + 2m/r^2 A_θ[u] u + A_θ[u]^2/r^2 u + A_0[u] u) _L_t^2 L_x^1 P_≤N/2 u _L_t^2 L_x^∞ P_> N/2 (ψ(r/R)) _L^∞
    ≲ RN inf_t ∈ [a, b]ϵ(t) _L^2^2 + RN/T^10.

Integrating by parts, the same estimate holds for

    ∫ Re[P_≤ N u·∂_r (P_> N (-g |u|^2 u + 2m/r^2 A_θ[u] u + A_θ[u]^2/r^2 u + A_0[u] u) ))] R ψ(r/R).

Meanwhile, recalling (<ref>), and using (<ref>)–(<ref>) along with the fact that ψ(r/R) is smooth,

    R ∫ Re[𝒩̅∂_r P_≤ N u] ψ(r/R) dx dt + R ∫ Re[(P_≤ N u) ∂_r𝒩] ψ(r/R) ≲ RN inf_t ∈ [a, b]ϵ(t) _L^2^2 + RN/T^10.

Therefore, ∫_a^bℰ dt ≲ RN inf_t ∈ [a, b]ϵ(t) _L^2^2 + RN/T^10. Since by Proposition <ref>,

    R ∫ψ(r/R) Im[P̅_̅≤̅ ̅N̅ϵ̅∂_r P_≤ Nϵ] ≲ RN ϵ_L^2^2≲ T^1/50 RN E[u(t)] ≲T^1/50 RN/T∫_a^bϵ(t) _L^2^2λ(t)^-2 dt + 1/T^9,

and therefore (<ref>) holds.












§ AN L_S^P BOUND ON Ε(S) _L^2 WHEN P > 1

As in the case of the two-dimensional mass-critical problem, Proposition <ref> implies that ϵ(s) _L^2 lies in L_s^p for any p > 1.

Let u be a solution to (<ref>) that satisfies u _L^2 =  Q _L^2, and suppose

    sup_s ∈ [0, ∞)ϵ(s) _L^2≤η_∗,

and ϵ(0) _L^2 = η_∗. Then

    ∫_0^∞ϵ(s) _L^2^2 ds ≲η_∗,

with implicit constant independent of η_∗ when η_∗≪ 1 is sufficiently small.


Furthermore, for any j ∈ℤ_≥ 0, let

    s_j = inf{ s ∈ [0, ∞) : ϵ(s) _L^2 = 2^-jη_∗}.

By definition, s_0 = 0, and the continuity of ϵ(s) _L^2 combined with sequential convergence of blowup solutions implies that such an s_j exists for any j > 0. Then,

    ∫_s_j^∞ϵ(s) _L^2^2 ds ≲ 2^-jη_∗,

for each j ≥ 0, with implicit constant independent of η_∗.



Set T_∗ = 1/η_∗ and suppose that T_∗ is sufficiently large such that Proposition <ref> holds. Then by (<ref>), for any s' ≥ 0,

    |sup_s ∈ [s', s' + T_∗]ln(λ(s)) - inf_s ∈ [s', s' + T_∗]ln(λ(s))| ≲ 1,

with implicit constant independent of s' ≥ 0. Let J be the largest dyadic integer that satisfies

    J = 2^j_∗≤ -ln(η_∗)^1/4.

By (<ref>) and the triangle inequality,

    |sup_s ∈ [s', s' + J T_∗]ln(λ(s)) - inf_s ∈ [s', s' + J T_∗]ln(λ(s))| ≲ J,

and therefore,

    sup_s ∈ [s', s' + 3 J T_∗]λ(s)/inf_s ∈ [s', s' + 3JT^∗]λ(s)≲ T_∗^1/500.

Rescale so that

    1 ≤λ(s) ≤  T_∗^1/500,     for any     s ∈ [s', s' + 3J T_∗].


Utilizing Proposition <ref> on [s', s' + J T_∗], for any s' ≥ 0,

    ∫_s'^s' + J T_∗ϵ(s) _L^2^2 ds ≲ϵ(s') _L^2 + ϵ(s' + J T_∗) _L^2 + O(1/J^8 T_∗^8).

Note that the left hand side of (<ref>) is scale invariant.

Moreover, for any s' > J T_∗,

    ∫_s'^s' + J T_∗ϵ(s) _L^2^2 ds ≲inf_s ∈ [s' - J T_∗, s']ϵ(s) _L^2 + inf_s ∈ [s' + J T_∗, s' + 2J T_∗]ϵ(s) _L^2 + O(1/J^8 T_∗^8).

In particular, for a fixed s' ≥ 0,

    sup_a > 0∫_s' + a J T_∗^s' + (a + 1) J T_∗ϵ(s) _L^2^2≲1/J^1/2 T_∗^1/2 (sup_a ≥ 0∫_s' + a J T_∗^s' + (a + 1) J T_∗ϵ(s) _L^2^2 ds)^1/2 + O(1/J^8 T_∗^8).

Meanwhile, when a = 0,

    ∫_s'^s' + J T_∗ϵ(s) _L^2^2≲ϵ(s') _L^2 + 1/J^1/2 T_∗^1/2 (sup_a ≥ 0∫_s' + a J T_∗^s' + (a + 1) J T_∗ϵ(s) _L^2^2 ds)^1/2 + O(1/J^8 T_∗^8).

Therefore, taking s' = s_j_∗,

    sup_a ≥ 0∫_s_j_∗ + a J T_∗^s_j_∗ + (a + 1) J T_∗ϵ(s) _L^2^2 ds ≲ 2^-j_∗η_∗ + O(2^-8 j_∗η_∗^8).

Then by the triangle inequality,

    sup_s' ≥ s_j_∗∫_s'^s' + J T_∗ϵ(s) _L^2^2 ds ≲ 2^-j_∗η_∗,

and by Hölder's inequality,

    sup_s' ≥ s_j_∗∫_s'^s' + J T_∗ϵ(s) _L^2 ds ≲ 1.


Repeating this argument, Proposition <ref> can be proved by induction. Indeed, fix a constant C < ∞ and suppose that there exists a positive integer n_0 such that for all integers 0 ≤ n ≤ n_0,

    sup_s' ≥ s_nj_∗∫_s'^s' + J^n T_∗ϵ(s) _L^2 ds ≤ C,     sup_s' ≥ s_nj_∗∫_s'^s' + J^n T_∗ϵ(s) _L^2^2 ds ≤ C J^-nη_∗.

Then for s' ≥ s_n j_∗,

    sup_s ∈ [s', s' + 3 J^n + 1 T_∗]λ(s)/inf_s ∈ [s', s' + 3 J^n + 1 T_∗]λ(s)≲ T_∗^1/500.



Then by Proposition <ref>,

    sup_s' ≥ s_(n + 1) j_∗∫_s'^s' + J^n + 1 T_∗ϵ(s) _L^2^2 ds ≤ C J^-(n + 1)η_∗,

and by Hölder's inequality,

    sup_s' ≥ s_(n + 1) j_∗∫_s'^s' + J^n + 1 T_∗ϵ(s) _L^2 ds ≤ C.

Therefore, (<ref>) holds for any integer n > 0.



Now take any j ∈ℤ and suppose n j_∗ < j ≤ (n + 1) j_∗. Then (<ref>) holds on [s_j + a J^n + 1 T_∗, s_j + (a + 1) J^n + 1 T_∗] for any a ≥ 0, so
by Proposition <ref>,

    sup_a ≥ 0∫_s_j + a J^n + 1 T_∗^s_j + (a + 1) J^n + 1 T_∗ϵ(s) _L^2^2 ds ≲ 2^-jη_∗,

and therefore by Hölder's inequality, for any s' ≥ s_j,

    sup_s' ≥ s_j∫_s'^s' + 2^j T_∗ϵ(s) _L^2 ds ≲ 1,

with bound independent of j. Inequalities (<ref>) and (<ref>) imply that the conditions of Proposition <ref> hold on [s', s' + 3 · 2^j J T_∗] for any s' ≥ s_j, so

    ∫_s_j^s_j + 2^j J T_∗ϵ(s) _L^2^2≲ 2^-jη_∗,

and therefore, by the mean value theorem,

    inf_s ∈ [s_j, s_j + 2^j J T_∗]ϵ(s) _L^2≲ 2^-jη_∗ J^-1/2,

which implies

    s_j + 1∈ [s_j, s_j + 2^j J T_∗].

Therefore, by (<ref>) and Hölder's inequality,

    ∫_s_j^s_j + 1ϵ(s) _L^2^2 ds ≲ 2^-jη_∗,     and    ∫_s_j^s_j + 1ϵ(s) _L^2 ds ≲ 1,

with constant independent of j. Summing in j gives (<ref>) and (<ref>).




Now then, for any 1 < p < ∞, (<ref>) implies

    (∫_s_j^s_j + 1ϵ(s) _L^2^p ds) ≲η_∗^p - 1 2^-j(p - 1),

which implies that ϵ(s) _L^2 belongs to L_s^p for any p > 1, but not L_s^1.

Comparing (<ref>) to the pseudoconformal transformation of the soliton, for 0 < t < 1,

    λ(t) ∼ t,     and    ϵ(t) _L^2∼ t,

so

    ∫_0^1ϵ(t) _L^2λ(t)^-2 dt = ∞,

but for any p > 1,

    ∫_0^1ϵ(t) _L^2^pλ(t)^-2 dt < ∞.

For the soliton, ϵ(s) ≡ 0 for any s ∈ℝ, so obviously, ϵ(s) _L^2∈ L_s^p for 1 ≤ p ≤∞.






§ MONOTONICITY OF Λ

Now prove monotonicity of λ, as in the mass-critical problem.


For any s ≥ 0, let

    λ̃(s) = inf_τ∈ [0, s]λ(τ).

Then for any s ≥ 0,

    1 ≤λ(s)/λ̃(s)≤ 3.




Suppose there exist 0 ≤ s_-≤ s_+ < ∞ satisfying

    λ(s_+)/λ(s_-) = e.

Then u is a soliton solution, which contradicts (<ref>). Recall that

    ϵ(t, x) = e^i γ(t)λ(t) u(t, λ(t) x) - Q(x).

Taking the derivative of (<ref>) in time and plugging in

    ∂_t u = i Δ u + i g |u|^2 u - i A_0[u] u - 2mi/r^2 A_θ[u] u - i A_θ[u]^2/r^2 u,

and using the formula Λ = 1 + x ·∇,

    ϵ_s = i γ_s [ϵ + Q] + λ_s/λΛ [ϵ + Q] + i Δ [ϵ + Q] + gi |ϵ + Q|^2 (ϵ + Q) - i A_0[ϵ + Q](ϵ + Q) 
     - 2mi/r^2 A_θ[ϵ + Q](ϵ + Q) - i/r^2 A_θ[ϵ + Q]^2 (ϵ + Q).

Now then,

    Δ Q + g |Q|^2 Q - i A_0[Q] Q - 2m/r^2 A_θ[Q] Q - 1/r^2 A_θ[Q] Q = α Q.

Plugging (<ref>) into (<ref>),

    ϵ_s = i γ_sϵ + i (γ_s + α) Q + λ_s/λΛ Q + λ_s/λΛϵ + i Δϵ + gi { |ϵ + Q|^2 (ϵ + Q) - |Q|^2 Q }
     - i { A_0[ϵ + Q](ϵ + Q) - A_0[Q] Q } - 2mi/r^2{ A_θ[ϵ + Q](ϵ + Q) - A_θ[Q] Q }
     - i/r^2{ A_θ[ϵ + Q]^2 (ϵ + Q) - A_θ[Q] Q }.

Now decompose ϵ into its real and imaginary parts, ϵ = ϵ_1 + i ϵ_2. Taking the real parts of both sides of (<ref>),

    ∂_sϵ_1 = -γ_sϵ_2 + λ_s/λΛ Q + λ_s/λΛϵ_1 - Δϵ_2 - g Q^2ϵ_2 + O(Q ϵ^2 + ϵ^3) 
      + 1/r^2 A_θ[Q] ϵ_2 + 2m/r^2 A_θ[Q] ϵ_2 + A_0[Q] ϵ_2
     + O_m(1/r^2∫_0^r{ |ϵ| Q + |ϵ|^2} s ds ·ϵ_2) + O(ϵ_L^2 |ϵ|).
 

Now compute the virial identity from <cit.>,

    d/ds (ϵ, |x|^2 Q) = -γ_s (ϵ_2, |x|^2 Q) + λ_s/λ (Λ Q, |x|^2 Q) - (Δϵ_2 + g Q^2ϵ_2 - 1/r^2 A_θ[Q] ϵ_2 - 2m/r^2 A_θ[Q] ϵ_2 - A_0 [Q] ϵ_2, |x|^2 Q) 
     + O(ϵ_L^2^2 + ϵ_L^∞^2).

Since

    Δ Q + g Q^3 - 1/r^2 A_θ[Q] - 2m/r^2 A_θ[Q] - A_0 [Q] Q = α Q,

then integrating by parts,

    (Δϵ_2 + g Q^2ϵ_2 - 1/r^2 A_θ[Q] ϵ_2 - 2m/r^2 A_θ[Q] ϵ_2 - A_0 [Q] ϵ_2, |x|^2 Q) = (4 ϵ_2, Λ Q) + α (ϵ_2, Q).

Therefore,

    d/ds (ϵ, |x|^2 Q) = -(γ_s + α) (ϵ_2, |x|^2 Q) + λ_s/λ (Λ Q, |x|^2 Q) - 4 (ϵ_2, Λ Q) + O(ϵ_L^2^2 + ϵ_L^∞^2).


Using Proposition <ref>, (<ref>), the fundamental theorem of calculus, and the fact that (|x|^2 Q, Λ Q) = - xQ _L^2^2,

    x Q _L^2^2 + 4 ∫_s_-^s_+ (ϵ_2, Q + x ·∇ Q)_L^2 = O(η_∗).

Therefore, there exists s' ∈ [s_-, s_+] such that

    (ϵ_2, Q + x ·∇ Q)_L^2 < 0.


Since s' ≥ 0, there exists some j ≥ 0 such that s_j≤ s' + T_∗ < s_j + 1. Using the proof of Proposition <ref>,

    ∫_s'^s_j + 1 + J |λ_s/λ| ds ≲ J.

Then by Proposition <ref>, (<ref>) implies

    ∫_s'^s_j + 1 + Jϵ(s) _L^2^2 ds ≲ 2^-(j + 1 + J)η_∗,

and therefore by definition of s_j + 1 + J,

    ∫_s'^s_j + 1 + Jϵ(s) _L^2 ds ≲ 1.


Arguing by induction, suppose that for some 1 ≤ k ≤ k_0,

    ∫_s'^s_j + kϵ(s) _L^2^2 ds ≲ 2^-j - kη_∗,

and

    ∫_s'^s_j + kϵ(s) _L^2 ds ≲ 1,

with implicit constant independent of k. By Proposition <ref>,

    ∫_s'^s_j + k + Jϵ(s) _L^2^2 ds ≲ 2^-j - kη_∗,

and

    ∫_s'^s_j + k + Jϵ(s) _L^2 ds ≲ J.

Then by Proposition <ref>,

    ∫_s'^s_j + k + Jϵ(s) _L^2^2 ds ≲ 2^-j - k - Jη_∗,

and

    ∫_s'^s_j + k + Jϵ(s) _L^2 ds ≲ 1,

for 1 ≤ k ≤ k_0 + J. Therefore, (<ref>) and (<ref>) hold for any k, with implicit constant independent of k.

Taking k →∞,

    ∫_s'^∞ϵ(s) _L^2^2 ds = 0,

which implies that ϵ(s) = 0 for all s ≥ s'. Therefore, u is a soliton solution.










§ RIGIDITY

To prove Theorem <ref>, we prove that if u is global solution, u is a soliton, but if u is a finite time blowup solution, then u is a pseudoconformal transformation of a soliton.


If u is a solution to (<ref>) that satisfies dist(u, ℳ) ≤η_∗ for all t ≥ 0, and furthermore, if

    sup(I) = ∞,

then u is equal to a soliton solution.


For any integer k ≥ 0, let

    I(k) = { s ≥ 0 : 2^-k + 2≤λ̃(s) ≤ 2^-k + 3}.

Then by Proposition <ref>,

    2^-k≤λ(s) ≤ 2^-k + 3,

for all s ∈ I(k). The fact that sup(I) = ∞ implies that

    ∑ 2^-2k |I(k)| = ∞.

If λ(s) → 0 as s →∞, then there exists a sequence k_n↗∞ such that

    |I(k_n)| 2^-2k_n≥1/k_n^2,     and     I(k) ≤ 2^2k_n k_n^-2,     ∀ k ≤ k_n.

If inf_s ≥ 0λ(s) > 0, then there exists some s_0 such that

    sup_s ≥ s_0λ(s)/inf_s ≥ s_0λ(s)≤ 2.


Now let I(k_n) = [a_n, b_n]. In the case of (<ref>), let a_n = s_0 and b_n = s_0 + 2^4 k_n. By (<ref>),

    |∫_a_n^a_n + b_n - a_n/4 (ϵ_2, Q + x ·∇ Q) ds ≲ 1|,      |∫_b_n - b_n - a_n/4^b_n (ϵ_2, Q + x ·∇ Q) ds ≲ 1|.

Therefore, there exists s_-∈ [a_n, a_n + b_n - a_n/4], s_+∈ [b_n - b_n - a_n/4, b_n] such that

    |(ϵ_2, Q + x ·∇ Q)(s_-)|,      |(ϵ_2, Q + x ·∇ Q)(s_+)| ≲ 2^-2k_n k_n^2.


Plugging (<ref>) into (<ref>),

    ∫_s_-^s_+ϵ(s) _L^2^2 ds ≲ 2^-2k_n k_n^2.

Again by the intermediate value theorem,

    inf_s ∈ [s_-, s_+]ϵ(s) _L^2^2≲ 2^-4k_n k_n^4.

After rescaling λ(s_-) = 1, plugging (<ref>) into Proposition <ref>, and then rescaling back,

    E(P_≤4/3 k_n u)(0) ≲ 2^8 k_n/3 2^-4 k_n k_n^4→ 0,     as     n →∞.

Therefore, E(u) = 0.


Now turn to a finite time blowup solution. Suppose without loss of generality that sup(I) = 0, and

    sup_-1 < t < 0ϵ(t) _L^2≤η_∗.

Then decomposing u,

    u(t,x) = e^-i γ(t)/λ(t) Q(x/λ(t)) + e^-i γ(t)/λ(t)ϵ(t,x /λ(t)).

Then apply the pseudoconformal transformation to u(t,x). For -∞ < t < -1, let

    v(t,x) = 1/tu(1/t, x/t) e^i |x|^2/4t = 1/te^i γ(1/t)/λ(1/t) Q(x /t λ(1/t)) e^i |x|^2/4t + 1/te^i γ(1/t)/λ(1/t)ϵ(1/t, x /t λ(1/t)) e^i |x|^2/4t.

Since the L^2 norm is preserved by the pseudoconformal transformation,

    lim_t ↘ -∞1/te^i γ(1/t)/λ(1/t)ϵ(1/t, x /t λ(1/t)) e^i |x|^2/4t_L^2 = 0,     and
        sup_-∞ < t < -11/te^i γ(1/t)/λ(1/t)ϵ(1/t, x/t λ(1/t)) e^i x^2/4t_L^2≤η_∗.



Since

    1/te^i γ(1/t)/λ(1/t) Q(x/t λ(1/t))

is in the form of e^i γ̃(t)/λ̃(t) Q(x/λ̃(t)), it only remains to estimate

    1/te^i γ(1/t)/λ(1/t) Q(x /t λ(1/t)) (e^i |x|^2/4t - 1) _L^2.




For any k ≥ 0, λ(s) ∼ 2^-k for all s ∈ I(k). Furthermore, ϵ(t) _L^2→ 0 as t ↗ 0 implies that there exists a sequence c_k↗∞ such that

    |I(k)| ≥ c_k,     for all     k ≥ 0.

Therefore, there exists r(t) ↘ 0 as t ↗ 0 such that

    λ(t) ≤ t^1/2 r(t),     so    λ(1/t) ≤ t^-1/2 r(1/t).

Therefore, since Q is rapidly decreasing,

    lim_t ↘ -∞1/t λ(1/t) Q(x /t λ(1/t)) |x|^2/4t_L^2 = 0,

as well as

    lim_t ↘ -∞1/t^d/2λ(1/t)^d/2 Q(x /t λ(1/t)) (e^i |x|^2/4t - 1)_L^2 = 0,

Therefore, v is a solution that blows up backward in time at inf(I) = -∞ and v _L^2 =  Q _L^2. Therefore, by Theorem <ref>, v is a soliton, and u is a pseudoconformal transformation of the soliton.




§ ACKNOWLEDGEMENTS

During the writing of this paper, the author was partially supported by NSF grant DMS-2153750.






alpha


 Paper Title: Sharp kernel bounds for parabolic operators with first order degeneracy.









Sharp kernel bounds  for  parabolic operators with first order degeneracy
    L. Negro Dipartimento di Matematica e Fisica“Ennio De Giorgi”, Università del Salento, C.P.193, 73100, Lecce, Italy.
e-mail:  luigi.negro@unisalento.it     C. Spina Dipartimento di Matematica e Fisica “Ennio De Giorgi”, Università del Salento, C.P.193, 73100, Lecce, Italy.
e-mail:  chiara.spina@unisalento.it
    August 5, 2024
=========================================================================================================================================================================================================================================================================================================================

§ ABSTRACT 
We prove sharp upper and lower estimates for the parabolic kernel of  the singular elliptic operator

    ℒ   =(AD^2)+(v,∇)/y,

in the half-space ^N+1_+={(x,y): x ∈^N, y>0} under Neumann or oblique derivative boundary conditions at y=0.

Mathematics subject classification (2020): 35K08, 35K67,  47D07, 35J70, 35J75.



Keywords: degenerate elliptic operators, singular elliptic operators, boundary degeneracy, kernel estimates.







§ INTRODUCTION

In this paper   we  study sharp upper and lower estimates for the parabolic kernel of the   singular   operator
 
    ℒ=(AD^2)+(v,∇)/y
    =∑_i,j=1^N+1a_ijD_ij+ d·∇_x+cD_y /y

in the half-space ^N+1_+={(x,y): x ∈^N, y>0}.  Here   A=(a_ij)∈^N+1,N+1 is a symmetric and positive definite 

matrix and   we suppose c/γ+1>0, where γ=a_N+1,N+1; the vector v=(d,c)∈^N+1 satisfies  d=0 if c=0 i.e. v is  oblique with respect to the boundary of ^N+1_+.  We    endow ℒ with   Neumann or  oblique derivative boundary conditions at y=0

    lim_y→ 0 D_y u=0  (if v=0),        lim_y→ 0y^c/γ  v ·∇ u=0  (if c≠ 0).

Up to some suitable linear transformations, the latter operator is equivalent to  the model operator 

    ℒ =Δ_x +2a·∇_xD_y+ B_y,       B_y=D_yy+ c/y D_y

were B_y is a one-dimensional Bessel operator, c+1>0  and  a∈^N satisfies the requirement |a| <1 which is equivalent to the ellipticity of the top order coefficients.



In the special case a=0 and  ℒ=Δ_x+B_y,  that is when the mixed second order derivatives do not appear, the   Bessel operator B_y and the Laplacian Δ_x commute and sharp kernel estimates  can be  proved  after an explicit description of the kernel of the Bessel operator (see <cit.>).   These operators  play a major role in the investigation of the fractional powers (-Δ_x)^s and  (D_t-Δ_x)^s, s=(1-c)/2, through the  “extension procedure" of Caffarelli and Silvestre, see <cit.>.


Unfortunately  adding the mixed second order derivatives term is an hard complication of the problem, 
one important reason being the loss of commutativity. However the study of the general operator (<ref>)   is   crucial  for treating degenerate operators in domains since mixed derivatives and oblique boundary conditions appear in  the localization procedure.
For this reason its properties have been  studied in a series of previous papers. Elliptic and parabolic solvability of the associated problems in weighted L^p spaces have been  investigated in <cit.> where we proved, under suitable assumptions on m and p and for c/γ+1>0, that  ℒ generates an analytic semigroup in L^p_m =L^p(^N+1_+; y^m dxdy).  In addition  we characterized its domain as a weighted Sobolev space and showed that it has the  maximal regularity.  



In addition we also refer to   <cit.> where the authors studied  operators of this form, even for variable coefficients, using tools form linear PDE and Muckhenoupt weights. In <cit.>, among other properties, kernel estimates for some second order degenerate operators in divergence form are considered.



Our main results are Theorems  <ref> and <ref>  where   we prove that ℒ generates in L^2_c/γ a contractive analytic semigroup (e^tℒ)_t∈Σ  on some  suitable sector  Σ of the complex plane,	and its heat kernel p_ℒ, written  with respect the measure y^c/γdz, satisfies the following two-sided  estimates 

    p_ℒ(t,z_1,z_2)
    	≃ C t^-N+1/2 y_1^-c/2γ(1∧y_1/√(t))^c/2γ y_2^-c/2γ(1∧y_2/√(t))^c/2γ exp(-|z_1-z_2|^2kt)

 where t>0, z_1=(x_1,y_1), z_2=(x_2,y_2)∈^N+1_+ and C,k>0 are some positive constants which may be different in the lower and upper bounds. We also point out here that the above inequalities can be written in the equivalent form (<ref>) (see Remark <ref>). 
 
We    proved   the upper  bound  of  (<ref>)   in <cit.>   by  revisiting the classical method based upon the equivalence between Gagliardo-Nirenberg type inequalities and ultra-contractivity estimates; we also proved that  the same estimate extends  to complex times  t∈Σ.  
 
 In this  the paper we   therefore focus on proving the    lower bounds in  (<ref>).
As a needed, but also of independent interest, result,   we prove  in Theorem <ref>  the following  gradient estimates 
	
    |∇ p_ℒ(t,z_1,z_2)|
    			≤ C |t|^-N+2/2  y_2^-c/γ(1∧y_2/√( |t|))^c/γ exp(-|z_1-z_2|^2k|t|),     t∈Σ.



Let us describe the  strategy of proof employed. Let us discuss, preliminarily and  in an abstract setting, a  well established  strategy (see e.g. <cit.>) for deducing lower heat kernel estimates referring to the discussion at  the beginning of Section <ref>  for further details.  Let A be  a non-negative self-adjoint operator  acting  on  L^2(X,μ) where  X is a homogeneous space i.e. a metric space (X,d) endowed with a doubling measure μ. Assume that the semigroup e^tA has a kernel p(t,z_1,z_2) which satisfies  the Gaussian upper bound
	
    p(t,z_1,z_2)
    	   ≤C/V(z_1,√(t))^1/2V(z_2,√(t))^1/2exp(-d(z_1,z_2)^2/κ t),  ∀ t>0,  z_1, z_2∈ X.
 where V(z,r)=μ(B(z,r)) is the measure of the X-balls. Then the first crucial step consists  in the derivation of the  on-diagonal lower  estimate 

    p(t,z,z)≥C/V(z,√(t)),     z∈ X, t>0,

which can be proved by  combining the conservation  property of the kernel and the upper estimate (<ref>) (see e.g. <cit.>). The general off-diagonal lower bound then is proved by using the Hölder continuity of p to go outside of the diagonal and an iterative argument based on the validity of the so-called chain condition  for X.




In our case, due to the non self-adjointness of ℒ, this strategy does not work. Then, inspired by the method of Nash <cit.>, we  prove  the on-diagonal lower estimate by proving, in Proposition <ref>, the so-called  “G-bound”

    ∫_^N+1_+log p_ℒ(1,z,z_2)e^-α|z_2|^2y_2^c/γ  dz_2≥ -C,     |z|≤ 1,

namely a lower bound of the integral of the logarithm of the heat kernel  with respect to the    weighted gaussian measure ν=e^-α|z|^2y^c/γ  dz, α>0.  This is not an easy task due to the presence of the weight y^c/γ  introduced to treat the degeneracy of the operator and which can be either singular or degenerate at y=0 and at infinity.  We emphasize that the  derivation of  (<ref>) relies on the  validity of some Poincaré-type inequalities in H^1_ν(^N+1_+) which require a separate study and which we prove in <cit.>. 



Another  difficulty appears: in order to compensate the non self-adjointness of ℒ, we need also  to  study in parallel both ℒ and ℒ^∗. In particular once (<ref>) is proved for both ℒ and ℒ^∗, then  by using  the reproducing property of the kernel and  Jensen inequality we derive, in Proposition <ref>, the on-diagonal lower estimate for  p_ℒ. Finally, thanks to the gradient estimates  proved in Section <ref>, we can go outside the diagonal proving the off-diagonal lower bound in  (<ref>).




We now describe the structure of the paper.


In  Section <ref>  we introduce the operator ℒ in (<ref>) and we collect the  results we need about generation of semigroups, maximal regularity and domain characterization in the weighted spaces L^p_m(_+^N+1):=L^p(_+^N+1; y^m dx dy)  proved in <cit.>. We also  recall the upper kernel estimates   proved  in <cit.>.



In  Section <ref>  we describe a simple change of variable which  allows to simplify the discussion to the model operator (<ref>) under Neumann boundary conditions.  In all the subsequent sections  we therefore deal, without any loss of generality, almost exclusively with the model case, leaving to Theorem <ref>  the  generalization of our results to the general case. 



In Section <ref> we write the forms associated with ℒ and the adjoint ℒ^*  in the weighted space  L^2_c, where the weighted measure  y^c dz takes into account the degeneracy of the operators. We state the main properties of  the forms which allow to get generation of analytic semigroups in L^2_c. 

In Section <ref> we prove gradient estimates for the heat kernels of ℒ and ℒ^∗. The strategy here relies on the analyticity of the semigroup and the Cauchy formula for holomorphic functions which allow to deduce a pointwise estimates for D_t p_ℒ=ℒ p_ℒ. Then the estimate for the gradient is proved by an interpolation argument  using  the solvability theory of Section <ref> for ℒ  in L^p(_+^N+1)  for p>N+1 and the  Morrey's embeddings. 




In Section <ref> we prove that the semigroups e^tℒ, e^tℒ^∗, initially defined on L^2_c, extend also  on L^1_c and we prove their  conservation property.



Section <ref>, which contains the main result of the paper, is devoted to prove the lower kernel  estimates. As explained before the crucial step consists in proving the   “G-bound” (<ref>) which allows to deduce  the lower bounds on the diagonal. Here we mainly focus on the range |x|/√(t), y/√(t)≤ 1  while the lower bound far away from the boundary is deduced  by domination with a uniformly elliptic operator.

In Appendix  we  characterize the boundedness  on  L^p_m of a family of  integral operators which are related to our operator.

Notation. For N ≥ 0, ^N+1_+={(x,y): x ∈^N, y>0}. We write ∇ u, D^2 u for the gradient and the Hessian matrix of a function u with respect to all x,y variables and ∇_x u, D_y u, D_x_ix_j u, D_x_i y u and so on, to distinguish the role of x and y.  For m ∈ we consider the measure y^m dx dy in ^N+1_+ and  we write L^p_m(_+^N+1), and often only L^p_m when ^N+1_+ is understood, for  L^p(_+^N+1; y^m dx dy). 
^+ stands for {λ∈: λ >0 } and, for |θ| ≤π, we denote by  Σ_θ  the open sector {λ∈: λ≠ 0,  |Arg (λ)| <θ}.  




We write often (x,y) or x· y to denote the inner product of ^N and, for A,B∈^N,N symmetric, (AB)=∑_i,ja_ijb_i,j. 
Given a and b ∈, a∧ b, a ∨ b  denote  their minimum and  maximum. We  write f(x)≃ g(x) for x in a set I and positive f,g, if for some C_1,C_2>0 

    C_1 g(x)≤ f(x)≤ C_2  g(x),   x∈ I.

Sometimes we also write C=C(α) to emphasize, for a positive  constant C>0, its dependence on the parameter α.

Acknowledgment.
The authors are members of the INDAM (“Istituto Nazionale di Alta Matematica”) research group GNAMPA (“Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni”).






§ THE OPERATOR  IN L^P_M AND UPPER KERNEL ESTIMATES

We consider the singular elliptic operator

    ℒ   =(AD^2)+(v,∇)/y,

in the half-space ^N+1_+={(x,y): x ∈^N, y>0}. Here   A=(a_ij)∈^N+1,N+1 is a symmetric and positive definite 

 matrix and we write (AD^2)=∑_i,j=1^N+1a_ijD_ij. The vector  v=(d,c)∈^N+1    satisfies 







    d=0      if        c= 0

i.e. v is  oblique with respect to the boundary of ^N+1_+.
 Writing

    A:=	(
    	[   Q q^t;   q   γ ])

where Q∈^N× N,  q=(q_1, …, q_N)∈^N and γ=a_N+1, N+1>0, the operator  ℒ takes the form

    ℒ   =(QD^2_x)+2(q,∇_xD_y)+γ D_yy+ d·∇_x+cD_y /y.


We work in the weighted spaces L^p_m(_+^N+1):=L^p(_+^N+1; y^m dx dy), where m∈ and we  often write only L^p_m when ^N+1_+ is understood. In this section we recall the  results about generation of semigroups, maximal regularity and domain characterization in L^p_m  proved in <cit.>. 




We    endow ℒ with   Neumann or  oblique derivative boundary conditions 

    lim_y→ 0 D_y u=0  (if v=0),        lim_y→ 0y^c/γ  v ·∇ u=0  (if c≠ 0).




The latter  conditions can be equivalently written in   integral form 

(see <cit.> and <cit.>:  we define accordingly the  weighted Sobolev spaces

    W^2,p(m)=   {u∈ W^2,p_loc(^N+1_+):    D^2u, ∇ u, u∈ L^p_m}

and we then impose the boundary conditions thus defining

    W^2,p_v(m)   ={u ∈ W^2,p(m): y^-1v·∇ u ∈ L^p_m},     (c≠ 0);
    
    W^2,p_𝒩(m)   ={u ∈ W^2,p(m): y^-1D_yu ∈ L^p_m},    (v= 0),

where, to keep consistency of the notation we write  when d=0,  W^2,p_(0,c)(m)=W^2,p_𝒩(m).



(<cit.> and <cit.>)
	Let v=(d,c)∈^N+1 with d=0 if c=0, and let 
	
    0<m+1/p<c/γ+1.
   
	Then the operator
	
    ℒ   =(AD^2)+(v,∇)/y

endowed with domain 	D(ℒ)=W^2,p_v(m)	generates a bounded analytic semigroup  in L^p_m which has maximal regularity.




	

(<cit.>)
	Under the assumptions of the previous theorem, the estimate
	
    D^2 u_L^p_m +y^-1 v·∇ u_L^p_m≤ Cℒu_L^p_m

	
	holds for every u ∈ W^2,p_v(m) (if c=0 replace y^-1 v·∇ u with y^-1 D_yu).

The aim of the paper is to prove  that, if  c/γ+1>0,  then the  heat kernel p_ℒ of ℒ, written  with respect the measure y^c/γdz, satisfies the upper and lower Gaussian estimates

    p_ℒ(t,z_1,z_2)
    	≃ C t^-N+1/2 y_1^-c/2γ(1∧y_1/√(t))^c/2γ y_2^-c/2γ(1∧y_2/√(t))^c/2γ exp(-|z_1-z_2|^2kt),

where t>0, z_1=(x_1,y_1), z_2=(x_2,y_2)∈^N+1_+ and C,k are some positive constants which may differ from the upper and the lower bounds. 

	We emphasize that the above estimate can be written  equivalently  as
	
    p_ℒ(t,z_1,z_2)
    		≃ C t^-N+1/2  y_i^-c(1∧y_i/√(t))^c exp(-|z_1-z_2|^2kt),     i=1,2

	for some possibly different constants C,k>0. This is a consequence of <cit.> which guarantees that for any ϵ>0 one has 
	
    y_1^-c/2(1∧ y_1)^c/2≃ 	y_2^-c/2(1∧ y_2)^c/2 exp(ϵ|y_1-y_2|^2),    ∀ y_1,y_2>0





In <cit.> we already  proved the  upper kernel estimate of (<ref>): we recall here the main results referring also to Section <ref> for the construction of ℒ in the space L^2_c/γ via form methods.

(<cit.>) 
Let c/γ+1>0.  Then for some  θ∈ (0,π/2), the operator  ℒ	generates  in L^2_c/γ a contractive analytic semigroup  (e^tℒ)_t∈Σ_π/2-θ which is positive for t>0. The semigroup   consists of integral operators i.e. there exists p_ℒ(t,·,·)∈ L^∞(^N+1_+×^N+1_+) 
such that  for t∈Σ_π/2-θ, z_1=(x_1,y_1), z_2=(x_2,y_2)∈^N+1_+

    e^tℒf(z_1)=
    	∫_^N+1_+p_ℒ(t,z_1,z_2)f(z_2) y_2^c/γ dz_2,   f∈ L^2_c/γ.

Moreover for every >0  there exist  constants C_, k_>0 such that if  t∈Σ_π/2-θ- then

    |p_ℒ(t,z_1,z_2)|
    	≤ C_ϵ |t|^-N+1/2 y_1^-c/2γ(1∧y_1/√(|t|))^c/2γ y_2^-c/2γ(1∧y_2/√(|t|))^c/2γ exp(-|z_1-z_2|^2k_ϵ|t|).

 




In view of the previous Theorem   the remaining part of the paper is then  devoted to   prove the  lower bound of (<ref>). 






