Minimizers for the thin one-phase free boundary problem

We consider the"thin one-phase"free boundary problem, associated to minimizing a weighted Dirichlet energy of the function in $\mathbb R^{n+1}_+$ plus the area of the positivity set of that function in $\mathbb R^n$. We establish full regularity of the free boundary for dimensions $n \leq 2$, prove almost everywhere regularity of the free boundary in arbitrary dimension and provide content and structure estimates on the singular set of the free boundary when it exists. All of these results hold for the full range of the relevant weight. While our results are typical for the calculus of variations, our approach does not follow the standard one first introduced in \cite{AltCaffarelli}. Instead, the nonlocal nature of the distributional measure associated to a minimizer necessitates arguments which are less reliant on the underlying PDE.


Introduction
This article is devoted to the study of the regularity properties of a weighted version of the thin one-phase problem. More precisely, we investigate even, nonnegative minimizers of the following functionals: denote x P R ng1 by x h .x H ; y/ P R n ¢ R, and for P . 1; 1/ we define where m stands for the n-dimensional Lebesgue measure. Here and throughout the paper the integration is done with respect to the .n g 1/-dimensional Lebesgue measure unless stated otherwise. This functional is finite for open sets, , and functions in the weighted Hilbert space, H 1 .; / h fv P L 2 .s jyj / rv P L 2 .s jyj /g; equipped with the usual weighted norm.
Our main concern is to investigate fine regularity properties of the free boundary of minimizers v of (1.1), that is, the set F .v/ h @ R n fv.x; 0/ > 0g : Since the free boundary lies on a codimension 1 subspace of the ambient space R ng1 , such a problem is called a thin one-phase free boundary problem. This type of free boundary problem has been investigated for the first time by Caffarelli, Roquejoffre, and the last author in [7] in relation to the theory of semipermeable membranes (see, e.g., [21]). As we will describe later,. this is an analogue of the classical one-phase problem (also called the Bernoulli problem) but for the fractional Laplacian. The Bernoulli problem was first treated in a rigorous mathematical way by Alt and Caffarelli in the seminal paper [2]: in the Bernoulli problem we consider minimizers of (1.1) where h 0, and the second term is replaced by L ng1 .fv > 0g / (where L ng1 stands for the Lebesgue measure in R ng1 ). In particular, for the Bernoulli problem, the free boundary fully sits in the ambient space, R ng1 .
In [2], the authors provided a general strategy to attack this type of problem. Out of necessity we needed to modify this blueprint in several substantial ways (see below for a more detailed comparison). For more information on the one-phase problem (and some of its variants) we refer to the book of Caffarelli and Salsa (and references therein) [8] and to the more recent survey of De Silva, Ferrari, and Salsa [14].
As noticed in [7], problem (1.1) is related in a tight way to the standard onephase free boundary problem but with the Dirichlet energy replaced by the Gagliardo seminorm u H for h 1 2 P .0; 1/. This connection suggests that the thin one-phase problem is actually intrinsically a nonlocal problem, though the energy in (1.1) is clearly local.

Connection with the Fractional One-Phase Problem
As previously mentioned, the functional J introduced by Caffarelli, Roquejoffre, and the last author in [7] is a local version of the following nonlocal free boundary problem: given a function f P L 1 loc .R n / with suitable decay at infinity, we can define its fractional Laplacian at x P R n by . / f .x/ h c n; p: v: .. x/¡.x// and where f satisfies a given "Dirichlet boundary condition" on the complement of .
As in the case of the classical Laplacian (see [2]), we are interested in obtaining equation (1.2) as the Euler-Lagrange equation of a certain functional. Given a locally integrable function f , consider its fractional Sobolev energy f H .R n / h R 2n jf .x/ f ./j 2 jx j ng2 d dx: Since we want to study competitors that vary only in a certain domain , it is natural to consider only the integration region that may suffer variations when changing candidates. Thus, we define the energy (1. 3) J.f; / h c n; R 2n n. c / 2 jf .x/ f ./j 2 jx j ng2 d dx g m.ff > 0g /: We say that f P L 1 loc is a minimizer of J in if J.f; / is finite and J.f; / J.g; / for every g satisfying that f g P H .R n / and such that f .x/ h g.x/ for almost every x P c . We say that f is a global minimizer if it is a minimizer for every open set & R n . Note that both terms in (1.3) are in competition, since a minimizer of the fractional Sobolev energy in is -harmonic and, thus, if it is nonnegative outside of , it is strictly positive inside of , maximizing the second term.
Consider now the Poisson kernel for fixed n P N and 0 < < 1, (1.4) P y ./ h P n; .; y/ h c n; jyj 2 j.; y/j ng2 for every .; y/ P R n ¢ R: The Poisson extension of f P L 1 loc .R n / is given by (1.5) u.x H ; y/ h f £ P y .x H / h R n P n; .; y/f .x H /d for every .x H ; y/ P R n ¢ R: By [9], with a convenient choice of the constant, one gets lim y80 y 1 2 u y .x H ; y/ h . / f .x H / in every point where f is regular enough. Moreover, the extension satisfies the localized equation r ¡.jyj ru/ h 0 weakly, away from R n ¢f0g. The whole point is that local minimizers of (1.3) can be extended via the previous Poisson kernel P y to (even) minimizers of (1.1) (see the Appendix for a precise statement). Therefore, the thin one-phase problem appears as a "localization" of the one-phase problem for the fractional Laplacian. Notice that-and this is of major importance for usthis localization technique does not carry over to other types of nonlocal operators besides pure powers of second-order elliptic operators. This is a major drawback of the theory, in the sense that, at the moment, it seems to be impossible to tackle onephase problems involving more general operators than the fractional Laplacian.
The main point is we do not know how to prove any kind of monotonicity for general integral operators. This connection between the nonlocal analogue of the Bernoulli problem and our thin one-phase problem allows us to simplify several arguments by working in the purely nonlocal setting. However, this underlying nonlocality is also the reason why several results, which came more easily in the setting of [2], are nontrivial or substantially harder for us. For example, perturbations of solutions need to take into account long-range effects that make classical, local perturbation arguments much more difficult.
In the paper [7], the authors proved basic properties of the minimizers for the functional J such as optimal regularity, nondegeneracy near the free boundary, and positive densities of phases. Also they provided an argument for n h 2 showing that Lipschitz free boundaries are C 1 . A feature of the functional J is that the weight jyj is either degenerate or singular at fy h 0g (except in the case h 0).
Such weights belong to the Muckenhoupt class A 2 and the seminal paper of Fabes, Kenig, and Serapioni [26] investigated regularity issues for elliptic PDEs involving such weights (among other things). After that, [19] proved an "-regularity result and [1] showed the existence of a monotonicity formula for this setting.
In the case h 0, the problem is still degenerate in the sense that derivatives near the free boundary blow up. The case h 0 has been thoroughly investigated in the series of papers by De Silva, Savin, and Roquejoffre [16][17][18].
The main goal of our paper is to provide a full picture of the regularity of the free boundary for any power P . 1; 1/, both in terms of measure-theoretic statements and partial (or full) regularity results. From this point of view our contribution is a complement to the paper by De Silva and Savin [18] for h 0. It has to be noticed that the standard approach to regularity of Lipschitz free boundaries as developed by Caffarelli (see the monograph [8]) does not seem to work in our setting.

