This is the submitted version of the article: Engelstein, Max; Kauranen, Aapo; Prats, Martí; [et al.]. «Minimizers for the thin one-phasefree boundary problem». Communications on pure and applied mathematics, Vol. 74 Núm. 9 (2021), p. 1971-2022. DOI 10.1002/cpa.22011 This version is available at https://ddd.uab.cat/record/250717 under the terms of the license Minimizers for the Thin One-Phase Free Boundary Problem MAX ENGELSTEIN University of Minnesota Twin Cities AAPO KAURANEN Universitat Autònoma de Barcelona-BGSMath MARTÍ PRATS Universitat Autònoma de Barcelona GEORGIOS SAKELLARIS Universitat Autònoma de Barcelona AND YANNICK SIRE Johns Hopkins University Abstract We consider the “thin one-phase" free boundary problem, associated to mini- mizing a weighted Dirichlet energy of the function in RnC1 C plus the area of the positivity set of that function in Rn . We establish full regularity of the free boundary for dimensions n  2, prove almost everywhere regularity of the free boundary in arbitrary dimension, and provide content and structure estimates on the singular set of the free boundary when it exists. All of these results hold for the full range of the relevant weight. While our results are typical for the calculus of variations, our approach does not follow the standard one first introduced by Alt and Caffarelli in 1981. In- stead, the nonlocal nature of the distributional measure associated to a minimizer necessitates arguments that are less reliant on the underlying PDE. © 2020 Wi- ley Periodicals LLC 1 Introduction This article is devoted to the study of the regularity properties of a weighted ver- sion of the thin one-phase problem. More precisely, we investigate even, nonnega- tive minimizers of the following functionals: denote x 2 RnC1 by x D .x 0 ; y/ 2 Rn  R, and for ˇ 2 . 1; 1/ we define Z (1.1) J .v; / WD jyjˇ jrvj2 dx C m.fv > 0g \ Rn \ /;  Communications on Pure and Applied Mathematics, 0001–0051 (PREPRINT) © 2020 Wiley Periodicals LLC 2 M. ENGELSTEIN ET AL. where m stands for the n-dimensional Lebesgue measure. Here and throughout the paper the integration is done with respect to the .n C 1/-dimensional Lebesgue measure unless stated otherwise. This functional is finite for open sets, , and functions in the weighted Hilbert space, H 1 .ˇ; / WD fv 2 L2 .I jyjˇ / W rv 2 L2 .I jyjˇ /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/ WD @Rn fv.x; 0/ > 0g \ : Since the free boundary lies on a codimension 1 subspace of the ambient space RnC1 , 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 min- imizers of (1.1) where ˇ D 0, and the second term is replaced by LnC1 .fv > 0g \ / (where LnC1 stands for the Lebesgue measure in RnC1 ). In particular, for the Bernoulli problem, the free boundary fully sits in the ambient space, RnC1 . 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 one- phase free boundary problem but with the Dirichlet energy replaced by the Gagli- ardo seminorm ŒuHP ˛ for ˛ D 1 2ˇ 2 .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, Roque- joffre, and the last author in [7] is a local version of the following nonlocal free boundary problem: given a function f 2 L1loc .Rn / with suitable decay at infinity, we can define its fractional Laplacian at x 2 Rn by f .x/ f ./ Z ˛ . / f .x/ D cn;˛ p: v: d : Rn jx jnC2˛ THE THIN ONE-PHASE PROBLEM 3 At the formal level, we are interested in solutions of the free boundary problem ( . /˛ f D 0 in  \ ff > 0g; (1.2) @˛ f D A on  \ F .f /; where @˛ f .x/ WD lim\ff >0g3!x ..f ./ f .x/ 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 jf .x/ f ./j2 “ Œf HP ˛ .Rn / WD d  dx: R2n jx jnC2˛ Since we want to study competitors that vary only in a certain domain , it is natu- ral to consider only the integration region that may suffer variations when changing candidates. Thus, we define the energy jf .x/ f ./j2 “ (1.3) J.f; / WD cn;˛ d  dx C m.ff > 0g \ /: R2n n.c /2 jx jnC2˛ We say that f 2 L1loc is a minimizer of J in  if J.f; / is finite and J.f; /  J.g; / for every g satisfying that f g 2 HP ˛ .Rn / and such that f .x/ D g.x/ for almost every x 2 c . We say that f is a global minimizer if it is a minimizer for every open set   Rn . 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 2 N and 0 < ˛ < 1, jyj2˛ (1.4) Py ./ WD Pn;˛ .; y/ D cn;˛ for every .; y/ 2 Rn  R: j.; y/jnC2˛ The Poisson extension of f 2 L1loc .Rn / is given by Z 0 0 u.x ; y/ WD f  Py .x / D Pn;˛ .; y/f .x 0 /d  (1.5) R n for every .x 0 ; y/ 2 Rn  R: By [9], with a convenient choice of the constant, one gets lim y 1 2˛ uy .x 0 ; y/ D . /˛ f .x 0 / y&0 in every point where f is regular enough. Moreover, the extension satisfies the localized equation r .jyjˇ ru/ D 0 weakly, away from Rn f0g. The whole point is that local minimizers of (1.3) can be extended via the previous Poisson kernel Py 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 4 M. ENGELSTEIN ET AL. for the fractional Laplacian. Notice that—and this is of major importance for us— this localization technique does not carry over to other types of nonlocal operators besides pure powers of second-order elliptic operators. This is a major drawback of the theory, in the sense that, at the moment, it seems to be impossible to tackle one- phase problems involving more general operators than the fractional Laplacian. The main point is we do not know how to prove any kind of monotonicity for general integral operators. This connection between the nonlocal analogue of the Bernoulli problem and our thin one-phase problem allows us to simplify several arguments by working in the purely nonlocal setting. However, this underlying nonlocality is also the reason why several results, which came more easily in the setting of [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 D 2 showing that Lipschitz free boundaries are C 1 . A feature of the functional J is that the weight jyjˇ is either degenerate or singular at fy D 0g (except in the case ˇ D 0). Such weights belong to the Muckenhoupt class A2 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 ˇ D 0, the problem is still degenerate in the sense that derivatives near the free boundary blow up. The case ˇ D 0 has been thoroughly investigated in the series of papers by De Silva, Savin, and Roquejoffre [16–18]. The main goal of our paper is to provide a full picture of the regularity of the free boundary for any power ˇ 2 . 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 ˇ D 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 THE THIN ONE-PHASE PROBLEM 5 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.jyjˇ 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 RnC1 cen- tered at the origin by Br , and Br0 WD Br \ Rn  f0g. Moreover, for the definition of Hˇ , see Section 2. We may then summarize our regularity results in the following theorem. T HEOREM 1.1. [Main regularity theorem] Let u 2 Hˇ .B1 / be a (nonnegative, 0 even) local minimizer of J in B1  RnC1 . Let B1;C .u/ WD fx D .x 0 ; 0/ 2 B1 W 0 u.x/ > 0g, let F .u/ be the boundary of B1;C .u/ inside of B10 , and assume that 0 2 F .u/. Then: 0 (1) B1;C .u/ (as a subset of Rn  f0g) is a set of locally finite perimeter in B10 . (2) We can write the free boundary as a disjoint union F .u/ D R.u/ [ †.u/, where R.u/ is open inside F .u/, and for x 2 R.u/ there exists an rx > 0 such that B.x; rx /\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, there- fore, of Hn 1 -measure zero). In particular, for n  2, †.u/ is empty, and moreover, if n D 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 ˇ but not on kukHˇ .B1 / . 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 6 M. ENGELSTEIN ET AL. boundary problems (in particular, the one-phase problem) by Edelen and the first author [22]. T HEOREM 1.2. Let u 2 Hˇ .B1 / be a (nonnegative, even) local minimizer of J in 0 B1 and 0 2 F .u/. Let B1;C .u/ WD fx D .x 0 ; 0/ 2 B1 W u.x/ > 0g and F .u/ be 0 the boundary of B1;C .u/ inside B10 . Then, there exists a k˛  3 such that †.u/ is  .n k˛ /-rectifiable and  Hn k˛ .†.u/ \ D/  Cn;˛;dist.D;@B1 / for every D b B1 : 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]. In- deed, 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 Lapla- cian 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  .jyjˇ ru/ D 0 weakly, away from Rn \ 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 Rn of a given minimizer u in   RnC1 , harmonic away from Rn  f0g and with asymptotic behavior u.x; y/ D O.j.x; y/j˛ /, is always a solution to (1.2) for A D A.˛/ at “nice” points of the free boundary. A brief summary of this paper follows. In Sections 3 and 4 we discuss com- pactness 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 R2C1 concluding the proof of Theorem 1.1. THE THIN ONE-PHASE PROBLEM 7 Once we have established the finite perimeter, in Section 7 we remove the de- pendence 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 exam- ple, 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 sub- scripts such as C0 retain their values along the text. For a; b  0, we write a . b if there is C > 0 such that a  C b. We write a  b to mean a . b . a. Let u be a continuous function in RnC1 . Then we write C .u/ WD \fu > 0g, and we denote the zero phase, the positive phase, and the free boundary by 0 .u/ WD fx 2 Rn  f0g W u.x/ D 0gı ; 0C .u/ WD C \ .Rn  f0g/ D fx 2 Rn  f0g W u.x/ > 0g; F .u/ WD F .u/ D @.C .u/ \ Rn  f0g/ \ ; respectively. Here both the boundary and the interior are taken with respect to the standard topology in Rn . Note that Rn  f0g is the disjoint union of 0 .u/, 0C .u/, and F .u/ whenever u is nonnegative. We also call Fred .u/ D Fred; .u/ the points of F .u/ where the free boundary is expressed locally as a C 1 surface. Finally, let †.u/ D † .u/ D F .u/ n Fred; .u/. In general, we will write 0 WD  \ .Rn  f0g/. Throughout the paper we will often fix ˇ 2 . 