§ SIMILARITY TRANSFORMATIONS AND THE MODEL OPERATOR

In this section we show that some simple change of variables allows to  work, in what follows  and without any loss of generality,  with the model operator

    Δ_x +2a·∇_xD_yu+D_yy+c/yD_y:=Δ_x u+2a·∇_xD_yu+ B_yu
 under Neumann boundary conditions. Here B_y=D_yy+c/yD_y is a Bessel operator and  a∈^N satisfies the requirement |a| <1 which is equivalent to the ellipticity of the top order coefficients. We briefly describe the strategy referring to <cit.>  and <cit.> for further details. Let 

    ℒ   =(AD^2)+(v,∇)/y=(QD^2_x)+2(q,∇_xD_y)+γ D_yy+ d·∇_x+cD_y /y

be the operator in (<ref>).
If d≠ 0 and c≠ 0, a  first transformation allows to transform ℒ into a similar operator with d=0 and Neumann boundary conditions. This is a consequence of the following lemma which follows by straightforward computation. Let us consider the following isometry of L^p_m

    T  u(x,y)   :=u(x-d/c y,y),   (x,y)∈_+^N+1.



 Let 1< p< ∞,  v=(d,c)∈^N+1,  c≠ 0. Then  for u∈ W^2,1_loc(_+^N+1) 
		
			
  (i) T^-1 (v·∇) T u=cD_yu;
			
			
  (ii) T^-1 ((AD^2)+ v·∇/y) T u=(ÃD^2u)+c/yD_y u.
		
		Here Ã  is a uniformly elliptic symmetric matrix defined by 
		
    Ã=
    			(
    			[ Q-2/c d⊗ q+γ/c^2 d⊗ d          q^t- γ/c d^t;              q- γ/c d                     γ ])
 
		and  γ= q_N+1,N+1.
	Proof. The proof follows by a straightforward computation.
	The latter lemma shows that the operator ℒ, endowed with the oblique  boundary condition lim_y → 0 y^c/γ  v ·∇ u=0 is unitarily equivalent to  the operator
	
    ℒ̃:=T^-1ℒ T =(ÃD^2)+c/yD_y=(Q̃D^2_x)+2(q̃,∇_xD_y)+γ D_yy+c/yD_y

	with the Neumann  boundary condition lim_y → 0 y^c/γ  D_yu=0.  Then through a linear change of variables in the x variables, the term (Q̃D^2_x) can be finally transformed into γΔ_x thus obtaining an operator which is similar to the one in (<ref>).


In view of the above simplification, in the next sections, without any loss of generality, we will work almost exclusively with the model operator  (<ref>), referring to Theorem <ref> for the generalization of our results to the general case. 

We start in the next section with the L^2-theory. In order to control the non self-adjointness of ℒ, we need to simultaneously study both ℒ and ℒ^∗.





§ THE OPERATOR ℒ=Δ_X +2A·∇_XD_Y+ B_YU IN L^2_C
 
Let c∈, a=(a_1, …, a_N) ∈^N such that c+1>0, |a|<1. In this section we study the L^2 theory related to  the  model degenerate   operator 

    ℒ :=Δ_x u+2a·∇_xD_yu+ B_yu

in L^2_c equipped with Neumann boundary condition.  The requirement |a| <1 is equivalent to the ellipticity of the top order coefficients. 


We use the Sobolev space H^1_c:={u ∈ L^2_c : ∇ u ∈ L^2_c} equipped with the inner product

    ⟨ u, v⟩_H^1_c:= ⟨ u, v⟩_L^2_c+⟨∇ u, ∇ v⟩_L^2_c.


The condition c+1>0 assures, by <cit.>, that the set 

    𝒞:={u ∈ C_c^∞(^N×[0, ∞)),  D_y u(x,y)=0  for y ≤δ   and  some δ>0},

is dense in H^1_c. Moreover,  by <cit.>, if a function u∈ H^1_c has  support in ^N×[0,b], then there exists a sequence (u_n)_n∈∈𝒞  such that u_n⊆^N×[0,b]  and  u_n→ u in H^1_c.


We consider the  form  in L^2_c 

    𝔞(u,v)
    	   :=
    	∫_^N+1_+⟨∇ u, ∇v⟩ y^c dx dy+2∫_^N+1_+ D_yu  a·∇_xv y^c dx dy,
    
    	D(𝔞)   :=H^1_c

and its adjoint 𝔞^*(u,v)=𝔞(v,u)  

    𝔞^*(u,v)=𝔞(v,u)   :=
    	∫_^N+1_+⟨∇ u, ∇v⟩ y^c dx dy+2∫_^N+1_+ a·∇_x u D_yv y^c dx dy.



(<cit.>) 
	The forms 𝔞, 𝔞^* are continuous, accretive and   sectorial.



	




	





We define the operators ℒ and ℒ^* associated respectively to the forms 𝔞 and 𝔞^*  by

    D( ℒ)   ={u ∈ H^1_c: ∃  f ∈ L^2_c  such that 𝔞(u,v)=∫_^N+1_+ f vy^c  dz  for every v∈ H^1_c},
    ℒu   =-f;


    D( ℒ^*)   ={u ∈ H^1_c: ∃  f ∈ L^2_c  such that 𝔞^*(u,v)=∫_^N+1_+ f vy^c  dz  for every v∈ H^1_c},
    ℒ^*u   =-f.

If u,v are smooth functions with compact support in the closure of _+^N+1 (so that they do not need to vanish on the boundary), it is easy to see integrating by parts that 
    -𝔞 (u,v)= ⟨Δ_x u+2a·∇_xD_yu+ B_yu, v⟩_L^2_c

if lim_y → 0 y^c D_yu=0. This means that 
ℒ is the operator 

    Δ_x +2a·∇_xD_y+ B_y

 with Neumann boundary conditions at y=0. On the other hand 

    -𝔞^* (u,v)= ⟨Δ_x u+2a·∇_xD_yu+2ca·∇_x u/y+ B_yu, v⟩_L^2_c

if lim_y → 0 y^c  ( 2a·∇_x u+D_yu )=0 and therefore the adjoint  of ℒ is the operator

    ℒ^*=Δ_x +2a·∇_xD_y+2ca·∇_x u/y+ B_y
 with the above oblique condition at y=0.  We refer the reader to <cit.> for further details about the Sobolev spaces H^1_c and their boundary conditions at y=0.
(<cit.>)
	ℒ and ℒ^*  generate contractive analytic semigroups e^t ℒ, e^t ℒ^*, t∈Σ_π/2-arctan|a|/1-|a|,  in L^2_c. Moreover the semigroups (e^tℒ)_t ≥ 0, (e^tℒ^*)_t ≥ 0 are positive and L^p_c-contractive for 1 ≤ p ≤∞.





	
	
	


	
	
	
		
	
	
	



The previous proposition is a special case of  Theorem <ref> which  shows that the  semigroups e^t ℒ, e^t ℒ^* extrapolate to the weighted spaces L^p(^N+1_+, y^mdxdy), m∈, when   0<m+1/p<c+1, and which describes  the domain of their generators as respectively the weighted Sobolev spaces defined in (<ref>)

    W^2,p_𝒩(m),         	W^2,p_v(m),   v=(2a,1).







Finally we collect in the following proposition some invariance properties of ℒ, e^tℒ and of of ℒ^∗, e^tℒ^∗ whose proofs follow after an easy computation.

 The following properties hold.
	
		
  (i) The scale homogeneity of ℒ is  2:
		
    s^2ℒ=I_s^-1ℒ I_s,      I_su(x,y)=u(sx,sy),      s>0.


  (ii)   ℒ is invariant under translation in the x-direction:

    ℒ=T_x_0^-1ℒ T_x_0,      T_x_0u(x,y)=u(x+x_0,y),      x_0∈^N.

		Moreover the semigroup e^tℒ generated by ℒ  in L^2_c
		satisfies for any t∈Σ_π/2-arctan|a|/1-|a|
		
    e^s^2 tℒ=I_s^-1e^tℒI_s,   s>0,     e^ tℒ=T_x_0^-1e^tℒT_x_0,   x_0∈^N.

The same results  hold for the adjoint operator ℒ^∗ and its generated semigroup. 

By Theorem <ref> , the semigroup e^tℒ,  t∈Σ_π/2-arctan|a|/1-|a|,   consists of integral operators and its heat kernel  p_ℒ, written with respect to the measure y^cdz, satisfies the upper estimate

    0≤ p_ℒ(t,z_1,z_2)
    	≤ C t^-N+1/2 y_1^-c/2(1∧y_1/√(t))^c/2 y_2^-c/2(1∧y_2/√(t))^c/2 exp(-|z_1-z_2|^2k t).

which is valid for any t>0, z_1,z_2∈^N+1_+ and for some positive constant C,k>0. By remark <ref> the above estimate can be written  equivalently  as

    p_ℒ(t,z_1,z_2)
    	≤  C t^-N+1/2  y_i^-c(1∧y_i/√(t))^c exp(-|z_1-z_2|^2kt),     i=1,2
 
The latter bounds extends also, for some possibly different C,k>0, for t in  any sub-sector Σ_π/2-arctan|a|/1-|a|-.





Moreover,  using  the invariance properties of Proposition <ref>, we have that  for any t∈Σ_π/2-arctan|a|/1-|a|, z_1, z_2∈^N+1_+, 

    p_ℒ(s^2t,z_1,z_2)=s^-(N+1+c) p_ℒ(t,s^-1z_1,s^-1z_2),     s>0,

and

    p_ℒ(t,z_1+x_0,z_2+x_0)=p_ℒ(t,z_1,z_2),     x_0∈^N,

where for z=(x,y)∈^N+1_+, we wrote,  with a little abuse of notation,  z+x_0=(x+x_0,y) to denote the translation in the x_0-direction. 







The same results  hold for the adjoint operator ℒ^∗ and its generated semigroup. In particular, if  p_ℒ^∗ is the heat kernel of ℒ^∗ , written with respect to the measure y^cdz, then by definition and by the positivity of the semigroups, the following relations hold for any z_1, z_2∈^N+1_+:

    p_ℒ^∗(t,z_1,z_2)   =p_ℒ(t,z_2,z_1),        ∀ t∈Σ_π/2-arctan|a|/1-|a|,
    
    	p_ℒ^∗(t,z_1,z_2)   =p_ℒ(t,z_2,z_1),        ∀ t>0.





§ GRADIENT ESTIMATES

In this section we prove gradient estimates for the heat kernels of ℒ and ℒ^∗. For simplicity in what follows we only work with ℒ but all the  results hold with similar proof also for ℒ^∗. 

We start by stating  the next lemma in which we specialize Theorem <ref> and Corollary <ref> to  the case of  L^p:=L^p(^N+1_+) with Lebesgue measure. In what follows, recalling  (<ref>), we write 

    W^2,p_𝒩:=W^2,p_𝒩(0)={u ∈ W^2,p(^N+1_+): y^-1D_yu ∈ L^p}.

Note that from Morrey's embedding, if p>N+1, then W^2,p_𝒩↪ C^1_b(^N+1_+).

	Let 0<1/p<c+1. 	Then ℒ endowed with domain 	D(ℒ)=W^2,p_𝒩	generates a bounded analytic semigroup  in L^p. Moreover there exists C>0 such that for λ>0 and  u ∈ W^2,p_𝒩 one has 
	
    λu_p+λ^1/2∇ u_p+D^2 u_p ≤ Cλ u- ℒu_p.

 
Proof. Recalling  Theorem <ref> and Corollary <ref>,   (ℒ,W^2,p_𝒩) generates a bounded semigroup on L^p(^N+1_+) and moreover we have for u∈ W^2,p_𝒩, λ >0

    λu_p≤ C( λ u-ℒ u_p),   D^2u_p ≤ Cℒu_p≤ C(λu_p+λ u-ℒu_p).

The estimate of the gradient term follows by the interpolative inequality ∇ u_p^2 ≤ Cu_p D^2u_p.


The following regularity result  is an immediate consequence of the holomorphy of the semigroup e^t ℒ.

 
For every fixed z_1, z_2∈^N+1_+ the kernel p_ℒ(t, z_1, z_2) is holomorphic with respect to t∈Σ_π/2-arctan|a|/1-|a|  and  p(t, · , z_2) ∈ C^1_b(^N+1_+).

Proof.  Let us fix  p>max{(c+1)^-1, N+1} and let us work, using Lemma <ref>,  in L^p:=L^p(^N+1_+). If s>0, by (<ref>), p_ℒ(s, ·, z_2) ∈ L^p and then,  since the semigroup is analytic, e^t ℒ p_ℒ belongs to the domain of ℒ,  D(ℒ)=W^2,p_𝒩. Since e^t ℒ p_ℒ(s,z_1,z_2)=p_ℒ(t+s,z_1,z_2), by the semigroup law, we have that p(t+s, ·, z_2) ∈ W^2,p_𝒩↪ C^1_b(^N+1_+). The analyticity with respect to t∈Σ_π/2-arctan|a|/1-|a|  follows again by the identity e^t ℒ p_ℒ(s,z_1,z_2)=p_ℒ(t+s,z_1,z_2), using the analyticity of the semigroup.


The Cauchy formula for the derivatives of holomorphic functions allows to estimate D_tp_ℒ and  ℒ p_ℒ.
	
			Let c+1>0, >0. Then   there exists C, k>0, such that, for every  t∈Σ_π/2-arctan|a|/1-|a|- and almost every z_1, z_2∈^N+1_+,
		
    |ℒ p_ℒ(t,z_1,z_2)|+	|D_tp_ℒ (t,z_1,z_2)|
    			≤ C |t|^-N+3/2  y_2^-c(1∧y_2/√( |t|))^c exp(-|z_1-z_2|^2k|t|).

	
Proof.
		Since the kernel p_ℒ  satisfies the equation D_tp_ℒ=ℒ p_ℒ,   it is sufficient to deal only with D_t p_ℒ.   Let  r be small enough such  that
		
    B(t_0,  r|t_0|)⊂Σ_π/2-arctan|a|/1-|a|,     ∀ t_0∈Σ_π/2-arctan|a|/1-|a|-.

			Using the Cauchy formula for the derivatives of holomorphic functions in the ball  B(t_0,  r|t_0|), we get 
		
    |D_t p_ℒ(t_0,y,ρ)|
    			≤1/r|t_0|max_|t-t_0|=r|t_0||p_ℒ(t,z_1,z_2)|,     z_1, z_2∈^N+1_+.

		Applying the estimate of  Theorem <ref> in the sector Σ_π/2-arctan|a|/1-|a| and recalling Remark <ref>, we obtain for suitable C', κ'
		
    |D_t p_ℒ(t_0,y,ρ)|≤ C'|t|^-N+3/2  
    			   y_2^-c (y_2/|t_0|^1/2∧ 1 )^cexp(-|z_1-z_2|^2/κ' |z_0|)

	which is equivalent to the statement. 
Before proving the estimates for  the gradient of  p_ℒ,  we recall that,  using  the scaling equalities of Proposition <ref>, we have that  for any t∈Σ_π/2-arctan|a|/1-|a|, z_1, z_2∈^N+1_+ 

    p_ℒ(t,z_1,z_2)   =|t|^-N+1+c/2p_ℒ(t|t|^-1,z_1/√(|t|),z_2/√(|t|)),
    ∇_z_1p_ℒ(t,z_1,z_2)   =|t|^-N+2+c/2∇ p_ℒ(t|t|^-1,z_1/√(|t|),z_2/√(|t|)).




Now we localize the  estimates of Lemma <ref>. For   z∈^N+1_+, r>0 we set B^+(z,r):=B(z,r)∩^N+1_+. 
	 Let c+1>0, p>max{(c+1)^-1, N+1}. Then there exists a constant C>0  such that  for every u ∈ W^2,p_𝒩, r>0,
	
    ∇ u _L^∞(B_r/2^+)≤ Cr^-N+1/p(r^2ℒ u_L^p(B_r^+)+u_L^p(B_r^+))


Proof.  Up to a scaling argument we can suppose, without any loss of generality, r=1. Set r_n=∑_k=1^n2^-k. Then r_1=1/2, r_∞=1, r_n+1-r_n=2^-(n+1).

Let  B^+_n=B^+(z,r_n), B^+= B^+(z,1), B_1/2^+= B^+(z,1/2)  and choose  cut-off functions η_n∈ C_c^∞(^N+1) such that  η_n(x,y)=η_n(x,-y),
0≤η_n≤ 1, 
η_n=1 in B_n^+, ( supp  η_n) ∩^N+1_+ ⊂ B_n+1^+,
|∇η_n| ≤C/r2^n,  |D^2 η_n| ≤C/r^24^n for some constant C>0 independent of n. Then also |y^-1D_y η_n| ≤C/r^24^n, since D_y η_n(x,0)=0.  If  u∈ W^2,p_𝒩 then  η_n u∈ W^2,p_𝒩   and we have

    ℒ(η_n u)=   η_n ℒ u +2∇_xη_n∇_xu+2 D_y η_n D_y u+2a·(∇_xη_nD_yu+D_yη_n∇_xu)
        +u(D_yyη_n+Δ_xη_n+2a·∇_xD_yη_n+cD_yη_n/y).

Applying Lemma <ref> with λ=1 to η_n u   we get 

    η_nu_p+∇ (η_nu)_p+D^2 (η_nu)_p
    	≤ C (ℒ (η_nu)_p+η_nu_p)
       ≤ C( ℒ u_L^p(B^+)+2^n∇(η_n+1u)_p+(4^n+1)u_L^p(B_r^+)).

Applying the interpolative inequality ∇ u_p ≤ CϵD^2u_p^2+1/ϵu_p we get for ϵ>0

    η_nu_p+∇ (η_nu)_p+D^2 (η_nu)_p 
       ≤ C(ℒ u_L^p(B^+)+ϵ 2^nD^2(η_n+1u)_p+2^n/ϵη_n+1u_p+(4^n+1)u_L^p(B^+))
       ≤ C(ℒ u_L^p(B^+)+ϵ 2^nD^2(η_n+1u)_p+(2^n/ϵ+4^n+1)u_L^p(B_r^+))

Setting
ξ:=C2^n, we get

    η_nu_p+∇ (η_nu)_p+D^2 (η_nu)_p 
       ≤ C(   u-ℒ u_L^p(B^+)+(4^n/ξ+4^n)u_L^p(B^+))+ξD^2(η_n+1u)_p.

It follows that 

    ξ^n( η_nu_p+∇ (η_nu)_p+D^2(η_nu)_p )
       ≤ C( ξ^n  u-ℒ u_L^p(B_r^+)+ξ^n(4^n/ξ+4^n)u_L^p(B^+))+ξ^n+1D^2(η_n+1 u)_p.

By choosing =_n  so that ξ=1/8 and summing up the  previous inequality over n∈ we get

    ∇ u _L^p(B_1/2^+)+∑_n=1^∞ξ^n D^2 (η_nu)_p
    	≤   C(u-ℒu_L^p(B^+)+u_L^p(B^+))
       +∑_n=1^∞ξ^n+1D^2 (η_n+1u)_p.

Cancelling equal terms on both sides it follows that

    ∇ u _L^p(B_1/2^+)   +1/8D^2 u_L^p(B_1/2^+)≤ C( u-ℒ u_L^p(B^+)+u_L^p(B^+)).

 Since p>N+1, using the  Morrey's embedding, the previous inequality implies
we get 

    ∇ u _L^∞(B_1/2^+)   ≤ C(∇ u _L^p(B_1/2^+)+D^2 u_L^p(B_1/2^+))≤ C( u-ℒ u_L^p(B^+)+u_L^p(B^+))

which implies the required claim for r=1.



		 We remark that the same results as those in  Lemmas <ref> and  <ref> and  Propositions <ref>  and  <ref>  hold also for the adjoint  operator ℒ^∗ under the oblique boundary  condition lim_y → 0 y^c  ( 2a·∇_x u+D_yu )=0.  To be concise  we do not state them explicitly but their proof follows by arguing as before and by using the generation and the  domain characterization for the operator ℒ^* endowed with domain 	D(ℒ)=W^2,p_v, v=(2a,1) (see Theorem <ref>  and Corollary <ref>). 

We need the following basic estimate.

	Let z_0, z_2 ∈^N+1_+. Then 
	
    sup_z_1∈ B(z_0,√(t)) exp(-|z_1-z_2|^2/t)≤ e^16exp(-9/16|z_0-z_2|^2/t)


Proof.  If |z_0-z_2|≤ 4√(t) then for every z_1∈^N+1_+

    exp(-|z_1-z_2|^2/t)≤  e^16exp(-|z_0-z_2|^2/t).

If |z_0-z_2|> 4√(t) and z_1∈ B(z_0,√(t)), then |z_1-z_2|≥ |z_0-z_2|-|z_1-z_0| ≥ |z_0-z_2|-1/4 |z_0-z_2|=3/4 |z_0-z_2| and 

    exp(-|z_1-z_2|^2/t)≤exp(-9/16|z_0-z_2|^2/t).



We can finally  deduce  pointwise estimates for  the gradient of the kernel p_ℒ(t,z_1,z_2). Recalling (<ref>), due  to the asymmetry of p_ℒ and  in order to control both derivatives with respect to  z_1, z_2∈^N+1_+, we state them also for the adjoint operator ℒ^∗. 

	Let c+1>0, >0.   Then  for  every ϵ>0,  there exist C,k>0, such that, for every  t∈Σ_π/2-arctan|a|/1-|a|- ϵ and almost every z_1, z_2∈^N+1_+,
		
    |∇_z_1 p_ℒ(t,z_1,z_2)|+	|∇_z_1 p_ℒ^∗(t,z_1,z_2)|
    			≤ C |t|^-N+2/2  y_2^-c(1∧y_2/√( |t|))^c exp(-|z_1-z_2|^2k|t|).



Proof. Let us firstly prove the required estimate for ∇_z_1 p_ℒ. In virtue of the validity of the scaling property (<ref>), we may  assume that |t|=1.  Let z_0,  z_2∈^N+1_+. Then applying Proposition <ref> with r=1  to the function  u=p_ℒ(t, · ,z_2) in  B^+(z_0,1)  we get 

    |∇ p_ℒ (t,z_0,z_2)|   ≤ C(ℒu_L^p(B^+(z_0,2))+u_L^p(B^+(z_0,2)))
       ≤ C(ℒu_L^∞(B^+(z_0,2))+u_L^∞(B^+(z_0,2))).
 
 Using (<ref>) and  Proposition <ref>,  we get for suitable  C, κ >0 
  
    u_L^∞(B^+(z_0,2))+ℒu_L^∞(B^+(z_0,2))   ≤  C y_2^-c (y_2∧ 1 )^csup_z∈ B^+(z_0,2) exp(-|z-z_2|^2/κ).

Lemma <ref>, combined with the previous inequalities, then implies (for suitable C', κ'>0)
 
    |∇ p_ℒ (t,z_0,z_2)|   ≤ C' y_2^-c (y_2∧ 1 )^cexp(-|z_0-z_2|^2/κ')

 which is the statement for |t|=1. The result for general t follows by the scaling property (<ref>).
 
The analogous estimates ∇_z_1 p_ℒ^∗  can be proved similarly using Remark <ref>. 


The following corollary is a direct consequence of Theorem <ref>. For simplicity we state it only for ℒ.

Let  1<p < ∞, 0< m+1/p <c+1, f∈ L^p_m.  Then for every  t∈Σ_π/2-arctan|a|/1-|a|- ϵ ,   e^tℒf is differentiable in ^N+1_+ and one has
	
    ∇ e^tℒf=∫_^N+1_+∇ p_ℒ(t,·,z_2)f(z_2) y_2^c dz_2.

Moreover, for  every ϵ>0, there exist C, k>0  such that,
	
    |∇ e^tℒf(z_1)|≤C/ |t|^N+2/2∫_^N+1_+(1∧y_2/√( |t|))^c exp(-|z_1-z_2|^2/k|t|)|f(z_2)| dz_2.

 
	
Proof. 
	Let z_0, r>0 such that B(z_0,r)∈^N+1_+. By Theorem <ref>,  for almost every z_1∈ B(z_0,r), z_2∈^N+1_+, one has 

    |∇ p_ℒ (t,z_1,z_2)|y_2^c≤C/ |t|^N+2/2(1∧y_2/√( |t|))^c exp(-|z_1-z_2|^2/k_ϵ|t|)|f(z_2)|.

	for  suitable  C and κ depending also  on r and z_0. Then (<ref>) follows by differentiating under the integral sign since 
the right hand side of (<ref>) belongs to L^p'_m. Finally, (<ref>) is consequence of Theorem <ref>.




	





§ THE SEMIGROUP IN L^1_C AND THE CONSERVATION PROPERTY

In this section we prove that the semigroups e^tℒ, e^tℒ^∗, initially defined on L^2_c, extend also   on L^1_c and we discuss the conservation property

    e^tℒ1=1,     e^tℒ^∗1=1,      for any   t≥ 0.


Following the notation introduced in the appendix, we start by  rewriting the upper  estimates of Theorems <ref> and <ref>  in terms of  the family of integral operators

    S^0,-c(t)f(z_1)=t^-N+1/2∫_^N+1_+ (y_2/√(t)∧ 1 )^cexp(-|z_1-z_2|^2/κ t)f(z_2)  dz_2,

defined for t>0 and z_1=(x_1,y_1), z_2=(x_2,y_2)∈^N+1_+. Here κ is a positive constant; to shorten the notation we omit the dependence on κ  which may vary in each occurrence. We recall that, by Proposition <ref> and by <cit.>, if c+1>0 then the family S^0,-c(t) is uniformly bounded on L^p_c for 1≤ p≤∞.


	Let c+1>0. Then, for some positive constant C>0 and  a  suitable κ>0, one has  for any  f∈ L^2_c and t>0
	
		
		
	
    |e^tℒf|+ √(t)|∇ e^tℒf|	≤ CS^0,-c(t)|f|,    |e^tℒ^∗f|+√(t)|∇ e^tℒ^∗f|≤ C S^0,-c(t)|f|.



	The required estimates follows from    Theorems <ref>,  Corollary <ref> and Remark <ref>.


We now prove that the semigroups e^tℒ, e^tℒ^∗ extend from L^2_c∩ L^1_c to  strongly continuous semigroups on L^1_c.

 
	Let c+1>0 and 1≤ p<∞.	Then the  semigroup e^tℒ  and  e^tℒ^* extend from L^2_c∩ L^p_c to  strongly continuous semigroups on L^p_c.
 

	We prove the statement for ℒ; the analogous claim for ℒ^∗ follows identically. By Proposition  <ref>, e^tℒ is  contractive in L^p_c for every 1≤ p≤∞: then e^tℒ  clearly extends, by density, from L^2_c∩ L^p_c to  L^p_c and the semigroup law is inherited  from the one of L^2_c.  Therefore we have only to prove the  strong continuity at 0.  
	Assume first that 1<p<∞ and let f, g ∈ C_c^∞ (^N+1_+). Then as t → 0
	
    ∫_^N+1_+ (e^tℒf)  g  y^c dy⟶∫_^N+1_+ fgy^cdy  ,

	by the strong continuity of e^tℒ in L^2_c. Since e^tℒ is contractive on L^p_c, a   density argument then proves that the previous limit holds for every f ∈ L^p_c, g ∈ L^p'_c. The semigroup is then weakly continuous, hence strongly continuous. Assume now p=1 and let us fix, preliminarily, f∈ L^1_c∩ L^2_c. For any r>0 we set B_r=B(0,r)× ]0,r[ and write f=f_|B_r+f_|B_r^c. Then we have, using the contractivity of e^tℒ on L^1_c,
	
    e^tℒf-f_L^1_c   ≤e^tℒ(f_|B_r)-f_|B_r_L^1_c+e^tℒ(f_|B_r^c)-f_|B_r^c_L^1_c
       ≤e^tℒ(f_|B_r)-f_|B_r_L^1_c+ 2 f_|B_r^c_L^1_c.

	Recalling that  the measure μ=y^cdz is locally finite  and using  the Hölder inequality we get
	
    e^tℒ(f_|B_r)-f_|B_r_L^1_c   =∫_B_r|e^tℒ(f_|B_r)-f_|B_r| y^cdz+∫_B_r^c|e^tℒ(f_|B_r)| y^cdz
       ≤(μ(B_r))^1/2e^tℒ(f_|B_r)-f_|B_r_L^2_c+∫_B_r^c|e^tℒ(f_|B_r)| y^cdz
       :=A_1+A_2

	which goes to 0 as t→ 0. Indeed  lim_t→ 0 A_1=0 by the strong continuity of  e^tℒ in L^2_c. For the second term we observe that, using again  the contractivity of e^tℒ on L^1_c 
	
    f_|B_r_L^1_c≥e^tℒ(f_|B_r)_L^1_c=	∫_B_r^c|e^tℒ(f_|B_r)| y^cdz+∫_B_r|e^tℒ(f_|B_r)| y^cdz.

	Then taking the limit in the above estimate and observing that, from the previous point, 
	
    ∫_B_r|e^tℒ(f_|B_r)| y^cdz→f_|B_r_L^1_c,

	we get lim_t→ 0 A_2=0. Then it follows from (<ref>) that
	
    lim sup_t→ 0e^tℒf-f_L^1_c≤ 2 f_|B_r^c_L^1_c⟶ 0,    as   r→∞

	which proves that 	lim_t→ 0e^tℒf-f_L^1_c=0 which is the required claim for   f∈ L^1_c∩ L^2_c. The general case follows  by a   density argument.




For the case p=∞ of the above Proposition we refer the reader to <cit.> for   generation of semigroup, in a special case, in spaces of continuous functions.




We can now prove the conservation property of the semigroups.
  
	Let us assume c+1>0. Then the heat kernel p_ℒ of ℒ satisfies  
	
    ∫_^N+1_+p_ℒ(t,z_1,z_2) y_2^c  dz_2=1
    		  ∀ t>0,  z_1∈^N+1_+.

	The same conservation property also holds for the kernel p_ℒ^∗ of the adjoint operator ℒ^∗.

Proof.  
We prove the statement for P_ℒ; the analogous claim for p_ℒ^∗ follows identically. We prove, equivalently, that for every f∈ L^2_c∩ L^1_c one has

    ∫_^N+1_+e^tℒf(z) y^c  dz=∫_^N+1_+f(z) y^c  dz
    	  ∀ t>0.

We fix a  cut-off function η(z)=ϑ(x)θ(y) where ϑ∈ C_c^∞( B(0, 2)) is a  cut-off function in the x-variables which  satisfies
0≤ϑ≤ 1, ϑ=1 in B(0,1)⊂^N and  θ∈ C_c^∞([0,2[) such that  0≤θ≤ 1, θ(y)=1 for y≤ 1. 
We also set, for every n>0, η_n(z)=η(z/n). 
Then  differentiating  under the integral sign and recalling Section <ref>,  we have for every t>0

    I_n(t)   =d/dt∫_^N+1_+e^tℒf(z)η_n(z) y^c  dz=∫_^N+1_+ℒ e^tℒf(z)η_n(z) y^c  dz=-𝔞(e^tℒ^*f, η_n)
       =-∫_^N+1_+∇ e^tℒf(z)·∇η_n(z) y^c  dz-2∫_^N+1_+D_y e^tℒf(z)a·∇_xη_n(z) y^c  dz.
 
Therefore, since |∇η_n|≤C/n, using Propositions <ref> and  <ref>  we get for some possibly different positive constants C>0

    |I_n(t)|≤C/n∇ e^tℒf_L^1_c≤C/n√(t)S^0,-c(t)|f|_L^1_c≤C/n√(t)f_L^1_c

which implies lim_n→∞I_n(t)=0. Then using Lebesgue's Theorem it follows that for every t, s>0

    ∫_^N+1_+e^tℒf(z) y^c  dz-∫_^N+1_+e^sℒf(z) y^c  dz
       =lim_n→∞(∫_^N+1_+e^tℒf(z)η_n(z) y^c  dz-∫_^N+1_+e^sℒf(z)η_n(z) y^c  dz)=lim_n→∞∫_s^tI_n(r)  dr=0

and hence 
    ∫_^N+1_+e^tℒf(z) y^c  dz=∫_^N+1_+e^sℒf(z) y^c  dz,     t, s>0.

Letting s go to 0, the claim follows by the strong continuity of  e^tℒ in L^1_c  proved in  Proposition <ref>.




§ KERNEL LOWER BOUNDS

This section, which contains the main result of the paper, is devoted to prove lower  estimates for the heat kernel of ℒ, defined in (<ref>), of the same form of the upper ones  of Theorem <ref>. We need some preparation.


We work in ^N+1_+ endowed with the standard euclidean metric  and, for z_0=(x_0,y_0)∈^N+1_+ and r>0, we consider the cylindric balls 

    Q(z_0,r):=B(x_0,r)× [y_0,y_0+r)
  in place of the standard balls B(z_0,r); we keep the same notation to denote Q(0,r):=B(0,r)× [0,r[. For c+1>0 we consider the weighted measure μ=y^c dx dy and we write V (z_0,r):=μ(Q(z_0,r)) to denote the measure  of the cylindric balls given by

    V (z_0,r)=∫_Q(z_0,r) y^c dz=w_Nr^N∫_y_0^y_0+r y^c dy.

We need the following  elementary lemma which guarantees that μ is a doubling measure and which is proved in  <cit.>. 


	Let c+1>0. 
	Then one has 
	
    V(z_0,r)   =r^N+1+c V(z_0/r,1),     
    		V(z_0,r)≃ r^N+1+c(y_0/r)^c(y_0/r∧ 1)^-c.

	In particular the function V satisfies, for some  constants C≥ 1, the doubling condition
	
	
	
	
	
    V(z_0,s)/V(z_0,r)≤ C (s/r)^N(1 ∨s/r)^1+c^+,    ∀  s,  r>0.














 
	The previous inequality  implies that, for some positive constant C>0, one has
	
	
    V(z_0,2r)≤ C V(z_0,r),    ∀ z_0∈ R^N+1_+, r>0.

	
	
		
		


In order to prove lower heat kernel estimates we also need to rewrite the upper  bounds of Theorem <ref> in terms of  the measure of the cylindric balls.
 Let us assume c+1>0. Then the heat kernel p_ℒ of ℒ, written with respect to the measure y^c dx dy, satisfies  
	
    p_ℒ(t,z_1,z_2)
    		   ≤C/V(z_1,√(t))^1/2V(z_2,√(t))^1/2exp(-|z_1-z_2|^2/κ t),  ∀ t>0,  z_1, z_2∈^N+1_+.



	The required estimates follows from    Theorems <ref> and Lemma <ref>.



In what follows we aim to prove that the similar lower bound

    p_ℒ(t,z_1,z_2)
    	   ≥C/V(z_1,√(t))^1/2V(z_2,√(t))^1/2exp(-|z_1-z_2|^2/κ t),  ∀ t>0,  z_1, z_2∈^N+1_+

holds for the heat kernel of ℒ.




Before to prove this result, let us discuss, in an abstract setting, a  well established  strategy (see e.g. <cit.>) for deducing lower kernel estimates  for a non-negative self-adjoint operator A acting  on an L^2-space L^2(X,μ); here X is a homogeneous space i.e. a metric space (X,d) endowed with a doubling measure μ. Assume that the semigroup e^tA has a kernel p(t,z_1,z_2) which satisfies the conservation property as the one in Proposition <ref> and  some Gaussian upper bounds of the same form as in  (<ref>) (in this setting V(z,r) is the measure of the X-balls and |z_1-z_2| is the X-distance d(z_1,z_2)). Then the analogous lower bound can be derived from the conservation property of the semigroup, the Gaussian upper bound and the Hölder continuity of p (or, a fortiori, from some gradient estimates on p). To be precise, the Gaussian lower bound   (<ref>) can be obtained by:

	
  (i) proving, preliminarily, the  on-diagonal lower  estimates 
	
    p(t,z,z)≥C/V(z,√(t)),     z∈ X, t>0,

	using the conservation  property of the kernel and the upper estimates (<ref>) (see e.g. <cit.>);
	
  (ii) proving the off-diagonal bound (<ref>), adding also the Gaussian term, using the Hölder continuity of p to go outside of the diagonal and an iterative argument based on the validity of the so-called chain condition  for X (see e.g. <cit.>).
 



In our case, due to the non self-adjointness of ℒ, this strategy does not work. Then, inspired by the method of Nash <cit.>, we  prove  the on-diagonal lower estimate (<ref>) by proving in Proposition <ref> the so-called  “G-bound”, namely a lower bound of the integral of the logarithm of the heat kernel  with respect to a  weighted Gaussian measure and  whose validity relies on some suitable Poincaré inequalities. Some difficulties appear: in order to compensate the non self-adjointness of ℒ we need to prove all the statements for both ℒ and ℒ^∗ and we need also to take control of the  weighted measure y^cdz whose density can be either singular or degenerate at y=0 and at infinity. Then, thanks to the gradient estimates  proved in Section <ref>, we can go outside the diagonal proving the off-diagonal lower bound (<ref>).



The proof is structured in many steps and requires some preliminary results.

	We point out that, by (<ref>) and Lemma <ref>     the  estimates 
	
    p_ℒ(t,z_1,z_2)
    		   ≃C/V(z_1,√(t))^1/2V(z_2,√(t))^1/2exp(-|z_1-z_2|^2/κ t)

	are equivalent, up to change the constants C,k>0,  to
	
    p_ℒ(t,z_1,z_2)
    		≃C/V(z_i,√(t)) exp(-|z_1-z_2|^2kt),     i=1,2.
	





We start by proving, preliminarily, the lower estimate (<ref>) far away from the boundary i.e. for   z_1=(x_1,y_1), z_2=(x_2,y_2)∈^N+1_+  satisfying y_1/√(t)≥ r, y_2/√(t)≥ r.

Its proof follows  by domination with a uniformly elliptic  operator with bounded coefficients.



	Let r>0 and let us consider the operator  𝒜=Δ_x+a·∇_xD_y+B_y acting in the L^2-space  L^2(^N×[r,+∞[,y^cdz) endowed with Dirichlet boundary conditions. Then 𝒜  generates an analytic   semigroup of integral operator which is positive for t>0 and  whose kernel p_0(t,z_1,z_2), taken with respect to the Lebesgue measure, satisfies, for some positive constants C,c>0,
	
    p_0(t,z_1,z_2)≥ C(1∧y_1-r/√(t))(1∧y_2-r/√(t))t^-N+1/2exp(-c|z_1-z_2|^2/ t)
 
	for any  t>0, y_1≥ r, y_2≥ r.