Our Approach to Regularity
In [2] (and many subsequent works), the minimizing property of the solution is used to prove that the distributional Laplacian of that solution is an Ahlfors-regular measure supported on the free boundary. This implies (among other things) that the free boundary is a set of (locally) finite perimeter, and thus almost every point on the free boundary has a measure-theoretic tangent. One can then work purely with the weak formula (i.e., the analogue of (1.2)) to prove a "flat implies smooth" result which, together with the existence almost everywhere of a measure theoretic tangent, has as a consequence that the free boundary is almost everywhere a smooth graph and the free boundary condition in (1.2) holds in a classical sense at the smooth points.
A similar "flat implies smooth" result exists in our context (this is essentially due to De Silva, Savin, and the last author [19]; see Theorem 2.4 below). However, showing that the free boundary is the boundary of a set of finite perimeter proves to be much more difficult. Due to the nonlocal nature of the problem, div.jyj ru/ (considered as a distribution) is not supported on the free boundary. Furthermore, the scaling of this measure does not allow us to conclude that the free boundary has the correct dimension (much less that it is Ahlfors regular).
To prove finite perimeter, we take the following approach inspired by the work of de Silva and Savin: after establishing some preliminaries we prove crucial compactness results. This, along with a monotonicity formula originally due to Allen [1] allows us to run a dimension reduction argument in the vein of Federer or (in the context of free boundary problems) Weiss [38]. With this tool in hand, we show that the set of points at which no blowup is flat is a set of lower dimension. Locally finite perimeter and regularity for the reduced boundary then follow from a covering argument and some standard techniques.
Here and throughout the paper, we will denote the ball of radius r in R ng1 centered at the origin by B r , and B H r h B r R n ¢f0g. Moreover, for the definition of H , see Section 2. We may then summarize our regularity results in the following theorem. (2) We can write the free boundary as a disjoint union F .u/ h R.u/ .u/, where R.u/ is open inside F .u/, and for x P R.u/ there exists an r x > 0 such that B.x; r x /F .u/ can be written as the graph of a C 1;s -continuous function.
(3) Furthermore, the set .u/ is of Hausdorff dimension n 3 (and, therefore, of H n 1 -measure zero). In particular, for n 2, .u/ is empty, and moreover, if n h 3 then .u/ is discrete.
The constants (implicit in the set of finite perimeter, and the Hölder continuity of the functions whose graph gives the free boundary) depend on n and but not on As usual .u/ & F .u/ is called the singular set of the free boundary: the set of points around which F .u/ cannot be parametrized as a smooth graph and all the blowups will be nontrivial minimal cones; see Theorem 2.4. Our second contribution concerns the structure and size of the singular set. It builds on recent major works on quantitative stratification [32] extended to free boundary problems (in particular, the one-phase problem) by Edelen and the first author [22]. ..u/ D/ C n;;dist.D;@B 1 / for every D b B 1 : In [15], De Silva and Jerison constructed a singular minimizer for the Alt-Caffarelli one-phase problem in dimension 7, giving the dimension bound k £ 8 in the previous theorem in this case (see [22]). This result is not known for the thin one-phase problem. The reason is that the one-phase problem, seen from the nonlocal point of view involving the fractional Laplacian, is related to the so-called nonlocal minimal surfaces introduced by Caffarelli, Roquejoffre, and Savin [6]. Indeed, in [33], the authors proved that a fractional version of Allen-Cahn equation converges variationally to the standard perimeter functional for ! 1=2 and to the so-called nonlocal minimal surfaces for < 1=2. We can then conjecture the bound k £ 8 for ! 1=2 by analogy with the result for the standard one-phase problem, but the bound for < 1=2 is not clear at all. However, one knows that there is no singular cone in dimension 2 for nonlocal minimal surfaces [34] and that the Bernstein problem is known for those in dimensions 2 and 3 [28]. We would like also to make a last remark about a result that is of purely nonlocal nature. In the case of the one-phase problem, one can show that the distributional Laplacian is a Radon measure along the free boundary. In the case of the thin one-phase free boundary problem, due to the nonlocality of the problem, such a behavior does not happen in the sense that we will show that the fractional Laplacian is an absolutely continuous measure with respect to n-dimensional Lebesgue measure with a precise behavior. This phenomenon is of purely nonlocal nature and similar to the fact that the fractional harmonic measure is of trivial nature.
More precisely, every minimizer u satisfies r ¡ .jyj ru/ h 0 weakly, away from R n fu 0g. Thus, equation (1.2) above can be understood as an Euler-Lagrange equation for the functional J in the sense that the restriction to R n of a given minimizer u in & R ng1 , harmonic away from R n ¢ f0g and with asymptotic behavior u.x; y/ h O.j.x; y/j /, is always a solution to (1.2) for A h A./ at "nice" points of the free boundary.
A brief summary of this paper follows. In Sections 3 and 4 we discuss compactness of minimizers and we recall Allen's monotonicity formula to derive some immediate consequences. In Section 5 we show that the positive phase is a set of locally finite perimeter, establishing the first part of Theorem 1.1 (modulo energy bounds), and we show that the singular set can be identified using the Allen-Weiss density. Section 6 is devoted to deducing full regularity of minimizers in R 2g1 concluding the proof of Theorem 1.1.
Once we have established the finite perimeter, in Section 7 we remove the dependence of the estimates on the energy of the minimizer in the previous theorems, using a rather subtle argument that combines results from all the previous sections. A crucial step is to analyze some basic properties of the distributional fractional Laplacian of our minimizer. As stated above this analysis will not be enough to establish that the positivity set of the minimizer is a set of locally finite perimeter. We believe that many of these results may be of independent interest. For example, corresponding results for the classical Bernoulli problem have been used to understand the free boundary problems for harmonic measure (see [31]).
Finally, Section 8 is devoted to the proof of Theorem 1.2.

Notation
We denote the constants that depend on the dimension n, , and perhaps some other fixed parameters that are clear from the context by C . Their value may change from one occurrence to another. On the other hand, constants with subscripts such as C 0 retain their values along the text. For a; b ! 0, we write a . b if there is C > 0 such that a Cb. We write a % b to mean a . b . a.
Let u be a continuous function in R ng1 . Then we write g .u/ h fu > 0g, and we denote the zero phase, the positive phase, and the free boundary by 0 .u/ h fx P R n ¢ f0g u.x/ h 0g ; H g .u/ h g .R n ¢ f0g/ h fx P R n ¢ f0g u.x/ > 0g; F .u/ h F .u/ h @. g .u/ R n ¢ f0g/ ; respectively. Here both the boundary and the interior are taken with respect to the standard topology in R n . Note that R n ¢ f0g is the disjoint union of 0 .u/, H g .u/, and F .u/ whenever u is nonnegative. We also call F red .u/ h F red; .u/ the points of F .u/ where the free boundary is expressed locally as a C 1 surface.
Finally, let .u/ h .u/ h F .u/ n F red; .u/. In general, we will write H h .R n ¢ f0g/. Throughout the paper we will often fix P . 1; 1/ but then refer to P .0; 1/ or vice versa. These two numbers are always connected by the relationship h 1 2 .