1; 1/ but then refer to ˛ 2 .0; 1/ or vice versa. These two numbers are always connected by the relationship ˛ D 1 2ˇ . 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 Rn  f0g, that is, u.x 0 ; y/ D u.x 0 ; y/. The function spaces that we will consider are the following Hˇ ./ WD fu 2 H 1 .ˇ; / W u is even and nonnegativeg and ˇ Hloc ./ WD u 2 L2loc ./ W u 2 Hˇ .B/ for every ball B b  : ˚ We will omit  in the notation when it is clear from the context. 8 M. ENGELSTEIN ET AL. ˇ D EFINITION 2.1. We say that a function u 2 Hloc ./ is a (local) minimizer of J in a domain  if for every ball B b  and for every function v 2 Hˇ .B/ such that the traces vj@B  uj@B , the inequality J .u; B/  J .v; B/ holds. As usual for several free boundary problems, it is a natural question to exhibit a particular (global) solution so that one gets an idea of the qualitative properties of general solutions. Let us consider the following function: for every x 2 Rn let fn;˛ .x/ WD cn;˛ .xn /˛C ; where aC D maxf0; ag. If n D 1, f1;˛ is a solution to (1.2) for a convenient choice of c1;˛ (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) . /˛ fn;˛ .x/ D cn;˛ .xn / ˛ ; where a D maxf0; ag. As a toy question we wonder whether the trivial solutions are minimizers. In- deed, this is the case, as we will see later in Section 4.1. P ROPOSITION 2.2. Let n 2 N and 0 < ˛ < 1. Then the trivial solution un;˛ WD fn;˛  Py is a minimizer of J in every ball B  RnC1 . Next we collect the main properties of minimizers in the unit ball proven in [7, theorems 1.1–1.4, prop. 3.3, and cor. 3.4]. T HEOREM 2.3. If u 2 Hˇ .B1 / is a minimizer of J in  D B1 with kukHP ˇ .B1 / WD krukL2 .B1 ;jyjˇ /  E0 and x0 2 F .u/ \ B 1 , then it satisfies 2 (P1) Optimal regularity (see [7, theorem 1.1]): kukCP ˛ .B1=2 /  C .1 C E0 /. (P2) Nondegeneracy (see [7, theorem 1.2]:) u.x/  C dist.x; F .u//˛ for x 2 0 B1=2;C . (P3) Interior corkscrew condition (see [7, prop. 3.3]:) there exists xC 2 Br0 .x0 / so that B 0 .xC ; C0 r/  0C .u/. (P4) Positive density (see [7, theorem 1.3]:) j0 \ Br0 .x0 /j & r n . (P5) Blowups are minimizers (see [7, cor. 3.4]): The limit of a blowup sequence uk .x/ WD u.x0C ˛ k x/ converging weakly in H 1 .ˇ; B1 / and uniformly is a k global minimizer. (P6) Normal behavior at the free boundary (see [7, theorem 1.4]): the boundary condition in (1.2) is satisfied at every point on the free boundary with a measure-theoretic normal (see [24]) for a prescribed value of A. All the constants depend on n and ˛, and also on E0 except for the ones in P1 and P2. THE THIN ONE-PHASE PROBLEM 9 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 overde- termined 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. T HEOREM 2.4 (-regularity). There exists  > 0 depending only on n, ˛, and E0 such that for every nonnegative, even minimizer u of the energy (1.1) on a ball n B  RnC1 with kukHˇ .B/  E0 r.B/ 2 and (2.2) f.x; 0/ 2 B W xn  g  B0 .u/  f.x; 0/ 2 B W xn  g; 1; we have that F .u/ 2 Cloc . 12 B/, with 0 < < 1. Note that the dependence on E0 will be removed in Section 7. P ROOF. We say that u is a viscosity solution to ( r  .jyjˇ ru/ D 0 in B1C .u/; (2.3) lim t !0C u.x0 Ctt .x ˛ 0 /;0/ D 1; for .x0 ; 0/ 2 F .u/ if (i) u 2 C.B1 /, u  0, 1;1 (ii) u 2 Cloc .B1;C .u//, u is even, and it solves r  .jyjˇ ru/ D 0 in the viscosity sense, and (iii) any strict comparison subsolution (resp., supersolution) cannot touch from below (resp., from above) at a point .x0 ; 0/ 2 F .u/. 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 suf- fices to prove condition (iii) above: that any strict comparison subsolution cannot touch u from below at a point .x0 ; 0/ 2 F .u/. The analogous claim for strict comparison supersolutions will follow in the same way. Let us recall (see, e.g., definition 2.2 in [19]) that w 2 C.B1 / is a strict compar- ison subsolution (resp., supersolution) to (2.3) if (a) w  0, (b) w is even with respect to fy D 0g, (c) w 2 C 2 .fw > 0g/, (d) div.j´jˇ rw/  0 in B1 n fy D 0g, (e) F .w/ is locally given by the graph of a C 2 function and for any x0 2 F .w/ we may write (2.5) w.x; y/ D aU..x x0 /.x0 /; y/Co.k.x x0 ; y/k˛ /; .x; y/ ! .x0 ; 0/: 10 M. ENGELSTEIN ET AL. Here U is the extension of the trivial solution (see [19]), and .x0 / is the unit normal to F .w/ considered as a subset of Rn pointing into fw > 0g and a  1. (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 D u at .x0 ; 0/ 2 F .u/. Since u.x0 ; 0/ D 0 it follows that .x0 ; 0/ 2 F .w/ and with a harmless rotation we can guarantee that ..x0 ; 0// D en . We want to show that en 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 .x0 ; 0/. It must then be the case that B  fu > 0g as well. Thus .x0 ; 0/ 2 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/ D U..x x0 /  .x0 /; y/ C o.k.x x0 ; y/k˛ /; .x; y/ ! .x0 ; 0/: If u  w, this implies that w must satisfy the expansion in (2.5) with a D 1 at the point x0 . This, in turn, implies that div.j´jˇ rw/ > 0 in B1 nfy D 0g (by the definition of a strict subsolution). Furthermore, since w 2 C 2 where fw > 0g, we can guarantee that div.j´jˇ rw/  0 in all of B1 \ fw > 0g. Let us return to the ball B that is a subset of fu > 0g and fw > 0g and for which x We know that w u ¤ 0 in B n fy D 0g (this is because w strictly .x0 ; 0/ 2 B. satisfies the differential inequality in B away from fy D 0g), and we know that w u is a subsolution in B. Furthermore, .x0 ; 0/ 2 B is a strict maximum, so by the Hopf lemma in [5, prop. 4.11] it must be that .w u/.x0 C t .x0 /; 0/ lim > 0: t #0C t˛ This contradicts the fact that u and w both satisfy (2.5) at .x0 ; 0/ with a D 1. Therefore, .x0 ; 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.  3 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. THE THIN ONE-PHASE PROBLEM 11 3.1 Caccioppoli Inequality First we want to show that the distribution  WD r  .jyjˇ ru/ is in fact a Radon measure with support in the complement of the positive phase as long as u is a min- imizer. In Section 7 we will come back to this measure to understand its behavior around the free boundary. 1;2 L EMMA 3.1. Let   RnC1 be an open set, and let u 2 Wloc .; jyjˇ / be such ˇ 1 that r  .jyj ru/ D 0 weakly in C .u/, i.e., for every  2 Cc .C .u//, Z ˇ (3.1) hr  .jyj ru/; i WD .jyjˇ ru/r D 0: Then  WD r  .jyjˇ ru/ is a positive Radon measure supported on fu D 0g and for every v 2 W 1;2 .; jyjˇ / \ Cc ./ Z Z (3.2) v d D jyjˇ ru  rv: P ROOF. Indeed, by (3.1) the quantity u o o Z Z  n n ˇ jyj ru  r D jyjˇ ru  r  max min 2 ;1 ;0 " Z "!0  jyjˇ jrujjrj ! 0 \f0 0g. Then 4 Z Z jyjˇ jruj2  2 jyjˇ u2 : 1 2B r 1 Bn 2 B 12 M. ENGELSTEIN ET AL. P ROOF. Let  be a Lipschitz function such that  1 B    B and with 2 jrj  1r . By Lemma 3.1 Z Z 0D u2 d  D jyjˇ ru  r.u2 /: B B By the Leibniz rule Z Z ˇ 2 2 jyj  jruj D jyjˇ 2uru  r; B B and using Hölder’s inequality we get 4 Z Z Z Z ˇ 2 ˇ 2 2 ˇ 2 2 jyj jruj  jyj  jruj  jyj 4u jrj  2 jyjˇ u2 : 1 2 B B B r Bn 1 2 B  L EMMA 3.3. Let u 2 Hˇ .Br / be a minimizer of J in B2r and 0 2 F .u/. Then n=2 ˛ r krukL2 . 1 Br Ijyjˇ /  r kukL1 .Br / 2 n=2   kukCP ˛ .Br /  C 1 C r krukL2 .B2r Ijyjˇ / : P ROOF. The first inequality is an immediate consequence of Caccioppoli, the middle estimate is trivial, and the last follows from P1 in Theorem 2.3.  3.2 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 nondegen- eracy 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 uk , we can get rid of this assumption in the light of the results of Section 7. ˇ L EMMA 3.4 (Compactness results). Let fuk g1 kD1  Hloc ./ be a sequence of minimizers in a domain   RnC1 with kuk kCP ˛ ./  E0 with nonempty free ˇ boundary. Then there exists a subsequence converging to some u0 2 Hloc ./ such that for every bounded open set G b  we have (1) uk ! u0 in C ˇ .G/ for every ˇ < ˛, (2) uk ! u0 in Lp .G/ for every p  1, (3) x ! @fu0 > 0g \ G @fuk > 0g \ G x in the Hausdorff distance, (4) 1 0 fuk >0g ! fu0 >0g in L .G /, and (5) ruk ! ru0 in Lp .GI jyjˇ / for every p  2. THE THIN ONE-PHASE PROBLEM 13 P ROOF. 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 2 Rn f0g we have  (3.3) d.x; F .u0 // >  ) d.x; F .uk // > 2 for large k: This implies that F .uk /  fxW d.F .u0 /; x/ < 2g for k large enough. Let B.x; /  F .u0 /c : If u0 is positive in B.x; /, then it is bounded from below by a positive number in B.x; =2/: In this case uk are also positive in B.x; =2/ for large k due to uniform convergence in G. Thus B.x; =2/  F .uk /c for large k: If u  0 in B 0 .x; /, then due to the uniform convergence we know that for k large enough uk < C  ˛ in B 0 .x; /, where C is a constant given by P2 in Theorem 2.3 so that uk has no free boundary points in B.x; =2/ for all large k: This proves (3.3). Next we will show that for all large k (3.4) F .u0 /  fxW d.F .uk /; x/ < g: If this was not true we could find a point x 2 F .u0 / and a subsequence of uk such that B 0 .x; /  F .uk /c for every k in the subsequence. If the subsequence contains infinitely many uk such that uk  0 in B.x; /, then also u0  0 due to uniform convergence. Otherwise the sequence contains infinitely many uk for which B.x; / is contained in the positive phase. In this case the nondegeneracy implies that in B.x; =2/ we have uk > C  ˛ ; with C independent of k: Again uniform convergence implies the same lower bound for u0 ; which contradicts our choice x 2 F .u0 /: To show the fourth claim we notice that F .u0 / has zero n-dimensional Lebesgue measure by the Lebesgue differentiation theorem and the positive density of the zero phase. Take an open set V  F .u0 / with m.V \ G 0 / < : For large k we have F .uk / [ F .u0 /  V \ G 0 , so kfuk >0g fu0 >0g kL1 .G 0 / < : Also the sequence is uniformly bounded in H 1;p .GI jyjˇ / by the Caccioppoli inequality. This implies by compactness [29, theorem 1.31] the weak convergence of ruk in Lp .GI jyjˇ /: To obtain strong convergence, use Lemma 3.5 below.  