Maria Colombo
email: maria.colombo@eth-its.ethz.ch

Luca Spolaor
email: lspolaor@mit.edu

Bozhidar Velichkov

A LOGARITHMIC EPIPERIMETRIC INEQUALITY FOR THE OBSTACLE PROBLEM

We study the regularity of the regular and of the singular set of the obstacle problem in any dimension.

Our approach is related to the epiperimetric inequality of Weiss (Invent. Math., 138 (1999), 23-50), which works at regular points and provides an alternative to the methods previously introduced by Caffarelli (Acta Math., 139 (1977), 155-184). In his paper, Weiss uses a contradiction argument for the regular set and he asks the question if such epiperimetric inequality can be proved in a direct way (namely, exhibiting explicit competitors), which would have significant implications on the regularity of the free boundary in dimension d > 2.

We answer positively the question of Weiss, proving at regular points the epiperimetric inequality in a direct way, and more significantly we introduce a new tool, which we call logarithmic epiperimetric inequality. It allows to study the regularity of the whole singular set and yields an explicit logarithmic modulus of continuity on the C 1 regularity, thus improving previous results of Caffarelli and Monneau and providing a fully alternative method. It is the first instance in the literature (even in the context of minimal surfaces) of an epiperimetric inequality of logarithmic type and the first instance in which the epiperimetric inequality for singular points has a direct proof. Our logarithmic epiperimetric inequality at singular points has a quite general nature and will be applied to provide similar results in different contexts, for instance for the thin obstacle problem.

Introduction

In this paper we study the regularity of the free-boundary of nonnegative local minimizers u of the functional E(u) := |∇u| 2 dx + max{u(x), 0} dx .

Our main result is a logarithmic epiperimetric inequality, which is a new tool for the study of the singular set of minimizers of variational energies. It is also an alternative approach to the regularity of the singular free boundary as proposed by Caffarelli [START_REF] Caffarelli | The regularity of free boundaries in higher dimensions[END_REF][START_REF] Caffarelli | The obstacle problem revisited[END_REF]. Before we state it we recall that, given u ∈ H 1 (B 1 ), the Weiss' boundary adjusted energy of u is defined by

W (u) = B 1 |∇u| 2 dx -2 ∂B 1 u 2 dH d-1 + B 1 max{u(x), 0} dx .
The class K of admissible blow-ups of u at singular points is defined by

K := {Q A : R d → R : Q A (x) = x • Ax, A symmetric non-negative with trA = 1 /4} . (1.1)
The energy W is constant on K, precisely we have W (Q A ) = ω d 8 d(d+2) , for every Q A ∈ K. We refer to this constant as to the energy density at the singular points and denote it by Θ.

Theorem 1 (Logarithmic epiperimetric inequality at singular points). There are dimensional constants δ > 0 and ε > 0 such that the following claim holds. For every non-negative function c ∈ H 1 (∂B 1 ), with 2-homogeneous extension z on B 1 , satisfying dist L 2 (∂B 1 ) (c, K) ≤ δ and

W (z) -Θ ≤ 1,
there is a non-negative function h ∈ H 1 (B 1 ) with h = c on ∂B 1 satisfying the inequality

W (h) -Θ ≤ W (z) -Θ 1 -ε W (z) -Θ γ , where γ = 0 if d = 2 γ = d-1 d+3 if d ≥ 3
.

(1.

2)

The adjective "logarithmic" for this new epiperimetric inequality is due to the fact that the presence of γ = 0 implies that the function decays to its blow up with a logarithmic rate and that the free boundary is contained in a C 1 curve with logarithmic modulus of continuity (see (1.5) and (1.6) below). At points where the classical epiperimetric inequality holds (namely (1.2) above with γ = 0) the decay and the modulus of continuity are of Hölder type.

At flat points we recover the Weiss' epiperimetric inequality with a direct proof. To state it, recall that the collection K + of possible blow-ups at flat points is defined by K + := q ν : R d → R : q ν (x) = (max{x • ν, 0}) 2 for some ν ∈ R d such that |ν| = 1 2 .

(1.

3)

The energy W is constant on K + , precisely we have W (q ν ) = ω d 16 d(d+2) , for every q ν ∈ K + . We will refer to this constant as the energy density at the flat points and denote it by Θ + .

Theorem 2 (Epiperimetric inequality at flat points). There are dimensional constants δ 0 > 0, δ > 0 and ε > 0 such that the following claim holds. For every non-negative function c ∈ H 1 (∂B 1 ) satisfying {x d < -δ 0 } ⊂ {c = 0} and c -q e d L 2 (∂B 1 ) ≤ δ , there exists a non-negative function h ∈ H 1 (B 1 ) such that h = c on ∂B 1 and

W (h) -Θ + ≤ (1 -ε) W (z) -Θ + , (1.4)
where z is the 2-homogeneous extension of c to B 1 .

Theorem 2 was already proved by Weiss in [START_REF] Weiss | A homogeneity improvement approach to the obstacle problem[END_REF] using a very elegant and innovative contradiction argument, later exploited also by Garofalo-Petrosyan-Garcia and Focardi-Spadaro in the context of the thin obstacle problem (see [START_REF] Garofalo | An epiperimetric inequality approach to the regularity of the free boundary in the Signorini problem with variable coefficients[END_REF][START_REF] Focardi | An epiperimetric inequality for the thin obstacle problem[END_REF]). However, the same proof works only at singular points of maximal and minimal dimension under some special assumptions on the projection of the trace on K, which can be verified only in dimension d = 2. Notice that the dimension of a singular point is the maximal dim(ker A) among all Q A ∈ K blow-ups of u at the singular point. Hence, no epiperimetric inequality was known in the literature for the whole singular set, as it happens in Theorem 1 and Weiss himself suggests that "...it should however be possible to give a direct proof of the epiperimetric inequality which would then also cover singular sets of intermediate dimension" (see [START_REF] Weiss | A homogeneity improvement approach to the obstacle problem[END_REF]). Theorems 1 and 2 answer affirmatively to this question, and in particular Theorem 1 is the first instance in the literature of an epiperimetric inequality of logarithmic type and the first instance in which the epiperimetric inequality for singular points has a direct proof. The methods developed to prove the epiperimetric inequality at singular points of any stratum have a quite general nature and will be applied to provide similar results in other problems, for instance in the case of the thin obstacle problem [START_REF] Colombo | Direct epiperimetric inequalities for the thin obstacle problem[END_REF].

The proof of Theorem 2 is the first adaptation of the Reifenberg and White's pioneering work to the classical obstacle problem (see [START_REF] Reifenberg | An epiperimetric inequality related to the analyticity of minimal surfaces[END_REF][START_REF] White | Tangent cones to two-dimensional area-minimizing integral currents are unique[END_REF] and [START_REF] Spolaor | Regularity of free boundary in the one-phase problem i: dimension 2[END_REF], where a similar argument is performed for the Alt-Caffarelli functional in dimension d = 2). In other words, it is based on a direct proof, producing explicit energy competitors: they are obtained by changing the homogeneity of the boundary datum c, after subtracting a suitable rotation of the blow up q ν . At singular points, namely in the setting of Theorem 2, it is remarkable in our opinion how the failure of Weiss' contradiction argument translates into a weakening of the epiperimetric inequality, that is the necessity of introducing the exponent γ in (1.2). Correspondingly, the energy competitor cannot be constructed by simply changing the homogeneity the extension of c, after possibly subtracting a rotation of the blow up; a more refined construction is necessary and the gain in the energy is logarithmic rather than being a fraction of the energy of the 2-homogeneous extension. An estimate of the form (1.2) is essentially the best one can get for general nonnegative traces c (we discuss this in detail in Subsection 1.1) and the estimate in Theorem 1 is optimal, in view of the example constructed by Figalli and Serra [START_REF] Figalli | On the fine structure of the free boundary for the classical obstacle problem[END_REF]Appendix A].

It is well known that Theorem 2 leads to the uniqueness of the blow-up at every flat point and also to the C 1,α regularity of the regular part of the free-boundary (see [START_REF] Weiss | A homogeneity improvement approach to the obstacle problem[END_REF]). We show that Theorem 1 yields the uniqueness of the blow-up and the C 1 regularity of the singular set, with an explicit logarithmic modulus of continuity. This is an improvement on the results of Caffarelli and Monneau, where such a modulus arises by contradiction arguments and is therefore not explicit (see [START_REF] Caffarelli | The regularity of free boundaries in higher dimensions[END_REF][START_REF] Monneau | A brief overview on the obstacle problem[END_REF]). The method of the present paper is flexible enough to cover more general and nonlinear functionals, such as the area. The stratification of the singular set for the area functional, even in the context of Riemannian manifolds, and the C 1 regularity of the strata were recently obtained in [START_REF] Focardi | The classical obstacle problem for nonlinear variational energies[END_REF][START_REF] Focardi | Monotonicity formulas for obstacle problems with Lipschitz coefficients[END_REF]. Before giving the precise statements, we need some additional definitions.