Preliminaries
In this section, we provide the known results concerning the problem under consideration. We say that a function u is even if it is symmetric with respect to the hyperplane R n ¢ f0g, that is, u.x H ; y/ h u.x H ; y/. The function spaces that we will consider are the following H ./ h fu P H 1 .; / u is even and nonnegativeg and H loc ./ h¨u P L 2 loc ./ u P H .B/ for every ball B b © : We will omit in the notation when it is clear from the context. DEFINITION 2.1. We say that a function u P H loc ./ is a (local) minimizer of J in a domain if for every ball B b and for every function v P H .B/ such that the traces vj @B uj @B , the inequality As usual for several free boundary problems, it is a natural question to exhibit a particular (global) solution so that one gets an idea of the qualitative properties of general solutions. Let us consider the following function: for every x P R n let f n; .x/ h c n; .x n / g ; where a g h maxf0; ag. If n h 1, f 1; is a solution to (1.2) for a convenient choice of c 1; (see [4, theorem 3.1.4]). In fact, one can see that the same is true for n ! 1 using Fubini's theorem conveniently, with (2.1) . / f n; .x/ h c n; .x n / ; where a h maxf0; ag.
As a toy question we wonder whether the trivial solutions are minimizers. Indeed, this is the case, as we will see later in Section 4.1.
PROPOSITION 2.2. Let n P N and 0 < < 1. Then the trivial solution u n; h f n; £ P y is a minimizer of J in every ball B & R ng1 . measure-theoretic normal (see [24]) for a prescribed value of A. All the constants depend on n and , and also on E 0 except for the ones in P1 and

P2.
A major tool in the present paper is an -regularity result, i.e., in the language of free boundaries a statement of the type "flatness implies smoothness." In [19], the authors proved such an -regularity result for viscosity solutions to the overdetermined system associated to minimizers of J . Here we establish that all local minimizers are in fact viscosity solutions. While this verification may be standard for experts in the field, we include it here for the sake of completeness. f.x; 0/ P B x n g & B 0 .u/ & f.x; 0/ P B x n g; we have that F .u/ P C 1; loc . 1 2 B/, with 0 < < 1. Note that the dependence on E 0 will be removed in Section 7.
We claim that (2.4) every nonnegative even minimizer is a viscosity solution.
Conditions (i) and (ii) have been verified in [19,36]. To verify our claim it suffices to prove condition (iii) above: that any strict comparison subsolution cannot touch u from below at a point .x 0 ; 0/ P F .u/. The analogous claim for strict comparison supersolutions will follow in the same way.
(f) Furthermore, either the inequality is strict in (d) or a > 1 in (e).
So assume that w ! u where w is a strict comparison subsolution and u is some minimizer and that w h u at .x 0 ; 0/ P F .u/. Since u.x 0 ; 0/ h 0 it follows that .x 0 ; 0/ P F .w/ and with a harmless rotation we can guarantee that ..x 0 ; 0// h e n . We want to show that e n is also the measure-theoretic unit normal to F .u/.
Indeed, since F .w/ is C 2 there must exist a ball B & fw > 0g that is tangent to F .w/ at .x 0 ; 0/. It must then be the case that B & fu > 0g as well. Thus .x 0 ; 0/ P F .u/ has a tangent ball from the inside, which, by proposition 4.5 in [7] implies that u has the asymptotic expansion u.x; y/ h U..x x 0 / ¡ .x 0 /; y/ g o.k.x x 0 ; y/k /; .x; y/ 3 .x 0 ; 0/: If u ! w, this implies that w must satisfy the expansion in (2.5) with a h 1 at the point x 0 . This, in turn, implies that div.j´j rw/ > 0 in B 1 nfy h 0g (by the definition of a strict subsolution). Furthermore, since w P C 2 where fw > 0g, we can guarantee that div.j´j rw/ ! 0 in all of B 1 fw > 0g.
Let us return to the ball B that is a subset of fu > 0g and fw > 0g and for which .x 0 ; 0/ P x B. We know that w u ¤ 0 in B n fy h 0g (this is because w strictly satisfies the differential inequality in B away from fy h 0g), and we know that w u is a subsolution in B. Furthermore, .x 0 ; 0/ P B is a strict maximum, so by the Hopf lemma in [5, prop. 4.11] it must be that lim t50 g .w u/.x 0 g t.x 0 /; 0/ t > 0: This contradicts the fact that u and w both satisfy (2.5) at .x 0 ; 0/ with a h 1.
Therefore, .x 0 ; 0/ must not have been a touching point and u is indeed a viscosity solution.
Since, u is a viscosity solution, [19, theorem 1.1] applies and we get the desired "-regularity.

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

Caccioppoli Inequality
First we want to show that the distribution h r ¡ .jyj ru/ is in fact a Radon measure with support in the complement of the positive phase as long as u is a minimizer. In Section 7 we will come back to this measure to understand its behavior around the free boundary.
be an open set, and let u P W 1;2 loc .; jyj / be such that r ¡ .jyj ru/ h 0 weakly in g .u/, i.e., for every P C I c . g .u//, (3.1) hr ¡ .jyj ru/; i h .jyj ru/r h 0: Then h r ¡ .jyj ru/ is a positive Radon measure supported on fu h 0g and for every v P W 1;2 .; jyj / C c ./ The Caccioppoli inequality is the first step to proving convergence in a Sobolev sense. It will also be useful when we remove the a priori dependence of our results on the Sobolev norm of the minimizer. LEMMA 3.2 (Caccioppoli inequality). Let B & R ng1 be a ball of radius r centered on R n ¢ f0g, and let u P W 1;2 .B; jyj / be such that r ¡ .jyj ru/ h 0 weakly in B fu > 0g. Then  PROOF. The first inequality is an immediate consequence of Caccioppoli, the middle estimate is trivial, and the last follows from P1 in Theorem 2.3.