It remains to show that weak convergence implies strong convergence. L EMMA 3.5. Any sequence of minimizers fuk g1 kD0 in   RnC1 with uk ! u0 uniformly and ruk * ru0 weakly in L2loc .; jyjˇ / satisfies that ruk ! ru0 strongly in L2loc .; jyjˇ /. P ROOF. Let  2 Cc0;1 ./ be a nonnegative function. We claim that for every " > 0 there exists j0 so that Z jyjˇ jru ruj j2  " for j  j0 : 14 M. ENGELSTEIN ET AL. First we isolate the main difficulty Z Z Z ˇ 2 ˇ jyj jru0 ruj j D jyj .ru0 ruj /ru0 jyjˇ .ru0 ruj /ruj : By weak convergence, ˇZ ˇ ˇ jyjˇ .ru0 ˇ ˇ ˇ ruj /  ru0 ˇˇ  "=4 for j big enough. Note that this is true even if the uj are not minimizers. The bound on the second term, however, needs the minimization property. We observe that Z jyjˇ .ru0 ruj /  ruj Z Z (3.5) D jyjˇ .ru0 ruj /  r.uj / jyjˇ uj .ru0 ruj /  r : „ ƒ‚ … „ ƒ‚ … DW I DW II To estimate I in (3.5), let j be the measures corresponding to uj from Lemma 3.1. By (3.2) we get that Z Z Z jyjˇ .ru0 ruj /  r.uj / D uj d 0 uj d j : Since j is supported on fuj D 0g, we have that Z uj d j D 0 for every j (including j D 0 as u0 is also a minimizer to J , see corollary 3.4 in [7]). To finish the estimate on I in (3.5), we observe that Z Z Z uj d 0 D .uj u0 /d 0  sup juj u0 j  d 0 : supp   By uniform convergence on compact subsets, supsupp  juj u0 j  4kkL1 . 0/ for j big enough. We turn towards estimating II in (3.5): ˇZ ˇ ˇ ˇ ˇ jIIj D ˇˇ jyj uj .ru0 ruj /  rˇˇ ˇZ ˇ ˇ ˇ ˇ (3.6)  ˇ jyj .ru0 ruj /  .u0 r/ˇˇ ˇ C sup juj u0 jkru0 ruj kL2 .;jyjˇ / krkL2 .;jyjˇ / : supp  THE THIN ONE-PHASE PROBLEM 15 The first term goes to zero by weak convergence of ruj to ru0 . The second term satisfies sup juj u0 jkru0 ruj kL2 .supp ;jyjˇ / krkL2 .;jyjˇ /  "=4 supp  for j big enough, by uniform convergence and the uniform bound of the norm kruj kL2 .supp ;jyjˇ / derived from the Caccioppoli inequality in Lemma 3.2 to- gether with uniform convergence.  Lemma 3.4 implies that minimizers converge to minimizers (which was ob- served in Corollary 3.4 in [7]), but also implies the stronger fact that the energy is continuous under this convergence: C OROLLARY 3.6. Let uk be a sequence of minimizers in   RnC1 with uk ! u0 locally uniformly and supk kuk kHˇ < 1. Then u0 is also a minimizer to J in  and for any B b  we have J .uk ; B/ ! J .u0 ; B/ after passing to a subsequence. 4 Monotonicity Formula and Some Immediate Consequences From [1] we have the following monotonicity formula: T HEOREM 4.1 (Monotonicity formula, see [1, theorem 4.3] ). Let u be a minimizer in Bı .x0 / for the functional J with x0 2 F .u/. Then the function J .u; Br .x0 // ˛ Z u r 7! ‰r .x/ WD ‰.r/ D jyjˇ u2 d Hn rn r nC1 @Br .x0 / is defined and nondecreasing in .0; ı/, and for 0 <  <  < ı, it satisfies 2j˛u.x/ .x x0 /  ru.x/j2 Z ‰. / ‰./ D jyjˇ dx  0: B .x0 /nB .x0 / jx0 xjnC2 As a consequence, the blowup limits are cones, in the sense of the following corollary. C OROLLARY 4.2. Let u be a minimizer in Bı .x0 / with x0 D .x00 ; 0/. Consider k!1 a decreasing sequence 0 < k ! 0 and the associated rescalings uk .x/ WD u.x0 Ck x/ r˛ . Then the Allen-Weiss density ‰0u .x0 / WD lim ‰ru .x0 / r&0 is well-defined. Furthermore, for every bounded open set D  RnC1 and k  k.D/ this subsequence uk is bounded in H 1;2 .DI jyjˇ / and, passing to a sub- sequence ukj , converges (in the sense of Lemma 3.4) to u0 , which is a globally defined minimizer of J that is homogeneous of degree ˛. The proof is the same as in [38, theorem 2.8]. 16 M. ENGELSTEIN ET AL. Remark 4.3 (Nonuniqueness of blowups). We call the function u0 appearing in Corollary 4.2 a blowup of u at x0 . A priori, the function u0 may depend on the subsequence ukj . However, a simple scaling argument shows that for all radii r  0 and all blowups u0 to u at x0 we have ‰ru0 .0/  ‰0u .x0 /: 4.1 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: ˇ L EMMA 4.4. Let u 2 Hloc .RnC1 / be an ˛-homogeneous minimizer of J and let x0 2 F .u/ n f0g. Then any blowup limit u0 at x0 is invariant in the direction of x0 ; i.e., for every x 2 RnC1 and every  2 R, u0 .x C x0 / D u0 .x/: P ROOF. Let x 2 RnC1 , and consider its decomposition x D xz C x0 with xz 2 hx0 i? . We only need to check that (4.1) u0 .x/ D u0 .z x /: Bk .x0 / P1 P P3 2 x0 O F IGURE 4.1. The distance dist.P1 ; P3 / D O.k2 /. Consider a ball B D B.0; r/  RnC1 so that xz; x 2 B. Let fk g be a se- quence of radii converging to 0 and such that uk .x/ WD u.x0C ˛ k x/ converges to u0 k uniformly on Br . For k big enough, kuk u0 kL1 .Br / < ". Then, (4.2) ju0 .x/ x /j  2" C juk .x/ u0 .z uk .z x /j: To control the last term above, we use the homogeneity of u. Writing P1 WD x0 Ck xz and P2 WD x0 Ck x, we have k˛ uk .z x / D u.P1 / and k˛ uk .x/ D u.P2 /. THE THIN ONE-PHASE PROBLEM 17 Let P3 be the intersection between the line through P1 and x0 and the line through the origin and P2 (see Figure 4.1). By homogeneity of u jP2 j ˛ jP2 P3 j ˛     u.P2 / D u.P3 / D u.P3 / 1 ˙ : jP3 j jP3 j Thus, k˛ juk .x/ u.P3 /.1 C O.k //˛ ˇ ˇ ˇ x /j  ˇu.P1 / uk .z  ju.P1 / u.P3 /j C ju.P3 /jO.k /: By Thales’ theorem, jP1 P3 j D jP1 Pjx 2 jjP3 x0 j 0j D O.k2 /, and using the CP ˛ character of u and the fact that u.x0 / D 0, we get x /j  kukCP ˛ jP1 P3 j˛ C jP3 j˛ O.k / D O.k2˛ / C O.k /; k˛ juk .x/ uk .z  and (4.1) follows by (4.2) since k ! 0.  We then recall that a minimizer with a translational symmetry is actually a min- imizer without that symmetry in one dimension less. This is known as “cone split- ting”: ˇ L EMMA 4.5. Let u 2 Hloc .RnC1 / be an ˛-homogeneous minimizer of J in RnC1 z.x 0 ; y/ WD u.x 0 ; 0; y/ is a minimizer that is invariant in the direction en . Then u of J in one dimension less. P ROOF. 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 triv- ial 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. P ROOF OF P ROPOSITION 2.2. Consider a nonzero minimizer u with nonempty free boundary (see [7, prop. 3.2] for its existence), choose a free boundary point x0 2 F .u/ and consider u0 to be a blowup weak limit at this point, which exists and is ˛-homogeneous by Lemma 4.2. Then u0 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 uj be an ˛-homogeneous global minimizer different from 0 such that it is invariant in a j -dimensional linear subspace Hj  Rn , i.e., for every v 2 Hj and every x 0 2 Rn , uj .x 0 ; y/ D u.x 0 C v; y/: Consider a point xj 2 F .uj / n .Hj  f0g/ that exists as long as j < n 1 by the interior corkscrew condition and positive density, and let uj C1 be a blowup limit at this point, which is again an ˛-homogeneous global minimizer. We claim that uj C1 is invariant in fact in the .j C 1/-dimensional subspace Hj C hxj0 i. 18 M. ENGELSTEIN ET AL. Indeed, uj C1 is invariant in hxj0 i by Lemma 4.4. On the other hand, since uj is invariant in Hj , so are the functions in the blowup sequence, and thus, uj C1 is invariant in Hj . Thus, for v 2 Hj , v0 2 hxj0 i, and x 2 RnC1 we get u.x C v C xj0 / D u.x C v/ D u.x/; and the claim follows. Thus, after n 1 steps, we obtain un 1 , which is an ˛-homogeneous global minimizer invariant in an .n 1/-dimensional space Hn 1 , with nonempty free boundary. Thus, un 1 .x 0 ; 0/ D Cn;˛ .xn0 /˛C ; where the constant is given by (P6). The proposition follows by Proposition A.1.  4.2 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: ˇ L EMMA 4.6. Let uj 2 Hloc ./ be minimizers of J in  and uj ! u0 in the sense of Lemma 3.4. Then, for xj ! x0 and r < dist.x0 ; @/, u j !1 ‰r j .xj / ! ‰ru0 .x0 /: P ROOF. Let " > 0. We want to check that for j big enough, ˇ uj ˇ‰r .xj / ‰ u0 .x0 /ˇ  ": ˇ r We will consider the three terms of the energy separately. For the first term, Z Z ˇ 2 jyj jruj j jyjˇ jru0 j2  r n "=3 Br .xj / Br .x0 / follows from the L2 convergence of the gradients. Indeed, if ıj WD jxj x0 j  ı for j big enough and BrCı  , then Z Z ˇ 2 jyj jruj j jyjˇ jru0 j2 Br .xj / Br .x0 / Z Z ˇ 2 2 jyjˇ jru0 j2   jyj jruj j jru0 j C Br .xj / Br .xj /Br .x0 / Z Z ˇ 2 2 jyjˇ jru0 j2   jyj jruj j jru0 j C BrCı .xj / .BrCıj nBr ıj /.x0 /  r n "=3: For the measure, we estimate ˇZ Z ˇ C .u0 / d mˇˇ  r n "=3 ˇ ˇ ˇ C .uj / d m 0 0 ˇ Br .xj / Br .x0 / THE THIN ONE-PHASE PROBLEM 19 for j big enough as a consequence of C .uj / ! C .u0 / in L1loc as before. The fact that ˇZ Z ˇ 2 u0 ˇˇ  r nC1 "=3 2 ˇ ˇ ˛ ˇˇ uj @Br .xj / @Br .x0 / for j big enough is a straight consequence of the uniform convergence and the continuity of u0 .  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. ˇ j !1 L EMMA 4.7. Let uj 2 Hloc ./ be minimizers of J in  and uj ! u0 in the sense of Lemma 3.4, with xj 2 F .uj / for j 2 N. Then, if xj ! x0 and rj ! 0, u u lim sup ‰0j .xj /  lim sup ‰rjj .xj /  ‰0u .x0 /: j j P ROOF. The first inequality comes from monotonicity. To see that u lim sup ‰rjj .xj /  ‰0u0 .x0 /; j it is enough to check that for every r > 0 u lim sup ‰rjj .xj /  ‰ru0 .x0 /; j or by using monotonicity, it suffices to show that for every " > 0 and j big enough, u ‰r j .xj / ‰ru0 .x0 /  ": 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.  