We split the free-boundary of a minimizer u in regular and singular part, defined as Reg(u) := {x ∈ ∂{u > 0} ∩ Ω : any blow up at x is of the form q ν ∈ K + } Sing(u) := {x ∈ ∂{u > 0} ∩ Ω : at least one blow up at x is not of the form

q ν ∈ K + }
Their regularity is the content of the following results.

Theorem 3 (Uniqueness of the blow up and logarithmic convergence). Let γ = d-1 d+3 , Ω ⊂ R d be an open set and u ∈ H 1 (Ω) a minimizer of E. Then the blow up of u at each point of the free boundary ∂{u > 0} ∩ Ω is unique. Moreover, the following convergence holds.

(1) For every x 0 ∈ Reg(u) ∩ Ω there exist r := r(x 0 ),

C := C(x 0 ) and ν(x 0 ) ∈ R d , with |ν(x 0 )| = 1 /2, such that ∂B 1 u x 1 ,r -q ν(x 1 ) dH d-1 ≤ Cr (n+2)ε 2(1-ε) , for every r ≤ r 0 and x 1 ∈ Reg(u) ∩ B r (x 0 ). (2)
For every open set Ω 0 Ω and x ∈ Sing(u) ∩ Ω 0 , there exist

C := C(Ω 0 ) and Q x ∈ K such that ∂B 1 |u x 0 ,r -Q x 0 | dH d-1 ≤ C(-log r) -1-γ 2γ , for every r ≤ dist(Ω 0 , ∂Ω).
(

The logarithmic modulus of continuity in (1.5) follows by the exponent γ = 0 in the epiperimetric inequality (1.2). Indeed, we will show that thanks to the epiperimetric inequality, for every x ∈ Sing(u), the quantity e(r) = W (u, r) -Θ u (0) satisfies, for r sufficiently small, the differential inequality d dr e(r) ≥ c r e(r) 1+γ , which corresponds to a logarithmic decay of e(r). Instead, when the classical epiperimetric inequality holds, we find the differential inequality d dr e(r) ≥ ce(r), which gives a Hölder decay of e(r).

The next regularity result recovers all the previously known results and improves the regularity of the singular set to C 1,log . Before stating it we need to make precise what we mean by singular points of intermediate dimension. Given k = 0, . . . , d -1, we define the singular set of dimension k (also called k-th stratum) S k (u) as

S k (u) := {x ∈ Sing(u) : dim(ker(A)) ≤ k for every blow-up Q A ∈ K of u at x} = k l=1 {x ∈ Sing(u) : dim(ker(A)) = l for the unique blow-up Q A ∈ K of u at x} ,
where the equivalence of the two definitions is guaranteed by Theorem 3. In the case of the stratum S 0 (u) the inequality (1.5) can be improved to C 1,β convergence.

Theorem 4 (Regularity of the free boundary). Let ε > 0 be the constant from Theorem 2,

β = (d+2)ε 2(1-ε) 1 + (d+2)ε 2(1-ε) -1 , Ω ⊂ R d be an open set and u ∈ H 1 (Ω) a minimizer of E. Then (1) Reg(u) is locally the graph of a C 1,β function; namely, for every x 0 ∈ Reg(u) ∩ Ω there exists r := r(x 0 ) such that Reg(u) ∩ B r (x 0 ) is a C 1,β -submanifold of dimension (d -1);
(2) For every k = 0, ..., d -1 and for every x 0 ∈ S k (u), there exists r 0 := r 0 (x 0 ) such that S k (u) ∩ B r 0 (x 0 ) is contained in a single submanifold of dimension k and class C 1,log ; moreover, for every open set Ω 0 Ω there exists C := C(Ω 0 ) such that for every x 1 , x 2 ∈ Sing(u) ∩ Ω 0 a logarithmic estimate holds

|Q x 1 -Q x 2 | ≤ C(-log |x 1 -x 2 |) -1-γ 2γ , (1.6)
where Q x is the unique blow up of u at x. (3) If the dimension d = 2, then we have the estimate

|Q x 1 -Q x 2 | ≤ C|x 1 -x 2 | β for any x 1 , x 2 ∈ S k (u) ∩ Ω 0 , (1.7)
for k = 1, 2, where β is the same as in [START_REF] Adams | Rates of asymptotic convergence near isolated singularities of geometric extrema[END_REF],

Ω 0 Ω is a open set, C := C(Ω 0 ).
In particular S 0 (u) consists of isolated points in Ω and S 1 (u) is locally contained in a 1-dimensional submanifold of class C 1,β .

Remark 1.1. Thanks to a result of Caffarelli and Riviére (see [START_REF] Caffarelli | Smoothness and analyticity of free boundries in variational inequalities[END_REF]) it is possible to improve (3) to the following result: the boundary of a connected component of the interior of the free-boundary is analytic except at finitely many singular points. After the completion of the present paper, a new higher regularity result for the singular part of the free boundary was proved by Figalli and Serra [START_REF] Figalli | On the fine structure of the free boundary for the classical obstacle problem[END_REF].

The stratification of the singular set for the area functional, even in the context of Riemannian manifolds, and the C 1 regularity of the strata were recently obtained in [START_REF] Focardi | The classical obstacle problem for nonlinear variational energies[END_REF][START_REF] Focardi | Monotonicity formulas for obstacle problems with Lipschitz coefficients[END_REF]. The method of the present paper is flexible enough to cover more general and nonlinear functionals, such as the area: Theorems 3 and 4 remain true if we consider a Hölder continuous weight function q : Ω → R + and more general functionals, for instance

E q (u) := Ω |∇u| 2 + q(x)|u| dx, A q (u) := Ω |∇u| 2 + 1 + q(x)|u| dx.
In this case, the regular and singular parts at a given point x are defined as for E, up to a constant which depends on q(x). Given u ∈ H 1 (B 1 ) positive minimizer of E, we define Reg q (u) := x ∈ ∂{u > 0} ∩ Ω : any blow up at x is of the form q ν for |ν| = q(x) 2 ,

Sing q (u) := x ∈ ∂{u > 0} ∩ Ω : at least one blow up at x is not of the form q ν , for |ν| = q(x) 2 , S q,k (u) := {x ∈ Sing q (u) : dim(ker(A)) ≤ k for every blow-up

Q A ∈ K of u at x}.
Theorem 5 (Hölder continuous weight functions and area functional). Let α > 0, Ω ⊂ R d be an open set and q ∈ C 0,α (Ω; R + ) be an Hölder continuous function such that q ≥ c q > 0, where c q is a given constant. Let u ∈ H 1 (Ω) be a minimizer of E q or A q . Then the blow up of u at each point of the free boundary ∂{u > 0} ∩ Ω is unique and

(1) there exists β > 0 such that Reg q (u) is locally the graph of a C 1,β function;

(2) For every k = 0, ..., d-1, S q,k (u) is contained in the union of countably many submanifolds of dimension k and class C 1,log ; namely for every x 0 ∈ Sing q (u) ∩ Ω there exists r 0 := r 0 (x 0 ) and C := C(x 0 ) such that a logarithmic estimate holds

|Q(x 1 ) -Q(x 2 )| ≤ C(log |x 1 -x 2 |) -1-γ 2γ for any x 1 , x 2 ∈ Sing q (u) ∩ B r (x 0 ). (1.8)
Compared to a similar result obtained from the epiperimetric inequality with indirect proof, here we have quantitative estimates as (1.8) and, for the regular set, an explicit Hölder regularity in terms of the dimension and the Hölder exponent of q.

1.1. Classical vs logarithmic epiperimetric inequality. The key ingredients in the proof of Theorems 1 and 2 are

• a Fourier decomposition of the trace c -q ν (resp. c -Q A ) onto the eigenfunctions of S d-1 ;

• an energy improvement with respect to z obtained by taking the harmonic extension of the modes with homogeneity larger than two; • a choice of ν (resp. A) to control the projection of c -q ν (resp. c -Q A ) onto the eigenfunctions of homogeneity one and two, with the projection on the higher modes.