Compactness
In the following lemma we prove the compactness of minimizers in the relevant Sobolev spaces. For convenience, we also detail several compactness results that were either already proven in [7] or are standard consequences of the nondegeneracy estimates in Theorem 2.3. Nevertheless, we include full proofs here for the sake of completeness. We note here (as we did above and will do again below) that while we currently need to assume the uniform bound on the Hölder norm of the functions u k , we can get rid of this assumption in the light of the results of Section 7.
PROOF. The first claim follows from uniform Hölder continuity and compact embeddings of Hölder spaces. The claim (2) follows from (1) easily.
We now prove the third claim. Let > 0: We will first show that for x P R n ¢f0g we have Next we will show that for all large k (3.4) F .u 0 / & fx d.F .u k /; x/ < g: If this was not true we could find a point x P F .u 0 / and a subsequence of u k such that B H .x; / & F .u k / c for every k in the subsequence. If the subsequence contains infinitely many u k such that u k 0 in B.x; /, then also u 0 0 due to uniform convergence. Otherwise the sequence contains infinitely many u k for which B.x; / is contained in the positive phase. In this case the nondegeneracy implies that in B.x; =2/ we have u k > C ; with C independent of k: Again uniform convergence implies the same lower bound for u 0 ; which contradicts our choice x P F .u 0 /: To show the fourth claim we notice that F .u 0 / has zero n-dimensional Lebesgue It remains to show that weak convergence implies strong convergence.
LEMMA 3.5. Any sequence of minimizers fu k g I kh0 in & R ng1 with u k 3 u 0 uniformly and ru k * ru 0 weakly in L 2 loc .; jyj / satisfies that ru k 3 ru 0 strongly in L 2 loc .; jyj /. PROOF. Let P C 0;1 c ./ be a nonnegative function. We claim that for every " > 0 there exists j 0 so that jyj jru ru j j 2 " for j ! j 0 : First we isolate the main difficulty jyj jru 0 ru j j 2 h jyj .ru 0 ru j /¡ru 0 jyj .ru 0 ru j /¡ru j : By weak convergence, jyj .ru 0 ru j / ¡ ru 0 "=4 for j big enough. Note that this is true even if the u j are not minimizers. The bound on the second term, however, needs the minimization property.
We observe that To estimate I in (3.5), let j be the measures corresponding to u j from Lemma Since j is supported on fu j h 0g, we have that u j d j h 0 for every j (including j h 0 as u 0 is also a minimizer to J , see corollary 3.4 in [7]).
To finish the estimate on I in (3.5), we observe that By uniform convergence on compact subsets, sup supp ju j u 0 j 4kk L 1 . 0 / for j big enough.
We turn towards estimating II in (3.5): jIIj h jyj u j .ru 0 ru j / ¡ r jyj .ru 0 ru j / ¡ .u 0 r/ g sup supp ju j u 0 jkru 0 ru j k L 2 .;jyj / krk L 2 .;jyj / : The first term goes to zero by weak convergence of ru j to ru 0 . The second term satisfies sup supp ju j u 0 jkru 0 ru j k L 2 .supp ;jyj / krk L 2 .;jyj / "=4 for j big enough, by uniform convergence and the uniform bound of the norm kru j k L 2 .supp ;jyj / derived from the Caccioppoli inequality in Lemma 3.2 together with uniform convergence.
Lemma 3.4 implies that minimizers converge to minimizers (which was observed in Corollary 3.4 in [7]), but also implies the stronger fact that the energy is continuous under this convergence: COROLLARY 3.6. Let u k be a sequence of minimizers in & R ng1 with u k 3 u 0 locally uniformly and sup k ku k k H < I. Then u 0 is also a minimizer to J in and for any B b we have J .u k ; B/ 3 J .u 0 ; B/ after passing to a subsequence.

Monotonicity Formula and Some Immediate Consequences
From [1] we have the following monotonicity formula:  k.D/ this subsequence u k is bounded in H 1;2 .Ds jyj / and, passing to a subsequence u k j , converges (in the sense of Lemma 3.4) to u 0 , which is a globally defined minimizer of J that is homogeneous of degree .
The proof is the same as in [38, theorem 2.8].
Remark 4.3 (Nonuniqueness of blowups). We call the function u 0 appearing in Corollary 4.2 a blowup of u at x 0 . A priori, the function u 0 may depend on the subsequence u k j . However, a simple scaling argument shows that for all radii r ! 0 and all blowups u 0 to u at x 0 we have u 0 r .0/ u 0 .x 0 /:

Dimension Reduction
We use the homogeneity of the blowups to obtain dimension estimates on the points in the free boundary for which there exists a nonflat blowup. This process is known as "dimension reduction" and has been applied to a variety of situations (see [38] for its application to the Bernoulli problem).
The first lemma shows that blowup limits have additional symmetry: LEMMA 4.4. Let u P H loc .R ng1 / be an -homogeneous minimizer of J and let x 0 P F .u/ n f0g. Then any blowup limit u 0 at x 0 is invariant in the direction of x 0 ; i.e., for every x P R ng1 and every P R, PROOF. Let x P R ng1 , and consider its decomposition x h z x g x 0 with z x P hx 0 i c . We only need to check that Consider a ball B h B.0; r/ & R ng1 so that z x; x P B. Let f k g be a sequence of radii converging to 0 and such that u k .x/ h u.x 0 g k x/ k converges to u 0 uniformly on B r . For k big enough, ku k u 0 k L I .B r / < ". Then, (4.2) ju 0 .x/ u 0 .z x/j 2" g ju k .x/ u k .z x/j: To control the last term above, we use the homogeneity of u. Writing P 1 h x 0 g k z x and P 2 h x 0 g k x, we have k u k .z x/ h u.P 1 / and k u k .x/ h u.P 2 /.
Let P 3 be the intersection between the line through P 1 and x 0 and the line through the origin and P 2 (see Figure 4.1). By homogeneity of u u.P 2 / h u.P 3 / jP 2 j jP 3 j h u.P 3 / 1 ¦ jP 2 P 3 j jP 3 j : Thus, k ju k .x/ u k .z x/j u.P 1 / u.P 3 /.1 g O. k // ju.P 1 / u.P 3 /j g ju.P 3 /jO. k /: By Thales' theorem, jP 1 P 3 j h jP 1 P 2 jjP 3 x 0 j jx 0 j h O. 2 k /, and using the C character of u and the fact that u.x 0 / h 0, we get We then recall that a minimizer with a translational symmetry is actually a minimizer without that symmetry in one dimension less. This is known as "cone splitting": PROOF. The proof is a slight variation of [38, proof of lemma 3.2].
Next we provide a nonstandard proof of Proposition 2.2, to show that the trivial solution is a minimizer. We use (P5) in a sequence of conveniently chosen blowups and a dimension reduction argument. Note that the proposition could also be proven via a classical dimension reduction argument. PROOF OF PROPOSITION 2.2. Consider a nonzero minimizer u with nonempty free boundary (see [7, prop. 3.2] for its existence), choose a free boundary point x 0 P F .u/ and consider u 0 to be a blowup weak limit at this point, which exists and is -homogeneous by Lemma 4.2. Then u 0 is also a global minimizer by (P5) and not null by the nondegeneracy condition.
Next we argue by induction: given 0 j n 2 let u j be an -homogeneous global minimizer different from 0 such that it is invariant in a j -dimensional linear subspace H j & R n , i.e., for every v P H j and every x H P R n , u j .x H ; y/ h u.x H g v; y/: Consider a point x j P F .u j / n.H j ¢f0g/ that exists as long as j < n 1 by the interior corkscrew condition and positive density, and let u j g1 be a blowup limit at this point, which is again an -homogeneous global minimizer. We claim that u j g1 is invariant in fact in the .j g 1/-dimensional subspace H j g hx H j i.
and the claim follows.
Thus, after n 1 steps, we obtain u n 1 , which is an -homogeneous global minimizer invariant in an .n 1/-dimensional space H n 1 , with nonempty free boundary. Thus, u n 1 .x H ; 0/ h C n; .x H n / g ; where the constant is given by (P6). The proposition follows by Proposition A.1.

Upper Semicontinuity
Next we show that Allen-Weiss energy at a fixed radius is continuous both with respect to the minimizer and with respect to the point: For the measure, we estimate for j big enough as a consequence of g .u j / 3 g .u 0 / in L 1 loc as before. The fact that for j big enough is a straight consequence of the uniform convergence and the continuity of u 0 .
It is well-known that the limit of a decreasing sequence of continuous functions is upper semicontinuous (see [11, theorem 1.8]). The monotonicity formula also implies the following result. or by using monotonicity, it suffices to show that for every " > 0 and j big enough, u j r .x j / u 0 r .x 0 / ": But this is true for j big enough because the left-hand side converges to 0 by the continuity of the energy from Lemma 4.6.