5 Measure-Theoretic Properties 5.1 Finite Perimeter We will show that 0C .u/ is a set of locally finite perimeter. Then Fred .u/ will coincide with the measure-theoretic reduced boundary by the -regularity theorem, see [2, secs. 4.6 and 4.7]. D EFINITION 5.1. For every 0 < ˛ < 1 we can define k˛ as the infimum of ˇ k 2 N W 9 an ˛-homogeneous minimizer u 2 Hloc .RkC1 / s.t. †.u/ D f0g : ˚ Note that, to the best of our knowledge, there is no result showing that k˛ needs to be finite. L EMMA 5.2. Let u be an ˛-homogeneous minimizer of J in RnC1 with n < k˛ . Then u is a rotation of the trivial solution. 20 M. ENGELSTEIN ET AL. See [38, sec. 3] for the proof. From the positive density properties, we know that k˛  2. From the homogene- ity of the blowups we find out that the free boundary in R1C1 is in fact a collection of isolated points. Later in Theorem 6.1 we will show that in fact k˛  3: ˇ L EMMA 5.3 (Isolated singularities). Let u 2 Hloc ./ for   R1C1 be a mini- mizer of J in . Then F .u/ has no accumulation points in . P ROOF. Arguing by contradiction, we assume that F .u/ has an interior accu- mulation point which, without loss of generality, we assume to be the origin. Let .xk ; 0/ be a sequence of singular points converging to 0 with xk > 0. Con- sider the blowup rescaling uk .x/ WD u.xxk˛ x/ . Note that uk .0; 0/ D uk .1; 0/ D 0. k Moreover, by the interior corkscrew condition, there exist ´k 2 .1=2; 3=2/ such that uk jBc0 .´k ;0/ > 0, so u.´k ; 0/ & C by the nondegeneracy condition. Choosing a subsequence, we may assume that ´k ! ´0  1=2, and uk ! u0 in the sense of Lemma 3.4. In particular, u0 is homogeneous by Corollary 4.2, reaching a contradiction with the fact that u0 .1; 0/ D 0 and u0 .´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  Rn satisfies the property (P t ) if the following holds: for every x 2 A there exists an rx > 0 such that for every 0 < r < rx , every subset S of B.x; r/ \ A can be covered with a finite number of balls B.xi ; ri / with xi 2 S such that X (5.1) ri  r t =2: i L EMMA 5.4. If Ht .†.U //D 0 for every minimal cone U in RnC1 , Ht .†.u// D 0 for every minimizer u of J defined on   RnC1 . P ROOF. We first show that †.u/ satisfies the property (P t ). If (P t ) does not hold, we find a point y 2 †.u/ for (P t ) is violated for a sequence rk ! 0: We consider the blowup sequence (5.2) urk .x/ D rk ˛ u.y C rk x/: By Corollary 3.6 we may assume, by taking a subsequence, that urk converges to a minimal cone U . By our assumptions we may cover †.U / \ B.0; 1/ with a finite i k collection of balls fB.xi ; 10 /gi D1 with X 1 it  : 2 i By Lemma 3.4 we know that free S boundaries converge in the Hausdorff sense and thus the set F .urk / \ B.0; 1/ n i B.xi ; i =5/ is flat for all large k: From THE THIN ONE-PHASE PROBLEM 21 Theorem 2.4 we infer that all singularities must be covered by the same balls; that is, for all k  k0 , [ (5.3) †.urk / \ B.0; 1/  B.xi ; i =5/: i After rescaling we see that u satisfies the condition for property (P t ) in the ball B.y; rk /; which is a contradiction. Therefore the property (P t ) holds as claimed. Consider the set Dk WD fy 2 †.u/ W ry  1=kg: Fix a point y0 2 Dk : By property (P t ) applied to r0 D 1=k, we find a finite cover of Dk \ B.y0 ; r0 / with balls B.yi ; ri /; yi 2 Dk ; satisfying X rit  r0t =2: i Similarly, for each ball B.yi ; ri / in the cover we use the property (P t ) to find a finite number of balls B.yij ; rij /; yij 2 Dk ; which cover Dk \ B.yi ; ri / and satisfy X rijt  rit =2; j r0t =4. P and thus i;j rij  By repeating the argument N times we obtain a cover of Dk \ B.y0 ; r0 / by balls B.´l ; rl / that satisfies X rlt  2 N r0t : l This implies that Ht .B.y t 0 ; r0 / \ Dk / D 0 and thus H .Dk / D 0: By countable additivity we obtain the claim.  L EMMA 5.5. If Ht .†.U // D 0 for some t > 0 and for every minimal cone in RnC1 , we then have that Ht C1 .†.V // D 0 for every minimal cone V in R.nC1/C1 : P ROOF. Without loss of generality we may assume †.V / ¤ f0g: Let x 2 †.V / n f0g: By Corollary 3.6 the blowups at any point of †.V / \ @B converge to a minimal cone in dimension .n C 1/ C 1 up to a subsequence. Let Vx be a blowup at x: By Lemma 4.5, Vx is a minimal cone that is invariant in at least one direction. By Lemma 4.5 , by using our assumption, this implies that Ht C1 .†.Vx // D 0, and thus the singular set of every possible blowup cone of any minimizer V has zero Ht C1 -measure. Arguing as in Lemma 5.4 we obtain Ht C1 .†.V // D 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. C OROLLARY 5.6. Every minimizer satisfies Hn 1 .†.u// D 0: 22 M. ENGELSTEIN ET AL. L EMMA 5.7. Let u 2 Hˇ .2B/ be a minimizer of J in 2B with kukCP ˛ .2B/ < E0 : Then there exists a constant C depending on n, ˛, and E0 and a finite collection of balls fB.Xi ; ri /g s.t. m ! /\ (5.4) Hn 1 .F .u/ \ B/ B.Xi ; ri /  C i D1 and m X 1 (5.5) rin 1  : 2 i D1 P ROOF. The proof is by contradiction. For k 2 N assume kuk kCP ˛ .2B/ < E0 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 uk is bounded in Hˇ .B/: Taking a subsequence we may assume that uk converges locally uniformly to a minimizer u (see Corollary 3.6). By Corollary 5.6 the set of singularities †.u/ has Hn 1 -measure zero, and thus they can be covered with finitely many balls Bi satisfying (5.5). Since F .u/ n †.u/ is a C 1; -surface by Theorem 2.4, using the Hausdorff con- vergence of the free boundaries we apply again Theorem 2.4 to see that F .uk / \ B1 n M 1; -surfaces converging to F .u/\B n M B uniformly S S i D1 Bi are also C 1 iD1 i in the C 1 -norm. This is a contradiction with the assumption that the Hausdorff measure blows up as k goes to 1:  The fact that the free boundary has finite perimeter follows now from the same iteration argument as [18, lemma 5.10]. L EMMA 5.8. Let u be as in Lemma 5.7. Then for some constant C depending only on E0 , (5.6) Hn 1 .F .u/ \ B/  C: P ROOF. By Lemma 5.7 we find a finite collection of balls Bri such that [ (5.7) F .u/ \ B  € [ Bri ; with Hn 1 .€/  C and rin 1  12 : P Applying Lemma 5.7 again for each ball Bri , we have [ (5.8) F .u/ \ Bri  €i [ Brij ; with Hn 1 .€i /  C rin 1 and rijn 1  12 rin 1 : Moreover, we have P  [  X Hn 1 .F .u/ \ B1 / \ Brij  Hn 1 .€/ C Hn 1 .€i / i;j    X  1 C 1C rijn 1  C 1 C : 2 i;j THE THIN ONE-PHASE PROBLEM 23 Continuing inductively, after k steps we have that N [ (5.9) F .u/ \ B1  € 0 [ Brq ; qD1 with k X  n 1 0 i H .€ /  C 2  2C; i D0 rqn 1 k: P and 2 This gives the claim.  Finally, the fact that fu > 0g [  has locally finite perimeter in  follows from the previous lemma and well-known results of Federer; see, for example, [3, prop. 3.62] or [27, 4.5.11]. 5.2 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]). ˇ L EMMA 5.9 (See [1, prop. 3.4]). Let u 2 Hloc ./ be a minimizer to (1.1) in . For every B b  we have Z Z ˇ 2 (5.10) jyj jruj D jyjˇ uru   d Hn : B @B Let u be a minimizer and x0 2 F .u/. If we consider a blowup u0 at x0 , then ‰0u .x0 / D ‰1u0 .0/ Z Z D jyjˇ jru0 j2 C m.fu0 > 0g \ Rn \ B1 / ˛ jyjˇ u20 d Hn : B1 @B1 By Lemma 5.9 we get Z ‰1u0 .0/ D jyjˇ u0 ru0   d Hn C m.fu0 > 0g \ Rn \ B1 / @B1 .x0 / Z ˛ jyjˇ u20 d Hn : @B1 ˛ Since ru0 .x/  .x/ D jxj u0 .x/ almost everywhere on the sphere, the first and the third terms cancel out, and we obtain ‰1u0 .0/ D m.fu0 > 0g \ B10 /: Thus, the density ‰0u at a free boundary point is given by the area of the positive phase of any blowup at the same point. We write !n WD m.B10 / for the volume of the n-dimensional ball. 24 M. ENGELSTEIN ET AL. ˇ P ROPOSITION 5.10. Every homogeneous minimizer u 2 Hloc .RnC1 / has density !n ‰1u .0/ D m.fu > 0g \ B10 /  ; 2 and equality is only attained when u is the trivial minimizer. P ROOF. Let u be a minimizer such that ‰1u .0/  !2n . Let x1 2 Fred .u/. Being a regular point, ‰0u .x1 / D !2n . On the other hand, by the homogeneity and the continuity in Lemma 4.6, !n lim ‰ u .x1 / D lim ‰1u .x1 =r/ D ‰1u .0/  : r!1 r r!1 2 Combining both assertions with the monotonicity of ‰ we get that ‰ru .x1 /  !2n . But using the second formula in Theorem 4.1, one can see that this is true only whenever ‰ is ˛-homogeneous with respect to x1 . Thus, u is 1-symmetric and invariant in the direction of hx1 i. By Corollary 5.6 Fred .u/ has full Hn 1 measure on F .u/. Thus, we can find x1 ; : : : xn 1 2 Fred .u/ linearly independent. By the previous discussion u is invari- ant on an .n 1/-dimensional affine manifold, and thus, it is the trivial solution.  C OROLLARY 5.11 (Energy gap). There exists  > 0 depending only on n and ˛ ˇ such that every minimizer u 2 Hloc ./ and every singular point x0 2 †.u/ satisfy !n ‰1u .x0 /  : 2 P ROOF. Assume the conclusion to be false. Then there exist uj minimizers in B1 with u !n ‰1j .0/  C 1=j: 2 Passing to a subsequence, uj ! u0 as in Lemma 3.4. Using Lemma 4.7 we get that u !n ‰1u0 .0/ D lim ‰1j .0/  : j 2 But then u0 is the trivial cone by Proposition 5.10. Since F .uj / ! F .u/ in the Hausdorff distance, using -regularity (see Theorem 2.4) we get that uj is the trivial cone for j big enough.  