At flat points and at every point in dimension d = 2, the estimate of the third bullet is linear, however in general dimension at singular points we can only prove a control of the form

P (c -Q A ) H 1 (∂B 1 ) ≤ C (Id -P )(c -Q A ) 1-γ H 1 (∂B 1 )
γ ∈ (0, 1) ,

where P denotes the projection on the modes relative to homogeneity two. The above estimate corresponds to (4.7) from the proof of Theorem 1 and is sharp in higher dimension (see Subsection 4.6). The reason for this different behavior in the flat and singular points is essentially the following: at the flat points we are able to eliminate the lower modes (the modes corresponding to homogeneity smaller than two) on a spherical cap by means of the choice of the vector ν; this is possible since the space of admissible functions q ν is an open manifold of the same dimension as the eigenspace corresponding to the lower modes, so we can apply an implicit function argument (see Lemma 3.2). At general singular points, we would like to eliminate the modes of homogeneity two, that is the modes corresponding to the eigenvalue 2d on the sphere and whose eigenspace can be identified with the space of d × d real symmetric matrices S d (R). However, the positivity constraint on the competitor forces the choice of the matrix A to be in the space of nonnegative symmetric matrices S + d (R). Now these two spaces have the same dimension, but, due to the non-negativity assumption, the set

S + d (R) ⊂ S d (R)
is not open, so we cannot apply the implicit function theorem here. Indeed, if we are in its interior, which corresponds to the singular points studied by Weiss, then the argument works and we can eliminate the second modes; but at the boundary of S + d (R) an implicit function argument only provides us with a matrix in the larger space S d (R). This leaves us to estimate the difference between the element of S d (R), corresponding to the second modes of the trace c, and its projection on S + d (R). We can do this by means of the additional condition that c is positive, which suggests that this difference should be comparable to the higher modes of the trace, but because of capacitary reasons the bound comes with an exponent γ = 0. Roughly speaking, if the negative part produced by the second modes is very small, concentrated on a set of small capacity, then it can be compensated by a function with very small energy, much smaller than the distance to S + d (R) in the space of symmetric matrices. In particular, it seems that this obstruction is of the same nature as the one that appears in [START_REF] Weiss | A homogeneity improvement approach to the obstacle problem[END_REF], where the strong convergence of the traces cannot see the nodal sets of small capacity. A similar phenomenon can be found in the theory of minimal surfaces. Indeed if we think about the collection of singular points of lower dimension as a minimal surface in codimension higher than one, then it is known the existence of non-integrable cones, that is cones with nonintegrable Jacobi fields. In this case the best possible rate of convergence to the blow-up is indeed logarithmic, as shown in [START_REF] Adams | Rates of asymptotic convergence near isolated singularities of geometric extrema[END_REF].

Organization of the paper. The paper is divided in four short sections. In Section 2 we fix notations and easy preliminary computations. In Section 3 we prove the Weiss epiperimetric inequality Theorem 2, while Section 4 is dedicated to Theorem 1. Finally, in Section 5 we apply these two theorems to deduce the various regularity results.

Preliminaries

In this section we fix some notations and we recall some known facts about the solutions of the obstacle problem, their blow-up limits, the decomposition of the free boundary in a regular and singular part and its realtion with the Weiss boundary adjusted functional. The final subsection is dedicated to the Fourier analysis on the unit sphere in R d , which will be useful for both Theorems 1 and 2.

2.1. Notations. We will use the following notations. B 1 is the d-dimensional unit ball centered in zero and ω d = |B 1 | is the Lebesgue measure of B 1 . We denote by S d-1 or ∂B 1 the unit (d -1)-dimensional sphere in R d equipped with the (d -1)-dimensional Hausdorff measure H d-1 . θ will denote the variable on the sphere ∂B 1 . For an open set Ω in R d or on the sphere S d-1 we will denote by H 1 (Ω) ⊂ L 2 (Ω) the Sobolev space of weakly differentiable functions on Ω with gradients in L 2 (Ω; R d ) and by H 1 0 (Ω) the space of functions H 1 (Ω) which are zero on ∂Ω. For a function f : R d → R we denote by f + its positive part, f + (x) = max{f (x), 0}. For instance, given a vector ν ∈ R d we will often use the notations

(x • ν) + = max{x • ν, 0} and (x • ν) 2 + = max{x • ν, 0} 2 ,
where x • ν is the scalar product of the vectors x and ν in R d .

2.2.

Weiss boundary adjusted energy. For a function u ∈ H 1 (Ω), with Ω ⊂ R d , we denote by W , W 0 and W the functionals

W 0 (u, x 0 , r) := 1 r d+2 Br(x 0 ) |∇u| 2 dx - 2 r d+3 ∂Br(x 0 ) u 2 dH d-1 , W (u, x 0 , r) := W 0 (u, x 0 , r) + 1 r d+2 Br(x 0 ) u(x) dx, W (u, x 0 , r) := W 0 (u, x 0 , r) + 1 r d+2 Br(x 0 ) max{u(x), 0} dx,
where x 0 ∈ ∂{u > 0} and 0 < r < dist(x 0 , ∂Ω), and we notice that for non-negative functions u ∈ H 1 (B 1 ) we have W (u, x 0 , r) = W (u, x 0 , r). In particular, we set

W (u, 0, r) = W (u, r) and W (u, 1) = W (u),
and we recall the scaling property

W (u, x 0 , r) = W (u r,x 0 ) , where u r,x 0 (x) = u(rx + x 0 ) r 2 .
For any u ∈ H 1 (Ω) the following identity holds for x 0 ∈ ∂{u > 0} and 0 < r < dist(x 0 , ∂Ω)

d dr W (u, x 0 , r) = d + 2 r W (z r,x 0 , 1) -W (u r,x 0 , 1) + 1 r ∂B 1 |x • ∇u r,x 0 -2u r,x 0 | 2 dH d-1 , (2.1)
where z r,x 0 (x) := |x| 2 u r,x 0

x |x| (see for instance [START_REF] Weiss | A homogeneity improvement approach to the obstacle problem[END_REF]).

2.3.

Global homogeneous solutions of the obstacle problem. Wa say that the function

u 0 : R d → R is a blow-up limit of u in the point x 0 , if u 0 = lim n→∞ u rn,x 0 for some sequence (r n ) n with lim n→∞ r n = 0,
where the converegnce is locally uniform and strong in H 1 loc (R d ). Thanks to work of Caffarelli (see [START_REF] Caffarelli | The obstacle problem revisited[END_REF]), it is well known that u 0 is a global homogeneous solution of the obstacle problem. Precisely, u 0 ∈ K ∪ K + (introduced in (1.1) and (1.3)). Moreover, we claim that

W (Q) = ω d 8 d(d + 2) =: Θ for every Q ∈ K , and W (q) = ω d 16 d(d + 2)
=: Θ + for every q ∈ K + .

(2.2) Indeed, for every Q A ∈ K we have ∆Q A = 2 trA = 1 2 and so an integration by parts gives

W 0 (Q A ) = B 1 |∇Q A | 2 -2 ∂B 1 Q 2 A = - B 1 Q A ∆Q A = - 1 2 B 1 Q A .
Since Q A is positive and denoting (a ij ) ij the coefficients of the matrix A, we get

W (Q A ) = W (Q A ) = W 0 (Q A ) + B 1 Q A = 1 2 B 1 Q A = 1 2 B 1 d i=1 a ii x 2 i dx = trA 2 B 1 x 2 d dx = Θ.
Analogously, for any q ν ∈ K + we have ∆q ν = 2 |ν| 2 = 1 2 on the set {x • ν > 0}, so that

W (q ν ) = W (q ν ) = W 0 (q ν ) + B 1 q ν = 1 2 B 1 q ν = |ν| 2 2 B 1 ∩{x d >0} x 2 d dx = ω d 16 d(d + 2) = Θ + .
2.4. Regular and singular free boundaries. We recall that, as observed by Weiss [START_REF] Sebastian | Partial regularity for a minimum problem with free boundary[END_REF], a consequence of (2.1) is that if u ∈ H 1 (Ω) is a nonnegative minimizer of E in the open set Ω ⊂ R d and x 0 ∈ Ω, then the function r → W (u, x 0 , r) is nondecreasing (in its domain of definition 0 < r < dist(x 0 , ∂Ω)) and there exists the limit

Θ u (x 0 ) := lim r→0 W (u, x 0 , r) = inf r>0 W (u, x 0 , r) = lim r→0 W (u r,x 0 ) . (2.3)
Moreover, if q is a blow-up limit of the minimizer u in x 0 , then

W (q) = lim n→∞ W (u rn,x 0 ) = Θ u (x 0 ).
Since we have that q ∈ K ∪ K + , there are only two possible values for the energy density Θ u (x 0 ):

Θ u (x 0 ) = Θ + or Θ u (x 0 ) = Θ.
Hence we can redefine the regular and the singular part of the free boundary ∂{u > 0} ∩ Ω as

Reg(u) = {x ∈ ∂{u > 0} ∩ Ω : Θ u (x) = Θ + }, Sing(u) = {x ∈ ∂{u > 0} ∩ Ω : Θ u (x) = Θ}.
By definition the free boundary ∂{u > 0}∩Ω is a disjoint union of Reg(u) and Sing(u). Moreover, by the definition of the density (2.3) and the fact that x 0 → W (u, x 0 , r) is continuous, the function x 0 → Θ u (x 0 ) is upper semicontinuous. This, together with (2.2) and the fact that all the blow-up limits are in K ∪ K + , finally gives the following characterization of Reg(u) and Sing(u):

• the set Reg(u) is a relatively open subset of the free boundary ∂{u > 0}, and every blow-up limit at a point of Reg(u) is of the form q ν , for some q ν ∈ K + ; • the set Sing(u) is closed, and every blow-up limit at a point of Sing(u) is of the form Q A , for some Q A ∈ K.

2.5.

Eigenvalues and eigenfunctions on subdomains of the sphere.

Let S ⊆ S d-1 be an open set. Let 0 < λ 1 ≤ λ 2 ≤ • • • ≤ λ j ≤ .
. . be the eigenvalues (counted with multiplicity) of the spherical Laplace-Beltrami operator with Dirichlet conditions on ∂S and {φ j } j≥1 be the corresponding eigenfunctions, that is the solutions of the problem

-∆ S d-1 φ j = λ j φ j in S, φ j = 0 on ∂S, S φ 2 j (θ) dH d-1 (θ) = 1. (2.4)
Any function ψ ∈ H 1 0 (S) can be decomposed as ψ(θ) = ∞ j=1 c j φ j (θ). The following lemma compares the energies of 2-homogeneous and α-homogeneous functions by means of the Fourier decomposition of their common values on ∂B 1 .