Finite Perimeter
We will show that H g .u/ is a set of locally finite perimeter. Then F red .u/ will coincide with the measure-theoretic reduced boundary by the -regularity theorem, From the positive density properties, we know that k £ ! 2. From the homogeneity of the blowups we find out that the free boundary in R 1g1 is in fact a collection of isolated points. Later in Theorem 6.1 we will show that in fact k £ ! 3: PROOF. Arguing by contradiction, we assume that F .u/ has an interior accumulation point which, without loss of generality, we assume to be the origin.
Let .x k ; 0/ be a sequence of singular points converging to 0 with x k > 0. Consider the blowup rescaling u k .x/ h u.x k x/ x k . Note that u k .0; 0/ h u k .1; 0/ h 0. Moreover, by the interior corkscrew condition, there exist´k P .1=2; 3=2/ such that u k j B H c .´k;0/ > 0, so u.´k; 0/ & C by the nondegeneracy condition. Choosing a subsequence, we may assume that´k 3´0 ! 1=2, and u k 3 u 0 in the sense of Lemma 3.4. In particular, u 0 is homogeneous by Corollary 4.2, reaching a contradiction with the fact that u 0 .1; 0/ h 0 and u 0 .´0; 0/ & C .
We will prove the local finiteness of the perimeter of the free boundary adapting a proof of De Silva and Savin in [18]. Our proof is essentially the same, but we repeat it for the sake of completeness.
As in [18] we say that a set A & R n satisfies the property (P t ) if the following holds: for every x P A there exists an r x > 0 such that for every 0 < r < r x , every subset S of B.x; r/ A can be covered with a finite number of balls B.x i ; r i / with x i P S such that PROOF. We first show that .u/ satisfies the property (P t ). If (P t ) does not hold, we find a point y P .u/ for (P t ) is violated for a sequence r k 3 0: We consider the blowup sequence (5.2) u r k .x/ h r k u.y g r k x/: By Corollary 3.6 we may assume, by taking a subsequence, that u r k converges to a minimal cone U . By our assumptions we may cover .U / B.0; 1/ with a finite collection of balls fB. After rescaling we see that u satisfies the condition for property (P t ) in the ball B.y; r k /; which is a contradiction. Therefore the property (P t ) holds as claimed.
Consider the set D k h fy P .u/ r y ! 1=kg: Fix a point y 0 P D k : By property (P t ) applied to r 0 h 1=k, we find a finite cover of D k B.y 0 ; r 0 / with balls B.y i ; r i /; y i P D k ; satisfying PROOF. Without loss of generality we may assume .V / ¤ f0g: Let x P .V / n f0g: By Corollary 3.6 the blowups at any point of .V / @B converge to a minimal cone in dimension .n g 1/ g 1 up to a subsequence. Let V x be a blowup at x: By Lemma 4.5, V x is a minimal cone that is invariant in at least one direction. By Lemma 4.5 , by using our assumption, this implies that H tg1 ..V x // h 0, and thus the singular set of every possible blowup cone of any minimizer V has zero H tg1 -measure.
Arguing as in Lemma 5.4 we obtain H tg1 ..V // h 0: Combining Lemmas 5.3, 5.4, and 5.5 we obtain the following corollary. Notice that we will be able to replace n 1 by n 2 by Theorem 6.1. PROOF. The proof is by contradiction. For k P N assume ku k k C .2B/ < E 0 and the left-hand side of (5.4) is bounded below by k > 0 for every collection of balls satisfying (5.5). By Lemma 3.2 we know the sequence u k is bounded in H .B/: Taking a subsequence we may assume that u k converges locally uniformly to a minimizer u (see Corollary 3.6).
By Corollary 5.6 the set of singularities .u/ has H n 1 -measure zero, and thus they can be covered with finitely many balls B i satisfying (5.5).
Since F .u/ n .u/ is a C 1; -surface by Theorem 2.4, using the Hausdorff convergence of the free boundaries we apply again Theorem 2.4 to see that F .u k / B 1 n M ih1 B i are also C 1; -surfaces converging to F .u/B 1 n M ih1 B i uniformly in the C 1 -norm. This is a contradiction with the assumption that the Hausdorff measure blows up as k goes to I: The fact that the free boundary has finite perimeter follows now from the same iteration argument as [18, lemma 5.10]. Continuing inductively, after k steps we have that

Energy Gap
Next we will check that the Allen-Weiss density can also be used to identify singular points. First, let us state a useful identity for minimizers (which is also valid in the context of variational solutions in the sense of [37]). Combining both assertions with the monotonicity of we get that u r .x 1 / ! n 2 . But using the second formula in Theorem 4.1, one can see that this is true only whenever is -homogeneous with respect to x 1 . Thus, u is 1-symmetric and invariant in the direction of hx 1 i. By Corollary 5.6 F red .u/ has full H n 1 measure on F .u/. Thus, we can find x 1 ; : : : x n 1 P F red .u/ linearly independent. By the previous discussion u is invariant on an .n 1/-dimensional affine manifold, and thus, it is the trivial solution. But then u 0 is the trivial cone by Proposition 5.10. Since F .u j / 3 F .u/ in the Hausdorff distance, using -regularity (see Theorem 2.4) we get that u j is the trivial cone for j big enough.
The value above depends on the constants and on kuk C in a neighborhood of x 0 . In Section 7 we will show that does not depend on u at all.

Full Regularity in R 2C1
In the case of n h 2, we prove full regularity of the free boundary for minimizers of our functional. Note that this result does not depend on the previous sections except that we use dimension reduction and blowups to deduce regularity of the free boundary. THEOREM 6.1. Let n h 2. Then there is no singular minimal cone. In particular, the free boundary F .u/ of every minimizer u is C 1; everywhere.
The case h 0 has been considered in [18]. The idea is to construct a competitor by a perturbation argument. We note at this point that the argument is two dimensional in nature and does not generalize to higher dimensions. Recall the functional under consideration: J .u; / h jyj jruj 2 g m.fu > 0g R n /: Now since V is homogeneous of degree by assumption, the function g.x; y/ h jyj jrV j 2 is homogeneous of degree g 2 2 h 1. Therefore by a trivial change of variables on the sphere of radius r and using the fact that n h 2, we get the very same estimate B R 2 jyj jrU j 2 kAk 2 C ln.R/

R3I 3 0:
The rest of the proof follows verbatim [18, p. 1318], since this is only based on energy considerations, and we refer the reader to it.

Uniform Bounds Around the Free Boundary
The optimal regularity bound and the nondegeneracy described in Theorem 2.3 were obtained in [7] with bounds that depend on the seminorm kuk H .B 1 / . As a consequence, this dependence propagates to many of our estimates above. In this chapter we use the seminorm dependent estimates (e.g., Lemma 5.8) to prove seminorm independent nondegeneracy estimates. Re-running the arguments above yields the seminorm independent results presented in our main Theorem 1.1.
The question of seminorm independence may seem purely technical; however, independence allows the compactness arguments of the next section to work without additional assumptions on the minimizers involved.