The value  above depends on the constants and on kukCP ˛ in a neighborhood of x0 . In Section 7 we will show that  does not depend on u at all. 6 Full Regularity in R2C1 In the case of n D 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. THE THIN ONE-PHASE PROBLEM 25 T HEOREM 6.1. Let n D 2. Then there is no singular minimal cone. In particular, the free boundary F .u/ of every minimizer u is C 1;˛ everywhere. P ROOF. We follow closely the arguments in [18, theorem 5.5], building on [34]. The case ˇ D 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 dimen- sional in nature and does not generalize to higher dimensions. Recall the functional under consideration: Z J .u; / D jyjˇ jruj2 C m.fu > 0g \ Rn \ /:  Let V be a nontrivial minimal cone. Define, as in [18], the Lipschitz continuous function 8 <1; ˆ 0  t  R; ln.t / (6.1) R .t / D 2 ln.R/ ; R  t  R2 ; t  R2 : ˆ 0; : Define now the bi-Lipschitz change of coordinates Z.x 0 ; y/ D .x 0 ; y/ C R .j.x 0 ; y/j/e1 and set VRC .Z/ D V .x 0 ; y/. Clearly, one has D.x 0 ;y/ Z D Id C A where kAk  j R 0 .j.x 0 ; y/j/j << 1. Defining now VR exactly as VRC changing R into R , the very same computation as in [18] gives Z J .VRC ; BR2 / C J .VR ; BR2 /  2J .V; BR2 / C jyjˇ jrV j2 kAk2 : BR2 Now, we have Z Z R2Z jyjˇ jrV j2 kAk2 D jyjˇ jrV j2 kAk2 d Hn dr: BR2 R @Br Now since V is homogeneous of degree ˛ by assumption, the function g.x; y/ D jyjˇ jrV j2 is homogeneous of degree ˇ C 2˛ 2 D 1. Therefore by a trivial change of variables on the sphere of radius r and using the fact that n D 2, we get the very same estimate C R!1 Z jyjˇ jrU j2 kAk2  ! 0: BR2 ln.R/ 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.  26 M. ENGELSTEIN ET AL. 7 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 kukHP ˇ .B1 / . 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 with- out additional assumptions on the minimizers involved. 7.1 Uniform Nondegeneracy We will begin by showing uniform nondegeneracy from scratch to deduce uni- form 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. L EMMA 7.1. Let u be a minimizer in Br with 0 2 F .u/. Then sup@Br u  C r ˛ with C depending only on n and ˛. P ROOF. By rescaling we can assume that r D 1. Let Lu WD r  .jyjˇ ru/, consider €.x/ D jxjn1 2˛ , which is a solution of L€ D 0 away from the origin (or €.x/ D logjxj if n D 1 and ˛ D 1=2), and let maxf1 €.2x/; 0g vz.x/ WD ` where ` WD sup u: 1 €.2/ @B1 It follows that u  vz on @B1 and thus J .u; B1 /  J .minfu; vzg; B1 /; THE THIN ONE-PHASE PROBLEM 27 and observing that vz D 0 on B1=2 and vz > 0 on the annulus A WD B1 n B1=2 , we get Z jyjˇ jruj2 C m B1=2;C 0  .u/ B1=2 Z  jyjˇ .jr.minfu; vzg/j2 jruj2 / C m.A0C .minfu; vzg// m.A0C .u// A Z  2 jyjˇ r maxfu vz; 0g  r vz: A By Green’s theorem, writing d D jyjˇ d Hn we get Z   Z jyjˇ jruj2 C m B1=2;C 0 .u/  2 u@ vz d B1=2 @B1=2 (7.1) Z D Cn;˛ ` u d; @B1=2 with Cn;˛ > 0. Using the monotonicity formula and Proposition 5.10, we get that u .r/  u .0/  !.B1 / , and therefore 2 ˛ !.B1 /r d Z (7.2) u2 d C  Jr .u/; r @Br 2 so using Hölder’s inequality and the AM-GM inequality we obtain Z  12 1 1 1 Z Z 2 u d  2 u d Cn;˛  u2 d C Cn;˛ (7.3) @B1=2 @B1=2 2 @B1=2 2  Cn;˛ J1=2 .u/: Combining (7.1), (7.2), and (7.3) we obtain 0 < J1=2 .u/  Cn;˛ `J1=2 .u/; and therefore `  Cn;˛1 .  To show averaged nondegeneracy we need a mean value principle that is well- known, but we include its proof for the sake of completeness. L EMMA 7.2 (Mean value principle). Let u 2 H 1 .ˇ; / be a weak solution to Lu WD r  .jyjˇ ru/ D 0 in , and let x0 2 Rn  f0g with Br .x0 /  . Then − u.x0 / D u d! Br where the mean is taken with respect to the measure d! WD jyjˇ dx. 28 M. ENGELSTEIN ET AL. P ROOF. Changing variables, we have that 1 Z Z ˇ A./ WD ˇ CnC1 jyj u.x/dx D jyjˇ u.x C x0 /dx:  B .x0 / B1 On the other hand, set Z z WD A./ jyjˇ ru.x C x0 /  x dx B1  ˇ jyj ru.x/  .x x0 / dx Z D B .x0 /   nC1 1 Z D jyjˇ ru.x/  rjx x0 j2 dx: 2ˇ CnC2 B .x0 / Since u is a weak solution to r .jyjˇ ru/ D 0 in , we can apply Green’s formula twice to obtain 1 Z z A./ D ˇ CnC2 jx x0 j2 jyjˇ ru.x/   dx 2 @B .x0 / 1 Z D ˇ Cn jyjˇ ru.x/   dx D 0: 2 @B .x0 / Because u is absolutely Rr continuous on lines (see [24, theorem 4.21]), for almost every x we have  ru.tx C x0 /  x dt D u.rx C x0 / u.x C x0 /. Applying Fubini’s theorem we get Z r Z Z r z /dt D A.t jyjˇ ru.tx C x0 /  x dt dx  B1  Z D jyjˇ .u.rx C x0 / u.x C x0 //dx D A.r/ A./: B1 So A.r/ A./ D 0 for all  < r. On the other hand, taking the mean with respect to the measure d! WD jyjˇ dx and using the continuity of u (see [26, theorem 2.3.12]) we obtain ˇ ˇ ˇZ ˇ ˇ 1 ˇ 1 ˇ ˇ ˇu.x0 / lim A./ˇ D lim ˇ .u.x 0 / u.x//d!.x/ˇ ˇ !.B1 / !0 ˇ !0 !.B .x // ˇ  0 B .x0 / ˇ  lim o!0 .1/ D 0:  !0 ¬ Let u be a minimizer C OROLLARY 7.3. in Br with 0 2 F .u/ and let d D ˇ n ˛ jyj d H . Then @Br u d  C r with C depending only on n and ˛. P ROOF. Let v be the L-harmonic replacement of u in Br , that is, the solution to ( Lv D 0 in Br ; (7.4) vu on @Br I THE THIN ONE-PHASE PROBLEM 29 see [29, theorem 3.17]. After differentiating ¬ with respect to the radius, by the mean value principle we get that v.0/ D @Br u d. By the comparison principle and the Harnack inequality we get that − ˛ (7.5) C r  sup u  sup v  C u d:  Br=2 Br=2 @Br 7.2 Behavior of the Distributional Fractional Laplacian 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 in- troduction, 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. T HEOREM 7.4. Let u 2 Hˇ .B2r .x0 // be a minimizer of J in B2r .x0 /, and let x0 2 F .u/. Then, we have .Br .x0 //  C r n ˛ : In particular, .F .u// D 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. P ROOF. Without loss of generality we may assume that x0 D 0. Let Lu WD r  .jyjˇ ru/ and let v be the L-harmonic replacement of u in B2r ; see (7.4). ˇ n ¬ Write d D jyj d H and M WD @B2r u d. By Harnack’s inequality (see [7], for instance) and the mean value principle in Lemma 7.2, inf v  C v.0/ D CM: Br We have that 1 Z Z .Br / D d  v d : Br CM Br Since u  0 in the support of  and u is L-subharmonic (see [2, lemma 2.2]) we get Z Z Z v d D .v u/d   .v u/d : Br Br B2r By the properties of the measure , we obtain Z Z Z ˇ .v u/d  D jyj r.v u/  ru D jyjˇ .jruj2 jrvj2 /; B2r B2r B2r 30 M. ENGELSTEIN ET AL. and using the definition of the functional and the fact that u is a minimizer, we get Z jyjˇ .jruj2 jrvj2 / B2r C 0 /  C r n:  D J .u; B2r / m B2r .u/ J .v; B2r / C m.B2r Altogether, we have that 1 .Br /  C r n; CM and since uniform nondegeneracy (see Corollary 7.3) implies that M  C r ˛ , we can conclude the proof of the first statement. To show the second one, note that since the free boundary has locally finite .n 1/-dimensional Hausdorff measure, given a set E  F .u/ and k 2 N we can find a collection of balls Ik D fBik gi such that [ X E B; sup r.B/  1=k; and r.B/n 1  2Hn 1 .E/: B2Ik B2Ik B2Ik Thus, 1 k!1 X X X .E/  .B/ . r.B/n ˛  sup r.B/1 ˛ r.B/n ! 0: B2Ik B2Ik B2Ik B2Ik  Next we study the measure away from the free boundary. We should empha- size here that even though the estimates in Lemma 7.5 and Theorem 7.6 depend on E0 , they will be used to remove the dependence of our other estimates on E0 . 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. After proving Theorem 7.8, we may drop the hypothesis kukHˇ .B2 /  E0 from both Lemma 7.5 and Theorem 7.6. ˇ L EMMA 7.5. If u 2 Hloc .B2 / is a minimizer of J in the ball B2 with kukHˇ .B2 /  E0 and 0 2 F .u/, then for every x0 D .x 0 ; 0/ 2 B1;0 .u/ we get lim jyjˇ ˇuy .x 0 ; y/ˇ  C dist.x0 ; F .u// ˛ : ˇ ˇ y!0 Moreover, for every ball B centered at Rn  f0g with B 0 b B1;0 .u/, we have that jyjˇ ˇuy .x 0 ; y/ˇ  C dist.x0 ; F .u// ˛ ˇ ˇ for jyj < CB dist.x; F .u//, where the constant CB may depend on B. P ROOF. Let u be a minimizer, and let B WD Br .x0 / with B 0  B1;0 .u/. By [35, lemma 2.2], we can write u.x 0 ; y/ D jyj1 ˇ g.x 0 / C O.y 2 /, where g is a C 1Cˇ . 21 B 0 / function, with a uniform control on the error term in terms of kukL2 .B;jyjˇ / . In particular, limy!0 jyjˇ 1 u.x 0 ; y/ D g.x 0 /. THE THIN ONE-PHASE PROBLEM 31 Let us define ( 0 u.x 0 ; y/ if y  0; (7.6) z.x ; y/ WD u 0 u.x ; y/ if y < 0: u  0 in B. According to [36, lemma 3.26, cor. 3.29] v.x 0 ; y/ D It is clear that Lz ˇ jyj y u 1 z.x ; y/ is an even C 1 . 12 B/ function in H2 ˇ .B/ (note that 1 < 2 ˇ < 0 3 is out of the usual range of ˇ) and satisfying r  .jyj2 ˇ rv/ D 0. The mean value principle (see Lemma 7.2) applies also to this case, so 1 1 Z Z 0 2 ˇ g.x0 / D v.x0 / D R 2 ˇ 1 jyj v.x/ D C 2 ˇ CnC1 jyju.x/; B jyj r 1 2B 2B 1 2 and using P1–P3 if r D dist.x0 ; F .u//, we get g.x00 / D v.x0 /  C r ˇ 2C1C˛ D Cr ˛ : On the other hand, on the upper half-plane we have uy D .y 1 ˇ v/y D .1 ˇ/y ˇ v C y1 ˇ vy ; so y ˇ uy .x 0 ; y/ D .1 ˇ/v.x 0 ; y/ C yvy .x 0 ; y/ and lim y ˇ uy .x 0 ; y/ D .1 ˇ/g.x 0 /  r ˛ ; y!0C the limit being uniform on compact subsets of B.  ˇ T HEOREM 7.6. If u 2 Hloc .B2 / is a minimizer of J in B2 with kukHˇ .B2 /  E0 , then the measure  is absolutely continuous with respect to the Lebesgue measure, and for m-almost every x 2 B10 .u/ we have that d .x/ D 2 lim jyjˇ uy .x 0 ; y/  B1;0 .u/ .x/ dist.x; F .u// ˛ ; dm y!0 with constants depending on n, ˛, and E0 . P ROOF. By Theorem 7.4 we only need to show absolute continuity in B1;0 .u/[ 0 0 B1;C .u/. For x D .x 0 ; 0/ 2 B1;C .u/ by [9, lemma 4.2] we have that lim jyjˇ uy .x 0 ; y/ D 0; y!0 and, for x 2 B1;0 .u/ we have seen in Lemma 7.5 that lim jyjˇ uy .x 0 ; y/  dist.x; F .u// ˛ ; y!0 showing the second part of the statement. 