Lemma 2.1. Let ψ ∈ H 1 0 (S) and consider the 2-homogeneous extension ϕ(r, θ) = r 2 ψ(θ) and the α-homogeneous extension φ(r, θ) = r α ψ(θ) respectively of ψ to B 1 , for some α > 2. Set

ε α := α -2 d + α and λ α := α(α + d -2) . (2.5)
Then the following inequality holds

W 0 ( φ) -(1 -ε α )W 0 (ϕ) = ε α d + 2α -2 ∞ j=1 (-λ j + λ α )c 2 j . (2.6)
Proof. Since φ j L 2 (∂B 1 ) = 1 and ∇ θ φ j L 2 (∂B 1 ) = λ j for every j ∈ {0} ∪ N, the energy of the α-homogeneous function φ(r, θ) = r α ψ(θ) can be written as

W 0 ( φ) = ∞ j=1 c 2 j 1 0 r d-1 dr S dH d-1 α 2 r 2α-2 φ 2 j + r 2α-2 |∇ θ φ j | 2 -2 S φ 2 j dH d-1 = ∞ j=1 c 2 j α 2 + λ j d + 2α -2 -2 .
When α = 2 and ϕ(r, θ) = r 2 ψ(θ), we get

W 0 (ϕ) = ∞ j=1 c 2 j 4 + λ j d + 2 -2 .
We now notice that for every λ we have

α 2 + λ d + 2α -2 -2 -(1 -ε α ) 4 + λ d + 2 -2 = λ(2 -α) (d + α)(d + 2α -2) + (α -2)α(α + d -2) (d + α)(d + 2α -2) ,
which concludes the proof of Lemma 2.1.

The above lemma, in particular, shows that if the decomosition of ψ involves only eigenfunctions corresponding to eigenvalues λ j ≥ λ α , then the α-homogeneous extension φ has a strictly lower energy than the two-homogeneous extension ϕ. In order to choose appropriately α we will need some additional information on the spectrum of the Laplacian on S. We recall that the function φ j : S → R is a solution of the first equation in (2.4) if and only if its α j -homogeneous extension ϕ j (r, θ) = r α j φ j (θ) is harmonic in the cone {(r, θ) ∈ R + × ∂B 1 : θ ∈ S}, where the homogeneity α j is uniquely determined by the identity λ j = α j (α j + d -2).

The spectrum on the sphere S d-1 . By the fact that the homogeneous harmonic functions in R d are necessarily polynomials, we have that:

• λ 1 = 0 and the corresponding eigenfunction is the constant

φ 1 = |∂B 1 | -1/2 = (dω d ) -1/2 . • λ 2 = • • • = λ d+1 = d -1, the corresponding homogeneity constants are α 2 = • • • = α d+1 =
1 and the corresponding eigenspace coincides with the space of linear functions in R d .

• λ d+2 = • • • = λ d(d+3)/2 = 2d, the corresponding homogeneity constants are α d+2 = • • • = α d(d+3)/2 = 2.
The corresponding eigenspace has dimension d(d -1)/2 and is generated by the (restrictions to S d-1 of the) two-homogeneous harmonic polynomials:

E 2d = {Q A : R d → R : Q A (x) = x • Ax, A symmetric with trA = 0}. • If j > d(d + 3)/2 (that is λ j > 2d), then λ j ≥ 3(3 + d -2) = 3(d + 1
). The spectrum on the half-sphere

∂B + 1 = {x d > 0} ∩ ∂B 1 .
We notice that the odd extension (with respect to the plane {x d = 0}) of any eigenfunction φ j on the half-sphere ∂B + 1 is an eigenfunction on the entire sphere ∂B 1 , which is zero on the equator {x d = 0} ∩ ∂B 1 . Thus, one can easily deduce that:

• λ 1 = d -1 and the corresponding eigenfunction is

φ 1 (x) = x d √ ω d . • λ 2 = • • • = λ d = 2d, the corresponding homogeneity constants are α 2 = • • • = α d = 2
and the corresponding eigenspace E 2d has dimension (d -1) and is generated by the polynomials

Q j (x) = x d x j-1 , for every j = 2, . . . , d. • If j > d (that is λ j > 2d), then λ j ≥ 3(3 + d -2) = 3(d + 1).
The spectrum on the spherical cap S δ = ∂B 1 ∩{x d > -δ}. We first notice that the spectrum {λ j (δ)} j≥1 of the spherical cap S δ varies continuously with respect to δ. Thus, for δ > 0 small enough (smaller than some dimensional constant), we have

• λ 1 (δ) is simple (isolated) eigenvalue and λ 1 (δ) ≤ d -1; • d -1 < λ j (δ) < 2d
, for every j = 2, . . . , d;

• λ j (δ) ≥ 3d, for every j > d.

Moreover, a standard separation of variables argument gives that:

• the first eigenfunction φ 1 on S δ is positive and depends only on the first variable

x d , that is φ 1 (x) = φ 1 (x d ). • the eigenfunctions φ 2 , . . . , φ d correspond to the same eigenvalue λ 2 (δ) = • • • = λ d (δ) and
there is a function φ = φ(x d ) such that φ j (x) = x j-1 φ(x d ) for every j = 2, . . . , d.

3.

The epiperimetric inequality at flat points: proof of Theorem 2

In order to prove Theorem 2 we decompose the function z as

z = q ν + ϕ,
where q ν (x) = (x • ν) 2 + and ν ∈ R d to be chosen later. We then replace the 2-homogeneous function ϕ(r, θ) = r 2 φ(θ) by an α-homogeneous function φ(r, θ) = r α φ(θ), for some α > 2. The final competitor will be of the form h = q ν + φ, and ε = ε α will be given by

ε α := α -2 d + α . (3.1)
We notice that the competitor h is non-negative in B 1 , thus we only need to prove the inequality

W (h) -Θ + -(1 -ε) W (z) -Θ + ≤ 0. (3.2)
We divide the proof into three steps.

Step 1. Using the properties of q ν , we first reduce the inequality (3.2) to a comparison of the energy of φ to the one of ϕ. Precisely, in Subsection 3.1, we prove the inequality

W (h) -Θ + -(1 -ε) W (z) -Θ + ≤ W 0 ( φ) -(1 -ε)W 0 (ϕ). (3.3)
Step 2. In Subsection 3.2 we prove that we can choose q ν in such a way that the function φ := c-q ν does not contain modes of the first d eigenvalues on the spherical cap S δ 0 . Precisely, we prove the following claim. For every δ 0 > 0 there exists δ > 0 such that for every c ∈ H 1 0 (S δ 0 ) satisfying c -q e d /2 L 2 (∂B 1 ) ≤ δ there exists ν ∈ R d such that u λ ∈ H 1 0 (S δ 0 ) and

S δ 0 c(θ)φ j (θ) dH d-1 (θ) = S δ 0 q ν (θ)φ j (θ) dH d-1 (θ), for every j = 1, . . . , d, (3.4
) where φ 1 , . . . , φ d are the first d, orthonormal in L 2 (∂B 1 ), eigenfunctions of the Laplace-Beltrami operator on S δ 0 with Dirichlet boundary conditions on ∂S δ 0 .

Step 3. In Subsection 3.3 we use Lemma 2.1 and the choice of ν from Step 2 to prove the inequality

W 0 ( φ) -(1 -ε)W 0 (ϕ) ≤ 0, (3.5)
which together with (3.3) gives (3.2).

3.1. Decomposition of the energy. We prove (3.3) in the following lemma:

Lemma 3.1. Let α > 2, ε α as in (3.1), ν = (ν 1 , . . . , ν d ) ∈ R d and q ν (x) = (x • ν) 2 + . Suppose that φ ∈ H 1 (∂B 1 ), ϕ(r, θ) = r 2 φ(θ) and φ(r, θ) = r α φ(θ). Then W (q ν + φ) -Θ + -(1 -ε α ) W (q ν + ϕ) -Θ + ≤ W 0 ( φ) -(1 -ε α )W 0 (ϕ).
(3.6)

Proof. Suppose, without loss of generality that, q ν (x) = c 0 q(x), where for the sake of simplicity we set q := q e d /2 ∈ K + . Notice that for every ψ ∈ H 1 (B 1 ) we have

W (c 0 q + ψ)-Θ + = c 2 0 B 1 |∇q| 2 -2c 2 0 ∂B 1 q 2 + c 0 B 1 q - 1 2 B 1 q + 2c 0 B 1 ∇q • ∇ψ -2 ∂B 1 qψ + B 1 |∇ψ| 2 -2 ∂B 1 ψ 2 + B 1 ψ = - (c 0 -1) 2 2 B 1 q + 2c 0 - B 1 ∆q ψ + ∂B 1 ∂ r q ψ -2 ∂B 1 qψ + W 0 (ψ) + B 1 ψ = -(c 0 -1) 2 Θ + + W 0 (ψ) + B 1 ψ -c 0 B + 1 ψ ,
where we used that

Θ + = 1 2 B 1 q, ∆q = 1 2 χ B + 1
and ∂ r q = 2 q. If ψ = φ = r 2 φ, then we have

B 1 φ -c 0 B + 1 φ = 1 d + α ∂B 1 φ -c 0 ∂B + 1 φ =: 1 d + α β(φ),
and we can write the energy of c 0 q + φ in the form

W (c 0 q + φ) -Θ + = -(c 0 -1) 2 Θ + + W 0 ( φ) + 1 d + α β(φ).
Applying the above estimate to ϕ and φ and thanks to the definition of ε α , we get

W (c 0 q + φ) -Θ + -(1 -ε α ) W (c 0 q + ϕ) -Θ + = -ε α (c 0 -1) 2 Θ + + W 0 ( φ) -(1 -ε α )W 0 (ϕ) + 1 d + α - 1 -ε α d + 2 β(φ) = -ε α (c 0 -1) 2 Θ + + W 0 ( φ) -(1 -ε α )W 0 (ϕ),
which concludes the proof of Lemma 3.1.