Proof. 
We consider  in L^2(^N×[r,+∞[,y^cdz)  the  sesquilinear form 

    𝔞_0(u,v)
    	   :=
    	∫_^N ×[r,+∞[⟨∇ u, ∇v⟩ y^c dx dy+2∫_^N ×[r,+∞[ D_yu  a·∇_xv y^c dx dy , 
    
    	D(𝔞_0)   =H_c,0^1(^N ×[r,+∞[):=C_c^∞(^N ×]r,+∞[)







where, in the latter equality, the closure is taken in the norm of H^1_c. Recalling Section <ref>, 𝔞_0 is the restriction of 𝔞 to the space H_c,0^1(^N ×[r,+∞[)↪ H^1_c and therefore inherits all its properties stated in Proposition <ref>; moreover  𝒜 is the operator associated with 𝔞_0. The generation results for 𝒜 then follows by classical results on forms as in <cit.>.
The existence of the kernel as well as its  estimate follows by <cit.>.


 
	Let r>0, c+1>0. Then there exists a positive constant C=C(r)>0 such that 
	
	
    p_ℒ(t,z_1,z_2)≥C/V(z_1,√(t))^1/2V(z_2,√(t))^1/2exp(-c|z_1-z_2|^2/ t)

	for every t>0 and every z_1, z_2∈^N+1_+ satisfying y_1/√(t)≥ r, y_2/√(t)≥ r.

Proof. Let us suppose, for simplicity,  r=1. The general case follows similarly.
Let us set ^N+1_≥1/2:=^N×[1/2,+∞[. We consider the uniformly elliptic operator 𝒜 of Lemma <ref>, in the case r=1/2,  acting in L^2_c(^N+1_≥1/2):=L^2(^N+1_≥1/2,y^cdz) and endowed with Dirichlet boundary conditions. By construction the operators 𝒜 and ℒ are associated respectively with the sesquilinear forms 𝔞_0, 𝔞.  Moreover 

    𝔞_0=𝔞_|D(𝔞_0)    and     	D(𝔞_0)=H_c,0^1(^N+1_≥1/2)↪ H^1_c=D(𝔞).
 
Then, under the notation of  <cit.>, the previous relations imply that D(𝔞_0) is an ideal of D(𝔞)  and moreover we have, in particular,

    𝔞_0(u,v)   = 𝔞(u,v),    for any   0≤ u,v∈ D(𝔞_0).
 
Then, recalling that 𝒜 and ℒ generate positive semigroups, we  apply <cit.> thus obtaining  











    |e^t𝒜f|≤ e^tℒ|f|,    ∀ f∈ L^2_c(^N+1_≥1/2)
 
which in terms of their heat kernels rewrites as

    p_ℒ(t,z_1,z_2)≥ p_0(t,z_1,z_2)y_2^-c,     t>0, z_1,z_2∈^N+1_≥1/2,

(note that p_ℒ is written with respect to the measure y^cdz and  p_0 with respect to the Lebesgue measure). Then specializing the previous inequality for t=1 and z_1,z_2∈^N×[1,+∞[ and using  (<ref>),  we get for some positive constant C,c>0

    p_ℒ(1,z_1,z_2)≥ p_0(1,z_1,z_2)y_2^-c≥ Cy_2^-cexp(-c|z_1-z_2|^2).

Finally, using  the scaling property (<ref>) and Lemma <ref>, we get 

    p_ℒ(t,z_1,z_2)   =t^-N+1+c/2p_ℒ(1,z_1/√(t),z_2/√(t))≥ Ct^-N+1+c/2(y_2/√(t))^-cexp(-c|z_1-z_2|^2/ t)
       ≃C/V(z_2,√(t))exp(-c|z_1-z_2|^2/ t)

for every  z_1, z_2∈^N+1_+ such that y_1/√(t)≥ 1,  y_2/√(t)≥ 1. The required estimates then follows by Remark <ref>.





We now come to the main part of the section: we prove  the lower estimate (<ref>) near the boundary i.e. for   z_1, z_2∈^N+1_+  when at least one of z_1/√(t), z_2/√(t)  lies in ^N× ]0,1]. The main point consists in Proposition <ref> where we prove the so-called  “G-bound”, i.e. a lower bound of the integral of the logarithm of the heat kernel with respect to a  weighted Gaussian measure. We need some preparation.




We fix  α>0 and we consider the weighted  Gaussian measure ν :=y^ce^-a|z|^2dz on ^N+1_+ which, from the assumptions  α, c+1>0, is  finite on ^N+1_+. By standard computation one has

    ν(^N+1_+)=∫_^N+1_+ y^c e^-α |z|^2 dz= a^-N+1+c/2π^N/2Γ(c+1/2)
 
where Γ is the Gamma function (see <cit.>).  We consider the space L^2_ν(^N+1_+):=L^2(^N+1_+, y^ce^-a|z|^2dz) and we define also the Sobolev space 

    H^1_ν(^N+1_+):={u ∈ L^2_ν(^N+1_+) : ∇ u ∈ L^2_ν(^N+1_+)}

The following Poincaré-type inequality in H^1_ν(^N+1_+) is essential for the proof of Proposition <ref>.
[Poincaré inequality] 
	Let α>0,  c+1>0. Then,  for some positive constant C>0, one has
	
    u-u_L^2_ν(^N+1_+)≤  C ∇ u_L^2_ν (^N+1_+),    ∀ u∈ H^1_ν(^N+1_+),

	where u=1/ν(^N+1_+)∫_^N+1_+ u dν(z).


	See <cit.>.

From now on, without any loss of generality,  we fix α>0 in (<ref>)  such that 

    ∫_^N+1_+ y^c e^-α|z|^2 dz=1.

We define, for fixed θ∈ [1/2,1) and z_2∈^N+1_+, the function 

    u(t,z)   :=θ p_ℒ(t,z,z_2)+1-θ,    ∀ t>0,  z∈^N+1_+.

We collect in the next lemma the properties we need about u.

	Let u be  the function above defined. Then the following properties hold.
	
		
  (i) u(t,z)≥ 1-θ>0 for any t>0, z∈^N+1_+.
		
  (ii) For fixed t>0 one has e^-α|z|^2/u∈ H^1_c(^N+1_+) and log u∈ H^1_ν(^N+1_+).
		
  (iii) If z_2∈^N× ]0,1] the one has  
    sup_t∈[1/2,1]u(t,z)≤ K,    ∀ z∈^N+1_+

		for some positive constant K>0, independent on θ and on z_2. 
		
  (iv)  One has    
    lim_R→+∞∫_Q(0,R)u(t,z)y^c dz≥1/2

		  uniformly  on θ,  t∈[1/2,1] and  z
		_2∈ Q(0,1)=B(0,1)×[0,1[ .
	


	(i) follows from the positivity of p_ℒ.  (ii) follows from standard calculation, the previous property, the upper  estimates (<ref>) and of the gradient estimates of Theorem <ref>. To prove (iii) we use again  (<ref>) to write
	
    p_ℒ(t,z,z_2)
    		≤ C t^-N+1/2  y_2^-c(1∧y_2/√(t))^c exp(-|z-z_2|^2kt)

	which from the assumption y_2≤ 1  implies 
	
    sup_t∈[1/2,1]u(t,z)≤ K  exp(-|z-z_2|^2k')≤ K

	for some positive constants K,k'>0.
	
	 Let us finally prove (iv). Since by the  definition u(t,z)≥1/2 p_ℒ(t,z,z_2) then using the conservation property of Proposition <ref> we get 
	
    ∫_Q(0,R)  u(t,z) 
    		 y^c dz   ≥1/2∫_Q(0,R)  p_ℒ(t,z,z_2) y^c dz= 1/2(1-∫_Q(0,R)^c  p_ℒ(t,z,z_2)  y^c dz).

	We then observe that,  by the  upper kernel estimates (<ref>) and the assumption 1/2≤ t≤ 1,  |x_2|<1, y_2< 1, one has,  for some positive constants C,k>0 which may be different in each occurrence,
	
    p_ℒ(t,z,z_2)
    		   ≤ C t^-N+1/2  y^-c(1∧y/√(t))^c exp(-|z-z_2|^2kt)
       ≤ C y^-c(1∧ y)^c exp(-|z|^2k).

The last inequalities  then imply, by Lebesgue's Theorem,

    ∫_Q(0,R)^c  p_ℒ(t,z,z_2)  y^c dz≤  	C ∫_Q(0,R)^c(1∧ y)^c exp(-|z|^2k) dzR→+∞⟶ 0

which using (<ref>) proves (iv).

We can define now the Nash's G-function.

	For fixed θ∈ [1/2,1) and z_2∈^N+1_+, we define 
	
    G(t)   :=∫_^N+1_+log u(t,z)e^-α|z|^2y^c  dz,    ∀ t>0.

	
	We remark that, in virtue of Theorem <ref> and of (<ref>),  G(t) is well defined and finite.

We will estimate G(1) from below trough a differential inequality satisfied by G(t) which we prove in the following lemma.








	The following properties hold.
	
		
  (i) G(t)≤ 0 for any t>0. 
		
  (ii) For some positive constants A, B>0, independent of t, z_2 and θ, one has 
		
    G'(t)≥ -A+B∫_^N+1_+  |∇log u|^2 e^-α|z|^2 y^c dz.

		In particular  G(t)+At is increasing.
	


	To prove (i) we recall that, by Proposition <ref> one has  
    ∫_^N+1_+  p_ℒ(t,z,z_2)y^c  dz=1.

	Recalling (<ref>) and applying  Jensen's inequality   it follows that
	
    G(t)   =∫_^N+1_+log u(t,z)e^-α|z|^2y^c  dz≤log(∫_^N+1_+  u(t,z)e^-α|z|^2y^c  dz_2)
       =log(∫_^N+1_+θ p_ℒ(t,z,z_2)e^-α|z|^2y^c  dz+∫_^N+1_+ (1-θ) e^-α|z|^2y^c  dz)
       ≤log(θ∫_^N+1_+   p_ℒ(t,z,z_2) y^c  dz+(1-θ))=log 1=0.

	To prove (ii) we observe preliminarily that 
	
    ∂_t u(t,z)=θℒ p_ℒ(t,z,z_2)=ℒ u(t,z).
 
	Then  differentiating under the integral sign and using  Lemma <ref>  we   obtain
	
    G'(t)   =∫_^N+1_+∂_tlog u(t,z)e^-α|z|^2y^c  dz
       =∫_^N+1_+ℒu(t,z)e^-α|z|^2/uy^c  dz=-𝔞( u,e^-α|z|^2/u).

	Then recalling (<ref>) and by standard calculation we get
	
    G'(t)   =-∫_^N+1_+∇ u·∇(e^-α|z|^2/u)y^c  dz-2∫_^N+1_+  D_y  u a·∇_x(e^-α|z|^2/u) y^c  dz
       
    		=2α∫_^N+1_+∇log u· z  e^-α|z|^2 y^c  dz+∫_^N+1_+  |∇log u|^2e^-α|z|^2 y^c  dz
       +4α∫_^N+1_+  D_ylog u   a· x e^-α|z|^2 y^c  dz +2∫_^N+1_+ a·∇_x log u D_ylog u  e^-α|z|^2 y^c  dz.

	
	By Young's inequality we get for any  >0 and some positive constant C>0
	
    |2α∫_^N+1_+∇log u· z  e^-α|z|^2 y^c  dz|   ≤∫_^N+1_+  |∇log u|^2  e^-α|z|^2 y^c  dz
       +C ∫_^N+1_+  |z|^2  e^-α|z|^2 y^c  dz,

	
    |4α∫_^N+1_+ D_ylog u   a· x e^-α|z|^2 y^c  dz|   ≤∫_^N+1_+  |D_y log u|^2  e^-α|z|^2 y^c  dz
       +C ∫_^N+1_+  |x|^2  e^-α|z|^2 y^c  dz,

	
    |2∫_^N+1_+ a·∇_x log u D_ylog u  e^-α|z|^2 y^c  dz|   ≤ |a|∫_^N+1_+  |∇_x log u|^2  e^-α|z|^2 y^c  dz
       +|a|∫_^N+1_+  |D_y log u|^2  e^-α|z|^2 y^c  dz.

	Putting together the previous inequality and choosing ϵ sufficiently small we get
	
    G'(t)≥ -A+B∫_^N+1_+  |∇log u|^2 e^-α|z|^2 y^c dz,

	some positive constants A, B>0, independent of t, z_2 and θ. This proves the required claim.


	We can also consider, similarly, 
	
	
	
	
	
    u^∗(t,z)   :=θ p_ℒ^∗(t,z,z_2)+1-θ,     
    		G^∗ (t):=∫_^N+1_+log u^∗ (t,z)e^-α|z|^2y^c  dz.

	Then, adapting almost identically all the  proofs to the operator ℒ^∗,   the same results of Lemmas <ref> and <ref> keep to hold for u^∗ and G^∗.


 
	There exists a positive constant C>0 such that   one has
	
	
	
	
	
	
	
	
	
	
    ∫_^N+1_+log p_ℒ(1,z,z_2)e^-α|z_2|^2y_2^c  dz_2≥ -C,  ∫_^N+1_+log p_ℒ^*(1,z,z_2)e^-α|z_2|^2y_2^c  dz_2≥ -C.

	for any z∈ B(0,1)×]0,1[.

 Let θ∈ [1/2,1), z_2∈ B(0,1)×]0,1[ and let us consider, recalling Definition <ref> , the function 

    G(t)   =∫_^N+1_+log u(t,z)e^-α|z|^2y^c  dz,    ∀ t∈[1/2,1].
 
Our aim is to prove that 

    G(1)≥ -C
 
for some positive constant C>0 independent of θ and of z_2. Indeed once proved (<ref>), then  taking the limit for θ→ 1 and recalling that p_ℒ(1,z,z_2)=p_ℒ^*(1,z_2,z), one obtains

    ∫_^N+1_+log p_ℒ(t,z,z_2)e^-α|z|^2y^c  dz=∫_^N+1_+log p_ℒ^*(1,z_2,z)e^-α|z|^2y^c  dz≥ -C

which is the second required inequality for p_ℒ^*. The analogue inequality for  p_ℒ  follows in the same manner after replacing the function G with G^∗ and using Remark <ref>.

Let us then prove (<ref>).  Recalling that, by (ii) of Lemma <ref>,   log u∈ H^1_ν(^N+1_+),  we can combine  (<ref>) and  the weighted Poincaré inequality of Theorem <ref> applied to log u  to obtain  

    G'(t)≥ -A+B∫_^N+1_+  (log u-G(t))^2 e^-α|z|^2 y^c dz

for some positive constants A, B>0, independent of t, z_2 and θ. In particular the function   G(t)+At is increasing. 


We make, preliminarily, the following useful observation: the previous properties allow, without any loss of generality, to suppose the function  G very negative, i.e. satisfying

    G(t)<-D,    ∀ t∈[1/2,1],

for some  arbitrary large constant D>0 to be specified later. Indeed if (<ref>) fails  for some t_0∈[1/2,1] then  we  have, from the monotonicity of G(t)+At, 

    G(1)+A≥ G(t_0)+At_0≥ -D+A/2

which already   imply the required claim (<ref>).

 Assuming   (<ref>)  we are going now to estimate G(1) from below by proving the differential inequality (<ref>)  which we derive using Lemma <ref> and the  upper kernel estimates of Theorem <ref>.
We start by observing that by (iii) of  Lemma <ref> one has

    sup_t∈[1/2,1]u(t,z)≤  K,    ∀  z∈^N+1_+

for some positive constant K independent on θ and z_2.
Moreover the function u↦(log u-G(t))^2/u is decreasing as a function of u in [e^2+G(t),+∞[. Therefore, choosing in (<ref>) D sufficiently large,  we can then suppose e^2+G(t)< K  thus obtaining, since by  (<ref>)  u≤ k,

    G'(t)   ≥ -A+B∫_^N+1_+  (log u-G(t))^2 e^-α|z|^2 y^c dz
       ≥ -A+B∫_{u≥ e^2+G(t)}(log u-G(t))^2/uu  e^-α|z|^2 y^c dz
       ≥ -A+B(log K-G(t))^2/K∫_{u≥ e^2+G(t)}  u(t,z) e^-α|z|^2 y_2^c dz_2.

Next we prove that ∫_{u≥ e^2+G(t)}  u(t,z) e^-α|z|^2 y^c dz is bounded from below. Using (iv) of  Lemma <ref>  we get, choosing a sufficiently large R>0, 

    ∫_Q(0,R)  u(t,z) y^c dz≥1/4
 which implies

    ∫_u≥ e^2+G(t)  u(t,z) e^-α|z|^2 y^c dz   =	∫_^N+1_+  u(t,z) e^-α|z|^2 y^c dz-	∫_{u< e^2+G(t)}  u(t,z) e^-α|z|^2 y^c dz
       ≥ e^-α R^2∫_Q(0,R)  u(t,z) y^c dz-
    	e^2+G(t)
       ≥1/4 e^-α R^2-
    	e^2+G(t).

We then obtain

    G'(t)   ≥ -A+B(log K-G(t))^2/K( 1/4e^-α R^2-e^2+G(t)).


Choosing D large enough in (<ref>) in  order to have 1/4e^-α R^2-e^2+G(t)>0 and using  Young's inequality 

    2|log k||G(t)|<ϵ G(t)^2+|log k|^2/ϵ

 with ϵ=1/2, we get for some possibly different constants A, B>0

    G'(t)   ≥ -A+B  G(t)^2.

Choosing again  D large enough in (<ref>) in  order to have G(t)^2>2A/B, the previous inequality implies

    G'(t)   ≥B/2 G(t)^2.

Integrating between 1/2 and 1 and recalling that G(t)<0, we finally obtain G(1)≥ -4/B which is the required claim.






We can now deduce, from  the G-bound of  Proposition <ref>, the  lower estimates (<ref>) on the diagonal.

 
There exists a positive constant C>0 such that 
	
    p_ℒ(t,z,z)≥C/V(z,√(t)),
    		    ∀ t>0,  z∈^N+1_+.


Proof. Recalling the scaling  properties (<ref>) of the kernel 
 and of the function V of Lemma <ref>, we have



 
    p_ℒ(t,z,z)   =(t/2)^-N+1+c/2p_ℒ(2,√(2)z/√(t),√(2)z/√(t)),
    
    		V(z,√(t))   =(t/2)^-N+1+c/2 V(√(2)z/√(t),√(2)).

From the previous equalities we can then assume, without any loss of generality, t=2. 

If y≥ 1, the claimed bound    follows from Proposition <ref>. Let us assume, therefore,   y<1. By using the   invariance for translation in the x-variable of the kernel (<ref>) and of the function V of Lemma <ref> we have

    p_ℒ(t,z+x_0,z+x_0)=	p_ℒ(t,z,z),     	V(z+x_0,√(t))   =	V(z,√(t))

where for any z=(x,y)∈^N+1_+ and any x_0∈^N, we wrote,  with a little abuse of notation,  z+x_0=(x+x_0,y) to denote the translation in the x_0-direction. Choosing then  a suitable x_0∈^N,  we may also assume  |x|< 1. 
Let α>0 satisfying (<ref>); then by using  the reproducing property of the kernel we get

    p_ℒ(2,z,z)   =∫_^N+1_+p_ℒ(1,z,z_2) p_ℒ(1,z_2,z)y_2^c  dz_2
       =∫_^N+1_+p_ℒ(1,z,z_2)p^*_ℒ(1,z,z_2)y_2^c  dz_2
       ≥∫_^N+1_+p_ℒ(1,z,z_2)p^*_ℒ(1,z,z_2)e^-α|z_2|^2y_2^c  dz_2.

Applying  then Jensen inequality  with respect to the unitary measure ν=e^-α|z_2|^2y_2^cdz_2 and using Proposition <ref>,  it follows that, for some positive constant C>0, one has 

    log(p_ℒ(2,z,z))≥   ∫_^N+1_+log(p_ℒ(1,z,z_2)p^*_ℒ(1,z,z_2))e^-α|z_2|^2y_2^c  dz_2
    
    	 =   ∫_^N+1_+log p_ℒ(1,z,z_2)e^-α|z_2|^2y_2^c  dz_2
       +∫_^N+1_+log p^*_ℒ(1,z,z_2)e^-α|z_2|^2y_2^c  dz_2≥ -C

which implies

    p_ℒ(2,z,z)≥  e^-C.

Recalling  Lemma <ref>,  since we are assuming  y<1, we have 

    V(z,√(2))≃ V(z,1)≃ y^c(y∧ 1)^-c=1;

 the above inequality  is then equivalent to

    p_ℒ(2,z,z)≥C/V(z,√(2)).

which proves the required claim. 



We may now use  the gradient estimates  proved in Section <ref>, to extend the lower bounds of Proposition <ref> “near the diagonal” i.e. in the range  |z_1-z_2|<c√(t), for some c>0.
 
There exist some positive constants C,c>0 such that 
	
    p_ℒ(t,z_1,z_2)≥C/V(z_2,√(t)),
    		    ∀ t>0,  z_1, z_2∈^N+1_+   s.t.   |z_1-z_2|≤ c√(t).


Proof.
Using the mean value theorem we get 

    |p_ℒ(t,z_1,z_2)-p_ℒ(t,z_1',z_2)|≤ |∇ p_ℒ(t,z̃,z_2)||z_1-z_1'|,    ∀ z_1, z_1', z_2∈^N+1_+

where z̃ lies in the segment linking z_1 and z_1'. Then using  the gradient estimate of Theorem <ref> and Remark <ref> we get
	
    |∇ p_ℒ(t,z̃,z_2)|≤C/√(t) V(z_2,√(t)) exp(-|z_1-z_2|^2kt)≤C/√(t) V(z_2,√(t)),

which combined with the previous inequality yields

    |p_ℒ(t,z_1,z_2)-p_ℒ(t,z_1',z_2)|≤C/t^1/2V(z_2,√(t)) |z_1-z_1'|.

 Then, specializing  the above inequality to z_1'=z_2 and  using   Proposition <ref>, we get  for some positive constants C, C_1>0.

    p_ℒ(t,z_1,z_2)   ≥ p_ℒ(t,z_2,z_2)-C/t^1/2 V(z_2,√(t)) |z_1-z_2|
       ≥C_1/V(z_2,√(t))-C/t^1/2V(z_2,√(t)) |z_1-z_2|
    	= C/V(z_2,√(t))(c-t^-1/2 |z_1-z_2|)

where we set c=C_1/C. Then imposing  c-t^-1/2|z_1-z_2|≥c/2, that is  |z_1-z_2|/√(t)≤c/2, we get the required claim.

We can finally deduce the lower  estimate (<ref>) in the whole space using a classical iterative  argument based on  the doubling property of Lemma <ref> satisfied by the function V  and  the chain condition satisfied by ^N+1_+, namely the existence, for any z_1, z_2∈^N+1_+ of a sequence of point ω_0=z_1, ω_2, …, ω_n=z_2∈^N+1_+ joining z_1 and z_2 and satisfying

    |ω_i-ω_i+1|≤ C |z_1-z_2|/n,    ∀ i=0,…,n-1.

 
There exists a positive constant C such that the heat kernel p_ℒ of ℒ satisfies  
	
    p_ℒ(t,z_1,z_2)
    		   ≥C/V(z_1,√(t))^1/2V(z_2,√(t))^1/2exp(-|z_1-z_2|^2/κ t),  ∀ t>0,  z_1, z_2∈^N+1_+.


Proof. The proof follows identically as in the proof of  <cit.>.

 We end the section by  stating the final result for the  general operator (<ref>) of Section <ref> under  Neumann or oblique derivative boundary condition which is a direct consequence of  Theorems <ref>  and  <ref>   and   the transformation explained in  Section <ref>.
 
	Let v=(d,c)∈^N+1 with d=0 if c =0, A be 
	an  elliptic matrix and let us consider the operator
	
    ℒ =(AD^2u)+ v·∇/y
 endowed with the boundary conditions

    lim_y → 0 y^c/γ  v ·∇ u=0,    (if c≠ 0),        lim_y → 0 y^c/γ  D_y u=0,    (if c=0)

where γ=a_N+1,N+1. If   c/γ+1>0 then the semigroup (e^tℒ)_t> 0 consists of integral operators	and its heat kernel p_ℒ, written  with respect the measure y^c/γdz, satisfies for some  C,k>0, 
 
    p_ℒ(t,z_1,z_2)
     	≃ C t^-N+1/2 y_1^-c/2γ(1∧y_1/√(t))^c/2γ y_2^-c/2γ(1∧y_2/√(t))^c/2γ exp(-|z_1-z_2|^2kt).







§ APPENDIX




We consider on the weighted spaces L^p_m(^N+M):=L^p(^N+M,|y|^m,dxdy) the two-parameters  family of integral operators (S_α,β(t))_t>0  defined for α,β∈  and t>0 by

    S^α,β(t)f(z_1)=t^-N+M/2  (|y_1|/√(t)∧ 1 )^-α∫_^N+M (|y_2|/√(t)∧ 1 )^-βexp(-|z_1-z_2|^2/κ t)f(z_2)  dz_2,

where κ is a positive constant, z_1=(x_1,y_1), z_2=(x_2,y_2)∈^N+M. We omit the dependence on κ even though in some proofs we need to vary it.
We characterize in the proposition below the  boundedness of (S_α,β(t))_t>0 on L^p_m. The proof follows by using the strategy in <cit.> where the case 1<p<∞ has  been considered. For completeness we refer to <cit.> for the case p=∞.


We start by observing  that the  scale homogeneity of S^α,β is 2 since a simple change of variable in the integral yields

    S^α,β(t)(I_s f)=I_s (S^α,β(s^2t) f),     I_sf(z)=f(sz),     t, s>0,

which in particular gives 

    S^α,β(t)f =I_1/√(t)(S^α,β(1) I_√(t) f),     t> 0.

The boundedness of S^α,β(t)f in L^p_m(^N+M) is then equivalent to that for t=1 and S^α,β(t)_p= S^α,β(1)_p.



	Let m∈, θ≥ 0 and 1≤ p<∞. The following properties are equivalent.
	
		
  (i)  For every t>0 (S^α,β(t))_t ≥ 0 is  bounded from   L^p_m to L^p_m-pθ and
		
    S^α,β(t)_L^p_m→ L^p_m-pθ≤ Ct^-θ/2.

		
  (ii) S^α,β(1) is  bounded from  from   L^p_m to L^p_m-pθ.
		
  (iii) One of the following properties:
		
			
  * 1<p<∞ and α+θ <M+m/p < M-β;
  * p=1 and α+θ <M+m ≤ M-β.
		
		In  particular α+β<M-θ.
	


	The proof follows identically  as in  <cit.> where the case N=0 has been considered. 
	









































We refer also the reader to <cit.> for  the case p=∞ of the above Proposition.







 Paper Title: Constraint maps: singularities vs free boundaries.

























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Constraint maps: singularities vs free boundaries]
Constraint maps: singularities vs free boundaries


 

A. Figalli]Alessio Figalli
alessio.figalli@math.ethz.ch
Department of Mathematics, ETH Zürich,  Raemistrasse 101, 8092 Zürich, Switzerland 

A. Guerra]André Guerra
andre.guerra@eth-its.ethz.ch
Institute for Theoretical Studies, ETH Zürich,  CLV, Clausiusstrasse 47, 8006 Zürich, Switzerland

S. Kim]Sunghan Kim
sunghan.kim@math.uu.se
Department of Mathematics, Uppsala University, S-751 06 Uppsala, Sweden

H. Shahgholian]Henrik Shahgholian
henriksh@kth.se
Department of Mathematics, KTH Royal Institute of Technology, 100 44 Stockholm, Sweden





§ ABSTRACT 
Energy-minimizing constraint maps are a natural extension of the obstacle problem within a vectorial framework. Due to inherent topological constraints, these maps manifest a diverse structure that includes singularities similar to harmonic maps, branch points reminiscent of minimal surfaces, and the intricate free-boundary behavior of the obstacle problem. 
The complexity of these maps poses significant challenges to their analysis.


In this paper, we first focus on constraint maps with uniformly convex obstacles and establish continuity (and therefore higher-order regularity) within a uniform neighborhood of the free boundary. More precisely, thanks to a new quantitative unique continuation principle near singularities (which is new even in the setting of classical harmonic maps), we prove that, in the uniformly convex setting, topological singularities can only lie in the interior of the contact set. We then establish the optimality of this result.

Second, while exploring the structure of the free boundary, we investigate the presence of branch points and show how they lead to completely new types of singularities not present in the scalar case.






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









§ INTRODUCTION



Let M be a smooth domain with compact, connected complement in ^m, m ≥ 2, and let Ω be a bounded domain in ^n, with n≥ 1. In this paper we consider maps u∈ W^1,2(Ω;M) minimizing the Dirichlet energy

    [v] := ∫_Ω |Dv|^2 dx,

among all admissible maps v∈ W^1,2(Ω;M) with v - u ∈ W_0^1,2(Ω;^m). To emphasize the target constraint u ∈M as well as its energy minimality, we call such maps minimizing constraint maps. As shown in <cit.>, minimizing constraint maps are weak solutions to the Euler–Lagrange system

    Δ u = A_u(Du,Du)χ_u^-1(∂ M)  in Ω,

where A is the second fundamental form of the boundary ∂ M relative to M.  

We shall call M^c := ^m∖ M the obstacle, as the image of the constraint maps is not allowed to go inside it. The presence of the obstacle naturally induces the free boundary[The notion of free boundary here differs significantly from the one in the theories of harmonic maps and minimal surfaces, as our notion resides in the domain, while the latter appears in the target.]  Ω∩∂ u^-1(M), which is the interface between the non-coincidence set u^-1(M) and the coincidence set u^-1(∂ M). System (<ref>) shows that, when restricted to int(u^-1(∂ M))⊂Ω, constraint maps are harmonic maps into ∂ M. At the same time, each component of a constraint map is harmonic in int(u^-1(M)). This variational problem hence can be seen as a canonical extension of the scalar obstacle problem into the vectorial setting. 

Being vectorial maps, constraint maps develop in general discontinuities[Although the term “singularities” is more common in the Harmonic Maps literature, here we have chosen to use the word “discontinuities” to ensure that the Free Boundary community does not confuse them with singularities of the free boundaries.] 
due to topological differences between the domain and their image in the target.  Their singular set is denoted   throughout the paper by 

    Σ(u) := { x∈Ω: u is not continuous at x}.

Besides discontinuities, constraint maps can also exhibit branching, which is concerned with the regularity of the image of the map and is encapsulated by the branch set

    (u) := ⋃_k=0^min{n,m}-1_k(u),    _k(u) := {x∈Ω∖Σ(u): rank(Du(x))=k }.

Here, the word branching is motivated by the connection with minimal surfaces and geometric function theory,  e.g.  the map  u z↦ (z^2,  z^ℓ/2), defined for z∈=^2, is such that (u)= _0(u)= {0}. Points in (u) can either be true branch points,
such as when ℓ is odd, or false branch points, 
such as when ℓ is even.
More precisely, false branch points correspond to a degeneracy of the parametrization and not of the actual image that is being parametrized, while true branch points correspond to singularities of the image; see Figure <ref> for an illustration. We will be especially concerned with the case k=0 since the right-hand side in (<ref>) vanishes at points in _0(u), making the obstacle problem degenerate. 

 


Interestingly, constraint maps reveal an extremely rare interplay between ∂ u^-1(M), Σ(u), and (u) altogether. Despite the long history of constraint maps since their initial appearance in the literature, dating back to the 1970s by the work of Hilderbrandt <cit.> and Tomi <cit.>, the interaction between these apparently correlated objects was addressed only recently by the authors <cit.> in a different but related setting. 
The goal of this paper is to address the following fundamental question: 


    
    Does the free boundary meet the mapping singularities manifested by Σ(u) or (u)?
    
    
    
    
    


This is an inherently difficult question,  
as answering it requires an understanding of the interplay between local analytic aspects and global geometric and topological  constraints.



One of the major findings of this paper is that there are no discontinuities of the maps near their free boundaries, provided that the obstacle is uniformly convex[Uniform convexity of M^c means that there exists R>0 such that the following holds: for any point x ∈∂ M there exists a ball B_R(y)⊃ M^c such that x ∈∂ B_R(y).].
This condition is essentially optimal: we find a large class of general convex obstacles which produce discontinuities on the free boundaries. 
To our knowledge, and despite the significance of this problem, the first suggestion of a possible resolution of discontinuities by free boundaries was raised in the authors’ recent work <cit.> in a different but related setting. 

The second major finding of this paper is that, even when the map is regular and the obstacle is a ball, the map may exhibit branching
on the free boundary in such a way that the free boundary itself becomes singular. Indeed, it is  important to observe that the right-hand side in (<ref>) becomes zero at all points within _0(u). Consequently, at these points, the behavior of the free boundary may exhibit significant irregularities, which do not occur in the classical scalar obstacle problem.




As we shall explain later, once discontinuities and branch points are understood,  the free boundary regularity can be reduced to that for the scalar obstacle problem (cf. <cit.>).






 §.§ A brief overview of the literature


The study of minimizing constraint maps as solutions to vectorial obstacle problems dates back to the 1970s. Pioneering work by Hildebrandt <cit.>, and Tomi <cit.> independently addressed the regularity problems for constraint maps of two independent variables, employing different methodologies.
Early advances in higher dimensions can be attributed to the work of Hildebrandt-Widman <cit.>, who considered convex target constraints. The first comprehensive result in higher dimensions was established by Duzaar <cit.>, who proved an ε-regularity theorem. As a consequence, a partial regularity theorem was derived, affirming the W^2,p-regularity of mappings outside a closed singular set with dimension less than n-2. Subsequently, Duzaar–Fuchs <cit.> proved that the singular set has dimension at most n-3. Although this result is optimal in general, it was later observed in <cit.> that, under certain topological and geometric conditions on the target constraint, discontinuities are not present.
We note that the development of these works closely followed the higher-dimensional theory of minimizing harmonic maps, pioneered by Schoen–Uhlenbeck <cit.>: as for harmonic maps, 
the theory of constraint maps has been generalized to the setting of the p-Dirichlet energy in <cit.>, almost minimizing maps in <cit.>,  and weakly constraint maps as well as the associated heat flow problem in <cit.>. The regularity results in two space dimensions were recently extended to weakly constraint maps in <cit.>. We also refer the reader to <cit.> for results in the case where the obstacle is thin, in the sense that it has higher codimension.

It is worth noting that, in the simple case where the domain is one-dimensional, minimizing constraint maps are simply energy-minimizing geodesics on the manifold-with-boundary M. The regularity and (non-)uniqueness of geodesics on manifolds-with-boundary were studied in depth in <cit.> and also more recently in <cit.>. 

Quite surprisingly, and despite all of the above results, the theory did not progress much since the 1990s, until recently three of the authors revived the problem in <cit.> from the perspective of free boundary problems, opening up new and delicate issues on the exact regularity of the maps away from their singular set and around their free boundaries. Although the study of minimizing constraint maps had its origin as a vectorial obstacle problem, historically only basic properties of the free boundaries were shown in <cit.>. The result closest to this paper is, in fact, the very recent work <cit.> of ours concerning constraint maps minimizing a vectorial Alt–Caffarelli energy,  which corresponds to a vectorial version of the Bernoulli problem.

In recent years there has been a surge of interest from the free boundary community towards extending scalar problems into vectorial settings, see for instance <cit.> for Bernoulli-type problems from diverse angles,  <cit.> for obstacle-type problems,  and <cit.> for thin-obstacle-type problems. All the aforementioned works successfully extend aspects of the scalar theory and explore inherently interesting issues related to the branching of free boundaries from several components. 

However, our minimizing constraint maps exhibit very different characteristics compared to the solutions to the aforementioned vectorial problems, due to the possible development of discontinuities of the map itself around the free boundaries. Hence a complete understanding of constraint maps requires bridging the theory of harmonic maps with that of free boundary problems. 





 §.§ Main results


Our first result asserts that, whenever the obstacle is uniformly convex, minimizing constraint maps are uniformly C^1,1-regular in a universal neighborhood of the non-coincidence set (hence, in a set which strictly includes the free boundary). 
It should be noted that C^1,1-regularity is optimal, due to the classical theory of the obstacle problem. 
In addition, the result cannot be improved: solutions cannot be continuous everywhere neither in the interior of the coincidence set,  due to the possible presence of singularities coming from minimizing harmonic maps with values in ∂ M.


Let M⊂^m be a smooth domain with uniformly convex complement. There exist constants δ∈ (0,1/2) and c > 1, depending only on n, m, ∂ M, and Λ, such that for any minimizing constraint map u∈ W^1,2(B_2;M) satisfying u _W^1,2(B_2)≤Λ, we have u ∈ C^1,1(B_1∩ B_δ(u^-1(M))) with the estimate 

    u _C^1,1(B_1∩ B_δ(u^-1(M)))≤ c.



To describe the strategy and challenges in the proof of Theorem <ref>, it is fruitful to introduce the distance function to ∂ M, which we denote by ρ.
Note that the smoothness and convexity of M^c guarantee that 
ρ is smooth in M. Then (<ref>) 
yields an Euler–Lagrange equation for ρ∘ u, namely

    Δ(ρ∘ u )     = ρ_u(Du,Du) χ_u^-1(M),

where ρ_u denotes the Hessian of ρ evaluated at u. Noting that u^-1(M) is precisely the positivity set of ρ∘ u, we observe that (<ref>) is a scalar obstacle-type equation, but with an important caveat. Indeed, the nonlinearity ρ_u(Du,Du), which acts as a force in (<ref>), 
can become unbounded (when u is discontinuous) and can also vanish (when either ρ_u, Du, or their bilinear combination vanishes). Hence ρ∘ u can behave quite differently from a solution to the scalar obstacle problem, where one deals with forcing terms that are regular and bounded away from zero. In particular, ρ∘ u has no inherent non-degeneracy: as we show in Theorem <ref> below, it may in fact vanish to arbitrarily high order on the free boundary.