Uniform Nondegeneracy
We will begin by showing uniform nondegeneracy from scratch to deduce uniform Hölder character from this fact, reversing the usual arguments in the literature.
The following lemma was shown in [1, cor. 4.2] in a more general setting. Here we give a more basic approach based on [2, lemma 3.4]. The main difference is that where Alt and Caffarelli could use the energy to directly control the H 1 norm of the minimizer, in our case we need to find an alternative because the measure term of the functional is computed on the thin phase (as opposed to the H 1 norm which is computed on the whole space). To bypass this difficulty we will use Allen's monotonicity formula.
The drawback of our approach is that we need the ball to be centered on the free boundary, while in the original lemma, Alt and Caffarelli could center the ball in the zero phase, allowing for a slightly better result. LEMMA 7.1. Let u be a minimizer in B r with 0 P F .u/. Then sup @B r u ! C r with C depending only on n and .
PROOF. By rescaling we can assume that r h 1. and therefore`! C 1 n; .
To show averaged nondegeneracy we need a mean value principle that is wellknown, but we include its proof for the sake of completeness. Next we use an idea of [2] and investigate the behavior of the distributional -Laplacian of the minimizer introduced in Section 3. As mentioned in the introduction, in [2] this investigation immediately yields that the positivity set is a set of locally finite perimeter, and more precisely, that it is Ahlfors regular of the correct dimension. However, the nonlocal nature of this problem indicates that the distributional fractional Laplacian may not be supported on the free boundary, and thus we cannot expect to immediately gain such strong geometric information.
First we can bound the growth of the fractional Laplacian measure around a free boundary point. Note that this growth is the natural counterpart to the upper Ahlfors regularity in the case of Alt-Caffarelli minimizers. THEOREM 7.4. Let u P H .B 2r .x 0 // be a minimizer of J in B 2r .x 0 /, and let x 0 P F .u/. Then, we have .B r .x 0 // C r n : In particular, .F .u// h 0.
A glance at (2.1) will convince the reader that these estimates are sharp, for they cannot be improved even in the case of the trivial solution.
PROOF. Without loss of generality we may assume that x 0 h 0. Let Lu h r ¡ .jyj ru/ and let v be the L-harmonic replacement of u in B 2r ; see (7.4 3 0: Next we study the measure away from the free boundary. We should emphasize here that even though the estimates in Lemma 7.5 and Theorem 7.6 depend on E 0 , they will be used to remove the dependence of our other estimates on E 0 .
More precisely, Theorem 7.6 will play a role in establishing the continuity of the Green function in Lemma 7.9. This qualitative fact is used to prove the quantitative uniform Hölder character in Theorem 7.8. .B r / jyj ru ¡ r k .B rg.1=k/ /; and for every " > 0 we use the Green's theorem to get jyj ru ¡ r k h jyj " jyj ru ¡ r k jyjh" jyj k ru ¡ d m: Using the symmetry properties and taking limits, (7.8) jyj ru ¡ r k h 2 lim ." u y . ¡ ; "//d m: A consequence of our control of the behavior of is that we can establish the existence of exterior corkscrews. We should note that exterior corkscrews can also be obtained by a purely geometric argument given the nondegeneracy and positive density of Theorem 2.3 (see, e.g., the proof of proposition 10.3 in [13] which is equivalent to the exterior corkscrew condition.

Uniform Hölder Character
The uniform nondegeneracy of Section 7.1 lets us conclude uniform control on the Hölder norm of u.
THEOREM 7.8. Let u be a minimizer of J in B r with 0 P F .u/. Then ju.x/j C jxj for every x P @B r=2 with C depending only on n and .
PROOF. Again we set v to be the L-harmonic replacement of u inside of B r as in (7.4). Let z u h v u so that Lz u h Lv Lu h h r ¡ .jyj ru/ and z u P H 1;2 0 .B r s jyj /.
Consider the Green function G B r ¢ B r 3 R such that LG. ¡ ;´/ h ´and G. ¡ ;´/ P H 1;2 loc .B r n f´g/ with null trace on @B r (see [25, prop. 2.4] By the strong maximum principle, the Green function in the annulus is bounded by C r n 2 . This fact, together with Theorem 7.4, implies that v.0/ I cr 2 n B C t 1 .n 2/ dt g C r C I cr 2 n t n n 2 dt g C r h C r : By the mean value theorem we conclude that @B r v d C r ; where d h jyj d H n . The theorem follows by observing that, as in (7.5), the mean of v dominates u by sup @B r=2 u sup @B r=2 v C ¬ @B r v d.
In case n 2 h 0, which could only happen for n h 1 and h 1=2, estimate (7.12) reads as G.´; x/ % log r jx ´j ; and the proof follows the same steps.
In case n 2 < 0, then estimate (7.12) reads as G.´; x/ % r n 2 ; and the estimate is even better compared to the above. LEMMA 7.9.
PROOF. Let " < r=2 and let´1;´2 P B r=4 , with j´1 ´2j "=2. Then In case n 2 h 0 we obtain " j 0 2 j log r 2 j " on the right-hand side instead, and in case n 2 < 0 we obtain " n r 2 n j 0 2 j.n / : In every case, by fixing " small enough, this term can be as small as wanted. The same will happen with the last term on the right-hand side of (7.13).
Remark 7.10. In light of Theorem 7.8 and the Caccioppoli inequality (see Section 3.1), arguing as in [7, theorem 1.1] we obtain that every minimizer u in a ball B r with 0 P F .u/ has uniform C character in B r=2 and the same for the H norm. Moreover, using [7, theorem 1.2] we can find interior corkscrew points with constants not depending on these norms. This allows us to remove the a priori dependence on kuk H from all of our results above.

Lower Estimates for the Distributional Fractional Laplacian
Next we bound the growth of the measure around a free boundary point from below. None of these results will be used in the present paper, but we include them to give a complete picture of the tools under consideration. THEOREM 7.11. Let u P H .B 2r / be a minimizer of J in B 2r such that 0 P F .u/. On the other hand, note that u is continuous. By the Riesz representation theorem, there exists a probability measure !Ĺ such that v.´/ h @B r u.x/d!Ĺ.x/: We can choose r so that @B r intersects a big part of a corkscrew ball; i.e., assume that there exists a point 0 P @B H r that is the center of a ball B H . 0 ; cr/ where u has positive values. This can be done by the interior corkscrew condition, with all the constants involved depending only on n and . Then, changing the constant if necessary, all points P B. 0 ; cr/ satisfy that u./ ! C r by the nondegeneracy condition and the optimal regularity. Call U h @B r B. 0 ; cr/. Then v.´/ & r !Ĺ.U /: But !Ĺ.U / is bounded below by a constant by [29, lemma 11.21] and the Harnack inequality (use a convenient Harnack chain). All in all, we have that (7.17) v.´/ & r : Combining (7.16), (7.15), and (7.17) and choosing small enough, depending on n and , we get .B r / & z u.´0/ .cr/ 2 n ! C r C H .r/ .cr/ 2 n ! C n; r n for small enough.
In case n 2 h 0, that is, for n h 1 and h 1=2, using similar changes as in the proof of Theorem 7.8 we get z u.´/ . .B r / sup xB.´0;cr/ log r jx ´j % .B r /jlog j instead of (7.16). In case n 2 < 0, the proof is even easier than before.
Remark 7.12. Theorem 7.11 implies that the .n /-Hausdorff measure of the free boundary is locally finite. This does not suffice to show finite perimeter of the positive phase; therefore we had to use the approach in Section 5.
The following theorem summarizes the information that we have gathered so far about the measure . THEOREM 7.13. If u P H loc ./ is a minimizer of J in , then the measure is absolutely continuous with respect to the Lebesgue measure in H .u/. Moreover, given x 0 P F .u/ and r > 0 such that B 2r .x 0 / & , then (7.18) .B r .x 0 // % r n ; and for almost every x P B H r .x 0 / we have that d d m .