Choice of ν.

In this section we prove the claim (3.4), which is a straightforward consequence of the following lemma.

Lemma 3.2. Given δ 0 > 0, we denote by S δ 0 the set {x d > -δ 0 } ∩ ∂B 1 and by φ 1 , . . . , φ d the first d eigenfunctions on the set S δ 0 . Then the function

F : R d → R d , F (ν) = S δ 0 q ν φ 1 , . . . , S δ 0 q ν φ d , is a C 1 diffeomorphism in a neighbourhood U ⊂ R d of e d 2 .
Proof of Lemma 3.2. We first notice that F is a C 1 function in a neighborhood of e d /2, because the function

R d × R d (x, y) → (x • y) 2 + is C 1 .
We now calculate the partial derivatives of F = (F 1 , . . . , F d ) in e d /2. Using the fact that the first eigenfunction depends only on one varibale, φ 1 = φ 1 (x d ), and that the higher eigenfunctions can be written in the form φ j (x) = x j-1 φ(x d ), for every j = 2, . . . , d (see Subsection 2.5), we get that

F 1 (ν) = ∂B 1 q ν (x) φ 1 (x d ) dx and F j (ν) = ∂B 1 q ν (x) x j-1 φ(x d ) dx, ∀j = 2, . . . , d. Setting ∂B + 1 = {x d > 0} ∩ ∂B 1 we have ∂F 1 ∂ν d (e d /2) = ∂B + 1 x 2 d φ 1 (x d ) dx > 0, ∂F 1 ∂ν j (e d /2) = ∂B + 1 x d x j φ 1 (x d ) dx = 0, ∀j = 1, . . . , d -1.
where the positivity of the first term follows from the positivity of φ 1 , while the second term is zero since x j is odd. Moreover, for every j = 2, . . . , d and i = 1, . . . , d, we have

∂F j ∂ν i (e d /2) = ∂B + 1 x i x d x j-1 φ(x d ) dx = δ i(j-1) ∂B + 1 x 2 i x d φ(x d ) dx = δ i(j-1) d -1 ∂B + 1 (1 -x 2 d )x d φ(x d ) dx ,
where we used the fact that x i and x j are odd for the first equality, and

x 2 j = 1 d-1 d-1 j=1 x 2 j = 1 d-1 (1 -x 2 d )
. By the positivity of φ and the fact that lim δ 0 →0 φ -c 2 x + d L 2 (∂B 1 ) = 0, for dimensional constants c 1 and c 2 (which is due to the fact that on the half-sphere ∂B + 1 the eigenfunctions are of the form φ j (x) = c 2 x d x j-1 for j = 2, . . . , d) we get that for δ 0 small enough DF (e d /2) is an invertible matrix and so, by the inverse function theorem there is a neighborhood of e d /2 on which F is a C 1 diffeomorphism.

3.3.

Homogeneity improvement of ϕ. We prove (3.5). Indeed, by the fact that the Fourier expansion of φ(θ) := c(θ) -q ν (θ) does not contain the first d modes φ 1 , . . . , φ d on the spherical cap S δ 0 (claim (3.4)), we obtain that the function φ can be expanded in Fourier series as

φ(θ) = ∞ j=d+1 c j φ j (θ) on the spherical cap S δ 0 = ∂B 1 ∩ {x d > -δ 0 }.

Thus, by Lemma 2.1 we get

W 0 ( φ) -(1 -ε α )W 0 (ϕ) = ε α d + 2α -2 ∞ j=d+1 (-λ j + λ α )c 2 j ,
where λ j are the eigenvalue of the Dirichlet Laplacian on S δ 0 and λ α = α(α + d -2). On the other hand, for δ 0 > 0 small enough, we have that λ j ≥ 3d, for j ≥ d + 1 (see Subsection 2.5), so that -λ j + λ α ≤ 0 whenever α > 2 and α(α + d -2) ≤ 3d. . Thus, choosing for instance

α = 5 2 and ε = α -2 d + α = 1 2d + 5 ,
we conclude the proof of (3.5) and Theorem 2.

The epiperimetric inequality for singular points: proof of Theorem 1

We can assume without loss of generality that W (z) -Θ ≥ 0, since otherwise the statement is true with h = z. Given any two-homogeneous function z(r, θ) = r 2 c(θ), we can decompose it in Fourier series on the sphere ∂B 1 as

c(θ) = ∞ j=1 c j φ j (θ) = c 1 φ 1 + {j : λ j =d-1} c j φ j (θ) + {j : λ j =2d} c j φ j (θ) + {j : λ j >2d} c j φ j (θ) .
Therefore z can be decomposed in a unique way as

z = q ν + Q A + ϕ, (4.1)
where

(i) ν ∈ R d is such that q ν (x) = (x • ν) 2
+ contains in its Forurier expansion precisely the sum {j : λ j =d-1} c j φ j (θ) ;

(ii) A is a symmetric matrix depending on the coefficients c j , corresponding to the eigenvalues λ j = 0, d -1, 2d, and

Q A (x) = x • Ax;
(iii) ϕ is a two-homogeneous function, in polar coordinates ϕ(r, θ) = r 2 φ(θ), containing only higher modes on ∂B 1 , that is the trace φ can be written in the form

φ(θ) = {j : λ j >2d} c j φ j (θ), (4.2)
where {φ j } j∈N are the eigenfunctions of the spherical laplacian as in (2.4) with S = ∂B 1 . Indeed, in order to obtain the decomposition (4.1) we first notice that the term corresponding to the d -1 modes in the Fourier expansion of z on the sphere is a linear function. Precisely, there is ν ∈ R d such that

{j : λ j =d-1} c j φ j (θ) = θ • ν ,
for every θ ∈ ∂B 1 .

On the other hand, given ν ∈ ∂B 1 , the term corresponding to the linear modes in the Fourier expansion of q ν is of the form c d (x • ν), where c d > 0 is a dimensional constant. We now choose ν := ν /c d . Thus, the expansion of the function z -q ν does not contain linear modes and so, it can be uniquely decomposed as z -q ν = Q A + ϕ, where Q A contains the constant and quadratic modes and ϕ contains all the other higher modes. Thus, we obtain (4.1). Notice that, in the above representation A might not be positive definite. Let B be a symmetric positive definite matrix and Q B (x) = x • Bx. Then, z can be rewritten as

z = q ν + Q B + (Q A -Q B ) + ϕ .
We then replace the 2-homogeneous function ψ := (Q A -Q B ) + ϕ by an α-homogeneous function ψ with the same boundary values as ψ. We will choose α > 2 such that

ε α := α -2 d + α = ε C 4 ∇ θ φ 2 L 2 (∂B 1 ) γ , (4.3)
where C 4 is the dimensional constant from the inequality (4.8) and γ is the constant from (1.2). Subsequently we will choose ε to be small enough, but yet depending only on the dimension. Finally, the competitor h is given by

h = q ν + Q B + ψ.
Since inf{ψ, 0} ≤ inf{ ψ, 0} and q ν + Q B ≥ 0, by the choice of B, h is non-negative in B 1 and so we only need to prove the inequality

W (h) ≤ W (z) -ε W (z) -Θ 1+γ .
The proof of Theorem 1 will be carried out in four steps.