In our previous work <cit.> (in a slightly different context) we observed that the distance map ρ∘ u is continuous everywhere, even across Σ(u), and that the non-coincidence set u^-1(M) has null density at any point of Σ(u): at any (small) scale r, we have 

    |B_r(x_0)∩ u^-1(M)| = o(r^n),  ∀ x_0∈Σ(u).

Interestingly, the assertion (<ref>) is proved via the unique continuation principle (UCP) for constraint/harmonic maps. However, to show that Σ(u) is completely disjoint from the free boundary, we need the much stronger assertion that ρ∘ u  vanishes identically in a full neighborhood of Σ(u), i.e. there is some small r_0>0 such that 

    B_r_0(x_0)⊂ u^-1(∂ M),  ∀ x_0∈Σ(u).
 
Our idea is then as follows. We look at the distance map ρ∘ u, which is supported exactly on u^-1(M), and is subharmonic for convex obstacles M^c. A key property of ρ∘ u is that it vanishes continuously at any point of Σ(u) whence, upon translating and rescaling, we may assume that 0≤ρ∘ u ≤ in B_2 and 0∈Σ(u), for a given > 0. Then, by a lemma reminiscent of the De Giorgi argument (Lemma <ref>), we observe that ρ∘ u = 0 in B_1, provided that  is sufficiently small and  

    |B_r∩ u^-1(M)|≤c/(R-r)^βρ∘ u_L^∞(B_R)^α,  ∀  1 ≤ r< R ≤ 2,

for some α, β, and c that do not depend on the scales r and R. Condition (<ref>) can be considered as the reverse of the local maximum principle for subharmonic functions. Since we are free to choose the smallness parameter  for the sup-norm of ρ∘ u over B_2, our task boils down to verifying (<ref>). 

What appears to be intriguing is the involvement in (<ref>) of the vectorial character of u implied by the presence of 0∈Σ(u) (note that u^-1(M) is simply the set where ρ∘ u is positive). To exploit the hidden vectorial character in (<ref>), we resort to the Euler-Lagrange equation (<ref>) for ρ∘ u. Integrating both sides of (<ref>), and invoking the uniform convexity of M^c, we obtain around 0∈Σ(u),[That 0∈Σ(u) is needed to get ρ_u(Du,Du)≈ |Du|^2 in the L^1-sense. In general, we always have ρ_u(Du,Du)≈ |D(Π∘ u)|^2 a.e., with Π being the nearest point projection to ∂ M (by (<ref>) and (<ref>)). However, ρ∘ u becomes negligible around any discontinuity point of u, so we obtain the last inequality in (<ref>).] along a chain of elementary tools (see (<ref>) – (<ref>)), that 

    1/(R-r)^2∫_B_R (ρ∘ u) dx ≳∫_B_r∩ u^-1(M)ρ_u(Du,Du)  dx ≳∫_B_r∩ u^-1(M) |Du|^2  dx

for every 1≤ r<R≤ 2. 

Our analysis would have been softer if |Du| ≳ 1 uniformly in B_r, as the rightmost side of (<ref>) would then readily yield (<ref>) with w = ρ∘ u. Since |Du| blows up at 0∈Σ(u) possibly with a rate 1/r in B_r, |Du|≳ 1 is expected to be true at least for a large portion of the neighborhood of 0∈Σ(u).
However, |Du| may not behave so uniformly near its singularity. In principle, |Du| may become very small in a set of tiny measure (and even vanish somewhere inside), and such a tiny set may just happen to be the non-coincidence set u^-1(M) as we are close to Σ(u) (recall (<ref>)): e.g., u^-1(M) is a cusp with its tip on Σ(u), such that one of the tangent maps ϕ of u is of rank zero along the direction of the cusp. The latter would certainly be the worst case scenario for us, as the rightmost side of (<ref>) sees u^-1(M) only. 

Thus, our analysis calls for a more robust approach that works in the absence of pointwise control of the vanishing behavior of |Du|. For this purpose, our new key result is an L^-γ-estimate of |Du| near Σ(u). Note that the next statement does not require convexity of the obstacle.



  
 
     Let M⊂^m be a smooth domain with compact complement and let u∈ W^1,2(B_4;M) be a minimizing constraint map satisfying u _W^1,2(B_4)≤Λ. There exist _0, γ > 0, both depending only on n, m and ∂ M, and a constant c>1, depending further on Λ, such that if ∫_B_1(y) |Du|^2 dx ≥_0^2 for some y∈ B_2 then 

    ∫_B_1 |Du|^-γ dx ≤ c.

 

 

This estimate can be understood as a quantitative version of the UCP for the constraint maps, as it shows that |Du| cannot vanish too fast around Σ(u). Once such an estimate is available, one can use the generalized Hölder inequality to verify the reverse link (<ref>) (see (<ref>)), which in turn yields (<ref>) as desired. Therefore, Theorem <ref> lies at the heart of our analysis for Theorem <ref>.

 
As already observed above, Theorem <ref> works for general smooth compact obstacles M^c without any condition on their principal curvatures. Moreover, this theorem also holds for minimizing harmonic maps into smooth compact manifolds without boundary, and it is even new in the latter context. 


 


    The above results of UCP-type are somewhat reminiscent of the classical quantization results for minimizing harmonic maps <cit.>, which assert  that there is a universal lower bound on the energy of smooth harmonic maps which are not homotopic to a constant.  The theorem  is also evocative of a result from <cit.>, where the authors prove an ^n-2-measure estimate on the set _0(u) for a minimizing harmonic map u; however,  this estimate is qualitatively different from (<ref>), and neither of them implies the other. 

    
    
    We emphasize that the structure of blow-ups of harmonic maps near their discontinuity points is only well-understood in some very specific instances, such as when n=3 and N = 𝕊^2 <cit.>,  or n=4 and N = 𝕊 ^3 <cit.>. In these cases, the structure of the blow-ups can be exploited to prove (<ref>), as we will see in Section <ref>. However, since blow-ups are poorly understood in general, we are forced to rely on a fundamentally different type of argument to prove Theorem <ref>. 


    The key observation for proving Theorem <ref> is the existence of a critical scale at every regular point such that,
    at this scale, we can control both the regularity and the frequency of |Du| uniformly and in our favor. The critical scale plays a  significant role in two ways. Above this scale, the effect of the discontinuous singularity dominates, so that |Du|≳ 1 in a large portion of the neighborhoods. Below the critical scale we have |Du| ≪ 1, but we can still control the doubling property of |Du| in a uniform way. This is because the regularity of u is uniformly controlled on such a scale. In other words, 1/|Du| does not grow so fast in a large portion of the neighborhoods on these scales. Finally, we can patch these two different scales altogether via a Calderón–Zygmund decomposition argument, which leads us to the desired L^-γ-estimate for |Du|. 
    
    Let us address that the notion of the critical scale here is closely related to that of the regularity scale introduced in <cit.> in the study of the rectifiability properties of Σ(u).  However, the analysis in these works is different from ours,  as the authors there are concerned with obtaining (optimal) L^p-estimates for |Du| with p > 1.


      


Returning to Theorem <ref>, let us now address the optimality of the uniform convexity condition. Our next result asserts that this condition is essentially optimal,  as the conclusion of Theorem <ref> no longer holds as soon as ∂ M contains a portion of a hyperplane, even when M^c is convex.   We remark that by <cit.>, the convexity of M^c ensures a well-defined free boundary even in the presence of discontinuities, as it implies that (u,∂ M) is continuous.  In fact,  we can show that for obstacles with flat sides, minimizing constraint maps whose boundary values have non-trivial topological degree (see Appendix <ref> for the definition) always develop discontinuities on their free boundaries:




Let Ω⊂^n be a  smooth uniformly convex  domain, and let M⊂^n be a smooth domain with convex complement such that  M^c⋐Ω and ∂ M contains a relatively open subset T of a hyperplane. Let n≥ 3 and  consider boundary data g∈ C^∞(∂Ω;∂Ω) with (g) ≠ 0. If u ∈ W_g^1,2(Ω;M) = { u ∈ W^1,2(Ω;M);  u = g  on ∂Ω} 
is a minimizing constraint map, then 

    ∂ u^-1(M)∩Σ(u) ≠∅.

In fact,   for every y∈ T there is {x_k}⊂Ω∩ u^-1(M)∖Σ(u) such that x_k→ x_0 ∈∂ u^-1(M)∩Σ(u) and u(x_k)→ y, and so we write

    T⊆ u( ∂ u^-1(M)∩Σ(u)).
  



The assumption n≥ 3 in Theorem <ref> is used only to guarantee the existence of a minimizing constraint map. See Remark <ref> below for further discussions.


The situation for non-convex obstacles is even more exotic than the one described above: as a byproduct of Theorem <ref>,  we show in Corollary <ref> that, for such obstacles,  the non-coincidence set u^-1(M) may not be open.[It is not immediately clear how to define u^-1(M) in general, since u may develop discontinuities. A robust definition, which is independent of the choice of representative for u∈ W^1,2(Ω;M), is to set 

    u^-1(M):={x∈Ω: lim inf_r→ 0ρ∘ u_L^∞(B_r(x))>0}.

This set contains those points in Ω∖Σ(u) which are mapped by (the smooth representative of) u|_Ω∖Σ(u) to M.]
This yields in particular the sharpness of <cit.>.


Producing examples as those in Theorem <ref> is challenging, since we do not know how to construct explicit minimizing maps  in non-perturbative regimes. Indeed,  due to the vectorial nature of the problem, minimizers generally do not inherit the symmetry of their boundary data; see e.g. <cit.> for a dramatic symmetry-breaking phenomenon in the classical case of harmonic maps from the unit ball 𝔹^3 to 𝕊^2.  In particular, our approach to Theorem <ref> is based on a topological degree argument and is necessarily global, in contrast to the typical local analysis performed in the context of free boundary problems.





The above results give a rather complete picture of the behavior of minimizing constraint maps near their free boundaries but do not address the structure of the free boundary itself.  The main difficulty in the analysis is due to the possible occurrence of branch points on the free boundary,  and in particular of points in _0(u): at such points, no matter what the obstacle is, the right-hand side in (<ref>) vanishes and so we are faced with a degenerate obstacle problem. On the other hand, if the obstacle is uniformly convex, then by Theorem <ref> and the classical theory for the obstacle problem, the structure of the free boundary is well-understood in ∂ u^-1(M)∖_0(u), see <cit.>. 
We thus arrive at the following questions: For uniformly convex obstacles, is the set ∂ u^-1(M)∩_0(u) non-empty? If so, how does the free boundary look like around such points?




The structure of the free boundary for degenerate free boundary problems is a difficult topic. In this direction, very little is known even in the scalar setting, except e.g. in the particular examples studied in <cit.>. However, the degeneracy in these references stems from an absorbing term acting as an additional force in the problem; see also the discussions in <cit.> for a forcing term given directly by the obstacle. Nevertheless, the potential appearance of branch points for vectorial maps is part of the intrinsic nature of the problem and is not dictated by external forces. 

Here we construct and analyze a large class of examples of mininimizing constraint maps u∈ W^1,2(Ω;^3\ B^3) with non-trivial free boundaries such that

    ∂ u^-1(^3\ B^3)∩_0(u)≠∅,

thus showing that branch points can occur on free boundaries, even when the obstacle is a ball. 
In order to construct such examples, we shall consider maps that minimize the Dirichlet energy in the more restricted class of k-axially symmetric maps;
see Definition <ref>. Briefly, a map is called k-axially symmetric if it maps each cylinder centered on the z-axis to another such cylinder while rotating it k-times. Our analysis shows that, for k≥ 2, each free boundary point on the z-axis is simultaneously a branch point and a singular free boundary point: thus, branching can occur and can be responsible for free boundary singularities.







    Fix  k≥ 2, λ > 1, and a k-axially symmetric map g∈ C^2(^2;^2) with |(g)| = k. Let u∈ W_λ g^1,2( B^3;^3\ B^3) be an energy minimizer among k-axially symmetric maps. For any x_0∈∂{|u|>1}, there is a ball B⊂ B^3 centered at x_0 such that u∈ C^1,1(B) is the unique minimizing constraint map in W^1,2_u(B;^3\ B^3). Moreover, the following hold for the set _0(u) of rank zero points and the free boundary ∂{|u|>1}:
    
        
        
  *  _0(u)∩∂{|u|>1}∩∂( int{|u|=1})≠∅;
        
  *  |Du| vanishes exactly with order k-1 at every point of _0(u);
        
  *  ∂{|u|>1} cannot be represented as a C^1-graph in any neighborhood of _0(u) on the z-axis.
    
    In particular, (<ref>) and (<ref>) show that the free boundary is non-trivial and singular.


We refer the reader to Section <ref>, and in particular to Theorem <ref> therein, for a much more precise description of the structure of the free boundary near points in _0(u). As we shall see, in the setting above we can give a complete classification of the possible blow-ups appearing in the free boundary analysis.














 §.§ Organization of the paper


The paper is organized as follows. In Section <ref>, we describe the setting and notation of this paper and collect preliminary results on minimizing constraint maps. In Section <ref>, in preparation for our first main result, we prove the uniform continuity of the distance of a minimizing constraint map to the convex hull of the obstacle. Section <ref> is devoted to the proof of Theorem <ref>. In Section <ref> we prove Theorem <ref>. In Section <ref>, we study minimizing constraint maps for non-uniformly convex obstacles, and in particular we prove Theorem <ref>. 
Section <ref> collects some remarks and examples concerning branch points and their interaction with free boundaries. In Section <ref> we develop the regularity theory of axially symmetric constraint maps, and in particular we prove analogues of Theorems <ref> and <ref>  in this setting. Finally, in Section <ref> we study branch points on free boundaries for axially symmetric maps and prove Theorem <ref>. 

The paper also contains four appendices consisting of auxiliary results that are either known or follow with some relatively mild modifications from standard arguments. In Appendix <ref> we recall the definition and some basic properties of the topological degree, and its local variant. In Appendix <ref> we recall standard facts about the Almgren frequency of a solution to a regular linear elliptic system. In Appendix <ref> we prove a version of Luckhaus' lemma in the setting of axially symmetric constraint maps. Finally, in Appendix <ref> we prove the uniqueness and regularity of weakly constraint maps into small geodesic balls.









 §.§ Notation


For the reader's convenience, we gather here a list of notation to be used throughout this paper.


[l]l l 
    Ω        a bounded domain in ^n (n≥ 1)  
 
    Ω_η        { x ∈Ω:(x,∂Ω)>η}  (η≥ 0)    
 
    M         smooth domain in ^m (m≥ 2)  
 
    u         (u^1, ⋯ , u^m)    

    W^1,p(Ω;M)       {u∈ W^1,p(Ω;^m): u∈M a.e. in Ω}      

    W^1,p_g(Ω;M)       the Dirichlet class W^1,p(Ω;M)∩ (g+W^1,p_0(Ω;^m)), for g∈ W^1,p(Ω;M)   

    B^m         the open unit ball in ^m  
  
    S^m-1         the unit sphere in ^m, S^m=∂ B^m 
 
    A_p (· , ·)         the second fundamental form of ∂ M at p, see (<ref>)   
   
    (∂ M)         tubular neighborhood  of ∂ M 
    
   
    ρ         signed distance function to ∂ M, smooth on (∂ M)    
   
    ν=∇ρ         inward unit normal to M, extended to (∂ M) 
   
    Π          nearest point projection: (∂ M)→∂ M
       
 
    ν⊗ν          the tensor product matrix with entries (ν^iν^j)_ij 
   
    ξ^⊤         (I - ν⊗ν)ξ,    orthogonal projection of ξ into T(∂ M), see (<ref>)      
 
    [u]         ∫_Ω |Du|^2   dx   
 
    E(u,x_0,r)         r^2-n∫_B_r(x_0) |Du|^2 dx,  normalized energy     

    N(f,x_0,r)         r∫_B_r(x_0) |Df|^2 dx/∫_∂ B_r(x_0) |f - (f)_x_0,r|^2, the frequency function     

    Σ(u)          { x∈Ω: u is not continuous at x}    
   
    _k(u)          {x∈Ω∖Σ(u):Du has rank k at x}  
 
    ^k         k-dimensional Hausdorff measure    
   
    dim_         Hausdorff dimension     
    
    F:G         f_α^i g_α^i,  where  F = (f_α^i) and  G= (g_α^i) are matrices in ^m× n    
 
    D^k        the k-th order differential operator in the ambient space, D_α := ∂/∂ x_α 

    ∇^k        the k-th order differential operator in the target space, ∂_i := ∂/∂ y^i 

    f_y(ξ,ζ)         ∂_ij f(y)ξ_α^iζ_α^j, for a function f from the target space, and ξ,ζ∈^m× n 
 








§ PRELIMINARIES
 





 §.§ Problem setting


Throughout this paper, Ω⊂^n is a bounded domain and M⊂^m is a smooth domain; we always assume that n,m≥ 2. The  complement of M, M^c, is called the obstacle, and we always assume that it is compact. We denote by ρ the (signed) distance function to ∂ M that takes positive values in M, and by Π:^m →∂ M the nearest point projection onto ∂ M. We also write (∂ M) for the largest open neighborhood of ∂ M where ρ and Π are smooth; in particular, in the case where M^c is convex, the set (∂ M) strictly contains M.  We set ν:=∇ρ on (∂ M) and note that, on ∂ M, ν is the inward-pointing unit normal to ∂ M. See Figure <ref> for an illustration.




We denote by B_r(x_0) the ball of radius r centered at the point x_0, and more generally B_r(F) denotes the neighborhood of width r around the set F.  If B=B_r(x_0) is a ball, we write λ B := B_λ r(x_0)
for the dilation of B by a factor λ>0 around its center; we use a similar notation for cubes. We also denote by _(S) the Hausdorff dimension of a set S, by ^k(S) the k-dimensional Hausdorff measure of S, and by |S| the Lebesgue measure of S. We write D and ∇ for the usual differential operators in the ambient space and target spaces, respectively.  
Also, for a function f, we write f_y(ξ,ζ):= ∂_ij f(y)ξ_α^iζ_α^j.






 §.§ Known results for minimizing constraint maps


To keep our exposition as self-contained as possible,  we gather here some basic results about minimizing constraint maps. These can be seen as generalizations of classical theorems concerning minimizing harmonic maps into closed manifolds <cit.>. 

A basic quantity in the study of minimizing constraint maps is the normalized energy

    E(u,x_0,r) := r^2-n∫_B_r(x_0) |Du|^2 dx.

Minimizing constraint maps are stationary upon inner variations, and this yields a monotonicity formula for the normalized energy. A more general result is proved in <cit.>; here, we only present the statement adapted to the current setting. 


Let u∈ W^1,2(Ω;M) be a minimizing constraint map. For all x_0∈Ω we have

    E (u, x_0,s) - E (u, x_0,r) ≥ 2 ∫_B_s(x_0)∖ B_r(x_0) |x-x_0|^2-n|∂ u/∂ N|^2 dx 
        ∀   0<r<s<(x_0,∂Ω),

where ∂/∂ N denotes the directional derivative in the radial direction. In particular, if E(u,x_0,s) = E(u,x_0,r) then u is 0-homogeneous around x_0 inside B_s(x_0)∖ B_r(x_0), i.e., 

    (x-x_0) · Du = 0   a.e. in  B_s(x_0)∖ B_r(x_0).

  



A crucial tool in our analysis is the compactness of minimizing constraint maps, originally established in <cit.> within a broader context. Since we will frequently reference it here, we provide the adapted statement for the reader's convenience.


Let {u_k}_k=1^∞⊂ W^1,2(Ω;M) be a bounded sequence of minimizing constraint maps. Then there exists a minimizing constraint map u ∈ W^1,2(Ω;M) such that, up to a subsequence, u_k → u strongly in W_^1,2(Ω;^m). 



The monotonicity of the normalized energy yields a well-defined limit E(u,x_0,0^+) for every x_0∈Ω. 
As an immediate consequence of Lemmas <ref> and <ref>, we obtain the upper semicontinuity of the energy density. 


Let {u_k}_k=1^∞⊂ W^1,2(Ω;M) be a sequence of minimizing constraint maps such that u_k→ u strongly in W^1,2(Ω;^m). Then for any sequence {x_k}_k=1^∞⊂Ω with x_k → x∈Ω we have 

    E(u,x,0^+) ≥lim sup_k→∞ E(u_k,x_k,0^+).



We will also make use of the -regularity theorem, which states that minimizing constraint maps are regular in the vicinity of points with small energy. The optimal version of this result is due to the recent work <cit.> (see also <cit.> for a previous version of this statement).


 
There exist a small constant _0 > 0 
depending only on n, m, and ∂ M, such that the following holds: if u∈ W^1,2(Ω;M) is a minimizing constraint map with
 
    E(u,x_0,2r) ≤_0^2,    where  B_2r(x_0)⋐Ω,
 
 then u∈ C^1,1(B_r(x_0)), with the estimate  
 
    D^j u_L^∞(B_r(x_0))≤1/r^j,   j=1,2.

 

Recall from (<ref>) that Σ(u) consists of all discontinuity points of the map u in the prescribed domain. A useful corollary to Theorem <ref> is the energy quantization at points of Σ(u):
 


There exists a constant _0 > 0, depending only on n, m, and ∂ M, such that for any minimizing constraint map u∈ W^1,2(Ω;M) we have that x_0∈Σ(u) if and only if E(u,x_0,0^+) ≥_0^2. 


Through the usual dimension-reduction principle,  the above results yield an optimal partial regularity theorem, see <cit.>.





Let u∈ W^1,2(Ω;M) be a minimizing constraint map. Then u∈ C_^1,1(Ω∖Σ(u)) and _(Σ(u))≤ n-3.






The following lemma can be found in <cit.>, and will be used frequently in our subsequent analysis.




Let u∈ W^1,2(Ω;M) be a minimizing constraint map such that u(B)⊂(∂ M) a.e. in a ball B⊂Ω. Then 
    
    Δ (ρ∘ u) =  ρ_u(Du,Du)χ_u^-1(M)  in B,

    in the weak sense. 






 §.§ Vector calculus identities and geometric bounds


In this subsection we prove some elementary identities and estimates related to the geometry of ∂ M. 
Since ν = ν∘Π in (M), we have the decomposition 

    𝕀 = Π+ ρν = Π+ρ(ν∘Π).

In particular, differentiating both sides of this identity, we have - ν⊗ν = ( + ρ∇ν)∇Π; note that -ν⊗ν is the orthogonal projection onto the tangent bundle T(∂ M).
For a vector ξ∈^m, we thus write
    
    ξ^⊤ := ξ - (ν·ξ)ν= [( + ρ∇ν)∇Π]ξ

 for the orthogonal projections of ξ into T(∂ M).
 
The second fundamental form of ∂ M, appearing in (<ref>), can be expressed at points in ∂ M as


    ρ(ξ,ξ) = ρ(ξ^⊤,ξ^⊤) = -ν· A(ξ^⊤,ξ^⊤).

When the obstacle M^c is uniformly convex, we obtain a simple relation between ρ and the differential |∇Π|:





Let M be a smooth domain with uniformly convex complement. Then 

    ρ(ξ,ξ) ≥κ/1+κρ |(∇Π)ξ|^2  ∀ ξ∈^m,

where κ is the curvature lower bound for ∂ M. 







As M^c is convex, ρ, ν, and Π are well-defined and smooth in a neighborhood of M. Let {τ^1,⋯,τ^m-1,ν} be the principal coordinate system of ∂ M, and κ_i be the i-th principal curvature. As ν points outward to M^c, it follows from <cit.> that 

    ρ(τ^i,τ^i) = κ_i∘Π/1 + (κ_i∘Π)ρ  in M.

Also, as M^c is uniformly convex, κ_i∘Π≥κ in M, for some κ>0 which depends only. Since the map t↦t/1+ at is strictly increasing in t>0, for each a ≥ 0, we deduce from (<ref>) that

    ρ(ξ,ξ)≥κ/1+κρ |ξ^⊤|^2  in M  ∀ ξ∈^m.
 
Since ∇ν = ρ (as a consequence of the relation ν = ∇ρ), using again (<ref>) we see that

    ( + ρ∇ν)(τ^i,τ^i) = 1 + 2(κ_i∘Π)ρ/1+ (κ_i∘Π)ρ∈ [1,2]  in M.

In particular, combining (<ref>) with (<ref>) yields 

    |ξ^⊤|^2≥  |(∇Π)ξ|^2  in M  ∀ ξ∈^m.

Thus, (<ref>)  follows from (<ref>) and (<ref>). 










 §.§ Existence and examples of constraint maps


Consider boundary data g∈ C^∞(∂Ω; M). In this setting, the direct method yields existence of minimizing constraint maps in W^1,2_g(Ω;M), provided this space is non-empty. 
We discuss whether this holds in the next remark and for simplicity we take Ω= B^n to be the open unit ball.


In studying W^1,2_g( B^n;M), there are two different cases to consider.
    
    
  * If n≥ 3, then W_g^1,2( B^n;M)≠∅. Indeed, it suffices to consider the 0-homogeneous extension x↦ g(x/|x|) to obtain a map in that class. The crucial calculation is that the map x↦ x/|x| is in W^1,p near the origin if and only if n>p. Thus, in order to be able to take p=2, we require that n≥ 3.
    
  * If n=2, the situation is more subtle and it can be concisely expressed in terms of the homotopy extension property: W^1,2_g( B^2;M)≠∅ if and only if
    
    g  is homotopic to  G|_ S^1 for some continuous map  G B^2→M.

    Note that this characterization holds for boundary data g that are not necessarily smooth, but are just in the natural trace space H^1/2( S^1;M) <cit.>.
    
    In the simplest possible case where 
    g( S^1) is diffeomorphic to S^1, (<ref>) can be characterized elegantly in terms of the topological degree of g (we refer the reader to Appendix <ref> for a brief introduction to this notion). Indeed, according to Hopf's Theorem[The Hopf Theorem is a result within differential topology asserting that the topological degree stands as the sole homotopy invariant among continuous mappings taking values in a sphere.], in this case
    
    (<ref>)     (g) = 0.

    For data g∈ H^1/2( S^1;M) that are not continuous, such equivalence still holds, but one needs to interpret the degree in the sense of the VMO-degree of Brezis and Nirenberg <cit.>.



We now construct an explicit example of a radially symmetric constraint map with a discontinuity at the origin.


Given n=m≥ 3 and a radius a ∈ (0,1), we aim to find a map u_a∈ C^1,1(B_1∖{0}; B_a^c)∩ W^1,2(B_1;B_a^c) such that u_a = 𝕀 on ∂ B_1
and which solves
        
    Δ u_a = - |Du_a|^2 u_aχ_{|u_a|=a}  in B_1,
    
i.e., the Euler–Lagrange system for constraint maps with  obstacle B_a, see (<ref>).
   
   We look for a map u_aB_1∖{0}→ B_a^c of the form 
    
    u_a (x) := w_a(|x|) x/|x|.

The above system yields an overdetermined ODE for w_a, namely
     
    ẅ_a + n-1/rẇ_a = n-1/r^2 w_a    for r ∈ (r_a,1),
    
            w_a(r_a) = a,   ẇ_a(r_a) = 0,    w_a(1) = 1, 
    
            w_a(r)=a   for r ∈ [0,r_a],

    where r_a ∈ (0,1) is a free parameter corresponding to the free boundary constraint. 
Ignoring the  boundary conditions, the above ODE  has two  particular solutions, namely  r and r^1-n.
We thus  look for a solution of the form
  
    w_a (r) = 
        {[                    a     for r ∈ [0,r_a],; t_a r + (1-t_a)r^1-n    for  r ∈ (r_a,1), ].

    where the parameters t_a,r_a ∈ (0,1) satisfy

    t_ar_a+(1-t_a)r_a^1-n=a,     t_a+(1-n)(1-t_a)r_a^-n=0.

One can easily check that the system above has a unique solution. In addition, one can observe that r_a→ 0^+  as a→ 0^+, while r_a→ 1^- as a→ 1^-.  
Returning to the original map u_a,  the coincidence set is given by u_a^-1(∂ B_a) = B_r_a, with free boundary ∂ u_a^-1((B_a)^c) = ∂ B_r_a. 




An interesting problem is to understand whether the map u_a is a globally minimizing constraint map. We recall that the radial solution[Note that the function u_1(x)= x/|x| corresponds to the limit of the functions u_a as a→ 1^-.] u_1(x)= x/|x| of the Dirichlet problem for harmonic maps into a sphere with identity boundary values is a minimizer, i.e.

    [u_1]≤[v]     for all  v∈ W^1,2_𝕀(B^n;𝕊^n-1),

see e.g. <cit.> and the references therein; here, as before, we assume that n≥ 3.   However, despite the intuitive nature of this result, it is quite non-trivial to show the minimality of u_1.
  By analogy with the case of harmonic maps, it seems plausible  that the maps u_a above should be minimizers. 
  


We now give a few more different examples of minimizing constraint maps.


    As mentioned in the introduction, (energy-minimizing) geodesic curves in Riemannian manifolds-with-boundary are (minimizing) constraint maps, see e.g. <cit.> for a study of their properties.




    Consider a smooth boundary condition g∂Ω→∂ M,  suppose that the obstacle M^c is convex, and let u∈ W^1,2_g(Ω;M) be a minimizing constraint map. Then, according to <cit.>, see also Corollary <ref> below, the non-negative function ρ∘ uΩ→ [0,+∞) is subharmonic and vanishes on ∂Ω. Therefore, by the maximum principle, ρ∘ u=0 in Ω and thus u^-1(∂ M)=Ω. It follows that u is a minimizing harmonic map into ∂ M.


An alternative  geometric  proof (which avoids subharmonicity of ρ∘ u)  is to take any supporting plane H(y) = (y-y_0) ·ν_y_0, with y_0 ∈∂ M, with 
M ⊂{ H < 0 },  
and consider the restriction of the map to 
{ H ≥ 0}. Applying the minimum principle to 
the harmonic function H(u (x)) in the set 
{ H > 0 } allows us to conclude that u^-1(∂ M) = Ω.






 In analogy with parametric minimal surfaces, one can show that 
    if the convex hull of the image of the boundary satisfies (u(∂Ω )) ⊂ M then u(Ω )⊂ M.   
Indeed,  as in Example <ref>, the distance function to the convex hull is subharmonic (as u is harmonic in each component), so the maximum principle shows that the distance function vanishes everywhere, proving that u(Ω)⊂(u(∂Ω))⊂ M.


We conclude this section with a comparison between minimizing constraint maps and solutions to the scalar obstacle problem.



 §.§ Comparison with the scalar obstacle problem

Given a scalar function ∈ C^∞(Ω), the classical obstacle problem consists in minimizing the Dirichlet energy  among all functions v ∈ W^1,2(Ω) satisfying the constraint v≥φ in Ω. Minimizers of this problem are precisely the solutions of the equation

    Δ v = (Δφ)χ_{v=}  in Ω.

This problem and many variations of it have been the subject of intensive research over several decades and from various perspectives. Several influential works that have shaped the recent landscape of the scalar theory are, for instance,
<cit.>.



Here, we consider graphical obstacles and compare the solutions of the classical scalar obstacle problem to minimizing constraint maps. As we shall prove below, these two objects are always different unless the solution does not “actively” touch the obstacle.  Informally speaking, the reason for this difference comes from the fact that the graph of a solution to the obstacle problem is pushed upwards by the obstacle, while the hypersurface parametrized by a minimizing constraint map is pushed away from the obstacle.

To begin our analysis, 
we denote by M_ the subgraph of , i.e., M_ = {(y̅,y_n+1)∈^n+1: (y̅) < y_n+1}, fix a solution v∈ W^1,2(Ω) to (<ref>), and consider the vectorial map u∈ W^1,2(Ω;M_) defined by 

    u(x) := (x,v(x)).


Assume by contradiction that u is a minimizing constraint map. Since ∈ C^2(Ω), we have v∈ C_^1,1(Ω), whence u∈ C_^1,1(Ω;M_). Since Dv = D on Ω∩{v = } = Ω∩ u^-1(∂ M), we also have 

    D_α u = e_α + (D_α) e_n+1  on Ω∩ u^-1(∂ M).

for every α∈{1,2,⋯,n}, where e_α is the standard α-th basis vector in ^n+1. Thus, denoting by A_y(·,·) the second fundamental form of ∂ M_ (which is precisely the graph of ) at y, 

    A_u(Du,Du) = (-D,1)/1 + |D|^2Δ  on Ω∩ u^-1(∂ M),

with the specific choice u as in (<ref>)[In fact, setting ψ:Ω→^n+1 as the canonical parametrization of the graph, ∂ M_, of , (i.e., ψ(x) := (x,(x))) we obtain, for any 1≤α,β≤ n,

    A_ψ(D_αψ, D_βψ) = (-D, 1 )/1 + |D|^2D_αβ  in Ω.
] and using (<ref>).
Thus, if u is a minimizing constraint map, it follows from the Euler-Lagrange system (<ref>) as well as the definition (<ref>) of u that 

    (Δ v)e_n+1 = Δ u = -Δ/1 + |D|^2((D_α)e_α - e_n+1)χ_{v = }  in Ω;

here, as usual, we used the summation convention for the repeated index α running through {1,2,⋯,n}. Comparing (<ref>) with (<ref>), it follows that 

    |D|^2Δ = 0   a.e. on Ω∩{v=}.

Note that (<ref>) is a very strong restriction.
Indeed, since D^2φ=0 (and so Δφ=0) a.e. on the set {Dφ=0}, (<ref>) is equivalent to implies

    Δ = 0   a.e. on Ω∩{v=}.

Combining this information with (<ref>), we deduce that Δ v=0 a.e., thus v is harmonic.

In other words, except in the trivial case when  v is harmonic (which means that the obstacle is not acting on the function v), the graph of the solution to the obstacle problem does not coincide with the corresponding minimizing constraint map.









§ REGULARITY OF THE DISTANCE TO THE CONVEX HULL OF THE OBSTACLE


In this section, we establish the uniform regularity of the distance of a minimizing constraint map to the convex hull of the given obstacle. This result significantly improves <cit.>.
For convenience, we state the result below with Y being an arbitrary closed convex set containing  M^c, but the result is particularly relevant when Y is the convex hull of M^c.


Let M be a smooth domain in ^m with compact complement, Y  a closed convex set containing  M^c, and u∈ W^1,2(Ω;M) a minimizing constraint map with u _W^1,2(Ω)≤Λ. Then for every δ > 0 there exists a modulus of continuity ω, depending only on n, m, ∂ M, Λ, and δ, such that 

    _B_r(x_0)(u,Y) ≤ω(r)    ∀ r ∈(0,δ),

whenever B_2δ(x_0)⊂Ω. 


We begin by showing that the distance function is subharmonic.


 Let Y be a closed convex set containing  M^c.   Then the function (u,Y) is weakly subharmonic in Ω.



  Define the function w:Ω→,  w (x)  :=  (u(x)  ,Y). Since {w  > 0}∖Σ(u)⊂ u^-1(M)∖Σ(u), it follows by (<ref>) that  Δ u = 0 inside int({ w > 0})∖Σ(u). Hence, since y↦(y,Y) is convex (by the convexity of Y), we conclude that 
   Δ w ≥ 0   inside  int({ w > 0})∖Σ(u).
 As _Σ(u) ≤ n-3 by Theorem <ref> and w∈ W^1,2(Ω), by a capacity argument it follows that Δ w ≥ 0 weakly in Ω. 


As a corollary, we obtain an interior L^∞-estimate for u:


Under the hypothesis of Theorem <ref>
there is  a∈ M^c such that |u - a| ∈ L_^∞(Ω) and 

    u -  a _L^∞(Ω')≤ c M^c +  c u - a_L^2(Ω)/(Ω',∂Ω)^n/2

for every subdomain Ω'⋐Ω, where c depends only on n.  



    Since M^c is compact, there is a∈ M^c such that M^c⊂B_R(a), with R :=  M^c. Set Y := B_R(a), so that (u,Y) = (|u-a| - R)_+. By Lemma <ref>, (u,Y) is weakly subharmonic in Ω. The conclusion then follows from the local L^∞-estimate for weakly subharmonic functions. 


We also immediately obtain the following conclusion on the range of constraint maps:



Let u ∈ W^1,2(Ω;M) be a minimizing constraint map. If Y is the closed convex hull of u(∂Ω)∪ M^c, then u(x)∈ Y for a.e. x∈Ω.



This follows from the weak maximum principle by applying Lemma <ref> to (u,Y).



In what follows, we shall fix a closed convex set Y containing M^c. We shall denote by _0= _0(n,m,∂ M) the small energy threshold from Corollary <ref>. In addition, c will denote a generic positive constant depending at most on n, m, ∂ M, and Λ, for a given Λ > 1. 

The proof of Theorem <ref> will be based on the two subsequent lemmas. The first lemma asserts that if a map mostly takes values outside an η-neighborhood of the convex hull of the obstacle, then the mapping is uniformly regular in the interior of the domain. 




    Let u∈ W^1,2(B_4;M) be a minimizing constraint map such that u _W^1,2(B_4)≤Λ. For every η > 0, there exists a constant σ≡σ(n,m,∂ M,Λ,η)∈(0,1), such that if |{(u,Y) ≤η}∩ B_2|≤σ, then u ∈ C^1,1(B_1) with the estimate 
    
    u _C^1,1(B_1)≤ c,

    where c > 1 depends only on n, m, ∂ M, Y, and Λ. 







In view of the -regularity theorem, Theorem <ref>, it suffices to find, under the assumption of the statement, a radius r_0≡ r_0(n,m,∂ M,Λ) > 0, such that 

    E(u,x,2r_0) < _0^2    ∀  x_0∈ B_1.