Rectifiability of the Singular Set
In this section we use the Rectifiable-Reifenberg and quantitative stratification framework of Naber-Valtorta [32] to prove Hausdorff measure and structure results for the singular set. Recall that k £ is the first dimension in which there exists nontrivial -homogeneous global minimizers to (1.1) defined in Section 5. THEOREM 8.1. Let u P H loc ./ be a minimizer of (1.1) in a domain . Then .u/ is .n k £ /-rectifiable, and for every D b we have H n k £ ..u/ D/ C n;;dist.D;@/ : Part of the power of this framework is that it is very general. One needs certain compactness properties on the minimizers and a connection between the drop in the monotonicity formula and the local flatness of the singular set (see Theorem 8.14 below). To avoid redundancy and highlight the original contributions of this article, we omit many details here and try to focus on the estimates needed to apply this framework to minimizers of (1.1). Whenever we omit details, we will refer the interested reader to the relevant parts of [22].
The key first step is to introduce the appropriate formulation of quantitative stratification. First introduced by Cheeger and Naber [10] in the context of manifolds with Ricci curvature bounded from below, this is a way to quantify the intuitive fact that F .u/ should "look" .n k £ /-dimensional near a point x 0 P F .u/ at which the blowups have .n k £ /-linearly independent translational symmetries.

Quantitative Stratification for Minimizers to J
We have seen in Section 4.1 that homogeneous functions have linear spaces of translational symmetry. Here we want to quantify (both in terms of size and stability) how far a function is from having no more than k directions of translational symmetry. DEFINITION 8.2. We write V k for the collection of linear k-dimensional subspaces of R n . A function u is said to be k-symmetric if it is -homogeneous with respect to some point, and there exists a L P V k so that u.x g v/ h u.x/ for every v P L: A function u is said to be .k; /-symmetric in a ball B if for some k-symmetric z u we have r 2 n B jyj ju z uj 2 dy < : Next we define the k-stratum S k .u/, the .k; /-stratum S k .u/, and the .k; ; r/stratum S k ;r .u/. A key insight here is to define these strata by the blowups having k or fewer symmetries as opposed to exactly k symmetries. DEFINITION 8.3. Let 0 k n, 0 < " < I, and 0 < r < d .x/ h dist.x; @/, let u be a continuous function in , and let x P F .u/. We say that: x P S k .u/ if u has no .k g 1/-symmetric blowups at x; x P S k .u/ if u is not .k g 1; /-symmetric in B s .x/ for 0 < s minf1; d .x/g; x P S k ;r .u/ if u is not .k g 1; /-symmetric in B s .x/ for r s minf1; d .x/g. If it is clear from the context we will omit u from the notation.
We now detail some standard properties of the strata defined above and how they interact with the free boundary F .u/. While the proofs are mostly standard, we give the details as the scaling associated to the problem (1.1) adds some technical difficulties. This proof also provides a blueprint for fleshing out the details in Sections 8.3 and 8.4. LEMMA 8.4. Let 0 j k n, 0 < " < I, 0 < r s < dist.x; @/, and let u P H loc ./ be a minimizer in . Then:  Therefore, for every " there exists a ball small enough so that u is .kg1; "/symmetric in it. In particular, S k ' To see the converse, assume that x S k . Then for every i P N there exist a .kg1/-symmetric function z u i , invariant with respect to L i P V kg1 , and r i < minf1; dist.x; @/g such that 1 r ng2 i B r i jyj ju.x/ z u i .x/j 2 dx < 1 i : In the case when r i stays away from 0, since r i < 1, we can take a subsequence converging to r 0 P .0; 1/, and one can see that u is .k g 1/symmetric in the ball B r 0 .x 0 /. Otherwise, consider u i h u.x 0 g r i x/ r i and z u i;i h z u i .x 0 g r i x/ r i : By taking subsequences, we can assume that L i 3 L 0 locally in the Hausdorff distance, and that u i 3 u 0 locally uniformly. One can check also using the Hölder character of u that fz u i;i g is uniformly bounded in L 2 .Bs jyj /, so taking subsequences again, we can assume the existence of z u 0 so that z u i;i 3 z u 0 in L 2 .Bs jyj /. This function will be .k g 1/- (5) Assume that z u i is invariant with respect to L i P V kg1 and jyj ju i z u i j 2 i : Consider a subsequence fu i g so that the varieties L i 3 L locally in the Hausdorff distance. Using the triangle inequality as in (4), it follows that u is .k; i /-symmetric with i 3 0. .n k £ g 1/-symmetric approximants. By rescaling we can assume that r i h 1. Passing to a subsequence we can assume that L i 3 L 0 P V n k £ g1 locally in the Hausdorff distance and x i 3 x 0 . By the compactness results in Lemma 3.4 we have a uniform limit u 0 that is a minimizer as well, and it is .n k £ g 1/symmetric with invariant manifold L 0 . By Lemma 4.4 any blowup u 0;0 at x 0 will be .n k £ g 1/-symmetric as well. Applying Lemma 4.5 .n k £ g 1/ times, we find that the restriction of u 0;0 to the orthogonal manifold L c 0 is a .k £ 1/dimensional minimal cone, which, by Lemma 5.2, is the trivial solution, and so is u 0;0 . Thus, x 0 is a regular point for u 0 .
On the other hand, the Hausdorff convergence of Lemma 3.4 together with the improvement of flatness of Theorem 2.4 imply that for i big enough x i P F red .u i /, reaching a contradiction.