Step 1. In Subsection 4.1 we set c 0 = 4 d j=1 ν 2 j and b = 4 trB and we prove the identity

W (h) -Θ -(1 -ε α ) (W (z) -Θ) (4.4) = - ε α 2 (1 -b -c 0 ) 2 + (1 -b) 2 Θ + W 0 ( ψ) -(1 -ε α )W 0 (ψ).
Step 2. We now choose Q B . If A is positive definite (Q A ≥ 0), then we choose B = A. If Q A changes sign, then up to a change of coordinates we can assume that there exist a j ≥ 0 for every j = 1, . . . , d such that

Q A (x) = - k j=1 a j x 2 j + d j=k+1 a j x 2 j , a d ≥ 1 4d > k j=1 a j ,
where the last inequality being due to the fact that Q A is L 2 (∂B 1 )-close to the set of admissible blow-ups K. We set

Q B (x) := d j=k+1 a j x 2 j - k j=1 a j x 2 d ≥ 0 (4.5)
where the last inequality, that is the positive definiteness of B, depends on a d ≥ 1 4d > k j=1 a j . In Subsection 4.2 we then prove that there exists a dimensional constant C 2 > 0 such that

W 0 ( ψ) -(1 -ε α )W 0 (ψ) ≤ ε 2 α C 2 k j=1 a 2 j - ε α 6(d + 1) ∇ θ φ 2 L 2 (∂B 1 ) . (4.6)
Step 3. In Subsection 4.3 we prove that there are dimensional constants C 3 > 0 and γ ∈ [0, 1) such that

k j=1 a 2 j ≤ C 3 ∇ θ φ 2(1-γ) L 2 (∂B 1 ) . (4.7)
In the supplementary Subsection 4.5 we show that this estimate can be improved in several ways:

-in dimension two (4.7) holds with γ = 0; -the dimensional constants γ can be replaced by a (smaller) constant γ k , this time depending on d and k. In the two extremal cases k = 0 and k = d -1 the constant is zero.

Step 4. In Subsection 4.4 we prove that there is a dimensional constant C 4 , such that

W (z) -Θ ≤ C 4 ∇ θ φ 2 L 2 (∂B 1 ) . ( 4
W 0 ( ψ) -(1 -ε α )W 0 (ψ) ≤ ε 2 α C 2 k j=1 a 2 j - ε α 6(d + 1) ∇ θ φ 2 L 2 (∂B 1 ) ≤ ε 2 α C 2 C 3 ∇ θ φ 2(1-γ) L 2 (∂B 1 ) - ε α 6(d + 1) ∇ θ φ 2 L 2 (∂B 1 )
.

By the definition of ε α and (4.8) we get

W 0 ( ψ) -(1 -ε α )W 0 (ψ) ≤ ε 2 C 2γ 4 ∇ θ φ 4γ L 2 (∂B 1 ) C 2 C 3 ∇ θ φ 2(1-γ) L 2 (∂B 1 ) - ε 6(d + 1) C γ 4 ∇ θ φ 2γ L 2 (∂B 1 ) ∇ θ φ 2 L 2 (∂B 1 ) = εC γ 4 εC 2 C 3 C γ 4 - 1 6(d + 1) ∇ θ φ 2+2γ L 2 (∂B 1 ) ≤ 0. (4.9)
The last inequality is justified since the previous quantity is negative for ε small enough depending only on the dimension. Finally, by (4.4), (4.9), the definition of ε α and (4.8) we obtain

W (h) -Θ ≤ (1 -ε α ) W (z) -Θ = 1 -ε C γ 4 ∇ θ φ 2γ L 2 (∂B 1 ) W (z) -Θ ≤ 1 -ε W (z) -Θ γ W (z) -Θ ,
which is precisely (1.2). We now proceed with the proof of (4.4), (4.6), (4.7) and (4.8).

4.1. Decomposition of the energy. We prove the following lemma, which implies easily (4.4).

Lemma 4.1. Let α > 2 and ε α = α-2 d+α ; let 0 = ν = (ν 1 , . . . , ν d ) ∈ R d , q ν (x) = (x • ν) 2 + , c 0 = 4 d j=1 ν 2 j ;
let B be a symmetric matrix with and b = 4 trB = 0 and Q B (x) = x • Bx. Suppose that φ ∈ H 1 (∂B 1 ), ϕ(r, θ) = r 2 φ(θ) and φ(r, θ) = r α φ(θ). Then

W (q ν + Q B + φ) -Θ -(1 -ε α ) W (q ν + Q B + ϕ) -Θ = - ε α 2 (1 -b -c 0 ) 2 + (1 -b) 2 Θ + W 0 ( φ) -(1 -ε α )W 0 (ϕ).
Proof. We Q = 1 b Q B . Thus, we have Q ∈ K. In particular, ∆Q = 1/2 in B 1 and W (Q) = Θ. We notice that for every function η ∈ H 1 (B 1 ), we have

W (bQ + η) -Θ = b 2 W 0 (Q) + W 0 (η) + 2b B 1 ∇Q • ∇η -2 ∂B 1 Qη + b B 1 Q + B 1 η -Θ = b 2 W 0 (Q) + W 0 (η) -2b B 1 η∆Q + b B 1 Q + B 1 η -Θ, which gives W (bQ + η) -Θ = -(1 -b) 2 Θ + W 0 (η) + (1 -b) B 1 η.
(4.10)

We set q = 1 c 0 q ν . Thus q ∈ K + and W (q) = Θ + = 1 2 B 1 q. Setting η = c 0 q + ψ in (4.10) we obtain

W (bQ + c 0 q + ψ) -Θ = -(1 -b) 2 Θ + W 0 (c 0 q + ψ) + (1 -b) B 1 (c 0 q + ψ) = -(1 -b) 2 Θ + c 2 0 W 0 (q) + W 0 (ψ) + 2c 0 B 1 ∇q • ∇ψ -2 ∂B 1 qψ + (1 -b) B 1 (c 0 q + ψ) = -(1 -b) 2 Θ -c 2 0 Θ 2 + W 0 (ψ) -2c 0 B 1 ψ∆q + (1 -b)c 0 Θ + (1 -b) B 1 ψ = - (1 -b -c 0 ) 2 + (1 -b) 2 2 Θ + W 0 (ψ) + β(ψ), (4.11)
where in the last line we set

β(ψ) := (1 -b) B 1 ψ -c 0 B + 1 ψ.
Taking ψ to be the α-homogeneous extension of ψ, we get that β( ψ) -(1 -ε α )β(ψ) = 0, which concludes the proof of the lemma.

4.2.

Homogeneity improvement of ψ. In this subsection we prove the inequality (4.6). We first notice that if Q A is non-negative, then we can choose Q B = Q A and k j=1 a 2 j = 0. Thus, (4.6) follows directly by Lemma 2.1 and the fact that for the eigenvalues on the sphere λ j > 2d implies λ j ≥ 3(d + 1).

In the rest of this subsection, we assume that Q A changes sign and Q B is given by (4.5). In particular, Q A -Q B is a homogeneous polynomial of second degree with ∆Q A -Q B = 0, so it is an element of the eigenspace E 2d , corresponding to the eigenvalue 2d. We choose φ 2 ∈ E 2d and c 2 ∈ R such that

Q A -Q B = c 2 φ 2 , where ∂B 1 φ 2 2 (θ) dH d-1 (θ) = 1.
Thus, on ∂B 1 we can write ψ as

ψ(θ) = c 2 φ 2 (θ) + φ(θ) = c 2 φ 2 (θ) + {j : λ j >2d} c j φ j (θ).
Applying Lemma 2.1 we have

W 0 ( ψ) -(1 -ε α )W 0 (ψ) = ε α d + 2α -2 (α -2)(d + α)c 2 2 + {j : λ j >2d} (-λ j + λ α )c 2 j = ε 2 α (d + α) 2 (d + 2α -2) c 2 2 + ε α d + 2 {j : λ j >2d} (-λ j + λ α )c 2 j
Choosing the constant ε small enough, the equation (4.3) implies that 2 < α ≤ 5 2 ; by the fact that λ j > 2d ⇒ λ j ≥ 3(d + 1) (see Subsection 2.5) we have

1 - λ α λ j ≥ 1 - 1 3(d + 1) 5 2 d + 1 2 > d + 2 6(d + 1)
, whenever λ j > 2d.

Hence, the right-hand side in the previous equality can be estimated by

W 0 ( ψ) -(1 -ε α )W 0 (ψ) ≤ ε 2 α (d + 3) 2 (d + 2) c 2 2 - ε α 6(d + 1) {j : λ j >2d} λ j c 2 j = ε 2 α (d + 3) 2 (d + 2) ∂B 1 (Q A -Q B ) 2 - ε α 6(d + 1) ∂B 1 |∇ θ φ| 2 . (4.12)
It order to estimate the first term in the right-hand side, we notice that

Q A -Q B = k j=1 a j x 2 d - k j=1 a j x 2 j
, hence its L 2 -norm is a degree 2 homogeneous polynomial in (a 1 , ..., a k ) (with coefficients depending only on d). Hence there exists a dimensional constant C d such that

∂B 1 (Q A -Q B ) 2 ≤ C d k j=1 a 2 j .
Together with (4.12), this gives (4.6).