It suffices to check this claim when Y is the convex hull of M^c, as the convex hull is the smallest (closed) convex set containing M^c. 

Suppose that the claim is not true for the convex hull Y, for some η > 0. Then, for each k=1,2,⋯, there exists a minimizing constraint map u_k ∈ W^1,2(B_4;M) such that u_k _W^1,2(B_4)≤Λ and 

    |{(u_k,Y) ≤η}∩ B_2|≤σ_k → 0,

but for some x_k∈ B_1 and r_k → 0^+ it holds

    E(u_k,x_k,2r_k) ≥_0^2.

By Lemma <ref>,  up to a subsequence u_k→ u strongly in W^1,2(B_3;^m) as k→∞, for some minimizing constraint map u∈ W^1,2(B_3;M). In particular, extracting a further subsequence if necessary, we have u_k→ u a.e. in B_3, so 

    (u_k,Y)→(u,Y)  a.e. in B_3.

Passing to the limit in (<ref>), we observe that

    (u,Y) ≥η  a.e. in B_2.

In particular, B_2∩ u^-1(∂ M) = ∅, which combined with  (<ref>) yields Δ u = 0 in B_2, and therefore

    Σ(u)∩ B_2 = ∅.

However, by (<ref>), it follows from Corollary <ref> that 
    E(u,x_0,0^+)≥_0^2 > 0,
 for some x_0∈B_1; by Corollary <ref> this implies x_0∈Σ(u), a contradiction to (<ref>). This finishes the proof. 



In the setting of Lemma <ref>, if σ is chosen sufficiently small (for instance, if σ≪η^n/2), then in fact Δ u =0 in B_1. Indeed, for such σ we must have (u,Y)>0 in B_1, since otherwise by (<ref>) we would have (u,Y)≤η in a ball of radius c√(η), contradicting our assumption.


Our second lemma shows the uniform decay of the distance map at a point of high energy.


    Let u∈ W^1,2(B_4;M) be a minimizing constraint map, with u _W^1,2(B_4)≤Λ. Then for every η >0, there exist radii 0 < s < r ≪ 1, both depending only on n, m, ∂ M, Λ, and η, such that   
    
    E(u,0,s) ≥_0^2     (u,Y) ≤η in B_r.




    Since u _L^2(B_4)≤Λ, we have (u,Y)_L^2(B_4)≤ c. By Lemma <ref>, (u,Y) is weakly subharmonic in B_4, therefore (u,Y) ≤ c_0c in B_3 for some c_0 = c_0(n), cf. also Corollary <ref>.


    Let η∈ (0,1/2c_0c) be given. Let ℓ be a positive integer, and suppose that 
    
    λ_ℓ := (u_ℓ,Y)_L^∞(B_4) > η,

    where u_ℓ(x):= u(2^-ℓx) is still  a minimizing constraint map in W^1,2(B_4;M). By the monotonicity of the normalized energy, Lemma <ref>, we have 
    
    E(u_ℓ,0,4) = E(u,0, 2^2-ℓ) ≤ E(u,0,4) ≤ 4^2-nΛ^2.
 Furthermore, since (u,Y)≤ c_0c in B_3, (u_ℓ,Y) ≤ c_0c in B_4 as 2^2-ℓ≤ 3, for any ℓ≥ 1.   
    
    Let c̅ be the C^1,1-bound in (<ref>), so c̅ depends only on  the parameters n, m, ∂ M and max{Λ^2,c_0c}, and choose σ to be the measure threshold in the same lemma, determined further by 1/2η. Since we assume E(u,0,s) ≥_0^2, we have E(u_ℓ,0,2^ℓ s) ≥_0^2. Hence, choosing s sufficiently small so that 
    
    2^2ℓ+1 s^2|B_1|c̅ < _0^2,

    there must exist a set F⊂ B_1 of positive measure such that |Du_ℓ| > 2c̅ on F. In particular, the C^1,1-estimate (<ref>) cannot hold for u_ℓ. Then it follows as the contraposition of Lemma <ref> that |{(u_ℓ,Y) ≤1/2η}∩ B_2| > σ. Combining this with (<ref>) gives us |{(u_ℓ,Y) ≤1/2λ_ℓ}∩ B_2| > σ.
    As (u_ℓ,Y) is weakly subharmonic in B_4 and (u_ℓ,Y) ≤λ_ℓ in B_4, the De Giorgi oscillation lemma (see e.g. <cit.>) now implies that 
    
    λ_ℓ+2 = (u_ℓ,Y) _L^∞(B_1)≤ (1-μ)^2λ_ℓ,

    for some μ∈ (0,1/2), depending only on n and σ_η, hence on n, m, ∂ M, Y, Λ and η. 

    Iterating the implication (<ref>)–(<ref>), and recalling that λ_ℓ≤λ_1 ≤ c_0c, we obtain 
    
    λ_ℓ≤max{ 4(1-μ)^ℓ c_0c, η},

    for every ℓ∈ for which (<ref>) holds. Taking ℓ̅:= min{ℓ∈: 4(1-μ)^ℓ c_0c ≤η}, and selecting s to be sufficiently small so that (<ref>) holds for all ℓ≤ℓ̅, we obtain λ_ℓ̅≤η, that is, 
    
    (u,Y)≤η  in B_4· 2^-ℓ̅.

    Finally, we choose r := 4· 2^-ℓ̅; it is not hard to see from (<ref>) that r > s. Tracking down the dependence of these radii, we also verify that both r and s depend only on n, m, ∂ M, Y, Λ and η as desired. 


We are now ready to prove the universal modulus of continuity of (u,Y).


It suffices to prove the theorem when Ω = B_4, x_0 = 0, δ = 1. The general case follows by a simple scaling argument. 

    Let η > 0 be given and let λ = 1/8. We fix a constant c(n)  to be fixed later, and we choose radii 0<s<r≪ 1 as in Lemma <ref>, corresponding to the parameters n, m, ∂ M, c(n)Λ, and η.     
    
    If E(u,0,2λ s) ≥_0^2, then Lemma <ref>, applied to u_2λ(x) := u(2λ x), shows that 0≤(u,Y) ≤η in B_2λ r, whence 
    
    _B_2λ r(u,Y)≤η.

    This application of Lemma <ref> determines the choice of c(n).

    Alternatively, if E(u,0,2λ s) < _0^2, then by the -regularity theorem, Theorem <ref>, we have u∈ C^1,1(B_λ s) and |Du|≤c_0/λ s in B_λ s, for some constant c_0≡ c_0(n,m,∂ M). Combined with the fact that (·,Y) is 1-Lipschitz, we find that
    
    _B_ηλ s/(2c_0)(u,Y)≤η.

    By virtue of (<ref>) and (<ref>), we derive in either case that the oscillation of (u,Y) is bounded by η in B_t(x_0), for some radius t ≡ t(n,m,∂ M,Λ,η). This finishes the proof. 


A direct consequence of Theorem <ref> is that, when the obstacle M^c is convex, there exists a universal continuity estimate on ρ∘ u, thus improving <cit.>.


    Let M⊂^m be a smooth domain with compact convex complement, and let u∈ W^1,2(Ω;M) be a minimizing constraint map with u _W^1,2(Ω)≤Λ. Then ρ∘ u is subharmonic and continuous in Ω. In addition, its modulus of continuity depends on n, m, ∂ M, Λ, and δ only.



 



§ UNIVERSAL L(-EPSILON)-ESTIMATE NEAR DISCONTINUITIES



In this section we establish a universal L^-γ-estimate of the energy density of minimizing constraint maps near effective discontinuity points,[By effective discontinuity points, we mean the points which have normalized energy above the -regularity threshold. The set of effective discontinuity points necessarily includes a neighborhood of the set Σ(u) of the actual discontinuity points, due to  the monotonicity of the energy. For further discussion, we refer the reader to <cit.>.] see Theorem <ref>.
As mentioned before, this result also applies to classical harmonic maps, and we believe it to be of its own interest.


Throughout this section, we shall fix _0 ≡_0 (n,m,∂ M) as the energy threshold provided by Corollary <ref>. Let us begin our analysis by finding the critical scale, labeled  r_u (x) below.




Let  u∈ W^1,2(B_4;M) be a minimizing constraint map satisfying u _W^1,2(B_4)≤Λ. 
There exists a large integer ℓ_0 > 1, depending only on n, m, ∂ M, and Λ, such that the following holds:  setting 

    r_u(x) := sup{ r∈ (0,2]: E(u,x,r)≤ℓ_0^-2_0^2}   if x∈ B_2∖Σ(u),
    
    0    if x∈ B_2∩Σ(u),

we have that Σ(u) = r_u^-1(0) and that if E(u,y,1)≥_0^2 for some y∈ B_2, then r_u(x) ≤ 1 for all x∈ B_2.


The assertion Σ(u) = r_u^-1(0) follows immediately from Corollary <ref>, as _0 is chosen as the energy threshold there.

Assume now that E(u,y,1)≥_0^2 for some y∈ B_2. To prove that r_u(x)≤ 1 for all x∈ B_2, we claim that 

    E(u,x,1) >1/ℓ_0^2_0^2,  ∀ x∈ B_2,

for some positive integer ℓ_0 ≡ℓ_0(n,m,∂ M,Λ). Once the claim holds, the conclusion follows from the monotonicity of the normalized energy, Lemma <ref>. 

Suppose that the claim does not hold. Then there is, for every k∈, a minimizing constraint map u_k ∈ W^1,2(B_4;M) with u_k _W^1,2(B_4)≤Λ such that E(u_k,y_k,1)≥_0^2 for some y_k ∈ B_2, but 

    E(u_k,x_k,1) ≤1/k^2_0^2,

for some x_k ∈ B_2. We may assume without loss of generality that x_k→ x_∞ and y_k→ y_∞ as k→∞, for some x_0,y_0∈B_2.
Up to a subsequence, the compactness of minimizing constraint maps (Lemma <ref>) implies that u_k converges strongly to u in W^1,2(B_3;M), where u is a minimizing constraint map. Given that B_1(y_k) ∪ B_1(y_∞) ⊂ B_3, this strong convergence implies that 

    E(u,y_∞,1) ≥_0^2 > 0.

However, since B_1(x_k)∪ B_1(x_∞)⊂ B_3, we also have

    E(u,x_0,1) = 0,

thus |Du| = 0 a.e. in B_1(x_0). 

Note now that, in view of (<ref>), we have |Δ u|≤ c|Du|^2. Also, u is locally Lipschitz in B_3∖Σ(u). Furthermore, Theorem <ref> implies that _Σ(u)≤ n-3 < n-1, and so by <cit.> the open set B_3∖Σ(u) is connected. Thus, the classical unique continuation principle (see <cit.>) yields |Du| = 0 everywhere outside Σ(u), whence also |Du|=0 a.e.  in B_3; in particular E(u,y_0,1) = 0,  contradicting (<ref>). This finishes the proof.




In what follows, we shall also fix the large constant ℓ_0 ≡ℓ_0(n,m,∂ M,Λ) chosen in Lemma <ref>. Given a smooth non-constant map fΩ→^m and a ball B_r(x_0)⊂Ω, let us write N(f,x_0,r) for the frequency of f in the ball B_r(x_0), i.e.,
 
    N(f,x_0,r) := r∫_B_r(x_0) |Df|^2 dx/∫_∂ B_r(x_0) |f - (f)_x_0,r|^2 ,

 where (f)_x_0,r denotes the integral average of f over the (n-1)-dimensional sphere ∂ B_r(x_0). When x_0 = 0, we shall write (f)_r ≡ (f)_0,r for simplicity. 

We are ready to make the key observation of this section: the frequency is uniformly bounded below the critical scale, provided that the mapping has discontinuities nearby (or, more generally, points of sufficiently large energy).
 
 

Let u∈ W^1,2(B_4;M) be a minimizing constraint map satisfying u _W^1,2(B_4)≤Λ. There exist a frequency threshold λ_0 > 1, depending only on n, m, ∂ M, and Λ, such that if E(u,y,1)≥_0^2 for some y∈ B_2, then 
     
    N(u,x,r_u(x)) ≤λ_0,

     for any x∈ B_1∖Σ(u), where r_u(x) is as in (<ref>).
 

  
Throughout the proof, c will denote a generic constant depending at most on n, m and ∂ M. Assume that the conclusion of the lemma is false. Then for every k ∈, there must exist a minimizing constraint map u_k ∈  W^1,2(B_4;M) such that u_k _W^1,2(B_4)≤Λ, E(u_k,y_k,1)≥_0^2 for some y_k∈ B_2, but at some x_k ∈ B_1∖Σ(u_k) it satisfies, with r_k := r_u_k(x_k),
     
    N(u_k,x_k,r_k) ≥ k.
    
By Lemma <ref>, r_k ≤ 1 for all k. This, along with x_k∈ B_2, yields 
     
    B_3/2r_k(x_k)⊂ B_3/2(x_k)⊂ B_4.

In what follows, let us write  
     
    v_k (y) := u_k(1/2r_ky + x_k),

 which is a minimizing constraint map in W^1,2(B_3;M). By Lemma <ref>, (<ref>) and the a priori bound ∫_B_1|Du_k|^2 dx ≤Λ^2, we deduce that
     
    E(v_k,0,3) = E(u_k,x_k,3/2r_k)≤ E(u_k,x_k,3/2) ≤ c∫_B_4 |Du_k|^2 dx ≤ cΛ^2.

In addition, since M^c is compact and u_k _L^2(B_4)≤Λ, Corollary <ref> implies that |u_k|≤ c (1+Λ) a.e. in B_7/2, thus
 
    |v_k|≤ c(1 + Λ)  a.e. in B_5/2.

In particular {v_k}_k=1^∞ is bounded in W^1,2(B_5/2;M), so Lemma <ref> yields a minimizing constraint map v ∈ W^1,2(B_2;M) such that 
     
    v_k→ v  strongly in W^1,2(B_2;^m),

after potentially extracting a subsequence. 

However, as (<ref>) yields N(v_k,0,2) = N(u_k,x_k,r_k) ≥ k, it follows from (<ref>) that
     
    ∫_∂ B_2 |v_k - (v_k)_1|^2 d^n-1≤c/kΛ^2,

for every k∈ .
By (<ref>), the boundedness of the trace operator <cit.>   implies 
     
    v_k → v   strongly in L^2(∂ B_2;^m),

and so we find, via (<ref>), that  
     
    v  = a  a.e. on ∂ B_2,

for some point a∈∂ M. 

We now observe that (<ref>) implies v = a a.e. in B_3: taking : B_3→^m defined by = a in B_2 and = v in B_3∖ B_2,  is an admissible map and so the minimality of v implies 

    ∫_B_2|Dv|^2 dx ≤∫_B_2 |D|^2 dx = 0,

i.e. |Dv| = 0 a.e. in B_2. As in the proof of Lemma <ref>, the  unique continuation principle <cit.> allows us to deduce that v = a in 
B_3∖Σ(v), which then readily implies that v = a in B_3 and Σ(v) = ∅. This is however a contradiction, as the choice of r_k and (<ref>) yield

    E(v,0,2) = lim_k→∞ E(v_k,0,2) = lim_k→∞ E(u_k,x_k,r_k) ≥1/ℓ_0^2_0^2.

This finishes the proof. 
 

Next we show that, above the critical scale, the differential of the map has a uniform lower bound on a universal fraction of each cube.

 
 
For any σ∈ (0,1), there are a small constant δ > 0, depending only on n, m, ∂ M, Λ, and σ, and a large constant c_0 > 1, depending solely on n, m, and ∂ M, such that for any minimizing constraint map u∈ W^1,2(B_8;M) satisfying u _W^1,2(B_8)≤Λ, if E(u,0,4) ≥1/ℓ_0^2_0^2, then
     
    |{|Du| ≥δ |Q|^-1/n}∩ Q| ≥ (1-σ)|Q|,

where Q=[-1/√(n),1/√(n)]^n is the largest cube contained in B_1.
 

 



Fix σ∈(0,1). Assume, for the sake of contradiction, that for each k∈, there exists a minimizing constraint map u_k ∈ W^1,2(B_8;M) satisfying u_k_W^1,2(B_8)≤Λ and E(u_k,0,4) ≥1/ℓ_0^2_0^2, but 
 
    |{ |Du_k| >δ_k |Q|^-1/n}∩ Q| < (1-σ)|Q|,

 along a sequence δ_k→ 0. 

 
 As in the proof of Lemma <ref>, one can find, via Lemma <ref>, a minimizing constraint map u ∈ W^1,2(B_4;M) such that 

    u_k → u   strongly in W^1,2(B_4;^m),

along a subsequence. To simplify our notation, let us assume without loss of generality that the convergence holds along the full sequence. 
Therefore, by Egorov's theorem, there exists a closed set F⊂ Q, with 

    |F| < 1/2σ|Q|,

such that 

    Du_k → Du   uniformly on Q∖ F.

Thus, by (<ref>) and (<ref>),

    |{ |Du_k| ≤δ_k |Q|^-1/n}∩ (Q∖ F)| ≥σ |Q| - |F| ≥1/2σ |Q|.

Recalling that δ_k→ 0, we obtain from (<ref>) and (<ref>) that 

    |{ |Du| = 0 }∩ (Q∖ F)}| ≥1/2σ |Q|.

Arguing as in the proof of Lemma <ref>, we again deduce from the minimality of u and the unique continuation principle that |Du| = 0 in B_4. However, by (<ref>),

    E(u,0,4) = lim_k→∞ E(u_k,0,4) ≥1/ℓ_0^2_0^2,

which is a contradiction. This completes the proof.

 


Next, we look below the critical scale r_u(x). There, our system (<ref>) can be considered as a linear system with  bounded coefficients for the first-order term, due to the -regularity theorem. Therefore, the energy density Du admits a uniform doubling property depending on the frequency of the mapping u.  For the reader's convenience, we recall these standard arguments in Lemma <ref> in the appendix.

 
 
Let λ > 1. Then there are constants δ = δ(n,m,∂ M,λ) and σ = σ(n,m,∂ M,λ), 0 < δ,σ < 1, such that if u∈ W^1,2(B_4;M) is a minimizing constraint map satisfying E(u,0,4) ≤_0^2 and N(u,0,4) ≤λ, then 
 
    |{ |Du| ≥δ Du_L^∞(3Q)}∩ Q|≥ (1-σ)|Q|,

 for every cube Q such that 3Q⊂ B_2. 
 

 
By our assumptions and Theorem <ref>, we deduce from (<ref>) that 
 
    Δ u^i = a_ij^α D_α u^j  in B_2,

with a_ij^α_L^∞(B_2)≤ c, where c depends only on n, m and ∂ M. Hence, the assertion follows immediately by applying Lemma <ref> with  f = u. 
 

 Combining the last three lemmas, we arrive at the following conclusion.

 
 
There are constants δ_0,σ_0 ∈ (0,1), both depending only on n, m, ∂ M, and Λ, such that for any minimizing constraint map u∈ W^1,2(B_4;M) satisfying u _W^1,2(B_4)≤Λ, if E(u,y,1)≥_0^2 for some y∈ B_2, then 
 
    |{ |Du| ≥δ_0 min{|Q|^-1/n,  Du_L^∞(3Q)}}∩ Q| ≥ (1-σ_0)|Q|,

 for every cube Q centered in B_2 such that Q≤1/8. 
 

 
We may assume without loss of generality that Λ > 1. Let λ_0 be as in Lemma <ref>, and choose δ_1 and σ_0 as in Lemma <ref> corresponding to λ =λ_0. We then select δ_2 as in Lemma <ref> corresponding to the chosen σ_0 (and with cΛ in place of Λ for some c≡ c(n,m,∂ M)). Finally, we set δ_0 := min{δ_1,δ_2} > 0. 

Let Q be a cube centered in B_2 with Q≤1/8. Denote by s its sidelength and by x_0 its center, so that Q = Q_s(x_0), √(n)s ≤1/8, and |x_0|≤ 2. Let B_r(x_0) be the smallest ball containing Q, i.e. r=1/2√(n)s ≤1/16. Then B_8r(x_0)⊂ B_1(x_0)⊂ B_3. Hence, the monotonicity of the normalized energy, Lemma <ref>, along with our assumption u _W^1,2(B_4)≤Λ, yields  
 
    E(u,x_0,8r)≤ E(u,x_0,2) ≤ 2^2-nDu_L^2(B_4)^2 ≤ 2^2-nΛ^2.

Moreover, by the local L^∞-estimate, Corollary <ref>, we have 

    u _L^∞(B_8r(x_0))≤ u_L^∞(B_3)≤ c + c u_L^2(B_4)≤ cΛ;

in the last inequality, we also used Λ > 1. 
Recall the critical scale r_u(x_0) defined in (<ref>). By Theorem <ref> and Lemma <ref>, it holds 0 ≤ r_u(x_0) ≤ 1.  
 Let us divide the proof into two cases.
 

 
 
 r ≥1/4r_u(x_0).
 
In view of (<ref>), we have E(u,x_0,4r) ≥1/ℓ_0^2_0^2. Also, by virtue of (<ref>) and (<ref>), we have 

    u_x_0,r_W^1,2(B_8)≤ cΛ,

where u_x_0,r(y) := u(ry + x_0) with c≡ c(n,m,∂ M). Hence, Lemma <ref> (with σ = σ_0 and δ = δ_2 there) applies to u_x_0,r. Rescaling back and recalling that Q is the largest cube contained in B_r(x_0), since δ_0 ≤δ_1 we obtain  
 
    |{ |Du| ≥δ_0 |Q|^-1/n}∩ Q| ≥ (1-σ_0)|Q|.

 Thus, the conclusion of the lemma follows.  


 
 r < 1/4r_u(x_0).
  
In this case, we have x_0∉Σ(u) and E(u,x_0,4r) < 1/ℓ_0^2_0^2. By Lemma <ref> we have N(u,x_0,r_u(x_0)) ≤λ_0, and by definition E(u,x_0,r_u(x_0)) ≤1/ℓ_0^2_0^2 < _0^2. Thus Lemma <ref> applies to u_x_0,r_u(x_0) (y) := u(r_u(x_0)y + x_0). Since 4r ≤ r_u(x_0), and Q⊂ B_r(x_0), we have 3Q⊂ B_3r(x_0)⊂ B_r_u(x_0)(x_0). Thus, we deduce (after rescaling back) that 
 
    |{ |Du| ≥δ_0Du_L^∞(3Q)}∩ Q| ≥ (1-σ_0)|Q|.

 Therefore, the conclusion of this lemma is verified for the alternative case as well. This finishes the proof. 
 

 We are now ready to prove Theorem <ref>, which is a direct consequence of the last lemma and the Calderón–Zygmund cube decomposition, whose statement we recall here for the reader's convenience (see e.g. <cit.> for a proof).

 
 Let A,B⊂ Q_1 be measurable sets, and let σ > 0 be given. Assume the following:  
 
 
  * |A| ≤σ.
 
  * If |A∩ Q| > σ |Q| for some dyadic cube Q, then Q̃⊂ B, where Q̃ is the predecessor of Q. 
 
 Then |A| ≤σ |B|. 
 



Consider a collection of disjoint open cubes Q^i, centered in B_2, with the same side-length s, i.e.  n s^2 = ( Q^i)^2, such that B_1⊂⋃_i=1^i_0Q^i and Q^i≤1/8 for each i. We can take the collection in such a way that its cardinality i_0 is bounded by a constant depending only on n. By the assumption Q^i≤1/8, we must have 64ns^2 ≤ 1. 

Take _0, ℓ_0, δ_0, and σ_0 as in Corollary <ref>, Lemma <ref> and Lemma <ref>. We claim that 
 
    |{|Du| ≤ t}∩ Q^i| ≤min{ct^2γ,1}s^n,

 for every t > 0, for some c > 1 and γ > 0, both depending only on n, m, ∂ M, and Λ, but not on i. Then by (<ref>) and elementary real analysis (integrating over all volume of super-level sets), we deduce that 
 
    ∫_B_1 |Du|^-γ dx    ≤∑_i=1^i_0∫_Q^i |Du|^-γ dx = γ∑_i=1^i_0∫_0^∞τ^-1 + γ |{ |Du| ≤τ^-1}∩ Q^i| dτ
       ≤γ i_0 s^n [∫_0^1 τ^-1+γ dτ + ∫_1^∞ cτ^-1-γ dτ] ≤ c,

where the last inequality follows from our choice (specified above) of both i_0 and s, in such a way that they are bounded above by a constant depending only on n.

To prove (<ref>),  let us define, for each k ∈ℕ, the set
 
    A_k^i := { |Du| ≤δ_0^k}∩ Q^i.

 It is clear that A_k+1^i⊂ A_k^i. Moreover, Lemma <ref>
 implies that |A_1^i|≤σ s^n, therefore 
 
    |A_k^i|≤σ s^n,   ∀ k ∈ℕ.

 Next we claim that if Q is a member of the dyadic cube decomposition of Q^i then 

    |A_k+1^i∩ Q| > σ_0 |Q|     Q̃⊂ A_k^i,
 where Q̃ is the predecessor of Q.
 
To prove this, assume by contradiction that Q̃∖ A_k^i≠∅. By the definition of A_k^i, we have 
 
    D u_L^∞(Q̃) > δ_0^k.

 Since Q̃⊂ 3Q ⊂ 3Q^i⊂ B_2 and |Q|^-1/n≥ |Q^i|^-1/n = s^-1 > √(n) > 1 > δ_0^k+1, Lemma <ref> yields that 
 
    |{ |Du| ≥δ_0^k+1}∩ Q| ≥ (1-σ_0)|Q|.

This implies that |A_k+1^i∩ Q| ≤σ_0 |Q|, a contradiction that proves the result.

Thanks to (<ref>) and (<ref>), it follows from Lemma <ref> that |A_k+1^i| ≤σ_0 |A_k^i|. Since k was an arbitrary positive integer and |A_1^i|≤σ_0s^n, we  conclude that 
 
    |A_k^i| ≤σ_0^k s^n=s^n (δ_0^k)^2γ,  ∀  k∈,

 with γ:=logσ_0/2logδ_0.
Due to this bound, (<ref>) follows. Since both δ_0 and σ_0 depend only on n, m, Λ, and ∂ M, so do the constants γ and c. This completes the proof. 
 

 







 


§ REGULARITY NEAR FREE BOUNDARIES


The purpose of this section is to prove Theorem <ref>, which asserts that,  whenever the obstacle is uniformly convex, minimizing constraint maps are C^1,1 in a universal neighborhood of the non-coincidence set. Therefore, throughout this section, we assume that M is a smooth domain with uniformly convex complement. Moreover, we fix _0 as the energy threshold chosen as in Lemma <ref>, which depends only on n, m, and ∂ M. 



As we shall see, the result follows from Theorem <ref>, proved in the previous section,  with the following elementary   De Giorgi-type lemma, which we will apply to the subharmonic function ρ∘ u.

 
 Let w∈ W^1,2(B_2) be a nonnegative subharmonic function, and suppose that there are positive constants a, p, q such that 
 
    |{ w > 0}∩ B_r|≤a/(R-r)^q w _L^∞(B_R)^p,

 whenever 1≤ r <R ≤ 2. Then there is a constant η∈(0,1), depending only on n, a, p, and q, such that if w ≤η a.e. in B_2, then w = 0 a.e. in B_1. 
  

 
 Let r_k := 1 + 4^-k and s_k := 1 + 2· 4^-k-1, for each k∈. Then 
 r_k - s_k = 2· 4^-k-1 and s_k - r_k+1 = 4^-k-1. Denote δ_k := |{ w > 0}∩ B_s_k| and λ_k := w_L^∞(B_r_k). Then our hypothesis (<ref>) yields
 
    δ_k ≤ a (2· 4^k+1)^q λ_k^p.

This, combined with the local L^∞-estimate for subharmonic functions, implies that 
 
    λ_k+1≤c/(s_k-r_k+1)^nw_L^2(B_s_k)≤ c· 4^n(k+1)λ_k δ_k ≤  C a · 4^ k(n+q)λ_k^1+p,

 where c= c(n) and C=2^3 β + 2n c. Thanks to this recursive relation, it follows that
 
    λ_0 ≤η  ⟹  lim_k→∞λ_k = 0,

 if η is sufficiently small; the smallness condition can be determined a priori by n, a and γ only. Since w _L^∞(B_1)≤λ_k for all k, we conclude that w = 0 a.e. in B_1.
 


We can now prove the following key step towards Theorem <ref>. 


There exists a constant η>0, depending only on n, m, ∂ M, and Λ, such that for any minimizing constraint map u ∈ W^1,2(B_4;M) satisfying ∫_B_4 |Du|^2 dx ≤Λ^2 and ρ∘ u ≤η in B_4, 

    E(u,y,1)≥_0^2 for some y∈ B_2     B_1⊂ u^-1(∂ M).

 

 
Fix a pair of radii with 1 ≤ r<R ≤  2, and let ψ∈ C_0^∞(B_R) be a smooth cutoff function such that ψ = 1 in B_r, |D^2ψ|≤ c/(R-r)^2, and 0≤ψ≤ 1 in B_R, with c = c(n). In view of the equation (<ref>) and by the geometric bounds (<ref>) with ξ = D_α u(x) (note that, by chain rule, ∇Π(u(x)) · D_α u(x) = D_α (Π∘ u)(x)), we find
 
    c/(R-r)^2∫_B_R (ρ∘ u) dx    ≥∫_B_R (ρ∘ u)Δψ dx 
       = ∫_B_R∩ u^-1(M)ψρ_u(Du,Du) dx 
       ≥ c_0∫_B_r∩ u^-1(M) |D(Π∘ u)|^2  dx.

On the other hand, as ρ∘ u is weakly subharmonic in B_4 due to the convexity of M^c, the Caccioppoli inequality along with ρ∘ u ≤η < 1 in B_4 and |D(ρ∘ u)| = 0 a.e. in B_r∖ u^-1(M) shows that 

    c/(R-r)^2∫_B_R (ρ∘ u) dx ≥∫_B_r∩ u^-1(M) |D(ρ∘ u)|^2 dx.

Adding (<ref>) and (<ref>), and utilizing |Du|^2≤ c|D(Π∘ u)|^2 + c|D(ρ∘ u)|^2 a.e., we get

    c/(R-r)^2∫_B_R (ρ∘ u) dx ≥ c_0 ∫_B_r∩ u^-1(M) |Du|^2 dx,

with a possibly larger constant c>1. 

Let ,η >0 and c_1 > 1 be the constants determined a priori by n, m, ∂ M, and Λ, as in Theorem <ref>. Since by assumption E(u,y,1)≥_0^2 for some y∈ B_2 and ρ∘ u≤η in B_4, we invoke Theorem <ref> to obtain 
 
    ∫_B_2 |Du|^-γ dx ≤ c.

 Choosing p > 1 such that 1/p = 1/2 + 1/γ, Hölder inequality and (<ref>) yield
 
    |B_r∩ u^-1(M)|^2/p≤(∫_B_r∩ u^-1(M) |Du|^-γ dx)^2/γ∫_B_r∩ u^-1(M) |Du|^2 dx ≤ c^2/γ∫_B_r∩ u^-1(M) |Du|^2 dx.

 By (<ref>) and (<ref>), this implies
 
    |B_r∩ u^-1(M)| ≤c/(R-r)^pρ∘ u_L^∞(B_R)^p/2;

 note that p = 2γ/2+γ depends only on n, m, ∂ M, and Λ. Since u^-1(M) = {ρ∘ u > 0}, condition  (<ref>) is verified. By choosing η even smaller if necessary,  Lemma <ref> applies, so from the assumption ρ∘ u ≤η in B_4 we conclude that ρ∘ u = 0 in B_1. In other words,  B_1⊂ u^-1(∂ M), as desired. 
 


 With the help of Lemma <ref>  we establish the following rigidity theorem, which may be of independent interest.

 
 Let u∈ W_^1,2∩ L^∞(^n;M) be a locally minimizing constraint map. Suppose that 
    lim sup_k→∞[ k^2-n∫_B_k  |Du|^2  dx ] < + ∞  and   0∈Σ(u) .

 Then u ∈ W_^1,2(^n;∂ M) and in particular u is a locally minimizing harmonic map into ∂ M. 
 


  
By Lemma <ref>, Corollary <ref>, 0∈Σ(u), and (<ref>), there is a constant θ_0 > _0^2 such that 
 
    lim_k→∞[ (kR)^2-n∫_B_kR |Du|^2 dy ] = θ_0,

 for every R> 0.  Let us write u_k (y) := u(ky), which is a minimizing constraint map in W^1,2(B_8;M). By (<ref>) and the bound |u|∈ L^∞(^n), the sequence {u_k}_k=1^∞ is bounded in W^1,2∩ L^∞(B_8;M). Thus, by Lemma <ref>, there exists a minimizing constraint map u_∞∈ W^1,2(B_4;M) and a subsequence k_i→∞ such that u_k_i→ u_∞ strongly in W^1,2(B_4;^m).  Moreover, by Lemma <ref>, u_∞ is 0-homogeneous in B_4 since, by (<ref>), we have 
 
    R^2-n∫_B_R |Du_∞|^2 dy = lim_i→∞[ R^2-n∫_B_R |Du_k_i|^2 dy ] = lim_i→∞[ (k_i R)^2-n∫_B_k_i R |Du|^2 dx] = θ_0,

 for a.e. R∈(0,4). Since 0∈Σ(u_k), it follows from Corollaries <ref> and <ref> that 0∈Σ(u_∞). 

 As u_∞ is a minimizing constraint map, it follows from Corollary <ref> that ρ∘ u_∞ is weakly subharmonic and continuous in B_4. Then the 0-homogeneity of ρ∘ u_∞ implies that ρ∘ u_∞ is constant in B_4, cf. <cit.>. Since 0∈Σ(u_∞), in fact we must have
 
    ρ∘ u_∞ = 0  in B_4.

Indeed, if ρ∘ u_∞ =a for some a>0,  then we would have |u_∞^-1(∂ M)| = 0,  and so by (<ref>), Δ u_∞ = 0 in B_4, a contradiction to 0∈Σ(u_∞). We remark that this argument can also be found in the proof of <cit.>.

 Owing to (<ref>) and the strong convergence of u_k_i→ u_∞ in L^2(B_4;^m), we have ρ∘ u_k_i→ 0 in L^2(B_4). By the subharmonicity of ρ∘ u_k_i from Corollary <ref>, we have 
 
    ρ∘ u_k_i→ 0  in L^∞(B_4).

 Now we choose η to be a small constant, depending only on n, m, ∂ M, and Λ, as in Lemma <ref>. Then by (<ref>) we can find a large i_0 such that ρ∘ u_k_i≤η in B_4 for all i≥ i_0. Recall  that 0∈Σ(u_k_i), which implies E(u,0,1)≥_0^2 by Lemma <ref> and Corollary <ref>. Hence  Lemma <ref> implies that B_1 ⊂ u_k_i^-1(∂ M). Rescaling back, we have proved that 
 
    B_k_i⊂ u^-1(∂ M),

 for every i≥ i_0. Since k_i→∞, we deduce that u^-1(∂ M) = ^n, which proves u∈ W_^1,2(^n;∂ M). Finally, u is a minimizing harmonic map, since  W^1,2(D;∂ M)⊂ W^1,2(D;M) for any bounded domain D⊂^n. 
 


We are now ready to prove the main result of this section.


As M^c is convex, we can take Y = M^c in Theorem <ref>, which along with u _W^1,2(B_2)≤Λ yields a modulus of continuity ω, determined a priori by n, m, ∂ M, and Λ, such that for every r∈(0,1/2) and x_0∈ B_1, 

    _B_r(x_0) (ρ∘ u)≤ω(r).

By Corollary <ref>, the free boundary B_1∩∂ u^-1(M) is well-defined, so let us take an arbitrary point x_0 there. Now let η be as in Lemma <ref> (with cΛ^2 in place of Λ there), and take δ= δ(n,m,∂ M,Λ) ∈ (0,1/8) small such that ω(8δ) ≤η. By (<ref>) and the fact that ρ∘ u(x_0) = 0, we have 

    ρ∘ u ≤η  in B_8δ(x_0).

On the other hand, since B_8δ(x_0)⊂ B_1(x_0)⊂ B_2, Lemma <ref> yields

    E(u,x_0,8δ)≤ E(u,x_0,1) ≤∫_B_2 |Du|^2 dx ≤Λ^2.

Now, as x_0∈ B_1∩∂ u^-1(M), B_2δ(x_0)∖ u^-1(∂ M) ≠∅. By (<ref>) and (<ref>), the contraposition of Lemma <ref> (applied to u_x_0,2δ(y) := u(2δ y + x_0)) implies that 

    E(u,x_0,2δ) < _0^2.

Then, by Theorem <ref>, u∈ C^1,1(B_δ(x_0)), and since δ≡δ(n,m,Λ,∂ M), we have the estimate

    D^j u_L^∞(B_δ(x_0))≤ c,   j∈{1,2},

where c ≡ c(n,m,Λ,∂ M). 
This proves the validity of a uniform C^1,1 bound in a neighborhood of the free boundary.
On the other hand, since Δ u = 0 in B_1∩ u^-1(M), for any y_0∈ B_1∩ u^-1(M) with (y_0,B_1∩∂ u^-1(M)) > δ, we may apply the interior estimates for harmonic functions, and use the universality of δ as well as ∫_B_2 |Du|^2 dx ≤Λ^2, to obtain that 

    |D^j u(y_0)| ≤ c,    j∈{1,2}.

Combining the above two inequalities, we arrive at the desired conclusion. 








§ DISCONTINUITIES ON FREE BOUNDARIES PRODUCED BY DEGENERATE OBSTACLES



This section stems from the search for the minimal geometric conditions on the obstacle under which there are no discontinuities of u near the free boundary. In Theorem <ref> we proved the absence of discontinuities under the assumption that the obstacle is uniformly convex.
The main result in this section is Theorem <ref>, which shows the sharpness of the previous result: for convex obstacles, there may be discontinuities on the free boundary if we allow the obstacle to have flat sides. More precisely, we shall hereby call a subset T⊂∂ M a flat piece if T is relatively open, connected, and all the principal curvatures of T vanish.  Equivalently, T is a relatively open connected subset of a hyperplane.  