32 M. ENGELSTEIN ET AL. Consider a ball Br .x0 / with x0 2 Rn  f0g and a collection of even smooth functions Br  k  BrC 1 . Then k Z (7.7) .Br /  jyjˇ ru  r k  .BrC.1=k/ /; and for every " > 0 we use the Green’s theorem to get Z Z Z ˇ ˇ jyj ru  r k D jyj ru  r k jyjˇ k ru   d m: jyj" jyjD" Using the symmetry properties and taking limits, Z Z (7.8) jyjˇ ru  r k D 2 lim "ˇ k .x 0 ; "/uy .x 0 ; "/d m.x 0 /: "!0 Next we want to apply the dominated convergence theorem. Let us begin by con- sidering a ball Br .x0 /  B1 centered in the zero phase, with dist.Br0 .x0 /; F .u//  2r. In this case, by Lemma 7.5 we have (7.9) "ˇ uy .x 0 ; "/ . r ˛ ; with constants depending perhaps on u and Br as well. 0 If instead Br0 .x0 / b B1;C .u/, by [36, theorem 3.28] u is an even C 1 function on Br0 .x0 /, so jyjˇ uy D O.jyj1Cˇ /: Thus (7.10) "ˇ uy .x 0 ; "/ . r 2 2˛ : In both cases, the dominated convergence theorem applies, and Z Z ˇ ˇ lim " k uy d m D k lim ." uy .  ; "//d m; "!0 BrC.1=k/ \fyD"g 0 BrC.1=k/ "!0 and by (7.7) and (7.8), we obtain Z ˇ .Br /  2 k lim ." uy .  ; "//d m  .BrC.1=k/ /: 0 BrC.1=k/ "!0 In particular, lim"!0 ."ˇ uy .  ; "// 2 L1loc .B1;0 .u/ [ B1;C 0 .u//, and taking limits in k we get Z .Br / D 2 lim ."ˇ uy .  ; "//d m:  Br0 "!0 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]). 0 C OROLLARY 7.7. If u 2 is a minimizer in B2 with kukHˇ .B2 /  E0 , then B1;C .u/ satisfies the exterior corkscrew condition, i.e. there exists a constant C1 such that THE THIN ONE-PHASE PROBLEM 33 for every x 2 F .u/ and every 0 < r < dist.x; @B1 / one can find x0 2 Br .x/ so that 0 B.x0 ; C1 r/ \ B1;C .u/ D ¿: P ROOF. This is a consequence of Theorems 7.4 and 7.6, and the positive density condition for the zero phase. Indeed, given a ball Br  RnC1 , combining both theorems we get Z n ˛ r & .B1;0 .u/ \ Br /  CE0 dist.x; @B1 / ˛ B1;0 .u/\Br   ˛ C sup dist.x; @B1 / jB1;0 .u/ \ Br j; B1;0 .u/\Br and the positive density condition implies that jB1;0 .u/ \ Br j  CE0 r n : Thus, sup dist.x; @B1 /  CE0 r; B1;0 .u/\Br which is equivalent to the exterior corkscrew condition.  7.3 Uniform Hölder Character The uniform nondegeneracy of Section 7.1 lets us conclude uniform control on the Hölder norm of u. T HEOREM 7.8. Let u be a minimizer of J in Br with 0 2 F .u/. Then ju.x/j  C jxj˛ for every x 2 @Br=2 with C depending only on n and ˛. P ROOF. Again we set v to be the L-harmonic replacement of u inside of Br as z WD v u so that in (7.4). Let u Lz u D Lv Lu D D r  .jyjˇ ru/ and uz 2 H01;2 .Br I jyjˇ /. Consider the Green function G W Br  Br ! R such that LG.  ; ´/ D ı´ and 1;2 G.  ; ´/ 2 Hloc .Br n f´g/ with null trace on @Br (see [25, prop. 2.4]). By [25, prop. 2.1, lemma 2.7] there exists p0 > 1 so that u z is the unique function in 1;p H0 0 .Br I jyjˇ / such that Lzu D , and moreover Z (7.11) z.´/ D u G.´; x/d .x/ Br for almost every ´ 2 Br . Below, in Lemma 7.9, weR will see that the equality (7.11) is in fact valid for every ´ 2 Br=4 , that is, u z D Br G.  ; x/d .x/. In particular, Z v.0/ D u z.0/ D G.0; x/d .x/: Br 34 M. ENGELSTEIN ET AL. Next we use the following estimate (see [25, theorem 3.3]): let ´; x 2 Br=4 . Then Z r s ds G.´; x/  ; jx ´j w.B.x; s// where w is the A2 weight w.x/ D jyjˇ . Computing, for x D .x 0 ; y/ we obtain Z yCs w.B.x; s//  s n jtjˇ dt  s n maxfjyj; sgˇ C1 : y s 0 First we assume that n 2˛ > 0. Thus, if x 2 Br=4 , then Z r n ˇ n ˇ C1 (7.12) G.´; x/  s ds  jx ´j D jx ´j2˛ n : jx ´j Note that .Br /  C r n ˛ by Theorem 7.4. Thus, writing A t;s WD Bs n B t , we have that Z v.0/ D G.0; x/d .x/ Br Z 1 Z  .fx 2 Br=4 W G.0; x/ > t g/dt C G.0; x/d .x/: cr 2˛ n Ar=4;r 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 Z 1   Z 1 n ˛ ˛ v.0/   B 1 dt C C r  C t n 2˛ dt C C r ˛ D C r ˛ : cr 2˛ n C t .n 2˛/ cr 2˛ n By the mean value theorem we conclude that − v d  C r ˛ ; @Br where d D jyjˇ d Hn . The theorem follows by observing ¬ that, as in (7.5), the mean of v dominates u by sup@Br=2 u  sup@Br=2 v  C @Br v d. In case n 2˛ D 0, which could only happen for n D 1 and ˛ D 1=2, estimate (7.12) reads as   r G.´; x/  log ; 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.  R L EMMA 7.9. Br G.´; x/d .x/ is continuous in ´ 2 Br=4 . THE THIN ONE-PHASE PROBLEM 35 P ROOF. Let " < r=2 and let ´1 ; ´2 2 Br=4 , with j´1 ´2 j  "=2. Then Z jG.´1 ; x/ G.´2 ; x/jd .x/ Br Z (7.13)  jG.´1 ; x/ G.´2 ; x/jd .x/ Br nB" .´1 / Z Z C G.´1 ; x/d .x/ C G.´2 ; x/d .x/: B" .´1 / B" .´1 / Next we use (7.12) and Theorems 7.4 and 7.6. By decomposing the domain on dyadic annuli, in case n 2˛ > 0 we get Z G.´1 ; x/d .x/ B" .´1 / XZ (7.14)  G.´1 ; x/d .x/ j 0 A2j 1 ";2j " .´1 / X X . .B2j " .´1 //.2j 1 "/2˛ n . "˛ 2j˛ : j 0 j 0 In case n 2˛ D 0 we obtain X  r  "˛ 2j˛ log 2j " j 0 on the right-hand side instead, and in case n 2˛ < 0 we obtain X "n ˛ r 2˛ n 2j.n ˛/ : j 0 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). On the other hand, by [26, theorem 2.3.12] Green’s function is uniformly con- tinuous on the set f.´; x/ 2 Br  Br W j´ xj > "g, so t !0 jG.´1 ; x1 / G.´2 ; x2 /j  ı" .j´1 ´2 j C jx1 x2 j/ with ı" .t / ! 0: Thus, Z jG.´1 ; x/ G.´2 ; x/jd .x/  ı" .j´1 ´2 j/.Br / ! 0: Br nB" .´1 / Assuming that j´1 ´2 j is small enough, we obtain that Z jG.´1 ; x/ G.´2 ; x/jd .x/ Br is as small as wanted and the claim follows.  36 M. ENGELSTEIN ET AL. 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 Br with 0 2 F .u/ has uniform C ˛ character in Br=2 and the same for the Hˇ 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 kukHˇ from all of our results above. 7.4 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. T HEOREM 7.11. Let u 2 Hˇ .B2r / be a minimizer of J in B2r such that 0 2 F .u/. Then we have .Br /  C r n ˛ : P ROOF. Let Lu WD r .jyjˇ ru/ and let v be the L-harmonic replacement of u z WD v u, and consider the Green function G W Br Br ! R in Br (see (7.4)). Let u as in the proof of Theorem 7.8. Let 0 <  < 1 to be fixed later. By P1–P3 in Theorem 2.3, there exists a point ´0 2 Br with (7.15) u.´0 /  .r/˛ ; with constants depending only on n and ˛ by Remark 7.10. By P1 there is a constant c such that for every ´ 2 B.´0 ; cr/ we have that u.´/  .r/˛ . Since  is supported on the zero phase of u, the ball B.´0 ; cr/ is away from its support, and Z z.´/ D u G.´; x/d .x/: Br nB.´0 ;cr/ Using the strong maximum principle (see [29, theorem 6.5]) and (7.12), for almost every ´ 2 B.´0 ; cr=2/ we get z.´/  .Br / u sup G.´; x/ D .Br / sup G.´; x/ x2Br nB.´0 ;cr/ x2Br=4 nB.´0 ;cr/  .Br / sup jx ´j2˛ n D .Br /.cr/2˛ n : x…B.´0 ;cr/ That is, (7.16) z.´/ . .Br /.cr/2˛ u n : On the other hand, note that u is continuous. By the Riesz representation theo- rem, there exists a probability measure !L´ such that Z v.´/ D u.x/d!L´ .x/: @Br We can choose r so that @Br intersects a big part of a corkscrew ball; i.e., assume that there exists a point 0 2 @Br0 that is the center of a ball B 0 .0 ; cr/ where u THE THIN ONE-PHASE PROBLEM 37 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  2 B.0 ; cr/ satisfy that u./  C r ˛ by the nondegeneracy condition and the optimal regularity. Call U WD @Br \ B.0 ; cr/. Then v.´/ & r ˛ !L´ .U /: But !L´ .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 z.´0 / u C r ˛ C 0 .r/˛ .Br / &   Cn;˛ r n ˛ .cr/2˛ n .cr/2˛ n for  small enough. In case n 2˛ D 0, that is, for n D 1 and ˛ D 1=2, using similar changes as in the proof of Theorem 7.8 we get   r z.´/ . .Br / u sup log  .Br /jlog j x…B.´0 ;cr/ jx ´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 . ˇ T HEOREM 7.13. If u 2 Hloc ./ is a minimizer of J in , then the measure  is absolutely continuous with respect to the Lebesgue measure in 0 .u/. Moreover, given x0 2 F .u/ and r > 0 such that B2r .x0 /  , then (7.18) .Br .x0 //  r n ˛ ; and for almost every x 2 Br0 .x0 / we have that d .x/ D 2 lim jyjˇ uy .x 0 ; y/  0 .u/ .x/ dist.x; F .u// ˛ ; dm y!0 with constants depending only on n and ˛. 8 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. 38 M. ENGELSTEIN ET AL. ˇ T HEOREM 8.1. Let u 2 Hloc ./ be a minimizer of (1.1) in a domain . Then †.u/ is .n k˛ /-rectifiable, and for every D b  we have  Hn k˛ .†.u/ \ D/  Cn;˛;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 strat- ification. 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 x0 2 F .u/ at which the blowups have .n k˛ /-linearly independent translational symmetries. 8.1 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 sta- bility) how far a function is from having no more than k directions of translational symmetry. D EFINITION 8.2. We write V k for the collection of linear k-dimensional subspaces of Rn . A function u is said to be k-symmetric if it is ˛-homogeneous with respect to some point, and there exists a L 2 V k so that u.x C v/ D u.x/ for every v 2 L: z A function u is said to be .k; /-symmetric in a ball B if for some k-symmetric u we have Z r 2 n jyjˇ ju uzj2 dy < : B Next we define the k-stratum S k .u/, the .k; /-stratum Sk .u/, and the .k; ; r/- k .u/. A key insight here is to define these strata by the blowups having stratum S;r k or fewer symmetries as opposed to exactly k symmetries. D EFINITION 8.3. Let 0  k  n, 0 < " < 1, and 0 < r < d .x/ WD dist.x; @/, let u be a continuous function in , and let x 2 F .u/. We say that:  x 2 S k .u/ if u has no .k C 1/-symmetric blowups at x;  x 2 Sk .u/ if u is not .k C 1; /-symmetric in Bs .x/ for 0 < s  minf1; d .x/g;  x 2 S;rk .u/ if u is not .k C 1; /-symmetric in B .x/ for r  s  s minf1; d .x/g. If it is clear from the context we will omit u from the notation. THE THIN ONE-PHASE PROBLEM 39 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. L EMMA 8.4. Let 0  j  k  n, 0 < "   < 1, 0 < r  s < dist.x; @/, and ˇ let u 2 Hloc ./ be a minimizer in . Then: (1) S 0  S 1      S n 1 D S n D F .u/. Moreover, for the reduced  boundary, we have that Fred .u/  S n 1 n S n 2 and †.u/  S n k˛ . j (2) We have S  Sk  S k , and moreover, S k D >0 Sk . S j j k and moreover, S k D T k (3) Also S  S;r  S;s  r>0 S;r . L2loc .Ijyjˇ / (4) The sets Sk are closed, in both x and u: if ui ! u and xi ! x with xi 2 Sk .ui /, then x 2 Sk .u/. L2loc .Ijyjˇ / (5) If ui ! u, i ! 