The higher modes control

k j=1 a 2 j . In this section we prove the inequality (4.7) from Step 3. Since the trace c(θ) is positive on ∂B 1 and can be written as

c(θ) = - k j=1 a j θ 2 j + d j=k+1 a j θ 2 j + q ν (θ) + φ(θ) ≥ 0, we get that φ(θ) ≥ k j=1 a j θ 2 j - d j=k+1 a j θ 2 j + ≥ a 1 θ 2 1 - d j=2 θ 2 j +
, on the half-sphere ∂B 1 ∩ {q ν = 0}. Thus, we get

φ(θ) ≥ a 1 4 θ 2 1 on the set U a 1 ∩ {q ν = 0}, where U a 1 = θ ∈ ∂B 1 : a 1 θ 2 1 > 2 d j=2 θ 2 j .
Notice that that for a 1 small enough we have

1 2 (d -1)ω d-1 √ a 1 d-1 ≤ H d-1 (U a 1 ) ≤ 2(d -1)ω d-1 √ a 1 d-1 .
Thus, we obtain

∂B 1 φ 2 ≥ C d a 2 1 √ a 1 d-1 = C d a (d+3)/2 1
, for a dimensional constant C d > 0. Without loss of generality we can suppose that

a 2 1 ≥ 1 k k j=1 a 2 j ≥ 1 d k j=1 a 2 j ,
and so, we get

∂B 1 |∇ θ φ| 2 ≥ 2d ∂B 1 φ 2 ≥ 2d C d a (d+3)/2 1 ≥ C d k j=1 a 2 j (d+3)/4 , which gives (4.7) with γ = d -1 d + 3 and a dimensional constant C 3 .
4.4. The higher modes control W (z) -Θ. In this section we prove the inequality (4.8) from the final Step 4. Using the decomposition z = q ν + Q A + ψ and the identity (4.11) we get, with ψ = r 2 φ and using ∂B 1 φ = 0 since it contains only high modes,

W (z) -Θ = - (1 -b -c 0 ) 2 + (1 -b) 2 2 Θ + 1 d + 2 ∂B 1 |∇ θ φ| 2 -2dφ 2 - c 0 d + 2 ∂B + 1 φ ≤ - c 2 0 4 Θ + 1 d + 2 ∂B 1 |∇ θ φ| 2 -2dφ 2 - c 0 d + 2 ∂B + 1 φ,
where the last inequality follows by the fact that

(1 -b -c 0 ) 2 + (1 -b) 2 ≥ c 2 0 2
, for every b, c 0 ∈ R.

By the Cauchy-Schwarz inequality, we have

- c 0 d + 2 ∂B + 1 φ ≤ c 2 0 4 Θ + 1 (d + 2)Θ ∂B + 1 φ 2 ≤ c 2 0 4 Θ + |∂B 1 | (d + 2)Θ ∂B 1 φ 2 = c 2 0 4 Θ + 8d 2 ∂B 1 φ 2 .
Thus, we get

W (z) -Θ ≤ 1 d + 2 ∂B 1 |∇ θ φ| 2 -2dφ 2 + 8d 2 ∂B 1 φ 2 ≤ 1 d + 2 ∂B 1 |∇ θ φ| 2 + 8d 2 ∂B 1 φ 2 ≤ 1 d + 2 + 4d ∂B 1 |∇ θ φ| 2 ,
where the last inequality is due to the fact that φ contains only modes φ j corresponding to eigenvalues λ j > 2d. This gives (4.8) where the constant C 4 can be choosen as C 4 = 1 + 4d.

4.5.

Improvement of the decay rate. This subsection is dedicated to the improvement of the inequality (4.7). The main result, contained in the following lemma, is more general and holds in any dimension.

Lemma 4.2. Suppose that 0 ≤ k < d and

Q A (x) = - k j=1 a j x 2 j + d j=k+1
a j x 2 j , where 0 < a j for every j = 1, . . . , k, 0 ≤ a j ≤ 1 for every j = k + 1, . . . , d.

Let φ ∈ H 1 (∂B 1 ) be of zero mean, that is

∂B 1 φ(θ) dH d-1 (θ) = 0, and such that φ ≥ Q A on the half-sphere {ξ ∈ ∂B 1 : ξ • ν > 0},
determined by some unit vector ν ∈ ∂B 1 .

Then, there are dimensional constants C > 0 and δ > 0 such that if

k j=1 a 2 j ≤ δ, then k j=1 a 2 j ≤ C d ∇ θ φ 2(1-γ k ) L 2 (∂B 1 ) , (4.13)
where

γ k =        0 , if k = 0, d -k d -k + 4
, for every k = 1, . . . , d -2,

0 , if k = d -1.
Remark 4.3. The above lemma is to be applied to the traces c of the solutions u of an obstacle problem, which can be written in the form c(θ) = Q A (θ) + q ν (θ) + φ(θ). We notice that, although one might think that k corresponds precisely to point of the k-th stratum, we do not know a way to deduce the precise form of Q A just from looking at the blow-up limits of u. This means that even if the blow up Q B is such that dim(ker B) = k, we still cannot infer anything on the structure of B. It follows that this result cannot be applied to improve the regularity of the singular sets of ∂{u > 0}, except in dimension two, where γ 0 = γ 1 = 0. This corresponds to the assumption of Weiss on the projection of c on the set of admissible blow-ups K, which again finds application only in dimension two.

Proof of Lemma 4.2. If k = 0, then the inequality is trivial and so, we can suppose that k ≥ 1.

Suppose that 1 ≤ k < d -1. Without loss of generality we can suppose

a 2 1 ≥ 1 k k i=1 a 2 i .
Setting X = (x 1 , . . . , x k ), X = (x k+1 , . . . , x d ) and

• = • L 2 (∂B 1 ) , we have k i=1 a i x 2 i - d i=k+1 a i x 2 i + ≥ a 1 x 2 1 - d i=k+1 a i x 2 i + ≥ a 1 x 2 1 -|X | 2 + = 1 k k j=1 a 1 x 2 j -|X | 2 + ≥ 1 d a 1 |X | 2 -|X | 2 + .
We now notice that

a 1 |X | 2 -|X | 2 ≥ a 1 2 |X | 2 on the set U a 1 = X = (X , X ) ∈ ∂B 1 : a 1 2 |X | 2 ≥ |X | 2 ,
and for a 1 small enough we get

1 2 kω k √ a 1 d-k ≤ H d-1 (U a 1 ) ≤ 2kω k √ a 1 d-k .
In particular,

inf{Q A , 0} 2 L 2 (∂B 1 ) ≥ a 1 |X | 2 -|X | 2 + 2 L 2 (∂B 1 ) ≥ C d a d-k+4 2 1 ≥ C d k i=1 a 2 i d-k+4 4 . Now, since Q A is even and φ 2 L 2 (∂B 1 ) ≤ 1 d-1 ∇ θ φ 2 L 2 (∂B 1 )
, we obtain the claimed inequality

k i=1 a 2 i ≤ C d ∇ θ φ 8 d-k+4 L 2 (∂B 1 ) .
Suppose that k = d -1. We argue by contradiction. Suppose that there are a sequence of functions φ n : ∂B 1 → R of zero mean and vectors ν n ∈ ∂B 1 and (a 1 n , . . . , a k n ) such that φ n (θ) ≥ a 1 n θ 2 1 on the set {θ ∈ ∂B 1 :

θ d = 0 , θ • ν n > 0}, a 1 n ≥ 1 k k j=1 |a j n | 2 1/2 and k j=1 |a j n | 2 ≥ n ∇φ n 2 L 2 (∂B 1 ) .
Thus, the sequence of functions ψ n := φ n /a 1 n is such that lim

{θ d =0}∩∂B 1 ψ 2 n dH d-2 ≤ C ∂B 1 |∇ψ n | 2 + ψ 2 n dH d-1 .
4.6. On the sharpness of the non-homogeneous estimate in Theorem 1. We conclude this section with an example, which shows that in dimension higher than three one cannot estimate the distance to the cone K by just using the energy of the higher modes φ to the power one. Indeed, such an estimate would be in contradiction with inequality (4.14) below. In particular, Example 1 shows that for general traces in higher dimension our method cannot be improved.

Example 1. Consider the non-negative trace c : ∂B 1 → R + given by

c(θ) =   1 4(d -1) d-1 j=1 θ 2 j -εθ 2 d   + .
Notice that, since c is even its Fourier expansion on the sphere ∂B 1 does not contain linear terms.

As in the proof of Theorem 1, the trace c can be uniquely decomposed as c(θ) = Q(θ)+φ(θ), where Q is a homogeneous polynomial of second degree and φ contains only higher modes, that is

φ(θ) = {j : λ j >2d} c j φ j (θ).
We claim that

∇ θ φ 4 d+1 L 2 (∂B 1 ) dist L 2 (∂B 1 ) (Q, K). (4.14)
In order to prove (4.14) we set

P (θ) = 1 4(d -1) d-1 j=1 θ 2 j -εθ 2 d and R(θ) =   εθ 2 d - 1 4(d -1) d-1 j=1 θ 2 j   + ,
and we notice that c(θ) = P (θ) -R(θ). It is easy to check that the term R has the following asymptotic behavior when the parameter ε is small:

R L ∞ (∂B 1 ) = ε , H d-1 ({R > 0}) ∼ ε d-1 2 , ∇ θ R L ∞ (∂B 1 ) ∼ √ ε , R L 2 (∂B 1 ) ∼ ε d+3 4 and ∇ θ R L 2 (∂B 1 ) ∼ ε d+1 4 .
The function R can be decomposed as