Before beginning the main proofs of this section, we collect some useful remarks.



Consider the setting of Theorem <ref>. By the boundary regularity theory <cit.>, whenever both ∂Ω and u|_∂Ω are of class C^2, we have that u ∈ C^1,α({x∈Ω: (x,∂Ω)< δ}) for  every α∈(0,1), where δ >0 depends only on n, α, the C^2-character of ∂Ω, and u|_∂Ω; in particular, 

    (Σ(u),∂Ω)≥δ.

Now, if M^c is convex then Ω∩ u^-1(M)  is an open set according to Corollary <ref>, and hence Δ u =0 in Ω∩ u^-1(M). In particular, we have u∈ C^∞(Ω∩ u^-1(M)). In conclusion, we obtain 

    u∈ C_^1,α(Ω∩ u^-1(M))∩ C^∞(Ω∩ u^-1(M)).



The previous remark shows that there are no discontinuities in a neighborhood of the fixed boundary ∂Ω. As we now observe,  discontinuities are unavoidable in the interior.


As mentioned in the introduction, constraint maps, just as harmonic maps, generally develop discontinuities for topological reasons. For the convenience of the reader, we briefly recall why this is the case. Given Ω⊂^n, we  have the following topological fact:

    u∈ C^0(Ω)
    
        u=𝕀 on ∂Ω    Ω⊆ u(Ω).

In particular, if M^c⋐Ω and u∈ W^1,2(Ω;M) is a minimizing constraint map with identity boundary values, then the conclusion in (<ref>) does not hold, and so we must have Σ(u)≠∅. This is essentially the setting of Theorem <ref>.

Implication (<ref>) is most easily proved using the notion of topological degree, which is  a way of counting (with multiplicity) the number of solutions x∈Ω to an equation

    u(x)=y,

The topological degree shows that in the setting of Theorem <ref>, we must have Σ(u)≠∅, but it can also be used to yield information about the image of u, since  y∈ u(Ω) whenever (u,Ω,y)≠ 0. 




In the setting of Theorem <ref>, we have


    u|_Ω∩ u^-1(M) is surjective onto Ω∩ M.

The proof of this fact is somewhat involved and relies on degree theory. In fact, in Proposition <ref> below we will prove a much stronger degree theoretic result, which we believe to be independently significant. We also note that, from well-known properties of the topological degree, see in particular Theorem <ref>(<ref>), our assumption (g) ≠ 0 on the boundary data implies that g is surjective, namely g(∂Ω)=∂Ω.


Assuming for now that (<ref>) holds,  let us show how it implies Theorem <ref>.  First, we state a simple consequence of the maximum principle:


Let u∈ W^1,2(Ω;M) be a minimizing constraint map, and let M and Ω be as in Theorem <ref>.  If u=g on ∂Ω and g(∂Ω)⊆∂Ω,  then the set Ω∩ u^-1(M) is open and connected.




Note that our assumptions  ensure that (∂Ω,∂ M) > 0.  Thus, since u = g on ∂Ω and g(∂Ω) ⊆∂Ω,  we see that the open set u^-1(M) 
contains a neighborhood of ∂Ω, according to the continuity provided by Remark <ref>. 

If the set u^-1(M) was not connected, then there would be a connected component C⊂ u^-1(M) that is disjoint from ∂Ω, thus ∂ C⊂∂ u^-1(M)∩Ω. We would then have (u,∂ M)=0 on ∂ C and (u,∂ M)>0 on C. However, since the function (u,∂ M) is subharmonic in u^-1(M) (since u is harmonic in u^-1(M) and (·,∂ M) is convex), this would contradict the weak maximum principle.





Let T be a flat piece of ∂ M, that is, 

    T⊂{x∈^n:e· x=c}
  for some e∈^n and c∈,
and assume without loss of generality that {e· x> c}⊂ M. Let M'⊂ M be the open half-cylinder with base T, namely

    M':={x=t e+x̅∈^n: x̅∈ T  and  t> 0 }.

Fix any point y ∈ T.  According to the surjectivity asserted in (<ref>) (see Proposition <ref> below),  there is a sequence {x_k}_k=1^∞⊂ u^-1(M)∩Ω such that u(x_k)→ y as k→∞. We may also assume that x_k→ x_∞ for some x_∞∈u^-1(M)∩Ω. We claim that

    x_∞∈Σ(u);

once this is shown, we must have x_∞∈∂ u^-1(M), since u is harmonic in the open set u^-1(M) according to (<ref>) and Theorem <ref>. 

Suppose, for the sake of contradiction, that x_∞∉Σ(u),  so there is r>0 with B_r(x_∞)⊂Ω∖Σ(u). Since T is a relatively open set in ∂ M and u∈ C^0(B_r(x_∞)), we have u(B_r(x_∞))⊂ M'∪ T, by taking r smaller if necessary. In particular, (<ref>) shows that u is harmonic in B_r(x_∞). Hence, 
e· u ≥ c  and Δ(e· u) = 0  in  B_r(x_∞).
Also, we have

    e·  u(x_∞) = lim_k→∞  e· u(x_k) = e· u(y) = c,
 so the strong minimum principle asserts that e · u = c in B_r(x_∞).  


Since e· u is a harmonic function in Ω∩ u^-1(M) and Ω∩ u^-1(M) is connected (see Lemma <ref>),  the unique continuation principle shows that e· u=c in Ω∩ u^-1(M).  On the other hand, u is continuous near ∂Ω and u(∂Ω)=g(∂Ω)=∂Ω (since (g)≠ 0, see Remark <ref>), so a contradiction is reached.  
 




The rest of this section is dedicated to proving  (<ref>). In fact, we will show a much more general result concerning the topological degree of minimizing constraint mappings, valid for non-convex obstacles, such as the one in Figure <ref>. We believe this more general result is independently interesting, and it is relevant in Corollary <ref> below.





Let Ω⊂^n be a smooth uniformly convex domain, and  M⊂^n be such that M^c is connected and M^c⋐Ω. Let O be the complement of the convex hull of M^c. Consider g∈ C^2(∂Ω;∂Ω) with (g) ≠ 0 and let u ∈ W^1,2_g(Ω;M) be a minimizing constraint map. 

For any domain O'⋐ O,  we have

    (u,Ω',·) = (g)   on Ω∩ O',     where Ω' := Ω∩ u^-1( O').

In particular,  u|_Ω∩ u^-1(O) is a surjective componentwise-harmonic map onto Ω∩ O.



Before beginning the proof of Proposition <ref>, we prove a simple topological lemma. This lemma does not rely on the specific geometric structure assumed in Proposition <ref>, and we use only that O⊆ M.



Let uΩ→M be a map such that u|_u^-1(O)  is continuous and u(∂Ω)⊆∂Ω. For every O'⋐ O we have u(∂Ω')⊆∂Ω∪∂  O', where Ω'=Ω∩ u^-1(O') is as in (<ref>).





Note that ∂Ω'= ∂Ω∪ [Ω∩∂ u^-1(O')]. Since we assume u(∂Ω)⊆∂Ω, it suffices to prove that 
    u(Ω∩∂ u^-1(O'))⊆∂  O'.
 For this purpose,  given x_0 ∈Ω∩∂ u^-1(O'), there is a sequence x_k∈Ω∩ u^-1(O') such that x_k→ x_0, and since u(x_k)∈ O' we have u(x_0)∈O'.

Assume by contradiction that u(x_0) ∈  O'. Then x_0 ∈Ω∩ u^-1(O'), and since Ω∩ u^-1(O') is an open set, there is a ball B_r(x_0)⊂Ω∩ u^-1(O'), impossible.

This proves that u(x_0)∈∂ O', as desired. 





Throughout the proof, we write f for the signed distance function to ∂Ω, with f<0 in Ω.  Since Ω is smooth and uniformly convex,  there is a tubular neighborhood (∂Ω) of ∂Ω where f is smooth and satisfies 

    f(ξ,ξ)≥ c|ξ^⊤|^2   on (∂Ω),
 for some c>0 and all ξ∈^n. Here we write ξ^⊤:= ξ - (ν·ξ)ν as in (<ref>), where now ν (with a slight abuse of notation deviating from the introduction) is the outwards unit normal to ∂Ω. For δ>0 small enough, we write

    O_δ := {-δ < f≤ 0}⊆Ω∩(∂Ω).

We split the proof into four steps.




There is no ball B⊂ u^-1(O) such that f∘ u=0 in B.


As in the proof of Theorem <ref>, since we assume that (g)≠ 0 we have that g is a surjective map, i.e. g(∂Ω)=∂Ω.
Now assume towards a contradiction that f∘ u=0 in some ball B⊂Ω∩ u^-1(O). Since  Δ u=0 in Ω∩ u^-1(O), we have

    0 = Δ(f∘ u) =    f_u (Du,Du)    in  B.

Since u(B)⊆∂Ω, Du lies in the tangent bundle T(∂Ω), i.e. Du = (Du)^⊤. Thus, by (<ref>)  we deduce that Du=0 in B. According to Lemma <ref>, the open set Ω∩ u^-1(O) is connected, and as u is harmonic there, 

by unique continuation
we must have that u is constant in Ω∩ u^-1(O). This contradicts the fact that u(∂Ω)= g(∂Ω)= ∂Ω.




    For every sufficiently small δ>0, there exists η > 0 such that Ω∩ u^-1(O_η) ⊂Ω∩ O_δ. 


Suppose, for the sake of contradiction, that there are some small δ>0 and a sequence η_k→ 0 for which points y_k∈∂ O_η_k and x_k∈ u^-1(y_k)∩Ω∖ O_δ can be found. Without loss of generality, we can assume that x_k→ x_∞ and u(x_k)=y_k→ y_∞.  Clearly, x_∞∈Ω∖ O_δ and y_∞∈∂Ω = f^-1(0). Since Theorem <ref> applies with Y=O^c, we have 

    (u(x_∞),∂ O)    =(u(x_∞),O^c) =lim_k→∞(u(x_k),O^c)
       =lim_k→∞(u(x_k),∂ O) 
    = (y_∞,∂ O)≥(∂Ω,∂ O)>0,

and so x_∞∈Ω∩ u^-1(O). In particular,  u is continuous at x_∞ and u(x_∞)=y_∞. Since f is convex in (∂Ω),  by continuity and harmonicity of u in Ω∩ u^-1(O) we conclude that the function f∘ u is subharmonic in a ball B centered at x_∞.  However,  Corollary <ref> shows that

    f∘ u ≤ 0   in Ω

therefore x_∞ is an interior maximum for f∘ u in B, since f∘ u(x_∞)=f(y_∞)=0.  Therefore, by the strong maximum principle we deduce that f∘ u = 0 in B.  This contradicts Step 1,  completing the proof of Step 2.


There is d∈ such that    (u,Ω',·) = d on Ω∩ O' for every O'⋐ O.


Since u∈ C^0(Ω∩ u^-1(O)) by Remark <ref>,  we have u ∈ C^0(Ω') (recall Ω' = Ω∩ u^-1(O')).  By Lemma <ref> we have Ω∩ O' ⊂^n∖ u(∂Ω') and so,  by Lemma <ref>, Ω∩ O' is contained in a connected component of ^n∖ u(∂Ω').  By Theorem <ref>(<ref>), (u,Ω',·) is constant in each connected component of ^n∖ u(∂Ω'), and so there is d∈ such that

    (u,Ω',·) = d    in Ω∩ O',

as desired. 


We have d=(g). 


Note that f∘ u  is subharmonic in a neighborhood of ∂Ω inside Ω, and it achieves a maximum on ∂Ω.  Thus, the Hopf lemma combined with Step 1 shows that 

    D_ν (f∘ u) ≥ 2 c   on ∂Ω,
 for some constant c>0, where we recall that ν is the outwards unit normal to ∂Ω, extended to (∂Ω) via the formula ν=∇ f.  Since f and u are C^1 near ∂Ω,  we deduce that 

    D_ν(f∘ u) ≥ c   on Ω∩ O_δ,

provided that δ>0 is sufficiently small.  

Consider an oriented orthonormal basis ℬ_x:={τ^1_x,…,τ^n-1_x,ν_x} at a point x∈∂Ω, and write 𝒯_x :=I-ν_x⊗ν_x for the matrix with columns τ_x^1,…, τ_x^n-1. 
Recall that (g) is an integer which can be computed as follows: if y∈∂Ω is a regular value[Recall that y is a regular value of g if the differential dg_x is non-singular at all points x∈ g^-1(y). In particular, by the inverse function theorem, a regular value of g has only a finite number of pre-images.] of g and g^-1(y)={x_1,…, x_i_0} for some points {x_i}⊂∂Ω, then

    (g) = ∑_x_i∈ g^-1(y)( dg_x_i),

where we see dg_x_i T_x_i∂Ω→ T_y∂Ω as an invertible linear map between (n-1)-dimensional vector spaces. More precisely, 

since u=g on ∂Ω, for each x ∈∂Ω we have

    dg_x = 𝒯_u(x)^t Du(x)𝒯_x,

where the superscript “t” denotes the transpose.

Moreover, there are  relatively open subset U_i, V⊂Ω such that x_i∈ U_i and y∈ V, 

    g^-1(V∩∂Ω)=∪_i=1^i_0 U_i∩∂Ω ,   where  g U_i ∩∂Ω→ V∩∂Ω is a diffeomorphism,

for all i. Note that, on ∂Ω, we also have

    ν_u(x)^t D u(x) 𝒯_x = D_τ(f∘ u)(x)=0,     ν_u(x)^t Du(x) ν_x = D_ν(f∘ u)(x).

Thus, for x∈∂Ω, we can write Du(x) with respect to the bases ℬ_x and ℬ_u(x) as follows:

    Du(x) = 
    [         dg_x            *;            0 D_ν(f∘ u)(x) ],

where * denotes an unspecified entry of the matrix.
Since u is C^1,α near ∂Ω, by shrinking δ, U_i, and V if needed,  we see from (<ref>) that

    ( Du(x))=( dg_x_i)   for all  x∈ U_i∩Ω.
  

Let η be determined in terms of δ through Step 2. By shrinking U_i and V further if necessary, we can also assume that V⊂Ω∩ O_η, so u^-1(V)⊂Ω∩ O_δ, and  by (<ref>) and (<ref>) we have

    u^-1(V) = ∪_i=1^i_0 U_i,      u U_i → V  is a diffeomorphism.

Thus any point y'∈ V∩Ω is a regular value for u which is not in u(∂Ω)=∂Ω and so

    (u,Ω',y')=
    ∑_x̃_i∈ u^-1(y')( Du(x̃_i))=
     ∑_x_i∈ g^-1(y)( dg_x_i)= (g)

for every >0 sufficiently small, as wished.




For the surjectivity claim, note that Theorem <ref>(<ref>) and the previous two steps show that Ω∩ O'⊆ u(Ω') for all O'⋐ O. The conclusion follows by taking the union over O'.



We conclude this section by noting that, for general non-convex obstacles, the non-coincidence set u^-1(M) is not necessarily open:


Let Ω,M,O and u be as in Proposition <ref>. Assume that M^c is non-convex and that ∂ O contains a flat piece T. Suppose in addition that

    u(Ω)⊆O.

Then u^-1(M) is not an open set.



Note that, by (<ref>), u∈ W^1,2_g(Ω;O) is a minimizing constraint map. Since O^c is convex, according to Theorem <ref> there is a point x̅∈∂ u^-1(O)∩Σ(u) and a sequence of points x_k→x̅ such that u(x_k)∈ O⊂ M  and u(x_k)→y̅ for some y̅∈∂ O∖∂ M. In particular, since u is continuous over u^-1(O), we have 

    lim inf_r→ 0(u,∂ M)_L^∞(B_r(x̅)) >0,

thus x̅∈ u^-1(M). On the other hand,  since x̅∈Σ(u), there is no ball around x̅ that is contained in u^-1(M), as otherwise u would be harmonic near x̅, according to (<ref>). Thus, u^-1(M) is not open.



We note that assumption (<ref>) is satisfied for many non-convex obstacles, such as the one in Figure <ref>. Indeed, for such an obstacle the nearest-point projection map π M→ O is 1-Lipschitz and so, by comparing a minimizing constraint map u∈ W^1,2_g(Ω;M) with a competitor π∘ u, we can deduce that in fact u∈ W^1,2_g(Ω;O).







§ BRANCH POINTS AND FREE BOUNDARIES





 §.§ General remarks on branch points

Our goal now is to understand the interaction between branch points and the free boundary. We begin by collecting some general remarks. Similarly to (<ref>), we state the following definition.



Let uΩ→^m be a minimizing constraint map.  We call x_0∈Ω∖Σ(u) a branch point of u, and we write x_0∈(u), if Du(x_0) does not have full rank. This leads to a decomposition

    (u) = ⋃_k=0^min{m,n}-1_k(u),    _k(u) := {x∈Ω∖Σ(u): rank(Du(x))=k }.

Given x_0∈(u), we say that:


  * x_0 is a false branch point if u(B_r(x_0)) is a C^1 manifold, for all r>0 sufficiently small;

  * x_0 is a true branch point if it is not a false branch point.



In order to illustrate the above concepts, consider the maps 

    u_k≃^2→^3,    z↦  (z^2,Re z^k),

which are component-wise harmonic. In particular, if M⊂^3 is a smooth domain such that u()⊆M, then u is a minimizing constraint map in W^1,2(Ω;M) for any Ω⊂. 
These maps also satisfy (u_k)=_0(u_k)={0},  and moreover 0 is a true branch point if k is odd, and it is a false branch point if k is even, see Figure <ref> for an illustration.



Before proceeding further, we compare Definition <ref> with other definitions of branch points occurring in the literature; we believe that this will also clarify the nature of our definition.

 In the function theory literature <cit.>, one typically considers maps between domains of the same dimension, i.e., m = n. The branch set ^(u) is defined as the set of points near which a map is not a local homeomorphism.  Clearly, for a C^1 map u we have

    ^(u)⊆(u) ,

according to the Inverse Function Theorem. This inclusion is generally strict, and this highlights the fact that our definition of branching is analytical, rather than topological. In fact, for open mappings of class C^1, we always have that

    ^(u) ⊆⋃_k=0^n-2_k(u);

in other words, u is a local homeomorphism near any point in _n-1(u) <cit.>. 
For example, the homeomorphism x↦ x^3 of the real line has a rank zero point at the origin.



    In the theory of parametric minimal surfaces, one considers weakly conformal parametrizations u^2→^m of a (possibly branched) minimal surface S=u(^2). Precisely, u must satisfy:
    
        
  *  weak conformality: |u_x|=|u_y| and u_x· u_y=0;
        
  *  minimality: Δ u = 0.
    
    If u satisfies (<ref>) then (u)=_0(u). Also, it is known that if, in addition, (<ref>) holds and u is minimizing, then (u) = ∅, under appropriate conditions on the boundary data, see <cit.> and the references therein. However, this result is not applicable in our setting, since minimizing constraint maps generally do not satisfy (<ref>), a simple example being given by (<ref>).


 We also note that, for harmonic maps between two-dimensional domains, a fine description of their behavior at the branch points was presented in <cit.>.






We now want to obtain some information on the sets _k(u), in particular concerning their dimension and rectifiability properties. For reasons that will become apparent later, we will focus on the case k=0. The next result follows from essentially standard considerations, cf. <cit.> for the case of harmonic maps. As we shall explain below, the following result is essentially contained in <cit.>.


Let M⊂^m be a smooth domain and let u∈ W^1,2(Ω;M) be a non-constant minimizing constraint map.  For any ball B such that 2B⋐Ω∖Σ(u),  we have the estimates
    
    ^n-2(B∩_0(u))   ≤ c ,
    ^n-2/2 (u(B∩_0(u)))   ≤ c,
 
for some constant c depending only on n, ∂ M, Du_L^∞(2B), and ∫_2B |u-(u)_2B|^2 dx.
 


Recall that, by definition, _0(u) is a closed set disjoint from Σ(u).
By the Euler–Lagrange system (<ref>), any minimizing constraint map is a solution of an elliptic system of the type

    Δ u^i = a_ij^α D_α u^j  in Ω,      a_ij^α∈ L^2(Ω)∩ L_^∞(Ω∖Σ(u)),

where the regularity of the coefficients follows from Theorem <ref>. In particular, as remarked in <cit.> (and as we formulate more precisely in Appendix <ref>), due to the diagonal structure of this system, the standard scalar methods concerning the critical set of elliptic equations are applicable. Therefore the measure estimate follows as in the scalar case <cit.>.  We remark here that the methods of <cit.> are not directly applicable, since minimizing constraint maps are not smooth near regular points; their methods require increased smoothness of the solution as the frequency increases. Nonetheless, the work <cit.> deals precisely with elliptic equations with rough coefficients.

Concerning estimate (<ref>), we argue as follows. 
By (<ref>),  for any δ>0 we can take points x_i∈_0(u) and radii r_i ∈ (0,δ), for i∈ I, such that 
    B∩_0(u)⊂⋃_i∈ I B_r_i(x_i),     ∑_i∈ I r_i^n-2≤ c.

Since u∈ C^1,1(B) and |Du| = 0 on B∩_0(u), we can find a uniform λ>0 such that u(B_r_i(x_i))⊂ B_λ r_i^2(u(x_i)), for every i∈ I.  Since {B_λ r_i^2(u(x_i)) : i ∈ I} is a covering of u(B∩_0(u)), letting δ→ 0 we get 

    ^n-2/2(u(B∩_0(u)))≤λ^n-2/2c.

as desired. 



Note that the estimate (<ref>) on the image of _0(u) is stronger than what one could deduce from Sard's Theorem whenever n ≤ 3 ≤ m or n > m, since minimizing constraint maps are only C^1,1.



For a smooth harmonic map u, Sampson <cit.> proved the following unique continuation result: if _1(u) has non-empty interior, then _1(u)= Ω and u maps into a geodesic. Then, it was shown in <cit.> that if u does not map into a geodesic, the set _1(u) has a locally finite ^n-1-measure.  
Perhaps surprisingly, the same unique continuation type properties do not hold in general[An obvious exception is for smooth harmonic maps into analytic manifolds, since in that case (Du) is constant in an open dense set.] for the sets _k(u) whenever k≥ 2, see e.g. <cit.>. 




 §.§ Branch points at free boundary points



We now turn our attention to the interaction between (u) and the free boundary ∂ u^-1(M). We first make the observation that, under rather general circumstances, one can make degree-theoretic arguments to produce branch points on the free boundary. Recall that  𝔹^n⊂^n denotes the open unit ball.


Let n≥ 3 and let u∈ W^1,2_g( B^n;^n\ B^n) be a minimizing constraint map. Assume that g∈ C^2( S^n-1; S^n-1) satisfies |(g)|≥ 2, and that, for a.e. >0 sufficiently small, the set  {|u|≥ 1+} is a topological annulus. Then 

    ⋃_k=0^n-2_k(u) ∩∂{|u|>1}≠∅.




From Theorem <ref> and Sard's theorem, we deduce that a.e. >0 sufficiently small is a regular value of |u|. Thus 

    D|u|= (u· Du)/|u|≠ 0    on  N_ := ∂{|u|>1+},

 and N_ is then a smooth manifold.  Since | (g)|≥ 2, the map g S^2→ S^2 is surjective. Thus, our assumption that the set {|u|≥ 1+} is a topological annulus for a.e. >0 sufficiently small, essentially amounts to saying that the manifold N_ is connected. 

Let us suppose, for the sake of contradiction, that 

    ⋃_k=0^n-2_k(u) ∩ N_0 = ∅.

We write, as usual, Π∘ u = u/|u|, so that u=|u|(Π∘ u) and 

    Du = (Π∘ u)⊗ D|u| + |u| D(Π∘ u).

Since  1≤ |u| is C^1,1 by Theorem <ref>, and attains a minimum on N_0, we have 

    D|u|=0   on N_0.

Notice also that, since Π is 0-homogeneous, we have

    u· D (Π∘ u )=0

and so D(Π∘ u) is always a singular matrix. Hence, 
combining (<ref>)–(<ref>),  we have 

    (D(Π∘ u))=n-1   on  N_0,

and by continuity the same also holds on N_, for >0 small enough.
Let us now choose  so that (<ref>) holds.  It follows from (<ref>) and (<ref>) that u· Du= |u| D|u|, therefore

    (Du)=n  on  N_.

We have seen in Proposition <ref> that 

    (u, B^n ∩{|u|>1},y) = (g)

whenever 1<|y|<λ. Thus, since N_={|u|=1+} is connected, taking |y|=1+ we have u^-1(y)⊂ N_ and so, according to (<ref>) and the definition (<ref>) of the local degree, we see that

    (u, B^n∩{|u|>1},y) = ∑_x∈ u^-1(y)( Du(x)) = (g).

 We now argue similarly to  the last step in the proof of Proposition <ref>. For x∈ N_, let ℬ_x:={τ^1_x,…,τ^n-1_x} be an oriented orthonormal basis for T_x N_. Thus D_τ^i |u|=0 on N_ and, by (<ref>), we have

    D_τ u(x) = Du(x)τ_x=D(Π∘ u)(x)τ_x.

Note that ν_x:=D|u|(x)/| D|u|(x)| is the unit normal to N_, which is well-defined by (<ref>), and so

    (D_ν|u|)(x) = (D|u|)(x) ·ν_x = | (D|u|)(x)|>0.

Hence, if we consider the map p_ N_→ S^n-1 defined by p_ := u/(1+), the last two identities show that, on N_, we have

    ( Du) = ( dp_),

where as usual dp_ TN_→ T S^n-1 is the differential between tangent spaces.  In particular, we deduce that

    (p_) = ∑_x: p_(x)=y/(1+)( dp_(x)) = (g).

Note that, by (<ref>), p_ N_→ S^n-1 is a local diffeomorphism. Since n≥ 3, S^n-1 is simply connected and so, since N_ is connected, p_ is actually a global diffeomorphism; in particular, we must have p_=1. However,  this contradicts (<ref>), since we assumed that |(g)|≥ 2.



Under the assumption that the non-coincidence set {|u|>1}, or a small perturbation of it, is a topological annulus, Proposition <ref> produces branch points of low rank on the free boundary.  This topological condition seems hard to verify in general, but we expect it to hold in any sufficiently symmetric scenario. 

 In any case, we are especially interested in knowing whether there are points with zero rank on the free boundary, and it seems that the existence of such points cannot follow from purely topological considerations, such as those used in the proof of Proposition <ref>.
Rank zero points are especially relevant to us since, at those points, (<ref>) becomes a degenerate obstacle problem, regardless of what obstacle M^c is; furthermore, for uniformly convex obstacles,  these are the only points at which (<ref>) is degenerate.  To construct such points, the proof of Proposition <ref> naturally leads us to consider axially symmetric maps: if u B^3→^3\ B^3 is a C^1 map which rotates at least twice about the z axis, then a simple calculation reveals that the gradient of u along the (x,y) plane must vanish on the z axis, cf. (<ref>).  In other words, the differential of the map p_ constructed in the proof of Proposition <ref> vanishes on the z-axis. The construction and complete analysis of points in _0(u)∩∂{|u|>1} in the axially symmetric case is actually quite involved, and it will occupy the final sections of this paper.

We will now show that the existence of rank zero points on the free boundaries is substantially simpler if we allow ourselves to consider non-convex obstacles. 
In fact, we can construct an artificial obstacle that touches the image of the maps u_k defined in (<ref>) over a cone with vertex at 0, cf. Figure <ref>. This example has the caveat that the obstacle is not convex, but on the other hand it has two advantages:

    
  * it shows that the bound _(_0(u) ∩∂ u^-1(M))≤ n-2, which follows from Proposition <ref>, is in general optimal;
    
  * it shows that branch points can lead to cone singularities on the free boundaries; such singularities do not occur in the classical obstacle problem <cit.>.

In the next statement, [β] denotes the greatest integer less than or equal to β. 

 






    For each odd integer k≥ 3 and each α∈(0,1/2), there is a C^[k/2],α-domain M⊂^3 with compact  complement such that the following holds: the analytic map u_k, defined in (<ref>), satisfies 
    u_k(^2)⊂M and  
 ∂ u^-1(M) is a cone with a true branch point at its vertex.




We work with polar coordinates z=re^i θ with (r,θ)∈ (0,∞)×[-π,π].
Take a small angle θ_k ∈ (0,π/8) such that cos(k/2θ_k)≥1/2, and fix  λ≥ 1 to be chosen later. Let 




    C   :={r e^i θ: r>0, |θ|∈ [0,θ_k]},
    
        E_1   :={r e^i θ: r>0, |θ|∈ (θ_k,π 2 + θ_k]},
    
        E_2   :={r e^i θ: r>0, |θ|∈ (π 2 + θ_k,3 π4]},
    
        F   :={r e^i θ: r>0, |θ|∈ (3π4,π]}.

To define the obstacle, we set

    (z):=
    - z^k/2   if  z∈ C,
    
    p_k(π(z))-λ(z,∂ C)^[k/2]+α   if  z∈ E_1∪ F,

where p_k(z) is the [k/2]-th order Taylor polynomial of z↦ - z^k/2 at the origin, and π^2→C is the nearest point projection onto the convex set C. Note that ∈ C_^[k/2],α(C∪ E_1∪ F), and  (z) = -λ|z|^[k/2]+α for z∈ F. Now, one may easily extend  to E_2 so that ∈ C_^[k/2],α(^2). Finally, we let M be the epigraph of φ, namely 
    M={(z,x_3)∈^2×: φ(z)< x_3}.

Note that the image of u_k corresponds to the union of the graphs of the functions z↦±| z^k/2|.  Selecting λ > 1 large, corresponding to the choice of α∈(0,1/2), we observe that ∂ M intersects the graph of -| z^k/2| precisely on C, and lies below it otherwise.  Thus, u(^2)⊂M with u^-1(∂ M)= C.  


In the next remark, we observe another interesting aspect of the example constructed in Proposition <ref>.


Consider any integer k≥ 5; this restriction ensures that M is at least C^2. Note that the map u_k-u_k(0)=u_k vanishes to order 2 at the origin, since we can compute, in polar coordinates,

    |u_k(re^iθ)|=r^2 √( 1+ r^2k-4cos^2(kθ)).

However, if k≥ 5 is odd, the distance ρ∘ u_k of u_k to the obstacle vanishes at least to order k-1+α >  4, since

    u_k(r e^i π/2) = -r^2(1,0,0)    ρ∘ u_k(r e^iπ/2) = λ  (r^2)^[k/2]+α= λ r^k-1+α.
 This shows that, although u_k-u_k(0) vanishes to a fixed order at the origin, ρ∘ u_k can vanish to arbitrarily high order there. 


The above example is only possible because the obstacle M^c in Figure <ref> is not uniformly convex. In fact, the next proposition, which builds on the results of Sections <ref> and <ref> and will be useful in Section <ref>, shows that, 
 for uniformly convex obstacles, ρ∘ u vanishes twice as fast as u-u(x_0) at a free boundary point x_0; by the above remark, this is false for general obstacles.



    Let M⊂^m be a smooth obstacle, and let u∈ W^1,2(Ω;M) be a weakly constraint map such that u ∈ C(B;(∂ M)) in a ball B⊂Ω. Then for every x_0∈ B∩∂ u^-1(M), there exists k∈ such that u - u(x_0) vanishes exactly to order k at x_0, and ρ∘ u vanishes at least to order 2k at x_0. In addition, if all the principal curvatures of ∂ M ∩ u(B) are uniformly positive, such as when M^c is uniformly convex, then ρ∘ u vanishes exactly to order 2k at x_0. 



Since u∈ C(B;(∂ M)), it follows from <cit.> that u∈ C_^1,1(B). By (<ref>), we can apply the classical results concerning the frequency of solutions to elliptic systems (cf. Appendix <ref>). 
This yields the existence of k ∈ℕ and a non-trivial (vector-valued) harmonic polynomial P_k of degree k, which is homogeneous around x_0, and a radius r_0 > 0 small (such that B_4r_0(x_0)⊂ B),  such that 

    D^j(u - u(x_0) - P_k)_L^∞(B_r(x_0))≤ c_σ r^k - j + σ,   j∈{0,1},

for every r∈(0,4r_0) and for each σ∈ (0,1), where c_σ > 1 is independent of r.
In particular u-u(x_0) vanishes to order k∈, proving the first assertion of this proposition. Moreover, replacing r_0 with a smaller radius if needed, we also obtain 
        
    N(u, x,r)≤3/2 k,

        for all r∈(0,r_0) and any x∈ B_r_0(x_0). Here N(u,x,r) is the frequency formula defined in (<ref>). 



We now turn our attention to ρ∘ u. 
Note that, as a consequence of (<ref>), we have

    Du _L^∞(B_r(x_0))≤ cr^k-1

for every r∈(0,r_0). 
Also, thanks to <cit.>, (<ref>) holds in B_4r_0(x_0). Therefore, by (<ref>), (<ref>), (<ref>), Corollary <ref>, and (<ref>), we get
    
    Δ(ρ∘ u)_L^∞(B_r(x_0))≤ cr^2k-2

    for every r∈ (0,r_0). As ρ∘ u≥ 0 in Ω, a standard blow-up argument (see <cit.>) yields a constant c_0, which may depend only n and k, such that 
    
    ρ∘ u _L^∞(B_r(x_0))≤ c_0cr^2k,

    for every r∈(0,r_0). This proves that ρ∘ u vanishes at least to order 2k at x_0. Moreover, it follows from (<ref>) that Π∘ u - u(x_0) vanishes at x_0 exactly to order k. Therefore, the first part of the proposition is proved. 


Assume now that all the principal curvatures of ∂ M∩ u(B) are uniformly positive.
Taking a smaller r_0 if necessary, we obtain a constant κ > 0 such that 

    ρ(ξ,ξ) ≥κ |ξ^⊤|^2  in M∩ u(B_4r_0(x_0)),

cf. (<ref>).
The rest of the proof is similar in spirit to that of Lemma <ref>, in that we will show that there is a small η > 0 such that if ρ∘ u < η r^2k in B_4r(x_0) for some r > 0, then ρ∘ u = 0 in B_r(x_0). Once this claim is proved, from our choice of x_0∈ B∩∂ u^-1(M) we deduce that 

    sup_B_r(x_0) (ρ∘ u) ≥η r^2k
 
for all r∈(0,r_0/4), as desired. 

For the moment, let us fix a scale r∈(0,r_0/4), and consider the rescaling

    ũ_x_0,r(x) := 1/r^k (u(x_0 + rx) - u(x_0)).

Note from (<ref>) that (<ref>) holds with |a_ij^α|≤ |Du| a.e. in Ω. Thus, by (<ref>), 

    Δũ_x_0,r = ã_ij^α D_αũ_x_0,r^j   a.e. in B_4,

where, by (<ref>),  all coefficients satisfy

    ã_ij^α_L^∞(B_4)≤1/r^k-1 a_ij^α_L^∞(B_4r(x_0))≤c/r^k-1Du_L^∞(B_4r(x_0))≤ c.

By (<ref>), we also have 

    N(ũ_x_0,r,0,4)  = N(u , x_0,4r) ≤3/2k.

Hence, Theorem <ref> yields 
        
    1/r^n - γ(k-1)∫_B_2r(x_0) |Du|^-γ dx = ∫_B_1 |Dũ_x_0,r|^-γ dx ≤ c,

        for some constants γ>0 and c> 1, independent of r. 
        
        We now argue as in the proof of Lemma <ref>. Fix a pair 1≤ t≤ T≤ 2. We observe from (<ref>) that analogously to (<ref>) we have 

    c/(T-t)^2r^2∫_B_Tr(x_0) (ρ∘ u)  dx≥ c_0 ∫_B_tr(x_0)∩ u^-1(M) |Du|^2 dx,

with some c>1 large and 0 < c_0 < 1 small now depending on κ above, but independent of s, t, and r. Using (<ref>) in place of (<ref>), we deduce from the generalized Hölder inequality, just as in (<ref>) and (<ref>), that 

    |B_tr(x_0)∩ u^-1(M)|^2/p≤c^2/γ/r^2k-2∫_B_Tr(x_0)∩ u^-1(M) |Du|^2 dx ≤c/(T-t)^2 r^2k∫_B_Tr(x_0) (ρ∘ u)  dx,

where p> 1 is chosen such that 1/p = 1/2 + 1/γ. Since the last inequality holds for all 1≤ t< T≤ 2, we can deduce from Lemma <ref> that

    sup_B_2r(x_0) (ρ∘ u) < η r^2k    ρ∘ u = 0  in B_r(x_0),

for some sufficiently small η > 0. Here the smallness parameter η can be chosen independently of r, as all constants appearing in (<ref>) are independent of r (nor of t and T). The implication of (<ref>) leads us to x_0∉B∩∂ u^-1(M), which gives us a contradiction. Thus, we must have 

    sup_B_r(x_0) (ρ∘ u) ≥η r^2k.

This proves that the vanishing order of ρ∘ u is at most 2k, provided that all principal curvatures of ∂ M∩ u(B) are uniformly positive, as desired. 








§ AXIALLY SYMMETRIC CONSTRAINT MAPS: REGULARITY THEORY




In this section, we initiate the study of Dirichlet energy minimizers in the class of axially symmetric maps that are constrained to take values outside the unit ball.

The regularity theory of such maps is parallel to that of minimizing constraint maps, which we developed in the last sections, but it is independent of it. Indeed, as we will see in Corollary <ref> below, minimizers in axially symmetric classes are not minimizing constraint maps in general. Nonetheless, in a neighborhood of any regular point, minimizers in axially symmetric classes are also minimizers in the full Sobolev space of admissible maps, cf. Appendix <ref>.