0, and ui are .k; i /-symmetric in B1 , then u is k-symmetric in B1 . P ROOF. (1) The inclusions S k  S kC1 of the first property are trivial. The last equali- ties are consequences of the nondegeneracy. The fact that Fred .u/\S n 2 D ¿ can be deduced from the Hausdorff convergence of the free boundaries  described in Lemma 3.4 and Theorem 2.4. Finally, †.u/  S n k˛ is a consequence of Lemmas 4.5 and 5.2. j (2) The inclusions S  Sk of the second property come from the definitions: if x … Sk then there exist a ballR B ˇ  centered at x and a .k C 1/- z so that r.B/ 2 n 2 symmetric u B jyj ju zj dy <   : But u u z is also j .j C 1/-symmetric. Thus, x … S . The fact that Sk  S k is a consequence of the uniform convergence on Lemma 3.4: if x … S k , then u has a .k C 1/-symmetric blowup sequence ui ! u0 at x converging uniformly. Thus, x x0 ˇˇ2 Z ˇ  ˇ ˇˇ ˛ ˇ jyj ˇu.x/ i u0 ˇ dx Bi .x0 / i ˇ ˇ2 ˇ ˇ u.x0 C i x/ Z ˇ C2˛CnC1 ˇ ˇ D i jyj ˇ ˛ u0 .x/ˇˇ dx B1 i  inC2 !.B1 /kui u0 kL1 : That is, x0 ˇˇ2 ˇ  ˇ x Z n 2 ˇˇ i !1 ri˛ u0 ˇ i jyj ˇu.x/ ˇ dx ! 0: Bi .x/ i 40 M. ENGELSTEIN ET AL. Therefore, for every " there exists a ball S smallkenough so that u is .kC1; "/- k symmetric in it. In particular, S  >0 S . To see the converse, assume that x …  Sk . Then for every i 2 N there S exist a .kC1/-symmetric function u zi , invariant with respect to Li 2 V kC1 , and ri < minf1; dist.x; @/g such that 1 1 Z nC2 jyjˇ ju.x/ u zi .x/j2 dx < : ri Bri i In the case when ri stays away from 0, since ri < 1, we can take a sub- sequence converging to r0 2 .0; 1/, and one can see that u is .k C 1/- symmetric in the ball Br0 .x0 /. Otherwise, consider u.x0 C ri x/ zi .x0 C ri x/ u ui WD zi;i D and u : ri˛ ri˛ By taking subsequences, we can assume that Li ! L0 locally in the Hausdorff distance, and that ui ! u0 locally uniformly. One can check also using the Hölder character of u that fz ui;i g is uniformly bounded in 2 ˇ L .BI jyj /, so taking subsequences again, we can assume the existence of uz0 so that u z0 in L2 .BI jyjˇ /. This function will be .k C 1/- zi;i ! u symmetric, being invariant in the directions of L0 . By the triangle inequal- ity we get Z Z Z ˇ 2 ˇ 2 jyj ju0 u z0 j dx . jyj ju0 ui j dx C jyjˇ jui u zi;i j2 dx B1 B1 B1 Z C jyjˇ jz ui;i u z0 j2 dx: B1 The first and the last integrals converge to 0 by our choice of the subse- quence. For the middle term, just change variables as before: 1 Z Z ˇ jyj jui u 2 zi;i j dx D nC2 jyjˇ ju.x/ uzi .x/j2 dx ! 0: B1 ri Bri Thus we have that u0 D ue0 and therefore, x … Sk . j j k of the third property come from the (3) The inclusions S  S;r  S;s definitions and thus Sk  r>0 S;r k . The converse implication is also T trivial. (4) The closedness is obtained by a contradiction argument again. It is straight- forward but we write it here for the sake of completeness. Assume by contradiction that x … Sk .u/. Then there exist a .k C 1/- symmetric function u z and a radius r such that 1 Z 0 WD nC2 jyjˇ ju.x/ u z.x/j2 dx < : r Br .x/ THE THIN ONE-PHASE PROBLEM 41 Let  < 1 be fixed and consider i0 2 N so that B r .xi /  Br .x/ for every i  i0 . By the triangle inequality 1 Z jyjˇ jui .x/ u z.x/j2 dx . r/nC2 B r .xi / 1 2 0  nC2 kui ukL 2 .B .x /Ijyjˇ/ C nC2 : . r/  r i  0 We define  so that  nC2 D C 0 2 . Choose i0 big enough so that every 2 i  i0 satisfies that kui ukL 2 .B .x /Ijyjˇ / < . r/nC2  20 . Then xi … r i Sk .ui /, contradicting the hypothesis. (5) Assume that u zi is invariant with respect to Li 2 V kC1 and Z jyjˇ jui u zi j2  i : Consider a subsequence fui g so that the varieties Li ! L locally in the Hausdorff distance. Using the triangle inequality as in (4), it follows that u is .k; ıi /-symmetric with ıi ! 0.  ˇ P ROPOSITION 8.5. There exists .n; ˛/ > 0 such that if u 2 Hloc ./ is a mini- n  k˛ mizer of J in a domain   RnC1 , then †.u/  S .u/. P ROOF. It is enough to show that if u is a minimizer of J in B2 .0/, then †.u/\ n k B1 .0/  S ˛ .u/. By contradiction, let us assume that there is a sequence of positive numbers i !1 i ! 0, functions ui minimizing J in B2 .0/ and xi 2 †.ui / \ B1 .0/, ri 2 .0; 1, with ui being .n k˛ C 1; i /-symmetric in Bri .xi /, and let Li be an .n k˛ C 1/-dimensional subspace that leaves invariant one of the admissible .n k˛ C 1/-symmetric approximants. By rescaling we can assume that ri D 1.  Passing to a subsequence we can assume that Li ! L0 2 V n k˛ C1 locally in the Hausdorff distance and xi ! x0 . By the compactness results in Lemma 3.4 we have a uniform limit u0 that is a minimizer as well, and it is .n k˛ C 1/- symmetric with invariant manifold L0 . By Lemma 4.4 any blowup u0;0 at x0 will be .n k˛ C 1/-symmetric as well. Applying Lemma 4.5 .n k˛ C 1/ times, we find that the restriction of u0;0 to the orthogonal manifold L?  0 is a .k˛ 1/- dimensional minimal cone, which, by Lemma 5.2, is the trivial solution, and so is u0;0 . Thus, x0 is a regular point for u0 . 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 xi 2 Fred .ui /, reaching a contradiction.  42 M. ENGELSTEIN ET AL. 8.2 The Refined Covering Theorem Our estimates on the size and structure of the singular set †.u/ come from sim- ilar results concerning the Sk .u/. In particular, we prove the following covering result: T HEOREM 8.6. Let u 2 Hˇ .B5 / be a minimizer to (1.1) in B5 with 0 2 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 fBr .xi /gN i D1 with N  Cn;˛; r k such that [ k S;r .u/ \ B1  Br .xi /: i In particular, k jBr0 .S;r \ B1 /j  Cn;˛; rn k for every 0 < r  1 and Hk .Sk .u/ \ B1 /  Cn;˛; : From Proposition 8.5 and Theorem 8.6, we can conclude the following corollary, which comprises the second part of Theorem 8.1 above. C OROLLARY 8.7. If u 2 Hˇ .B5 / is a minimizer to (1.1) in B5 with 0 2 F .u/, then †.u/ is .n k˛ /-rectifiable and for every 0 < r  1 we have  jBr .†.u/ \ B1 /j  Cn;˛ r k˛ : In particular,  Hn k˛ .†.u/ \ B1 /  Cn;˛ : 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). T HEOREM 8.8. Let u be a nonnegative, even minimizer to (1.1) in a domain . Then Sk .u/ is k-rectifiable for every , and hence each stratum S k .u/ is k-recti- fiable 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]). T HEOREM 8.9. Let  > 0. There exists .n; ˛; / such that, for every minimizer u 2 Hˇ .B5 / of J in B5 with 0 2 F .u/ and 0 < R < 1=10, there is a finite k collection U of balls B with center xB 2 S;R and radius R  rB  1=10 that satisfy the following properties: (A) Covering control: [ k S;R \ B1  B: B2U THE THIN ONE-PHASE PROBLEM 43 (B) Energy drop: For every B 2 U, u either rB D R or sup ‰2rB  sup ‰2u : 2B B2 (C) Packing: X rBk  c.n; ˛; /: B2U 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 mea- sure 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 2 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. 44 M. ENGELSTEIN ET AL. 8.3 Tools for Bad Balls: Key Dichotomy T HEOREM 8.10 (Key dichotomy). Let ; ; ; 0 > 0 be fixed numbers with  < 2. There exists an 0 .n; ˛; ; ; ; 0 / < =100 such that for every   0 , every r > 0, every E > 0, and every minimizer u 2 Hˇ .B4r / of J in B4r with 0 2 F .u/ u and supBr ‰2r  E, then either  ‰ ur  E 0 on S;r k \ Br , or k 1 u  there exists ` 2 L so that fx 2 Br W ‰2r .x/  E g  Br .`/. 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 C 1/-symmetric blowups by Lemma 4.4. L EMMA 8.11. Let ; ; ; 0 > 0 be fixed numbers with  < 2. There exist 0 ;  > 0 such that for every  < 0 , every E > 0, and every minimizer u of J in B4 with 0 2 F .u/ and supB1 ‰2u  E, if there exist w0 ; : : : ; wk 2 B1 and affine manifolds Li WD hw0 ; : : : ; wi i 2 V i with wi … B .Li 1 / u and ‰2 .wi /  E  for every i 2 f0; : : : ; kg; then (8.1) ‰ u .x/  E 0 on B .Lk / \ B1 and k (8.2) S; \ B1  B .Lk /: 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: D EFINITION 8.12. Let x 2 B2 , 0 < R < r < 2, and u be a minimizer to J in B5 . We say that the ball Br .x/ is good if ‰ ur  E 0 k on S;R \ Br .x/; and otherwise we say that Br .x/ is bad. By Theorem 8.10 in any bad ball B there exists an affine .k 1/-manifold `B with u k 1 ˚ (8.3) w 2 B W ‰2r .w/  E   Br .`B /: 8.4 Tools for Good Balls: Packing Estimates and GMT In this section we control the local flatness of the singular strata by the drop in monotonicity. To do this we introduce a key tool from geometric measure theory that estimates the flatness of a set. Given a Borel measure , a point x, and a radius r, the beta coefficient is defined as follows: 1 dist.´; L/2 Z k (8.4) ˇ;2 .Br .x//2 WD ˇ;2 k .x; r/2 D inf k d.´/ L2Vak r Br .x/ r2 THE THIN ONE-PHASE PROBLEM 45 where Vak stands for the collection of k-dimensional affine sets of Rn . The beta coefficients are meant to measure in a scale invariant way how far a measure is from being flat, in this case in the L2 distance, although other Lp versions have been used in the literature for 1  p  1 quite often, dating back to [30] (for the L1 version) and David-Semmes [12] (for the Lp version). If we control the size of the ˇ 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. T HEOREM 8.13 (Discrete-Reifenberg Theorem; see [32, theorem 3.4]). Let fBrq .q/gq be a collection of disjoint balls, with q 2 B1 .0/ and 0 < rq  1, and let  be the packing measure  WD q rqk ıq , where ıq stands for the Dirac P delta at q. There exist constants DR ; CDR > 0 depending only on the dimension such that if Z 2rZ k ds ˇ;2 .´; s/2 d.