The Refined Covering Theorem
Our estimates on the size and structure of the singular set .u/ come from similar results concerning the S k .u/. In particular, we prove the following covering result: THEOREM 8.6. Let u P H .B 5 / be a minimizer to (1.1) in B 5 with 0 P F .u/. For given real numbers > 0, 0 < r 1, and every natural number 1 k n 1, we can find a collection of balls fB r .x i /g N ih1 with N C n;; r k such that S k ;r .u/ B 1 & i B r .x i /: In particular, jB H r .S k ;r B 1 /j C n;; r n k for every 0 < r 1 and H k .S k .u/ B 1 / C n;; : From Proposition 8.5 and Theorem 8.6, we can conclude the following corollary, which comprises the second part of Theorem 8.1 above. COROLLARY 8.7. If u P H .B 5 / is a minimizer to (1.1) in B 5 with 0 P F .u/, then .u/ is .n k £ /-rectifiable and for every 0 < r 1 we have jB r ..u/ B 1 /j C n; r k £ : In particular, H n k £ ..u/ B 1 / C n; : Rectifiability is encoded in the following result. We omit the details of proof here but it is a consequence of the packing result above, the Rectifiable-Reifenberg theorem of [32], and Theorem 8.14 below. For more details see sections 2 and 8 of [22] (particularly theorem 2.2 in the former and the proof of theorem 1.12 in the latter). THEOREM 8.8. Let u be a nonnegative, even minimizer to (1.1) in a domain . Then S k .u/ is k-rectifiable for every , and hence each stratum S k .u/ is k-rectifiable as well.
The proof of Theorem 8.6 follows from inductively applying the following, slightly more technical packing result (for details see section 4 of [22]). THEOREM 8.9. Let > 0. There exists .n; ; / such that, for every minimizer u P H .B 5 / of J in B 5 with 0 P F .u/ and 0 < R < 1=10, there is a finite collection U of balls B with center x B P S k ;R and radius R r B 1=10 that satisfy the following properties: ( We construct the balls of Theorem 8.9 using a "stopping time" or "good ball/bad ball" argument. Much of this argument uses harmonic analysis and geometric measure theory and is completely independent of the original problem (1.1). However, there are a few places in which we need to connect the behavior of minimizers to the geometric structure of the singular set. Here we will sketch the good ball/bad ball argument, taking for granted the estimates needed to apply this argument to our functional. In the next few subsections we will provide these estimates. For more details on the construction itself, we refer the reader to section 7 in [22].
Outline of the Construction in Theorem 8.9 To find this covering we define good and bad balls as follows: imagine our ball, B, has radius 1. We say that B is a good ball if at every point in x P S k " .u/B the monotone quantity centered at that point at some small scale, , is not much smaller than the monotone quantity on ball B (we say these points have "small density drop"). A ball B is a bad ball if all the points in S k " .u/ B with small density drop are contained in a small neighborhood of a .k 1/-plane. This dichotomy follows from Theorem 8.10 in Section 8.3.
In a good ball of radius r we cover S k " .u/ with balls of radius r, iterating the construction until we find a bad ball or until the radius of the ball becomes very small. In a bad ball, we cover S k " .u/ away from the .k 1/-plane without much care. Close to the .k 1/-plane we cover S k " .u/ with balls of radius r, iterating the construction until we reach a good ball or until the radius of the ball becomes very small. Inside long strings of good balls, the packing estimates follow from powerful tools in geometric measure theory (see Theorem 8.13 below) and the connection between the drop in monotonicity and the local flatness of the singular strata (see Theorem 8.14 below). We give more details in Section 8.4.
Inside long strings of bad balls, each of which is near the .k 1/-plane of the previous bad ball, we have even better packing estimates than expected (as we are effectively well approximated by planes that are lower dimensional). This leaves only points that are in many bad balls, and in most of those balls they are far away from the .k 1/-plane. However, at these points the monotone quantity drops a definite amount many times, which contradicts either finiteness or monotonicity. This implies that the points and scales inside the bad balls that are not close to the .k 1/-plane form a negligible set (the technical term is a Carleson set). We give more information about the bad balls in Section 8.3. THEOREM 8.10 (Key dichotomy). Let ; ; ; H > 0 be fixed numbers with < 2. There exists an 0 .n; ; ; ; ; H / < =100 such that for every 0 , every r > 0, every E > 0, and every minimizer u P H .B 4r / of J in B 4r with 0 P F .u/ and sup B r u 2r E, then either u r ! E H on S k ;r B r , or there exists`P L k 1 so that fx P B r u 2r .x/ ! E g & B r .`/.
The key dichotomy is a direct consequence of Lemma 8.11 below. The core idea is to make effective the following assertion: if u is k-symmetric, then along the invariant manifold the Allen-Weiss density is constant, and every point away from the manifold will have .k g 1/-symmetric blowups by Lemma 4.4. The proof follows (with only minor modifications) the proof in [22, lemma 3.3]. We end this subsection by formally defining the good/bad balls alluded to above: DEFINITION 8.12. Let x P B 2 , 0 < R < r < 2, and u be a minimizer to J in B 5 .  [30] (for the L I version) and David-Semmes [12] (for the L p version). If we control the size of the k 's we can conclude size and structure estimates on the measure . The following theorem says exactly this and represents a major technical achievement. It differs (importantly) from prior work in this area by the lack of a priori assumptions on the upper or lower densities of the measure involved. THEOREM 8.13 (Discrete-Reifenberg Theorem; see [32, theorem 3.4]). Let fB r q .q/g q be a collection of disjoint balls, with q P B 1 .0/ and 0 < r q 1, and let be the packing measure h q r k q q , where q stands for the Dirac delta at q. There exist constants DR ; C DR > 0 depending only on the dimension such that if To obtain the packing estimates required for the Discrete-Reifenberg theorem, we need to control the beta coefficients. The key estimate of this entire framework lies in the following theorem, which shows the drop in monotonicity at a given point and that a given scale controls the beta coefficient at a comparable scale. THEOREM 8.14. Let > 0 be given. There exist .n; ; / and c.n; ; / such that for every u P H .B 5r / minimizing J in B 5r .x/ with x P F .u/ and (8.5) @ u is .0; /-symmetric in B 4r .x/; u is not .k g 1; /-symmetric in B 4r .x/; where L k h X g spanhv 1 ; : : : ; v k i.
Next we find a relation between the eigenvalues of Q and Allen-Weiss' energy. PROOF. The argument follows as in [22, (18) and below]. In formula (18) one needs to change u.´/ by u.´/, which can be done with exactly the same argument.
Finally, using compactness, we bound the left-hand side of (8.16) from below. LEMMA 8.17. Let > 0 be given. There exists a .n; ; / and c.n; ; / such that, for every orthonormal basis fv i g n ih1 and every u P H .B 5r / minimizing J in B 5r .x/ with x P F .u/ and satisfying (8.5), we have that .v i ¡ Du.´// 2 d´: PROOF. The proof follows that of [22, (19) If u h f £ P y is a minimizer of J , then f is a minimizer for J . In particular, if u is a minimizer of J in every ball, positive outside the hyperplane fy h 0g, and u.x; y/ h O.j.x; y/j /, then uj R n ¢f0g is a minimizer for J in every ball.
LEMMA A.2. Let f; g satisfy that J 0 .f; B 1 / and J 0 .g; B 1 / < I, and suppose that f g is compactly supported in B 1 & R n . Then we have that J 0 .g; B 1 / J 0 .f; B 1 / h c n; inf jyj .jrv.x; u/j 2 jr.f £ P y /.x/j 2 /; where the infimum is taken among all the symmetric bounded Lipschitz domains with the property that .R n ¢ f0g/ & B 1 and among all symmetric functions v with trace g satisfying that v f £ P y is compactly supported on . PROOF OF PROPOSITION A.1. Let f be a minimizer of J in the unit ball of R n and let B r be a ball such that B H r b B H 1 . We want to show that u h f £ P y is a minimizer of J in B r .
Let v R ng1 3 R so that v u in R ng1 n B r and v P H 1 .; B r /. Let g be the trace of v in R n ¢ f0g. By Lemma A. The proposition follows combining (A.1) and (A.2) and letting " 3 0.
The converse follows the same sketch: every global minimizer can be expressed as the Poisson extension of its restriction to the hyperplane by Proposition B.1.
As a consequence of the previous proposition, all the results that we have proven for minimizers of J also apply to minimizers of J : COROLLARY A.3. If u R n 3 R is a minimizer to J in B 2 & R n and 0 P F .u/, then kuk C .B 1 / C , it satisfies the nondegeneracy condition u.x/ ! C dist.x; F .u// for x P B 1 , the positive phase satisfies the corkscrew condition, every blowup limit is -homogeneous, and the boundary condition in (1.2) is satisfied at F red .u/.
Moreover, the positive phase fu > 0gB 1 is a set of finite perimeter, the singular set is an .n 3/-rectifiable set, it is discrete whenever n h 3, and it is empty if n 2.
All the constants depend only on n and .