This section is divided into three parts. In Subsection <ref> we define the basic classes of maps under consideration. In Subsection <ref> we develop the partial regularity theory for minimizers in axially symmetric classes; the results of this subsection are the analogue of the classical results for minimizing constraint maps presented in Section <ref>. Finally, in Subsection <ref>, we prove the main results of this section, Theorems <ref> and <ref>, which are the analogue of Theorems <ref> and <ref>, respectively, in the axially symmetric setting. We emphasize that the proof of the L^-γ estimate in the axially symmetric setting is rather different from the one presented in Section <ref>, and we believe it is of independent interest.



 §.§ Setup and definitions


Throughout this section, we will use the following:


    Set e_3 := (0,0,1). We use cylindrical coordinates (r,θ,z) in the domain. We write 𝔻={(r,z): r^2 + z^2 < 1, r ≥ 0} for the half-disk in the (r,z)-plane. We shall write D' = (∂_r,∂_z) and define Δ' := ∂_rr + ∂_zz + 1/r∂_r, so that Δ= Δ' + 1/r^2∂_θθ  is the usual Laplacian in ^3. 


We now define axially symmetric maps, and the corresponding energy minimizers.



Fix k∈.

    
  *  A measurable map u  B^3→^3 is said to be k-axially symmetric if it is of the form
    
    u = ϱ (sinφcos kθ,sinφsin kθ,cosφ),

    for some pair (ϱ,φ):→ [0,∞)×[0,π].

  * A map u∈ W^1,2( B^3;^3\ B^3) is said to be energy minimizing among k-axially symmetric maps if u is itself k-axially symmetric and it satisfies 

    [u]≤[ũ]

for all k-axially symmetric maps ũ∈ W^1,2_u( B^3;^3\ B^3). 




    Note that (<ref>) implies 
    
    u∘ R_θ = R_kθ∘ u,

    for every θ∈[0,2π], where R_θ denotes the rotation about the z-axis by angle θ. However, (<ref>) is not sufficient to deduce the representation (<ref>). In fact, (<ref>) allows additional rotation of each horizontal circle in ^3 (i.e. a circle of form {r = r_0, z = z_0}) by an amount depending on (r_0,z_0). However, minimizers of the energy among all maps that verify (<ref>) subject to a k-axially symmetric boundary condition, in the sense of (<ref>), turn out to be k-axially symmetric as well.  We refer the reader to <cit.> for a similar argument in a slightly different context.


In what follows, given any pair of function (ϱ,)→ [1,∞)× [0,π], we define its energy through

    _k[ϱ,] := ∬_𝔻( |D'ϱ|^2 + ϱ^2( |D'φ|^2 + k^2/r^2sin^2)) r dr dz.

We call (ϱ,φ) admissible if in addition _k[ϱ,φ] < +∞. 


Note that u∈ W^1,2( B^3;^3∖ B^3) is k-axially symmetric if and only if the associated pair (ϱ,φ) is admissible, since 

    [u] = ∫_ B^3 |Du|^2 dx = 2π_k[ϱ,φ].

Thus u ∈ W^1,2( B^3;^3∖ B^3) is -minimizing among all k-axially symmetric maps if and only if the associated pair (ϱ,φ) is _k-minimizing among all admissible pairs,  i.e.

    _k[ϱ,φ] ≤_k[ϱ̃,φ̃]

for every admissible pair (ϱ̃,φ̃) satisfying (ϱ -ϱ̃)∪(φ -φ̃) ⋐. 

 



 §.§ Partial regularity theory



We now turn to the partial regularity theory for energy minimizers among k-axially symmetric maps.  Throughout this subsection we will always write u∈ W^1,2( B^3;^3\ B^3) for a fixed energy minimizer among k-axially symmetric maps.

Let us begin with the Euler–Lagrange system for the energy minimizers, and their regularity away from the z-axis, i.e. away from {r=0}.


Let u∈ W^1,2( B^3;^3∖ B^3) be an energy minimizer among k-axially symmetric maps.
Then u∈ C_^1,1( B^3∖{r=0}) and it is a weak solution of 
    
    -Δ u = |Du|^2 u χ_{|u|=1}  in  B^3.

Moreover, |u|-1 ∈ W^1,2( B^3) is nonnegative and weakly subharmonic. 





Throughout the proof we work on a fixed domain Ω⋐∖∂ = {(r,z): r^2 + z^2 < 1, r >0}. Note from Remark <ref> that (ϱ,φ) is a minimizer of the energy _k among all admissible pairs. With φ fixed, we see that ϱ minimizes _k[·,φ], i.e., _k[ϱ,φ] ≤_k[ϱ̃,φ] for every admissible pair (ϱ̃,φ) with (ϱ-ϱ̃) ⋐. Note that the coefficient of ϱ^2 in _k[ϱ,φ] is nonnegative and belongs to L^1(Ω), due to the constraint ϱ≥ 1. Let β∈ C^1((-∞,∞)) be such that  β = 0 in (-∞,1), β = 1 in [2,∞) and β' ≥ 0 in [0,∞). Then (ϱ + β((ϱ-1)/)ψ,φ) is an admissible pair, for all small > 0, for each ψ∈ C_c^1(Ω). Hence, the minimality of (ϱ,φ) implies that  

    Δ'ϱ≥ϱ(|D'φ|^2 + k^2/r^2sin^2φ)χ_{ϱ > 1}≥ 0   in Ω,

in the weak sense. Since the coefficients of Δ' are bounded away from zero and infinity in a neighborhood of Ω, the local L^∞-estimate <cit.> for weak subsolutions applies and so ϱ∈ L_^∞(Ω).


Let us now consider the variation of _k[ϱ,·] with ϱ fixed. Although an admissible φ̃ is constrained to taking values in [0,π], we can actually consider unconstrained variations  in deriving the Euler–Lagrange equation for φ. Indeed, given any φ̃→ satisfying _k[ϱ,φ̃] < ∞, the pair (ϱ,max{min{φ̃,π},0}) becomes admissible, and 
_k[ϱ, max{min{φ̃,π},0}] ≤_k[ϱ, φ̃].
Thus, the minimality of (ϱ,φ) implies _k[ϱ, φ] ≤_k[ϱ, φ + ψ] for any > 0 small, for each ψ∈ C_c^∞(Ω), from which we deduce that 

    D'· (rϱ^2 D'φ)  = k^2ϱ^2/2rsin 2  in Ω

in the weak sense. 
Note from our earlier discussion that ϱ∈ L_^∞(Ω). Moreover, since _k[ϱ,φ] < + ∞ and ϱ≥ 1 in , the right-hand side of (<ref>) belongs to L_^2(Ω). Thus, the higher integrability estimate <cit.> applies to (<ref>) and we see that ∈ W_^1,p(Ω) for some exponent p = p(Ω) > 2. Since Ω is a two-dimensional domain, the Sobolev embedding implies φ∈ C_^0,σ̅(Ω), with σ̅:= 1-2/p. 

Let us return to ϱ. Noting that (ϱ +ψ,φ) is an admissible pair for any >0 small, for each ψ∈ C_c^∞(Ω) nonnegative, we obtain from the minimality of (ϱ,φ) that  

    Δ'ϱ≤ϱ(|D'φ|^2 + k^2/r^2sin^2φ)  in Ω,

in the weak sense. Since φ∈ W_^1,p(Ω) with some p > 2 and ϱ∈ L_^∞(Ω), the right-hand side of (<ref>) belongs to L_^p/2(Ω). By (<ref>) and (<ref>), we obtain from <cit.> that ϱ∈ W_^2,p/2(Ω). 
Thus, since once again Ω is a two dimensional domain, the Sobolev embedding implies ϱ∈ C_^0,2σ̅(Ω) (recall σ̅= 1-2/p). Hence (ϱ +ψ,φ) is an admissible pair for any > 0 small and for each ψ∈ C_c^∞(Ω∩{ϱ > 1}), i.e. ψ can change sign in the region where ϱ>1. Thus, the minimality of (ϱ,φ) yields equality in (<ref>) in Ω∩{ϱ>1} in the weak sense. However, as  ϱ has weak second derivatives in Ω, we also have Δ'ϱ = 0 a.e. in Ω∩{ϱ = 0}. The last two observations combined yield  

    Δ'ϱ = ϱ(|D'φ|^2 + k^2/r^2sin^2φ)χ_{ϱ > 1}  a.e. in Ω.
 

We now return once again to .  Since ϱ∈ C_^0,2σ̅(Ω), we can apply the Hölder estimate for the first derivatives <cit.> to (<ref>) and deduce that φ∈ C_^1,2σ̅(Ω). This in turn yields that the right-hand side of (<ref>) belongs to L_^∞(Ω)∩ C_^0,2σ̅(Ω∖∂{ϱ>1}), so by the interior C^1,1-estimate <cit.> for the classical obstacle problem we obtain ϱ∈ C_^1,1(Ω).  Plugging this information into (<ref>), we deduce again from the higher-order regularity theory <cit.> that ∈ C_^2,σ(Ω) for every σ∈(0,1). In particular, the Euler–Lagrange equation (<ref>) for φ holds in Ω in the classical sense. 

Since Ω was an arbitrary domain compactly contained in ∖∂, the above observations show that (ϱ,φ) ∈ C_^1,1(∖∂)× C_^2,σ(∖∂). Thus, in light of (<ref>), we must have u∈ C_^1,1( B^3∖{r=0}). Moreover, as the Euler–Lagrange equations (<ref>) and (<ref>) hold a.e. in ∖∂, a direct computation yields that the Euler–Lagrange system (<ref>) holds in B^3∖{r=0} in the strong sense. Since u∈ W^1,2( B^3;^3) and the z-axis has zero hamronic capacity, we deduce that (<ref>) holds in the weak sense. Finally, the fact that |u| - 1 is weakly subharmonic in B^3 follows either directly from (<ref>) or from Lemma <ref>.



    It is an interesting problem to decide whether, in the setting of Lemma <ref>, we have φ∈ C^2,1_(𝔻), see <cit.> for the one-dimensional case and  <cit.> for the two-dimensional case.


Lemma <ref> shows that u is regular away from the z-axis. We now prove the following -regularity theorem on the z-axis, which is the analogue of Theorem <ref> and <cit.> in our setting:


    There are absolute constants _0 > 0 and c>1, such that if  B_4s(z_0e_3)⊂ B^3 and E(u,z_0e_3,2s) ≤_0^2, then u ∈ C^1,1(B_s(z_0e_3)) and 
    
    D^j u_L^∞(B_s(z_0e_3))≤c/s^j,   j=1,2.




The idea follows essentially from <cit.>. Translating and rescaling u in the domain, we can assume that z_0=0, s=1, and u is defined in 4 B^3. We first note that

    |D'(cos)| ≤|sin| (|∂_r φ|+ |∂_z |) ≤1/2(|D'|^2 + sin^2 /r^2) r

and so, since ϱ≥ 1 and k≥ 1, we have

    ∫_ D |D'(cos)|   d x ≤_k[ϱ, ] = E(u,0,1)/2π≤_0^2/2π.

By Fubini's Theorem and the fact that _k[ϱ,]<∞, for a.e. R>0 we have that |_S_R is continuous, where we write S_R = {(r,z)∈ D:r^2+z^2=R^2}.  Similarly, using (<ref>), we can further choose R so that

    _S_Rcos≤ c _0^2,

for a universal constant c. Since cos(0,[-1,1])⊆{0,π}, we can assume without loss of generality that (0,R)=(0,-R)=0, the case where (0,± R)=π being totally analogous. We thus see from (<ref>) that, if _0 is chosen small enough, then the image of B^3 under u lies in an upper half space, i.e.

    u( B^3)⊂ (^3\ B^3) ∩{x_3> 1 2}.

In other words, u( B^3) lies in a star-shaped region and so, by <cit.> (see also Appendix <ref> and in particular Theorem <ref> therein), we conclude that u∈ C^1,σ( B^3) for any σ∈ (0,1). The sharp regularity u∈ C^1,1( B^3) then follows from the standard theory for the scalar obstacle problem, as in <cit.>.


We now develop further properties of u. In analogy with Lemma <ref>, we have the following monotonicity formula:


     Let z_0∈ (-1,1) and 0<t<s < 1- |z_0| be given. Then 
     
    E(u,z_0e_3,t) ≤ E(u,z_0e_3,s),
 with equality if and only if u is 0-homogeneous about z_0e_3 in B_s(z_0e_3)∖ B_t(z_0e_3).



    Note that the minimizing property over the class of k-axially symmetric maps yields the stationarity over domain variations for balls centered on the z-axis. The rest of the proof is identical to the standard one, see e.g. <cit.>. 


As in Section <ref>, Lemma <ref> allows us to define the normalized energy density E(u,z_0 e_3, 0^+) through the limit E(u,z_0 e_3, r) as r↘ 0.

We now turn to the compactness properties of minimizers in the axially symmetric class, which is an analogue of Lemma <ref> in the axially symmetric setting.



    Let {u_i}_i=1^∞⊂ W^1,2( B^3;^3\ B^3) be a bounded sequence of energy minimizers among k-axially symmetric maps. Then there is a map u∈ W^1,2( B^3;^3\ B^3), which is energy minimizer among k-axially symmetric maps, such that u_i→ u strongly in W_^1,2( B^3;^3) along a subsequence.  


As in the classical case <cit.>, the key idea is to consider a suitable axially symmetric extension of the boundary values, see  <cit.> and also <cit.> in the setting of axially symmetric harmonic maps. Although the extension here gets slightly more involved due to the presence of the non-constant radial component ϱ, most of the argument is similar to the aforementioned literature. To maintain the technicalities to a minimum and keep the reading flow, we shall present the proof in Appendix <ref>. 



As a corollary of Lemma <ref> and Theorem <ref>, and using Lemmas  <ref> and <ref>, we can run the usual dimension reduction argument to deduce the following:


We have u∈ C^1,1_(Ω∖Σ(u)).    The singular set Σ(u) consists of discrete points on the z-axis, which are characterized by E(u,z_0 e_3, 0^+)≥_0^2.


From Theorem <ref> and Lemma <ref>, arguing just as in Section <ref>, we also obtain:


    The function |u|-1 is nonegative, subharmonic and  continuous on B^3, with a  universal modulus of continuity.   





 §.§ Regularity near free boundaries


We now turn to the regularity theory of energy minimizers among k-axially symmetric maps near their free boundaries. A crucial ingredient in our analysis is the uniqueness and structure of blow-ups in the axially symmetric class, see <cit.> and <cit.>.  In what follows,  we use the following standard terminology:


Given z_0∈ (-1,1), we call ϕ_z_0 a tangent map of u at z_0e_3 if it is a weak W_^1,2-limit of a sequence u_r_j,z_0e_3:= u(z_0e_3 + r_j·), where r_j→ 0 as j→∞. 

 Combining Lemmas <ref> and <ref> with Corollary <ref>,  it follows that z_0e∈Σ(u) if and only if there is a non-constant tangent map ϕ_z_0 of u at z_0e, which is necessarily 0-homogeneous and minimizing among k-axially symmetric maps. In our setting, there are only four different non-constant tangent maps:



Let u∈ W^1,2( B^3;^3\ B^3) be an energy minimizer among k-axially symmetric maps. 
For every point z_0e_3∈Σ(u), there exists a unique tangent map ϕ_z_0 which, when restricted to 𝕊^2, is one of the four maps 

    ± S^-1∘ψ_k ∘ S    or   ± S^-1∘ψ_k∘ S,

where ψ_k→ is the map w↦ w^k, and S𝕊^2→ is the stereographic projection from e_3.
Moreover, there is a modulus of continuity ω, depending only on k, (z_0e_3,∂Ω), and u_W^1,2(Ω), such that 

    |u(x) - ϕ_z_0 (x- z_0e_3)| ≤ω(|x-z_0e_3|),

for every x∈Ω with |x-z_0e_3|≤1/2(z_0e_3,∂Ω). 



According to Corollaries <ref> and <ref>,  we have Σ(u)⊂{|u|=1}∩{r=0}. Thus, by Lemmas <ref> and (<ref>), at any point in Σ(u), all tangent maps of u at that point are 0-homogeneous harmonic maps into S^2, which further minimize the energy among k-axially symmetric harmonic maps. The first claim now follows from the classification of such maps from <cit.>. The second claim, concerning the existence of a modulus of continuity, is a direct consequence of the compactness of minimizing maps provided by Lemma <ref>,  and the fact that there is only a finite number of tangent maps. Indeed, since the function r↦ u_r,z_0 e_3 is continuous into W^1,2( B^3), the closest tangent map to u_r,z_0 e_3 varies continuously with r. But since there is only a finite number of such maps, such a map must eventually always be the same for r sufficiently small.



Let k≥ 2 and let u∈ W^1,2( B^3;^3∖ B^3) be an energy minimizer among k-axially symmetric maps. If z_0 e_3 ∈Σ(u) then u is not energy minimizing in any neighborhood of z_0 e_3.



    As described before the proof of Proposition <ref>, to  any point z_0e_3∈Σ(u) one can associate tangent maps ϕ_z_0 of u at z_0e_3, which in particular are non-constant 0-homogeneous k-axially symmetric maps. If u was minimizing near z_0 e_3 (in the full Sobolev space of admissible maps), then according to Lemma <ref> so would ϕ_z_0. However, by k-axial symmetry it is easy to see that 
    ϕ_z_0 S^2→ S^2 has (ϕ_z_0)=k. Thus, according to <cit.>, ϕ_z_0 is not a minimizer.




We can now prove the first main result of this section, which is an extension of Theorem <ref> to the setting of energy minimizers among k-axially symmetric maps.




Let u∈ W^1,2( B^3;^3\ B^3) be energy minimizing among k-axially symmetric maps. There are positive constants s,γ, and c, which depend only on k and u_W^1,2( B^3), such that if z_0e_3∈Σ(u) for some z_0∈ (-1,1), then 
    
    ∫_B_s(z_0e_3) |Du|^-γ dx ≤ c.





Let us write Λ:=u _W^1,2( B^3) for simplicity. By pre-composing u with a translation along the z-axis, which of course preserves the k-axial symmetry, we can assume that z_0=0.
Furthermore, by Lemma <ref> and Corollary <ref>, we deduce that B_s̅∩Σ(u) = {0}, for some s̅ =s̅(k,Λ). 

Consider the 0-homogeneous rescalings u_s(y) := u(sy), and let ϕ_0 be as in Proposition <ref>. This proposition also yields a modulus of continuity ω, depending only on k and Λ, such that u_s - ϕ_0_L^∞( S^2)≤ω(s) on S^2 for all s < 1/2. Since ϕ_0|_ S^2 is smooth and u_s|_ S^2 is uniformly close to it, ε-regularity implies that it must also be continuous and therefore C^1,1. Hence, given α∈ (0,1),  by interpolation we can find a new modulus of continuity ω, which still depends only on k and Λ, such that

    u_s -ϕ_0_C^1,α( S^2)≤ω(s).
 
 In light of Proposition <ref>, we can find ,δ > 0, depending only on k, such that

    |Dϕ_0| ≥ 2δ  on  S^2∖ B_(± e_3).

Hence, combining the previous two estimates, we can take s̅ smaller if necessary, without changing its dependence on k and Λ, in such a way that 

    |Du_s| ≥δ  on  S^2∖ B_(± e_3)

for every s∈(0,s̅). Thus, taking the cone _ := {0}∪ (⋃_s > 0 B_s (± se_3)),

    ∫_B_s̅∖_ |Du|^-1 dx = ∫_0^s̅ s^2  ds ∫_ S^2∖ B_(± e_3) |Du_s|^-1 d^2 ≤ cδ^-1.

Therefore, by Jensen's inequality, it suffices to find some γ = γ(k,Λ) ∈ (0,1) such that 

    ∫_B_s̅∩_ |Du|^-γ dx ≤ c.

We now follow the arguments in Section <ref>. We first claim, analogously to Lemma <ref>, that

    sup_1≤ t≤ 4 N(u_s, ± te_3, 1) ≤λ_0,

uniformly for all s ∈ (0,2s̅), for some λ_0=λ_0(k,Λ), with possibly smaller s̅ = s̅(k,Λ). This claim can be verified by following the lines of the proof of Lemma <ref> without major modifications. We only need to employ Lemmas <ref>, <ref>, and Corollary <ref>  in place of Lemmas <ref>, <ref>, and Corollary <ref>, respectively. Note also that |u_s - ϕ_0| ≤ω(5s) in B_5 and |ϕ_0| = 1 replace Corollary <ref> in the proof; alternatively, this could also be inferred from Corollary <ref>. Apart from these changes the proof goes through unchanged, and so we shall not repeat the details here. 


Let us fix an absolute constant τ∈ [1,4) to be determined later. In light of Proposition <ref>, we have 

    |Dϕ_0| ≤ c   in B_4τ(± te_3)

uniformly for t∈[1,4].  Choosing >0 smaller if necessary, we also have 

    Du_s → Dϕ_0  uniformly in B_4τ(± te_3)

as s→ 0; here again the rate of convergence depends only on k and Λ. Thus, 

    |Du_s| ≤ 2c  in B_2τ(± te_3),

uniformly for all s∈(0,2s̅), with a possibly smaller s̅ = s̅(k,Λ). Thus, in view of (<ref>) and (<ref>), we can also employ Lemma <ref> to find δ,σ∈ (0,1), which depend only on k and Λ, such that 

    |{ |Du_s| ≥δDu_s_L^∞(3Q)}∩ Q|≥ (1-σ)|Q|,

for every cube Q with 3Q⊂ B_2τ(± t e_3). With (<ref>) at hand, we can proceed with the Calderón-Zygmund cube decomposition argument, just as in the proof of Theorem <ref>, to get that 

    ∫_B_τ(± te_3) |Du_s|^-γ dx ≤ c,

uniformly for all s∈(0,2s̅) and 1≤ t≤ 4. 

To conclude, we choose τ≥ 1 such that (B_4∖ B_1)∩_⊂∪_1≤ t≤ 4 B_τ(± te_3). By (<ref>), 

    ∫_(B_4∖ B_1)∩_ |Du_s|^-γ dx ≤ c,

uniformly for s∈ (0,2s̅). Choose ℓ̅ such that 2^-ℓ̅≥s̅ > 2^-ℓ̅-1. As s̅ depends only on k and Λ, so does ℓ̅. Then by (<ref>), 

    ∫_B_s̅∩_ |Du|^-γ dx    ≤∑_ℓ = ℓ̅^∞∫_(B_2^-ℓ+2∖ B_2^-ℓ)∩_ |Du|^-γ dx ≤∑_ℓ =ℓ̅^∞ 2^-ℓ(n + γ) c ≤ c.

This finishes the proof. 


Based on Theorem <ref>, and as shown  in Section <ref>, we can establish that energy minimizers among k-axially symmetric maps are regular around their free boundaries.


    Let u ∈ W^1,2( B^3;^3\ B^3) be an energy minimizer among k-axially symmetric maps. Then for every x_0 ∈ B^3 lying in the closure of {|u|>1}, u ∈ C^1,1(B_δ(x_0)) with the estimate 
    
    u _C^1,1(B_δ(x_0))≤ c,

    where δ > 0 and c>1 depend only on 1-|x_0|, k and u_W^1,2( B^3). 


 This result follows from Theorem <ref> by using the properties of energy minimizers in the axially symmetric class established in Subsection <ref>,  exactly as in the proof of Theorem <ref>: since the argument is identical, we omit it.





§ AXIALLY SYMMETRIC CONSTRAINT MAPS: BRANCH POINTS ON FREE BOUNDARIES


The goal of this section is to prove Theorem <ref>, concerning the structure of the free boundary points in _0(u), for an energy minimizer u∈ W^1,2( B^3;^3\ B^3) in the class of k-axially symmetric maps. 
Uunless otherwise stated, in what follows we assume that u is such a map and that
k≥ 2. As in (<ref>), we then write

    u=ϱ (sinφcos kθ,sinsin k θ, cos),

for some pair (ϱ,φ) D→[0,∞)× [0,π], with ϱ = ϱ (r, z) and φ = φ (r,z). We will also consistently use Notation  <ref> and,  to keep the notation as simple as possible, we will write v  B^3→^2 for the projection map of u onto the sphere ^2. Specifically,  v=Π∘ u, in the notation of Section <ref>:   

    v  := Π∘ u = u/ϱ = (sinφcos (kθ),sinφsin(kθ),cosφ)  in  B^3.

In particular, |Dv| is constant in θ, and 

    |Dv|^2 = (∂_rφ)^2 + (∂_zφ)^2 + k^2/r^2sin^2φ  in  B^3∖{r = 0}.

As mentioned in Section <ref>, our motivation for considering maps u as described above is that any free boundary point on the z-axis is necessarily a branch point in _0(u). Indeed, by Theorem <ref>, u is C^1,1 around the free boundary,  and thus Dϱ=0 there, since ϱ attains its minimum on the free boundary. On the other hand,  we have Dv=0 at any regular point on the z-axis. Indeed,  using the representation (<ref>), we see that, for a fixed z, the map f→ defined by f(w)=f(r e^iθ) :=(v^1+i v^2)(r,θ,z) satisfies

    ∂_w f = (cosφ ∂_r φ+k/rsinφ)e^i (k-1)θ,  ∂_w̅ f = (cosφ ∂_r φ - k/rsinφ)e^i (k+1)θ.

Since k≥ 2, both exponentials e^i(k∓ 1)θ cannot be extended continuously to  w=0 and so, at any point where v  (hence f)  is C^1, their coefficients in (<ref>) must converge to 0 as |w| = r→ 0^+. Since |cosφ(0,z)|=1 and sinφ(0,z)=0, we conclude that Dv vanishes at all regular points on the z-axis. Thus

    ∂{|u|>1}∩{r=0}⊆_0(u).

The results of this section can be seen as a much more precise version of the above calculation.  Indeed, we will perform a fine blow-up analysis of u near a free boundary point on the z-axis. The structure of u at such a point is quite rich and the next theorem, which is the main result of this section, summarizes the main conclusions of our analysis.



 Given k ≥ 2, let  u∈ W^1,2_g( B^3;^3\ B^3) be an energy minimizer in the class of k-axially symmetric maps. Set ϱ=|u| and v = u/|u| as in (<ref>), and assume that g∈ C^2( S^2;λ S^2) for some λ>1, and that g is k-axially symmetric with |(g)|=k.

Let z_0 e_3∈∂{ϱ>1} be an arbitrary free boundary point on the z-axis. Then:

    
  *  
    v∓ e_3 vanishes at z_0e_3 exactly to order k, and there are constants C,a,σ,s > 0 such that 
    
    | D^j( v - (ar^kcos k θ,ar^ksin kθ,± e_3))| ≤ C r^k - j +σ,    j = 0,1,2,

    whenever r^2 + (z-z_0)^2 < s^2;
    
  *  ϱ-1 vanishes exactly to order 2k at z_0e_3; also, if we define the sequence of rescalings
    
    ϱ_s(r,z):=ϱ(s r,z_0+s z)-1/a s^2k,

    then, for every sequence s_i→ 0^+, up to a subsequence there is a blow-up ϱ̃ such that ϱ_s_i→ϱ̃ in  W_^2,q(^3) for any q∈ [1,∞). Moreover, ρ̃ is a non-constant, k-axially symmetric, (2k)-homogeneous, non-negative solution of
    
    Δϱ̃= k^2 r^2k-2χ_{ϱ̃>0}  in ^3;

    
  *  
    there exists a modulus of continuity ω and a small constant s_0 > 0 such that 

    _(B_s(z_0e_3)∩∂{ϱ > 1}, {r = 0}∪ (Z_2k-1∖{z= 0})) < sω(s),

for all s ∈ (0,s_0), where Z_ℓ is the nodal set of the zonal harmonics of degree ℓ. In particular, the free boundary ∂{ϱ>1} cannot be represented as a C^1-graph near any point of _0(u).



Before proceeding with the blow-up analysis required to prove Theorem <ref>, let us show how it easily implies Theorem <ref>.


Let k≥ 2, λ > 1, and consider a k-axially symmetric map g ∈ C^2(^2; ^2) with |(g)|= k. Let u∈ W^1,2( B^3;^3∖ B^3) be energy minimizing among all k-axially symmetric maps in W^1,2_λ g( B^3;^3∖ B^3). Note that, by axial symmetry, the image of the z-axis under u is contained in the z-axis; on the other hand, the image of the z-axis under u is also disconnected, due to the constraint. Hence,  u is necessarily discontinuous, i.e. Σ(u)≠∅. 

By Corollary <ref>, Σ(u) is a discrete set on the z-axis. Fix any z_*e_3 ∈Σ(u). By Theorem <ref>, we can find a ball B_* centered at z_*e_3 such that B_*⊂{ϱ = 1}, where ϱ is as in (<ref>). Hence, we can consider the largest open and connected set U⊂{ϱ = 1} containing z_*e_3. Choose z_0e_3∈∂ U. Then z_0e_3∈∂{ϱ > 1} by the definition of U. By Theorem <ref> again, we can find a small ball B centered at z_0e_3 such that u ∈ C^1,1(B). 
Taking the radius of B smaller if necessary, we deduce from Corollary <ref> that u is the unique minimizing constraint map in W^1,2_u(B;^3∖ B^3). 

We now verify the claims in Theorem <ref>; note that the boundary condition |(g)|=k guarantees that u is necessarily non-constant. Claim (<ref>) follows from the choice of z_0e_3∈∂ U, which readily implies z_0e_3∈∂( int{ϱ = 1}) ≠∅. Claim
(<ref>) follows from Theorem <ref>(<ref>)-(<ref>). Finally, claim 
(<ref>) follows from Theorem <ref>(<ref>).


The proof of Theorem <ref> is rather involved and is divided over the next three subsections. 



 §.§ Proof of Theorem <ref>(<ref>)


We begin our analysis with a simple consequence of Lemma <ref>.


We have ϱ∈ C_^1,1( B^3∖Σ(u)), v∈ C_^2,σ( B^3∖Σ(u);𝕊^2), and φ∈ C_^2,σ( B^3∖Σ(u)) for every σ∈(0,1). Also, the pair (ϱ,v) solves 

    Δϱ = ϱ |Dv|^2χ_{ϱ>1}
    Δ v + 2/ϱ Dϱ· Dv = -|Dv|^2 v
      in  B^3,

in the strong sense. 



Note that (<ref>) follows directly from (<ref>); in fact, the computation works without k-axial symmetry. Concerning the regularity statements, let z_0e_3∈ B^3∖Σ(u). By Lemma <ref>, there is a ball B⊂ B^3 centered at z_0e_3 such that u∈ C^1,1(B). Since ϱ = |u|≥ 1 in B^3, we also have ϱ∈ C^1,1(B). We then have v∈ C_^2,σ(B) for every σ∈(0,1): this can be deduced from the second equation in (<ref>), cf. <cit.>. 

Finally, we deal with the regularity of φ. Writing v=(v^1,v^2,v^3) ∈ C_^2,σ(B;^3), we see that φ = arccos v^3 in B^3. Then, since arccos is continuous in [-1,1], and v^3 ∈ C^2,σ(B)⊂ C(B), we have φ∈ C(B). Since φ({r = 0}∩ B) ⊆{0,π}, shrinking the radius of B if necessary, we may assume without loss of generality that φ∈ [0,π/6]; the other case is symmetric by replacing φ with π-φ. Then we may also write φ = arcsin (v^1/(cos k θ)) in B∩{|θ|<π/2}. As arcsin is analytic in (-1,1), |v^1|≤sinφ≤ 1/2 in B, and v^1 ∈ C^2,σ(B), then φ∈ C^2,σ(B∩{|θ|<π/2) for every σ∈(0,1). But  does not depend on θ, so this last restriction on θ can be removed, completing the proof. 



Assume that z_0e_3∈ B^3∖Σ(u). By Lemma <ref> we have, for some s small, that ϱ∈ C^1,1(B_s(z_0e_3)) and v∈ C^2,σ(B_s(z_0e_3)) for every σ∈(0,1). Since ϱ≥ 1 in B^3 by the constraint,it follows that (1/ϱ)Dϱ∈ C^0,1(B_s(z_0e_3)). Moreover, by Lemma <ref>, 
|Dv|^2 ∈ C^1,σ(B_s(z_0e_3)) for every σ∈(0,1). 

Let ṽ B_s(z_0e_3) →^2 be the map consisting of the first two components of v, i.e., 

    ṽ := ( sinφcos kθ,sinφsin kθ).

Since the boundary datum g S^2→λ S^2 satisfies |(g) |=k, we see from Proposition <ref> that u is surjective onto the annulus λ B^3∖ B^3. In particular, we deduce that sin is not identically zero in B_s(z_0 e_3), since  satisfies the unique continuation property by (<ref>), and hence ṽ is also not identically zero in B_s(z_0 e_3).
We observe that each component of ṽ is a C^2,σ-solution of the second line of (<ref>), 
where we think of (1/ϱ)Dϱ and |Dv|^2 as coefficients of the equation. Therefore, it follows from <cit.> that each component of ṽ vanishes to finite order. Applying <cit.>  to each component of ṽ, we see that there is a non-trivial homogeneous harmonic polynomial P = (P^1,P^2) of degree d such that  

    |D^j (ṽ - P)| ≤ C (r^2 + (z-z_0)^2)^d-j + σ/2,   j∈{0,1,2},

whenever r^2 + (z-z_0)^2 < s^2; here we take s smaller if necessary, and the constant C may depend on σ but not on (r,z-z_0). 

Since P is homogeneous of degree d, (<ref>) implies that 

    P  = (pcos kθ,psin kθ),

where p is a homogeneous polynomial of degree d in (r,z). Moreover, since P is non-trivial, so is p. In addition, being φ a polar angle of v, it must satisfy sinφ≥ 0 in B^3. So, (<ref>), (<ref>), and (<ref>) altogether imply 

    p ≥ 0   in {r≥ 0}.
 
Then, the structure of homogeneous harmonic polynomials <cit.> shows that we must have d = k, and 

p (r,z) = ar^k,

for some a > 0; in other words, P is a zonal harmonic. Noting that ṽ = P = 0 on {r = 0}, it follows from (<ref>) that 

    |D^j (ṽ - P)| ≤ C r^k-j + σ,   j∈{0,1,2},

whenever r^2 + (z-z_0)^2 < s^2. 

Choosing s smaller if necessary, we can also assume that sin^2φ∈ [0,1/2] in B_s(z_0e_3). Thus, (<ref>) and (<ref>) together yield

    (∂_rφ)^2 + (∂_zφ)^2 + k^2/r^2sin^2φ≤ 2|Dṽ|^2 ≤ Cr^2k-2,

whenever r^2 + (z-z_0)^2 < s^2 and r>0. Also, since v(z_0e_3) = ± e_3, and sin^2φ∈ [0,1/2] in B_s(z_0e_3), we have

    |v^3 ∓ 1| ≤ Csin^2φ,    |Dv^3| ≤sinφ|Dφ|,  and   |D^2v^3| ≤ |D|^2 + sin^2φ |D^2φ|^2    in B_s(z_0e_3).

Therefore, (<ref>) along with φ∈ C^2,σ(B_s(z_0e_3)) yield

    |D^j (v^3 ∓ 1)| ≤ Cr^2k - j,    j =0,1,2,

whenever r^2 + (z-z_0)^2 < s^2. Combining (<ref>) with (<ref>), we arrive at (<ref>).




 §.§ Proof of Theorem <ref>(<ref>)


We begin by noting that, by Theorem <ref>(<ref>), the (2k-2)-homogeneous rescaling of the function |Dv|^2, which depends only on (r,z) as seen in (<ref>), converges to k^2a^2 r^2k-2, which is again (2k-2)-homogeneous. The following corollary of Theorem <ref>(<ref>) shows that this convergence occurs at an almost-Lipschitz rate. 


    Given z_0e_3∈ B^3∖Σ(u) and σ∈(0,1), there exist C,s_0 > 0  such that 
    
    |( r∂_r + (z-z_0)∂_z  - (2k-2))|Dv|^2 | ≤ Cr^2k-2 s^σ,

    whenever r^2 + (z-z_0)^2 ≤ s^2 and s∈(0,s_0). 


 
    By (<ref>), we have
    
    | D^j( |Dv|^2 - k^2 a^2 r^2k-2) | ≤ C r^2k - j -2 +σ,    j = 0,1,

    whenever r^2 + (z-z_0)^2 < s^2. Since (r∂_r + (z-z_0)∂_z) r^2k-2 = (2k-2)r^2k-2, the conclusion follows from (<ref>) and the triangle inequality. 


We now observe that, due to (<ref>), the function ϱ - 1, which corresponds to the distance map from B^3 to ^2, must vanish exactly to order 2k at every free boundary point. 


    Given z_0e_3∈∂{ϱ>1}, ϱ - 1 vanishes at z_0e_3 exactly with order 2k, and there exist s_0 > 0 small and C>1 large such that
    
    1/C s^2k≤sup_B_s(z_0e_3)(ϱ - 1 + s|Dϱ|) ≤ Cs^2k,

    for all s ∈ (0,s_0).  



    By Lemma <ref>, u∈ W^1,2( B^3;^3∖ B^3) is a weakly constraint map. Taking a small s_0 we have u∈ C^1,1(B_s_0(z_0e_3)). The conclusion then follows directly from Proposition <ref>.


Given any z_0∈ (-1,1), let us define the following Weiss-type <cit.> energy  

    W(z_0,s) := 1/s^4k+1∫_B_s(z_0e_3)(|Dϱ|^2+ 2(ϱ-1)ϱ|Dv|^2) dx - 2k/s^4k+2∫_∂ B_s(z_0e_3) (ϱ - 1)^2  d^2;

recall from (<ref>) that |Dv| is a function of (r,z).  In the scalar obstacle problem, the analogous energy is monotone. Here we have the following similar result: 


    Given any z_0e_3∈∂{ϱ>1} and every σ∈(0,1), there is s_0 > 0 such that s↦ W(z_0,s) + Cs^σ is non-decreasing in s ∈ (0,s_0), where C>1 may depend on s_0 and σ. In particular, W(z_0,0^+):=lim_s→ 0 W(z_0,s) exists. 