´/  DR r k for every x 2 B1 .0/; 0 < r  1; 0 Br .x/ s then X .B1 .0// D rqk  CDR : q 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. T HEOREM 8.14. Let  > 0 be given. There exist ı.n; ˛; / and c.n; ˛; / such that for every u 2 Hˇ .B5r / minimizing J in B5r .x/ with x 2 F .u/ and ( u is .0; ı/-symmetric in B4r .x/; (8.5) u is not .k C 1; /-symmetric in B4r .x/; and every Borel measure , we have that c.n; ˛; / Z k 2 u ‰ru .w/ d.w/:  (8.6) ˇ;2 .Br .x//  ‰4r .w/ rk Br .x/ We follow the proof of [22, theorem 5.1] closely. First, the authors give an explicit formula for the beta coefficients. L EMMA 8.15. Let X be the center of mass of a Borel measure  on B D Br .x/. Let fi gniD1 be the decreasing sequence of eigenvalues of the nonnegative bilinear form − Q.v; w/ WD .v  .´ X //.w  .´ X //d.´/; B 46 M. ENGELSTEIN ET AL. and let fvi gniD1 be a corresponding orthonormal sequence of eigenvectors, that is, vi  vj D ı ij and Q.vi ; v/ D i vi  v. Then 1 dist.´; Lk /2 .B/ .kC1 C    C n / Z k 2 ˇ;2 .B/ D k 2 d.´/ D k ; r B r r r2 where Lk WD X C spanhv1 ; : : : ; vk i. Next we find a relation between the eigenvalues of Q and Allen-Weiss’ energy. L EMMA 8.16. Under the hypothesis of Lemma 8.15, for every u 2 Hˇ .B5r / min- imizing J in B5r .x/ and every i  n, we have that 2 Z i nC2 jyjˇ .vi  Du.´//2 d´ r A2r;3r .x/ (8.7) − u .w/ ‰ru .w/ d.w/:  C ‰4r Br .x/ P ROOF. 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. L EMMA 8.17. Let  > 0 be given. There exists a ı.n; ˛; / and c.n; ˛; / such that, for every orthonormal basis fvi gniD1 and every u 2 Hˇ .B5r / minimizing J in B5r .x/ with x 2 F .u/ and satisfying (8.5), we have that kC1 1 Z X n ˇ (8.8) r jyj .vi  Du.´//2 d´: c.n; ˛; / A2r;3r .x/ i D1 P ROOF. The proof follows that of [22, (19)] and we omit it.  P ROOF OF T HEOREM 8.14. By Lemmas 8.15, 8.17, and 8.16 we get that k .B/ ˇ;2 .B/2  .n k/kC1 r kC2 kC1 .B/ i X Z  k .n k/c.n; ˛; / jyjˇ .vi  Du.´//2 d´ r r nC2 A2r;3r .x/ i D1 c.n; ˛; / Z u ‰ru .w/ d.w/:   ‰4r .w/  rk Br .x/ Appendix A Relation with the Nonlocal Bernoulli Problem As in [20, lemma 2.1], we see that the study of minimizers of J includes the study of minimizers of J . THE THIN ONE-PHASE PROBLEM 47 P ROPOSITION A.1. If f is a minimizer of J in the unit ball of Rn , then f  Py is a minimizer of J in every ball B such that B 0 b B10 . If u D f  Py 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 D 0g, and u.x; y/ D O.j.x; y/j˛ /, then ujRn f0g is a minimizer for J in every ball. We follow [20, lemma 2.1]; that is, we use the following result from [6, sec. 7]. L EMMA A.2. Let f; g satisfy that J0 .f; B1 / and J0 .g; B1 / < 1, and suppose that f g is compactly supported in B1  Rn . Then we have that Z J0 .g; B1 / J0 .f; B1 / D cn;˛ inf jyjˇ .jrv.x; u/j2 jr.f  Py /.x/j2 /;  where the infimum is taken among all the symmetric bounded Lipschitz domains  with the property that  \ .Rn  f0g/  B1 and among all symmetric functions v with trace g satisfying that v f  Py is compactly supported on . P ROOF OF P ROPOSITION A.1. Let f be a minimizer of J in the unit ball of Rn and let Br be a ball such that Br0 b B10 . We want to show that u WD f  Py is a minimizer of J in Br . Let v W RnC1 ! R so that v  u in RnC1 n Br and v 2 H 1 .ˇ; Br /. Let g be the trace of v in Rn  f0g. By Lemma A.2 we have that Z (A.1) J0 .g; B1 / J0 .f; B1 /  cn;˛ jyjˇ .jrvj2 jruj2 / BrC" for every " > 0. Since gj.B 0 /c  0, g is an admissible competitor for f and J.f; B1 /  J.g; B1 /, i.e., J0 .g; B1 / J0 .f; B1 /  m.fg > 0g \ B1 / C m.ff > 0g \ B1 / (A.2) D m.fu > 0g \ Br0 / m.fv > 0g \ Br0 /: The proposition follows combining (A.1) and (A.2) and letting " ! 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 : C OROLLARY A.3. If u W Rn ! R is a minimizer to J in B2  Rn and 0 2 F .u/, then kukC ˛ .B1 /  C , it satisfies the nondegeneracy condition u.x/  C dist.x; F .u//˛ for x 2 B1 , the positive phase satisfies the corkscrew condi- tion, every blowup limit is ˛-homogeneous, and the boundary condition in (1.2) is satisfied at Fred .u/. Moreover, the positive phase fu > 0g\B1 is a set of finite perimeter, the singular set is an .n 3/-rectifiable set, it is discrete whenever n D 3, and it is empty if n  2. All the constants depend only on n and ˛. 48 M. ENGELSTEIN ET AL. Appendix B Uniqueness of Extensions In Proposition A.1 we have used the following result, included in [7, prop. 3.1]. Here we provide a proof that is different than the one appearing in [7]. P ROPOSITION B.1. Let ˛ 2 .0; 1/ and ˇ D 1 2˛, and set Lu D div.jyjˇ ru/ in RnC1 . Suppose that v W RnC1 C ! R is nonnegative outside Rn , and it is a nC1 solution to Lv D 0 in RC with v.x 0 ; 0/ D 0 for all x 0 2 Rn and jv.x/j  C jxj˛ . Then v  0. 1 P ROOF. First, since jyjˇ is C 1 away from the hyperplane Rn , v 2 Cloc .RnC1 C /. Let now i 2 f1; : : : ng, and set 1  v xCm ei v.x/ fm .x/ D : 1=m Let Br D Br .x 0 ; 0/ be a ball centered at .x 0 ; 0/ 2 Rn  f0g with radius r, and let B2r be its double ball. Set also w.x/ D w.x 0 ; y/ D y ˇ for y > 0. Since fm is a solution of Lfm D 0 in BrC D B \ RnC1 C , [26, theorem 2.4.3] shows that !1=2 1 Z max jfm .x/j  C C jfm j2 w : BrC w.B 2r / B C 2r From convergence of difference quotients (similarly to [23, theorem 3, p. 277]), if C v 2 H 1 .ˇ; B2r /, the last estimate will imply that fm is uniformly bounded in BrC by a constant Cr . Therefore, from the boundary Caccioppoli estimate [26, (2.4.2)] we have that C C Z Z Z 2 2 jrfm j w  2 jfm j w  2 Cr2 w  Cn;r;w < 1I C Br=2 r C Br r C Br C hence ffm g is bounded in H 1 .ˇ; Br=2 /. From weak compactness, a subsequence C of ffm g converges to a solution of Lu D 0 in Br=2 , and since fm ! @i v pointwise, C C we obtain that @i v is an H 1 .ˇ; Br=2 / solution in Br=2 . Hence @i v is a solution to Lu D 0 in RnC1 C . Now, for x D .x 0 ; y/ 2 RnC1 C , let R D jxj. We distinguish between two cases: y > R=16 and y < R=16. In the first case, set BR to be the ball of radius R, centered at x. Note then that BR=16  RnC1 C . Then, from [26, theorem 2.3.1], Caccioppoli’s estimate, and the assumption jv.x/j  C jxj˛ , C C C Z Z 2 2 j@i v.x/j  j@i vj w  jvj2 w w.BR=32 / BR=32 w.BR=32 / R2 BR=16 C w.BR=16 /  2 sup jvj  CR2˛ 2 : R w.BR=32 / BR=16 THE THIN ONE-PHASE PROBLEM 49 In the second case, let BR be the ball centered at .x 0 ; 0/ with radius R, and C N C1 C denote BR D BR \ RC . Then x 2 BR=8 ; therefore from [26, theorem 2.4.3] and the boundary Caccioppoli estimate, C C C Z Z j@i v.x/j2  C j@ i vj2 w  C 2 jvj2 w w.BR=8 / BR=8 C w.BR=8 / R C BR=4 C C w.BR=4 /  2 C sup jvj  CR2˛ 2 : R w.BR=8 / BC R=4 So, in all cases, j@i v.x/j  C jxj˛ 1 . Letting R ! 1 and using the maximum principle, we find that @i v D 0 for any i D 1; : : : ; n. Therefore v does not depend on the first n variables, so v.x 0 ; y/ D v.y/. Hence, in RnC1 C , 0D div.y ˇ rv.y// D @y .y ˇ v 0 .y// ) y ˇ v 0 .y/ D cz; for some constant cz. From [26, theorem 2.4.6], v is Hölder continuous up to the boundary; therefore, for any y > 0, Z y Z y 0 cz v.y/ D v.y/ v.0/ D v D czs ˇ ds D y1 ˇ ; 0 0 1 ˇ which implies that c j D .1 ˇ/y ˇ jz 1 jv.y/j D .1 ˇ/y ˇ 1 jv.0; y/j  .1 ˇ/y ˇ 1 ˛ y D .1 ˇ/y ˛ for any y > 0. Letting y ! 1 we obtain that cz D 0; hence v 0 .y/ D 0 as well, which implies that v is a constant. Since v vanishes on Rn , this implies that v  0.  Acknowledgments. M.E. was partially supported by an National Science Foun- dation postdoctoral fellowship, National Science Foundation DMS 1703306 and by David Jerison’s grant DMS 1500771. A.K. acknowledges financial support from the Spanish Ministry of Economy and Competitiveness, through the María de Maeztu Programme for Units of Excellence in R&D (MDM-2014-0445). M.P. was funded by the European Research Council under grant agreements 320501- ANGEOM and 834728-QUAMAP. G.S. has received funding from the European Union’s Horizon 2020 research and innovation programme under Marie Skłodow- ska-Curie grant agreement no. 665919. A.K., M.P., and G.S. were also partially funded by 2017-SGR-0395 (Catalonia) and MTM-2016-77635-P (MINECO, Spain). Y.S. is partially supported by the Simons Foundation. M.E. would also like to thank Nick Edelen for many fruitful conversations re- garding the quantitative stratification and rectifiable Reifenberg framework. A.K., M.P., and G.S. would like to thank Xavier Cabré, Tomás Sanz, Matteo Cozzi, Albert Mas, Maria del Mar González, Luis Silvestre, and Stefano Vita for some conversations around [36]. They would also like to thank Mihalis Mour- goglou for conversations regarding the degenerate elliptic measure. 50 M. ENGELSTEIN ET AL. Bibliography [1] Allen, M. Separation of a lower dimensional free boundary in a two-phase problem. Math. Res. Lett. 19 (2012), no. 5, 1055–1074. doi:10.4310/MRL.2012.v19.n5.a8 [2] Alt, H. 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M AX E NGELSTEIN A APO K AURANEN University of Minnesota-Twin Cities University of Jyväskylä 206 Church St. SE, Room 127 Department of Mathematics Minneapolis, MN 55455 and Statistics USA P.O. Box 35 (MaD) E-mail: mengelst@umn.edu FI-40014 University of Jyväskylä FINLAND E-mail: aapo.p.kauranen@jyu.fi M ARTÍ P RATS G EORGIOS S AKELLARIS Universitat Autònoma de Barcelona Universitat Autònoma de Barcelona Facultat de Ciències Facultat de Ciències 08193 Bellaterra, Barcelona 08193 Bellaterra, Barcelona SPAIN SPAIN E-mail: mprats@mat.uab.cat E-mail: gsakellaris@mat.uab.cat YANNICK S IRE Johns Hopkins University Department of Mathematics Krieger Hall 3400 N. Charles Street 21218 Baltimore, MD USA E-mail: sire@math.jhu.edu Received July 2019.