By Theorem <ref>, we have z_0e_3∉Σ(u). Let s_0 be a small radius to be determined and note that, from Lemma <ref>, d/dsW(z_0,s) exists for every s∈ (0, s_0).

Let a > 0 be as in (<ref>). In what follows, we fix s∈(0,s_0). For every s > 0 small, consider 

    ϱ_s (r,z) := ϱ(sr,z_0 + sz) - 1/a^2s^2k,   F_s (r,z) := |Dv|^2(sr,z_0 + sz) /a^2s^2k-2.

In view of (<ref>), ϱ_s solves  

    Δϱ_s = (1 + s^2kϱ_s)F_sχ_{ϱ_s>0}  in B_1/s_0,

in the strong sense. Let us also write ϱ̇_s = d/dsϱ_s and Ḟ_s = d/dsF_s, so that

    ϱ̇_s  = (r∂_r + z∂_z- 2k)ϱ_s/s,  Ḟ_s = (r∂_r + z∂_z- (2k-2))F_s/s.

Note that, thanks to Corollary <ref> and Lemma <ref>,

    ϱ_s + s|ϱ̇_s| +  F_s + s^1-σ |Ḟ_s| ≤C/a^2  in  B^3,

for every s∈(0,s_0). Also, we can rewrite 

    1/a^4W(z_0,s)     = ∫_ B^3( |Dϱ_s|^2 + 2ϱ_s(1 + s^2kϱ_s)F_s)  dx  - 2k ∫_^2ϱ_s^2 d^2.

Differentiating both sides of (<ref>) in s, integrating by part, and using (<ref>) for ϱ_s, we compute  

    1/a^4d/dsW(z_0,s)    = 2∫_ B^3(Dϱ_s· Dϱ̇_s + ϱ̇_s (1 + s^2kϱ_s) F_s)  dx - 4 k ∫_^2ϱ_sϱ̇_s d^2 
          + 2∫_ B^3ϱ_sd/ds( (1 + s^2kϱ_s)F_s) dx 
        = 2∫_^2 (r∂_rϱ_s + z∂_zϱ_s - 2kϱ_s)ϱ̇_s  d^2 
          + 2 ∫_ B^3ϱ_s ((1+ s^2kϱ_s)Ḟ_s + s^2k-1(2kϱ_s + sϱ̇_s)F_s) dx
       ≥  2 ∫_^2 sϱ̇_s^2  d^2 - C/a^4 s^σ - 1≥ - C/a^4 s^σ - 1,

where, in the second last inequality, we used (<ref>) and (<ref>). By (<ref>), we obtain 

    d/ds (W(z_0,s) + Cs^σ) ≥ 0,

for every s∈(0,s_0), which finishes the proof. 



Let s_i→ 0^+ be given 

and define ϱ̃_i := ϱ_s_i, and F̃_i := F_s_i, where ϱ_s and F_s are as in (<ref>). By Theorem <ref>(<ref>), we have 

    F̃_i → k^2r^2k-2  in C_^1,σ(^3).

In addition, due to Theorem <ref>(<ref>) and Lemma <ref>, a standard compactness argument yields, up to a non-relabeled subsequence,

    ϱ̃_i →ϱ̃  in C_^1,σ∩ W_^2,q(^3),

for every σ∈(0,1) and q∈[1,∞), for some non-trivial ϱ̃∈ C_^1,1(^3). Since ϱ̃_i is axially symmetric and nonnegative, so is its limit ϱ̃. The fact that ϱ̃ satisfies 
 (<ref>) follows from (<ref>) and (<ref>), along with the regularity ϱ̃∈ C_^1,1(^3). 

Thus, it remains to prove the (2k)-homogeneity of ϱ̃, and this follows from the almost monotonicity of the Weiss-type energy, as we now detail. By Lemma <ref>, the limit W(z_0,0^+) exists. However, analogously to (<ref>), we also have, for each t > 0, that

    1/a^4 W(z_0,s_it) = 1/t^4k+1∫_B_t( |Dϱ̃_i|^2 + 2ϱ̃_i(1+(s_it)^2kϱ̃_i)F̃_i) dx - 2k/t^4k+2∫_∂ B_tϱ̃_i^2 d^2.

Thus, passing to the limit in (<ref>) along with (<ref>) and (<ref>), we deduce from Lemma <ref> that 

    1/a^4W(z_0,0^+) = lim_j→∞ W(z_0,s_it) = W̃(t),

for all t > 0, where the energy W̃ is defined as 

    W̃(t) := 1/t^4k+1∫_B_t (|Dϱ̃|^2 + 2ϱ̃k^2r^2k-2) dx - 2k/t^4k+2∫_∂ B_tϱ̃^2 d^2.

Using  (<ref>), one can following essentially verbatim the proof of Lemma <ref> to show that 

    d/dtW̃(t) = 2 ∫_^2 tϱ̇̃̇_t^2  d^2,    ϱ̃_t(r,z)=ϱ̃(tr,tz)/t^2k,

cp. (<ref>). Since (<ref>) implies that W̃(t) is constant, we deduce that ϱ̃ is (2k)-homogeneous, as desired.





 §.§ Proof of Theorem <ref>(<ref>)

To complete our analysis, it remains to analyze the free boundary for ϱ̃. According to Theorem <ref>(<ref>),  ϱ̃ satisfies (<ref>), and a crucial feature of this equation is that the forcing term on the right-hand side is independent of z. In particular, ∂_z ϱ̃ is a (2k-1)-homogeneous k-axially symmetric harmonic polynomial on each component of {ϱ̃>0}. In other words, ∂_zϱ̃ is a zonal harmonic of degree (2k-1) on each connected component of {ϱ̃>0}. 

Recall that the zonal harmonics of degree ℓ are constant multiples of

    x↦ |x|^ℓ P_ℓ(cos),

where P_ℓ is the Legendre polynomial of degree ℓ.  The polynomial t↦ P_ℓ(t) has ℓ distinct zeros  in (-1,1), arranged symmetrically about t = 0, and the nodal set of P_ℓ is given by 

    Z_ℓ={P_ℓ(cos)=0}.
 We are of course specially interested in the case where ℓ=2k-1 is odd: in this case, the intersection of  Z_ℓ with S^2 is  the union of ℓ-latitude circles. We refer the interested reader to <cit.> for further details.

The previous discussion leads us to consider the connected components of ^3\ Z_ℓ.  Thus, for -1 ≤ t_1 < t_2 ≤ 1,  consider the axially symmetric region 

    _t_1,t_2 := {t_1 √(r^2 +z^2)< z < t_2 √(r^2 + z^2)},
 

and an axially symmetric 2k-homogeneous solutions q to 

    Δ q = k^2 r^2k-2  in _t_1,t_2.

By the axial symmetry of q,  we see that 

    q(r,z) = (r^2 + z^2)^k y(z/√(r^2 +z^2)),

where y(t_1,t_2)→ is a solution to 

    (1 - t^2) ÿ - 2t ẏ + 2k(2k+1) y = k^2 (1-t^2)^k-1,   t_1 <t < t_2.

The following result gives a description of the solutions to this ODE.


  Let y(t) be a solution of (<ref>) and assume that k≥ 2.  Then
  
    y  = c_1 P_2k + c_2 Q_2k + p_k,

where P_2k is the Legendre polynomial of degree 2k, Q_2k is the Legendre function of the second kind of degree 2k, and p_k is the particular solution to (<ref>) given by

    p_k(t) = ∑_i = 0^k a_2it^2i,

under the recurrence relation

    a_0=0,      a_2i + 2 = -(2k-2i)(2k + 2i + 1)a_2i + (-1)^i k^2 k-1i/(2i+2)(2i+1),  i = 0,1,2,⋯.

Moreover ṗ_k(t) ≠ 0 for every t∈ (-1,1)∖{0} where p_k(t) = 0. 



The claims concerning (<ref>)–(<ref>) are elementary and are left to the interested reader to verify, so let us turn to the last claim.
    Suppose, towards a contradiction, that p_k(t_0) = ṗ_k(t_0) = 0 for some 0< |t_0| < 1. In view of (<ref>) and t_0≠ 0, we obtain  

    p̈_k(t_0) = k^2 (1-t_0^2)^k-2.

    Using the recurrence relation (<ref>), we can compute from (<ref>) (and k≥ 2) that 
    
    k^2 (1-t_0^2)^k-2    = ∑_i=0^k-1(2i+2)(2i+1)a_2i+2t_0^2i
       = ∑_i=0^k-1(2k - 2i)(2k+2i+1)a_2it_0^2i + k^2 ∑_i=0^k-1 (-1)^i k-1it_0^2i,

    whence the binomial theorem yields 
    
    k^2 t_0^2 (1-t_0^2)^k-2 = ∑_i=0^k-1(2k-2i)(2k+2i+1)a_2it_0^2i = ∑_i=1^k (2k-2i)(2k+2i+1)a_2it_0^2i;

    the last identity follows from the choice a_0 = 0 and (2k-2i)(2k+2i+1)a_2i = 0 when i =k. 
    However, p_k(t_0) = ṗ_k(t_0) = 0 along with (<ref>) and t_0≠ 0 yields 
    
    ∑_i=1^k a_2it_0^2i-2 = ∑_i=1^k 2i a_2it_0^2i-2 = 0.

    By (<ref>), we can first divide by t_0^2 ≠ 0 the leftmost and rightmost sides of (<ref>) and then proceed further as 
    
    k^2 (1-t_0^2)^k-2   = ∑_i=1^k(2k-2i)(2k+2i + 1)a_2i t_0^2i - 2
            = - ∑_i=1^k 2i(2i + 1) a_2it_0^2i-2
    
                = - ∑_i=1^k 2i(2i - 1) a_2it_0^2i-2 = - k^2 t_0^2(1-t_0^2)^k-2,

where the derivation of the last identity follows simply from the first line of (<ref>). However, (<ref>) implies k^2(1+t_0^2)(1-t_0^2)^k-2 = 0, which is incompatible with |t_0| < 1. This finishes the proof. 



Finally, the next lemma gives a description of the possible free boundaries in the blow-up limit.




Let ϱ̃ be any blow-up limit as in Theorem <ref>(<ref>). Then its free boundary ∂{ϱ̃>0} is contained either in the z-axis, i.e. {r = 0}, or inside {0}∪ (Z_2k-1∖{z =0}). 



The simplest case is when the harmonic function ∂_zϱ̃ vanishes identically in ^3. Then it is easy to compute that the only such non-trivial, 2k-homogeneous, axially symmetric solution to (<ref>) is 

    ϱ̃(r,z) = 1/4k^2 r^2k.
 This is precisely the case where {ϱ̃= 0} = {r = 0}. Hence, in what follows, we assume that ∂_zϱ̃≢0 in ^3. 


As we discussed above, by differentiating both sides of (<ref>), and noting that ϱ̃= |Dϱ̃| = 0 on {ϱ̃= 0} and that ϱ̃ is a non-trivial 2k-homogeneous axially symmetric function on ^3, we observe that ∂_zϱ̃ is a zonal harmonic of degree 2k-1 on each connected component of {ϱ̃> 0} that vanishes continuously on ∂{ϱ̃> 0}. Since ∂_zϱ̃≢0, the above observation along with |Dϱ| = 0 on {ϱ = 0} shows us that {ϱ̃= 0}⊆{∂_zϱ̃= 0}⊆ Z_2k-1. Therefore, it only remains for us to show that ∂{ϱ̃> 0}∩{z = 0}∖{0} = ∅.

Suppose by contradiction that ∂{ϱ̃> 0}∩{z = 0}∖{0}≠∅. By the homogeneity of ϱ̃, we must have {z = 0}⊆∂{ϱ̃>0}. Let U be the maximal open connected set in ^3 contained in {ϱ̃> 0} and such that ∂ U contains {z = 0}. Since ∂_zϱ is a zonal harmonic of degree 2k-1 in U that vanishes on ∂ U, U cannot be the entire half-space above or below {z = 0}. Thus, by the homogeneity of ϱ̃, U is enclosed between a cone {z = t_0 √(r^2 + z^2)}, for some 0< |t_0| < 1, and the plane {z = 0}; i.e. we have (following the notation in (<ref>))

    U = _0,t_0   or    U=_t_0,0.
 
Then, it follows from our discussions at the beginning of this subsection that

    0<  ϱ̃= q   in U,     ϱ̃= |Dϱ̃| = 0   on ∂ U,

where q satisfies (<ref>)–(<ref>) with either (t_1,t_2) = (0,t_0) or (t_1,t_2) = (t_0,0). We note that, as 0<|t_0|<1, the Legendre function Q_2k of the second kind is regular in [-|t_0|, |t_0|]. By our counter assumption and the 2k-homogeneity of ϱ̃, we must have ϱ̃= |Dϱ̃| = 0 at both (r,z) = (1,0) and (r,z) = (√(1-t_0^2),t_0). Then, by (<ref>) and (<ref>),

    c_1 P_2k(t) + c_2 Q_2k(t) + p_k(t) = 0 
    
    c_1 Ṗ_2k(t) + c_2 Q̇_2k(t) + ṗ_k(t) = 0
      at t = 0, t_0  .

By (<ref>) and (<ref>), we have p_k(0) = ṗ_k(0) = 0. On the other hand, 2k being an even number implies that P_2k(0) ≠ 0, Ṗ_2k(0) = 0, Q_2k(0) = 0, and Q̇_2k(0) ≠ 0. Using these elementary facts in (<ref>) for t = 0, we deduce c_1 = c_2 = 0. Using this information back inside (<ref>), it follows that p_k(t_0) = ṗ_k(t_0) = 0. Recalling that 0 < |t_0| < 1, we arrive at a contradiction to Lemma <ref>. This shows that ∂{ϱ̃> 0}∩{z = 0} = {0 }, as desired. 



The conclusion follows immediately from Lemma <ref>.














§ TOPOLOGICAL DEGREE



In this appendix, we recall some standard properties of the local topological degree, referring the reader to <cit.> for further details.

Given Ω⊂^n, u∈ C^0(Ω;^n), and a point y∉u(∂Ω), one can associate to it an integer, which we denote by (u,Ω,y) and refer to as the local topological degree of u at y with respect to Ω.  This integer can be defined as follows. If u∈ C^1(Ω) and y is a regular value[Recall that y is a regular value of u if Du(x) is invertible for all points x∈ u^-1(y).] of u,  then

    (u,Ω,y):= ∑_x∈ u^-1(y)( D u(x));

note that, by Sard's theorem, this defines (u,Ω,·) a.e. in ^n. For maps u that are just continuous, or at points y which are not regular values, one defines (u,Ω,·) by approximation, in such a way that y↦(u,Ω,y) is continuous and (u_j,Ω,y)→(u,Ω,y) if u_j→ u  uniformly.



The topological degree has the following properties:

    
  *  normalization: (𝕀,Ω,y) is 1 if y∈Ω and 0 if y∉Ω;
    
  * 
    decomposition: if Ω_1∩Ω_2=∅ then (u,Ω_1∪Ω_2, y) = (u,Ω_1,y) + (u,Ω_2, y);
    
  *  homotopy invariance: if h∈ C^0([0,1]×Ω;^n) and y∉h([0,1]×∂Ω) then 
    λ↦(h(λ,·),Ω,y)
 is constant in [0,1];
    
  *  existence of solutions: if (u,Ω,y)≠ 0 then y∈ u(Ω);
    
  *  constancy: (u,Ω,·) is locally constant in ^n∖ u(∂Ω).



In fact, properties (<ref>)–(<ref>) characterize the topological degree uniquely.

It is also useful to have a notion of topological degree for maps defined on manifolds without boundary. If N,Ñ⊂^m are connected oriented compact manifold without boundary, then we can define the degree of a map g∈ C^1(N;Ñ) exactly as in (<ref>), namely through

    (g) := ∑_x∈ u^-1(y)( d g_x),

where y is a regular value of g and we see dg_x T_xN→ T_y Ñ as an invertible linear map.  This definition is independent of y and it coincides with the usual algebraic-topological definition of the degree: in algebraic topology,  one says that the homomorphism induced by g between the top homology groups of N and Ñ corresponds to multiplication by (g). In this paper, we are content with the differential topology definition (<ref>), and we will only apply it in the case where N=Ñ=∂Ω is the boundary of a smooth domain in ^n.








§ THE FREQUENCY FUNCTION FOR ELLIPTIC SYSTEMS





In this appendix, we recall some standard properties of the Almgren frequency for solutions of elliptic systems and some of its consequences. In this paper we are often concerned with weak solutions f∈ W^1,2(Ω;^m) of regular elliptic systems of the type

    Δ f^i = a_ij^α D_α f^j   in Ω,

for each 1≤ i≤ m.
Indeed, according to Theorem <ref>, any solution of (<ref>) is a solution of such a system near regular points.

In Section <ref> we recalled the definition of the Almgren frequency function, but other variants of this function are also possible. Following <cit.>, we define the generalized frequency through

    N(f,x_0,r):=r∫_B_r(x_0)[|Df|^2+(f-f(x_0))·Δ f ]dx/∫_∂ B_r(x_0) |f - (f)_x_0,r|^2 .

The generalized frequency is essentially comparable to the standard frequency (see the notation in Section <ref>), since

    |N(f,x_0,r)-N(f,x_0,r)|/N(f,x_0,r)≤ c r,

where c depends only on n and a_ij^α_L^∞(Ω), cf. <cit.>. However, the generalized frequency has the following advantage:


Let f∈ W^1,2(B_2;^m) be a non-constant weak solution to (<ref>).
   There are constants c > 1 and r_0 ∈ (0,1), depending only on n and a_ij^α_L^∞(B_2), such that r↦ e^c r N(f,x_0,r) is non-decreasing for all r∈ (0,r_0) and for each x_0∈ B_1. 


Proposition <ref> is proved in the scalar case in <cit.>, i.e. when m=1; however, as pointed out in <cit.>, the same calculations go through for general systems of the type (<ref>).

As a consequence of Proposition <ref>, 
the limit N(f,x_0,0^+):=lim_r→ 0 N(f,x_0,r) exists. Moreover, by the existence and uniqueness results for tangent maps proved in <cit.>, this quantity coincides with the vanishing order d of f-f(x_0) at x_0, i.e. d is the integer characterized by
   
    lim sup_x→ x_0|f(x)-f(x_0)|/|x-x_0|^d<∞,     lim sup_x→ x_0|f(x)-f(x_0)|/|x-x_0|^d+1=∞.

   In fact, near x_0, we may write
   
    f(x)-f(x_0)= P_d(x-x_0)+R(x),

   for some non-zero d-homogeneous vector-valued harmonic polynomial P_d and a remainder term R which vanishes to higher order, and satisfies the estimates

    |D^j R(x)|≤ c|x-x_0|^d - j +,   j∈{0,1},

for every ∈ (0,1), where c here may depend on . 

In a standard way, Proposition <ref> and (<ref>) also yield the following result, cf. <cit.>:


    Let f ∈ W^1,2(B_2;^m) be a weak solution of (<ref>). There is a constant c>1, depending only on n and Λ, such that 
    
    N(f,x_0,r)≤ c N(f,0,2)   for all  r∈ (0,1),

    for every x_0∈ B_1. 


The frequency controls not only the vanishing order, but also doubling properties of f. Here we state and prove a variant of the doubling property of f, which is used in Section <ref>. 


Let f ∈ W^1,2(B_2;^m) be a weak solution of (<ref>).  For every λ > 1 there exist δ,σ∈(0,1), depending only on n, m, λ, and a_ij^α_L^∞(B_2), such that if N(f,0,2) ≤λ then 

    |{ |Df|≥δ Df_L^∞(3Q)}∩ Q| ≥ (1-σ) |Q|,

for every cube Q with 3Q⊂ B_1.



By Theorem <ref> and a compactness argument we can find a small constant δ, which may depend further on λ, such that for every cube Q with 3Q⊂  B_1, 
 
    Df _L^∞(3Q)≤1/2δ Df_L^∞(1/2Q).

 The rest of the proof is now classical. By the interior estimates <cit.>, Df^i ∈ C^1,γ(2Q), for each γ∈(0,1), with 
 
    [Df^i]_C^0,γ(2Q)≤ c_γ_3Q f^i ≤ 3c_γ ( Q) D f_L^∞(3Q),

 where c_γ depends only on n, m, a_ij^α_L^∞(B_2), and γ. On the other hand, by (<ref>), there exists a point y_0∈1/2 Q such that 
 
    |Df(y_0)| ≥ 2δDf _L^∞(3Q).

 Hence, by (<ref>) with γ = 1/2, we can find a small scale r > 0, depending only on c_1/2 and δ_λ, such that 
 
    |Df| ≥δDf _L^∞(3Q)  in  Q_s(y_0),

 where s = 1/√(n)r  Q. Since y_0 ∈1/2 Q, we can choose r small so that  Q_s(y_0)⊂ Q. Thus, the assertion of the lemma holds with σ = 1- 2^-nr^n ∈ (0,1). 
 

 As a result of Lemma <ref>, we obtain an L^-γ-estimate for the energy density of solutions to regular elliptic systems. This can be considered as a simpler version of Theorem <ref>. 


    Let f ∈ W^1,2(B_2;^m) be a weak solution of (<ref>).  For every λ > 1 there exist γ > 0 and c > 1, depending only on n, m, λ, and a_ij^α_L^∞(B_2), such that if N(f,0,2) ≤λ, then

    ∫_B_1 |Df|^-γ dx ≤ c.




    The proof is essentially the same as that of Theorem <ref>, but  simpler. In fact, as Σ(f) = ∅, the critical scale r_f as in Lemma <ref> becomes irrelevant. So, we only need to use Lemma <ref> instead of Lemma <ref>, and do not need to invoke Lemma <ref> in the argument. We shall omit the details. 






§ PROOF OF LEMMA <REF>


Here we prove Lemma <ref>, which asserts the strong W^1,2-convergence of the energy minimizers among k-axially symmetric maps. Our proof follows essentially from <cit.> (see also <cit.>), which deals with harmonic maps in the axially symmetric setting. The involvement of the radial component ϱ needs extra attention, and we shall present the full detail below. 


Throughout the proof, c will denote a positive constant independent of i, and it may differ at each occurrence. Moreover, we shall use the spherical coordinate systems in the target space, to simplify the exposition; e.g., in such a system, (<ref>) can be written as

    u(r,θ,z) = (ϱ(r,z), kθ,φ(r,z)).

Furthermore, let us write D' = (∂_r,∂_z).

Upon a 0-homogeneous rescaling, we may assume that {u_i}_i=1^∞⊂ W^1,2(Ω;^3\ B^3)  is a bounded sequence for a concentric ball Ω strictly containing B^3. Then 
there exists a map u∈ W^1,2(Ω;^3) such that 

    u_i⇀ u  weakly in W^1,2(Ω;^3),

along a subsequence. By the Rellich compactness theorem, up to non-relabelled subsequences, u_i→ u strongly in L^2(Ω;^3) and u_i→ u a.e. in Ω. The almost everywhere convergence ensures that u is k-axially symmetric and that |u|≥ 1 a.e. in Ω. 

By (<ref>) and  B^3⋐Ω, we deduce from the Fubini theorem (upon slightly changing the radius of Ω through a further 0-homogeneous rescaling) that

    ∫_ S^2(|D u_i|^2 + |D u|^2 + |u_i - u|^2/λ_i^2)  d^2 ≤ c,

for a sequence λ_i→ 0.  
Let (ϱ_i,_i) be the representation of u_i, and let (ϱ,) be that of u. By (<ref>), there is λ_i'→ 0 satisfying 

    1/2λ_i^3/4≤λ_i' ≤ 2λ_i^3/4,

such that at p_i^± := (λ_i', ±√(1 - (λ_i')^2))∈𝔻, we have 

    |D'(ϱ_i,_i)(p_i^±)|^2 + |D'(ϱ,)(p_i^±)|^2 + |(ϱ_i-ϱ,_i-)(p_i^±)|^2/λ_i^2≤ c.

On the other hand, due to Lemma <ref>, ϱ_i is weakly subharmonic in Ω and, since B^3⋐Ω and {u_i} is bounded in L^2(Ω),

    ϱ_i≤ c  a.e. in  B^3;

now since ϱ_i→ϱ a.e. in Ω, (<ref>) yields

    ϱ≤ c   a.e. in  B^3.


Let v∈ W^1,2( B^3;^3\ B^3) be an arbitrary k-axially symmetric map with v=u on S^2. As in the proof of <cit.>, let us define w_i  B^3→^3 as follows. Consider the annulus _i :=  B^3 ∖ (1-λ_i) B^3, and the polar region _i := {(r,θ,z) ∈_i: r≤λ_i'√(r^2+z^2)}. Set ϱ̃_i _i→ as the linear interpolation between ϱ_i|_ S^2 and ϱ|_ S^2, i.e., 

    ϱ̃_i(r,z) := λ_i^-1(1 - |(r,z)|)ϱ((r,z)/|(r,z)|) + λ_i^-1(|(r,z)| + λ_i - 1) ϱ_i ((r,z)/|(r,z)|),

and define _i _i→ in the same way. Clearly, ρ̃_i≥ 1 and 0≤_i≤π in _i. Next, we set w̃_i _i →^3 as the 0-homogeneous extension of (ρ̃_i,kθ,_i)|_∂_i with respect to the center (0,0,±(1-λ_i/2)). Finally, define w_i B^3→^3 by

    w_i(r,θ,z)  := 
        v((1-λ_i)^-1r,kθ,(1-λ_i)^-1z)    in  B^3∖_i, 
    
        (ϱ̃_i(r,z),kθ,_i(r,z))    in _i∖_i, 
    w̃_i (r,θ,z)    in _i,
    
        u_i(r,θ,z)    in  S^2.

We refer the reader to Figure <ref> for a depiction of this construction.



Note that w_i ∈ W^1,2( B^3;^3\ B^3). We claim that 

    ∫__i |D w_i|^2  dx ≤ c λ_i^1/4.

Once the claim is proved, then we may use the minimality of u_i over the class of k-axially symmetric maps in W^1,2( B^3;^3\ B^3) to deduce that 

    ∫_ B^3 |D u_i|^2 dx ≤∫_ B^3 |D w_i|^2 dx = (1 + o(1))∫_ B^3 |D v|^2 dx + ∫__i |D w_i|^2 dx = ∫_ B^3 |D v|^2 dx + o(1).

However, by (<ref>) and the lower semicontinuity of the energy, we also have

    ∫_ B^3|D u|^2 dx ≤lim inf_i→∞∫_ B^3|D u_i|^2 dx.

The last two inequalities show that u is energy minimizing among k-axially symmetric maps, since v was an arbitrary admissible k-axially symmetric map with v=u on S^2. In fact, taking v=u,  the last two inequalities show that

    ∫_ B^3 |D u|^2 dx=lim_i→∞∫_ B^3 |D u_i|^2 dx,

and so by the weak convergence u_i ⇀ u in W^1,2( B^3;^3), we deduce that u_i → u strongly in W^1,2( B^3;^3), as desired. 

Therefore, we are only left with justifying the claim in (<ref>). Denote by _i the domain in the (r,z)-plane for which _i is obtained by the revolution of _i about the z-axis. Following the first three general (in)equalities at <cit.>, we obtain 

    |D'ϱ̃_i|^2(r,z) ≤ c [|D'ϱ_i|^2 + |D'ϱ|^2 + |ϱ_i - ϱ|^2/λ_i^2]((r,z)/|(r,z)|),

for (r,z)∈_i, whence 

    ∬__i |D'ϱ̃_i|^2 r dr dz    ≤ cλ_i ∫_∂𝔻(|D'ϱ̃_i|^2 + |D'ϱ|^2 + |ϱ_i -ϱ|^2/λ_i^2)r d^1 
       ≤ cλ_i∫_∂ B(|D u_i|^2 + |D u|^2 + |u_i - u|^2/λ_i^2) d^2 ≤ cλ_i.

Analogously, we can also compute that 

    ∬__i |D'φ̃_i|^2 r dr dz ≤ cλ_i.

On the other hand, since (r,z) ∈_i implies r > λ_i'|(r,z)|≥λ_i'(1-λ_i), it follows from (<ref>) that 

    ∬__isin^2φ̃_i/r dr dz ≤ cλ_i^-3/4 |_i| ≤ c λ_i^1/4.

However, since (<ref>) and (<ref>) yield

    __iϱ̃_i ≤ c,

combining (<ref>), (<ref>), (<ref>), and (<ref>) altogether gives   

    ∫__i∖_i |D w_i|^2 dx = ∬__i( |D'ϱ̃_i|^2 + ϱ̃_i^2 ( |D'φ̃_i|^2 + k^2/r^2sin^2φ̃_i ) ) r dr dz ≤ cλ_i^1/4.


The L^2-estimate of |D w_i| over _i is similar to the second paragraph of <cit.>. Note that the lateral part of ∂_i is the revolution of the union of the line segments γ_i^± := {t p_i^±: 1-λ_i ≤ t≤ 1}. Hence, by the definition (<ref>) of ϱ̃_i,  (<ref>) and (<ref>), we get

    |D'ϱ̃_i|^2(r,z) ≤ c[|D'ϱ_i|^2 + |D'ϱ|^2 + 1/λ_i^2|ϱ_i-ϱ|^2](p_i^±) ≤ c,

^1-a.e. for (r,z)∈γ_i^±. A similar argument for _i, using also (<ref>), gives

    |D'φ̃_i|(r,z) + k^2/r^2sin^2φ̃_i (r,z)≤ c + cλ_i^-3/2,

^1-a.e. for (r,z)∈γ_i^±. Recalling from (<ref>) the construction of w_i|_∂_i, it follows from (<ref>), (<ref>), (<ref>), (<ref>), and (<ref>) that 

    ∫__i |D w_i|^2  dx ≤ |_i| ( c + c λ_i^-3/2 ) ≤ cλ_i'λ_i ( 1 +  λ_i^-3/2 ) ≤ cλ_i^1/4.

Consequently, the claim in (<ref>) follows from (<ref>) and (<ref>), which completes the proof. 






§ UNIQUENESS OF CONTINUOUS WEAKLY CONSTRAINT MAPS


The purpose of this appendix is to establish the uniqueness of weakly constraint maps into the exterior of a ball, provided that the images lie in a small neighborhood of a point on the boundary of the ball. Our analysis follows the classical argument of Jäger–Kaul <cit.> for harmonic maps, who consider very general target manifolds without boundary. Here we give a short, self-contained proof for weakly constraint maps, which exploits the symmetry of the ball, although we believe that the same uniqueness result should hold for general target constraints. 

To be precise, let Ω be a bounded domain in ^n, let B^m⊂^m be the open unit open ball centered at the origin, and let p be an arbitrary point on S^m-1. We say that a map u∈ W^1,2(Ω; ^m\ B^m) is a weakly constraint map if it verifies 

    Δ u = - |Du|^2uχ_{|u| = 1}  in Ω

in the weak sense. For sufficiently regular maps, this system is equivalent to saying that u is a solution of the variational inequality for the Dirichlet energy <cit.>.



    Let u_i ∈ W^1,2∩ C(Ω;^m\ B^m) be weakly constraint maps, for each i∈{1,2}, such that u_i(Ω)⊂ B_r(p) for some r∈(0,1), and p ∈ S^m-1. If u_1 - u_2 ∈ W_0^1,2(Ω;^m), then u_1 = u_2 in Ω. 



Our proof essentially follows from the idea of <cit.> on harmonic maps. However, we chose to present the argument since our system (<ref>) involves a characteristic term, which may behave badly when taking the difference between two solutions. Here, the exact structure of the obstacle being a ball turns out to resolve this issue quite smoothly.

    Let us introduce 
    
    ψ:= 1/2 |u_1 - u_2|^2   and  ψ_i := 1/2 |u_i - p|^2,

    and define 
    
    w := ψ/(1-ψ_1)(1-ψ_2).

    The basic idea of the proof is that, up to a conformal change of the  metric on the domain (which depends on ψ_1,ψ_2), the function ψ is subharmonic, and so it satisfies a maximum principle.
    
    As u_i∈ C(Ω;^m\ B^m), it follows from  Theorem <ref> that u_i∈ C_^1,1(Ω;^m\ B^m). Hence, we have ψ,ψ_1,ψ_2, w∈ W^1,2∩ C_^1,1(Ω). By (<ref>), a direct computation yields 
    
    Δψ = |D(u_1-u_2)|^2 - (1 - u_1· u_2)∑_i=1^2 |Du_i|^2χ_{|u_i|=1}

    a.e. in Ω. Recall that, if f is a Sobolev function, then for every k∈ we have D^kf=0 a.e. in f^-1(0), thus
    
    Δψ = 0   a.e. on ψ^-1(0).

    On the other hand, note that |Dψ|^2 ≤ 2ψ |D(u_1-u_2)|^2 in Ω. Also, by the constraint, we have ψ≥ 1 - u_1· u_2 in Ω. Using these two inequalities in the above equation, we derive that

    Δψ≥|Dψ|^2/2ψ - ψ∑_i=1^2 |Du_i|^2χ_{|u_i|=1}  a.e. in Ω∖ψ^-1(0).

Again by the constraint we have ψ_i ≥ 1 - p· u_i in Ω, and so by (<ref>) we similarly derive that  

    Δψ_i    = |Du_i|^2 - (1- p· u_i)|Du_i|^2χ_{|u_i|=1}≥ (1 - ψ_i) |Du_i|^2 χ_{|u_i|=1}  a.e. in Ω.

One may compare (<ref>), (<ref>), and (<ref>) with <cit.>. 

As in the proof of <cit.>, let us take 

    ϕ := -∑_i=1^2log(1-ψ_i),

so that w = e^ϕψ in Ω; note from the assumption u_i (Ω)⊂ B_r(p) for some r∈(0,1) that ϕ∈ C(Ω)∩ C_^1,1(Ω) and 0≤ϕ≤ c in Ω. Define the self-adjoint elliptic operator ≡ (e^-ϕ∇· ); the ellipticity is uniform over Ω as 1≥ e^-ϕ≥ e^-c there. 

As in the proof of <cit.>, we can compute that 

    w =  (e^ϕψ) = Δψ + ψΔϕ + Dψ· Dϕ  a.e. in Ω.

As above, we have

    w = 0  a.e. in ψ^-1(0).

On the other hand, away from ψ^-1(0), we have 

    ψΔϕ + Dψ· Dϕ   =∑_i=1^2 (ψ|Dψ_i|^2/(1-ψ_i)^2 + ψΔψ_i/1-ψ_i +  Dψ· Dψ_i/1-ψ_i)
    ≥  - |Dψ|^2/2ψ + ψ∑_i=1^2 Δψ_i/1-ψ_i,

where in the last inequality we used Young's inequality

    Dψ· Dψ_i/1-ψ_i≤ψ|Dψ_i|^2/(1-ψ_i)^2 + |Dψ|^2/4ψ.

Thus, we deduce from (<ref>), (<ref>), and (<ref>) that 

    w≥Δψ - |Dψ|^2/2ψ + ψ∑_i=1^2 Δψ_i/1-ψ_i≥ 0   a.e. in Ω∖ψ^-1(0).

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

    w ≥ 0  a.e. in Ω.

By the (uniform) ellipticity of , the weak maximum principle applies to w ∈ W^1,2∩ C_^1,1(Ω). Finally, if u_1 - u_2 ∈ W_0^1,2(Ω;^m), then w ∈ W_0^1,2(Ω), so the weak maximum principle implies w = 0 in Ω, i.e., u_1 = u_2 in Ω as desired. 


We also note here the following result, which is the analogue for constraint maps of the results of Hildebrant–Kaul–Widman for harmonic maps <cit.>:


    Let u∈ W^1,2(Ω;^m\ B^m) be a weakly constraint map.
    If u(Ω)⊂ B_r(p) for some r∈(0,1) and some p∈ S^m-1, then u is continuous.



Since 0<r<1, B_r(p)∩ S^m-1 is a graph in the direction of p. Thus we can employ <cit.> to deduce that u is continuous. Note that, although <cit.> is stated for minimizers, the short proof only relies on the Euler–Lagrange system (<ref>) and the results of <cit.>, which hold for solutions of elliptic systems.


As an immediate corollary of the above two results we obtain that a weakly constraint map which  takes values in a small neighborhood of p is the unique weakly constraint map with those boundary values: 


    Let u ∈ W^1,2(Ω;^m\ B^m) be a weakly constraint map such that u(Ω)⊂ B_r(p) for some r∈(0,1) and some p∈ S^m-1.  Then u is the unique weakly constraint map with its own boundary data, and so in particular it is minimizing.


By Theorem <ref>, u is continuous. 
    Let v∈ W^1,2_u(Ω; ^m\ B^m) be an arbitrary weakly constraint map. We claim that v∈ B_r(p) a.e. in Ω.  Indeed,  (<ref>) shows that f:= |v - p|^2 is a weakly subharmonic function in Ω. However, as v ∈ u + W_0^1,2(Ω;^m) and u(Ω)⊂ B_r(p), we have f≤ r^2 on ∂Ω, so the weak maximum principle implies that f ≤  r^2 a.e. in Ω, i.e. v∈ B_r(p) a.e. in Ω. The claim now follows from Proposition <ref>.






 

§ ACKNOWLEDGEMENTS

A.G. was supported by Dr. Max Rössler, the Walter Haefner Foundation, and the ETH Zürich Foundation. Additionally, he is thankful to Uppsala University and KTH for their hospitality during a visit where part of this research was conducted.
S.K. was supported by a grant from the Verg Foundation.
H.S. was supported by the Swedish Research Council (grant no. 2021-03700).

 



§ DECLARATIONS


Data availability statement: All data needed are contained in the manuscript.

Funding and/or Conflicts of interests/Competing interests: The authors declare that there are no financial, competing or conflict of interests.

 

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