R(θ) = c 0 H d-1 (∂B 1 ) + c 2 φ 2 (θ) -φ(θ),
where • c 0 ∈ R corresponds to the first (constant) mode of the Fourier expansion of R on ∂B 1 and can be estimated in terms of ε as

c 0 = 1 H d-1 (∂B 1 ) ∂B 1 R dH d-1 ≤ R L 2 (∂B 1 ) ε d+3 4 ;
• φ 2 (θ) is an eigenfunction of the Laplacian on the sphere corresponding to the eigenvalue 2d and φ 2 L 2 (∂B 1 ) = 1 and the constant c 2 ∈ R can be estimated as

|c 2 | ≤ ∂B 1 Rφ 2 ≤ R L 2 (∂B 1 ) ε d+3 4 ;
• the function φ is precisely the one from the decomposition of c, contains only higher modes and satisfies the following estimate:

∇ θ φ L 2 (∂B 1 ) ≤ ∇ θ R L 2 (∂B 1 ) + |c 2 | ∇ θ φ 2 L 2 (∂B 1 ) ε d+1 4 + ε d+3 4 2d ε d+1 4 .
On the other hand, the L 2 (∂B 1 ) distance from Q = P -c 0 -c 2 φ 2 to the cone K of nonnegative homogeneous polynomials of second degree has the behavior

dist L 2 (∂B 1 ) P -c 0 -c 2 φ 2 , K ∼ dist L 2 (∂B 1 ) P, K ∼ ε.
Thus, we finally get the claimed inequality (4.14)

∇ θ φ 4 d+1 L 2 (∂B 1 ) ε ∼ dist L 2 (∂B 1 ) P -c 0 -c 2 φ 2 , K .

Uniqueness of blow-up and regularity of free boundary

In this Section we prove Theorems 3 and 4, focusing on the statement 2 of each result. We show in detail how the logarithmic estimates follow from the "modified" epiperimetric inequality of Theorem 1 and we prefer to skip the analogous estimates on the Hölder continuity at regular points, since this is the main improvement of the present paper and since the proof of the latter is a simpler version of the estimates below and it is already contained in [16, Theorem 4 and 5].

Proposition 5.1. Let Ω ⊂ R d be an open set and u ∈ H 1 (Ω) a minimizer of E. Then for every compact set Ω 0 Ω, there is a constant C := C(d, Ω 0 , Ω) > 0 such that for every free boundary point x 0 ∈ Sing(u) ∩ Ω 0 , the following decay holds

u x 0 ,t -u x 0 ,s L 1 (∂B 1 ) ≤ C (-log(t)) -1-γ 2γ
for all 0 < s < t < dist(Ω 0 , ∂Ω) .

(5.1)

Proof.

Step 1 (closeness of the blow ups for a given point x 0 ). Let x 0 ∈ Ω 0 and let r 0 ∈ (0, dist(Ω 0 , ∂Ω)] be such that the epiperimetric inequality of Theorem 1 can be applied to the rescaling u x 0 ,r for every r ≤ r 0 . We claim that

u x 0 ,t -u x 0 ,s L 1 (∂B 1 ) ≤ C (-log(t/r 0 )) -1-γ 2γ
for all 0 < s < t < r 0 .

We assume x 0 = 0 without loss of generality and e(r) = W (u, r) -Θ u (0).

By the monotonicity formula (2.1) and the epiperimetric inequality of Theorem 1, there exists a radius r 0 > 0 such that for every r ≤ r 0

d dr e(r) ≥ d + 2 r W (c r ) -Θ u (0) -e(r) + f (r) ≥ c r e(r) 1+γ + 2f (r) (5.2)
where γ ∈ (0, 1) is a dimensional constant and

f (r) := 1 r ∂B 1 |x • ∇u r -2u r | 2 dH 1 .
We obtain that d dr

-1 γe(r) γ -c log r = 1 e(r) 1+γ d dr e(r) - c r ≥ 1 e(r) 1+γ f (r) ≥ 0 (5.3)
and this in turn implies that -e(r) -γ -cγ log r is an increasing function of r, namely that e(r) decays as e(r) ≤ (e(r 0 ) -γ + cγ log r 0 -cγ log r)

-1 γ ≤ (-cγ log(r/r 0 )) -1 γ . (5.4)
For any 0 < s < t < r 0 we estimate the L 1 distance between the blow ups at scale s and t through the Cauchy-Schwarz inequality and the monotonicity formula (2.1)

∂B 1 |u t -u s | d H d-1 ≤ ∂B 1 t s 1 r |x • ∇u r -2u r | dr dH d-1 ≤ dω d 1/2 t s 1 r 1 r ∂B 1 |x • ∇u r -2u r | 2 dH d-1 1/2 dr ≤ dω d 2 1/2 t s e (r) r 1/2 dr ≤ dω d 2 1/2 (log(t) -log(s)) 1/2 (e(t) -e(s)) 1/2 . (5.5) Let 0 < s 1/2 < t 1/2 < r 0 such that s/r 0 ∈ [2 -2 i+1 , 2 -2 i ), t/r 0 ∈ [2 -2 j+1 , 2 -2 j
) for some j ≤ i and applying the previous estimate (5.4) to the exponentially dyadic decomposition, we obtain

∂B 1 |u t -u s | d H d-1 ≤ ∂B 1 u t -u 2 -2 j+1 r 0 d H d-1 + ∂B 1 u 2 -2 i r 0 -u s d H d-1 + i-1 k=j+1 ∂B 1 u 2 -2 k+1 r 0 -u 2 -2 k r 0 d H d-1 ≤ C i k=j log 2 -2 k -log 2 -2 k+1 1/2 e 2 -2 k r 0 -e 2 -2 k+1 r 0 1/2 ≤ C i k=j 2 k/2 e 2 -2 k r 0 1/2 ≤ C i k=j 2 (1-1/γ)k/2 (5.6) ≤ C2 (1-1/γ)j/2 ≤ C(-log(t/r 0 )) γ-1 2γ ,
where C is a dimensional constant that may vary from line to line.

Step 2 (uniform smallness of monotonic quantity for x 0 ∈ Sing(u) ∩ Ω 0 ). We claim that for every ε > 0 there exists r 0 > 0 such that e(u x,r ) ≤ ε for every x ∈ Sing(u) ∩ Ω 0 , r ≤ r 0 .

Assume by contradiction that there exists a sequence x k → x 0 and r k → 0 such that ε < e(u x k ,r k ) for any k ∈ N. By the monotonicity of W , for any ρ > 0 and k large enough

ε < W (u, x k , r k ) -Θ u (0) ≤ W (u, x k , ρ) -W (u, x 0 , ρ) + W (u, x 0 , ρ) -Θ u (0).
In turn, the right-hand side can be made arbitrarily small by choosing first ρ sufficiently small (to make the difference of the last two terms small) and then k sufficiently large.

Step 3 (uniform scale for the application of the epiperimetric inequality at x 0 ∈ Sing(u) ∩ Ω 0 ). We claim that for every ε > 0 there exists r 0 > 0 such that dist L 2 (u x,r , K) ≤ ε for every x ∈ Sing(u) ∩ Ω 0 , r ≤ r 0 .

(notice that this statement holds also if in place of the L 2 -distance we consider the H 1 -distance). Assume by contradiction that there exists ε > 0 a sequence x k → x 0 and r k → 0 such that

ε < dist L 2 (u x k ,r k , K)
for any k ∈ N.

(5.7)

Since the sequence {u x k ,r k } k∈N is uniformly bounded in H 2,∞ , it converges strongly in H 1 up to a (not relabelled) subsequence to u 0 . Moreover, thanks to Step 2, the limit u 0 must satisfy W (u 0 , x 0 , 1) = Θ u (0), so that it belongs to K. This contradicts (5.7).

Step 4 (conclusion). We can now conclude the proof of the Proposition. We observe that for every r 0 > 0 and t ≤ r 2 0 , we have log(t/r 0 ) ≤ 2 log t. From Step 1 and 3, we deduce that there exists r 0 > 0 such that for all 0 < s < t < r 2 0 , x 0 ∈ Sing(u) ∩ Ω 0 u x 0 ,t -u x 0 ,s L 1 (∂B 1 ) ≤ C (-log(t))

-1-γ 2γ .

From (5.5) we have u x 0 ,t -u x 0 ,r 2 0 L 1 (∂B 1 ) ≤ C(-log(r 0 )) 1/2 e(dist(Ω 0 , ∂Ω)) 1/2 and the right hand side is estimated by C(-log(r 0 ))

-1-γ 2γ
for a constant C depending only on d, r 0 , e(dist(Ω 0 , ∂Ω)), dist(Ω 0 , ∂Ω).

As a consequence of the previous proposition we can prove the uniqueness of the blow up Theorem 3, with a logarithmic rate of convergence of the blow up sequence at each point of the singular set (and uniform in any compact set inside the domain).

Proof of Theorem 4. We notice that

|Q x 1 -Q x 2 | ≤ c(n) ∂B 1 |Q x 1 (x) -Q x 2 (x)| dH d-1 (x)
By the triangular inequality

Q x 1 -Q x 2 L 1 (∂B 1 ) ≤ u x 1 ,r -Q x 1 L 1 (∂B 1 ) + u x 1 ,r -u x 2 ,r L 1 (∂B 1 ) + u x 2 ,r -Q x 2 L 1 (∂B 1 )
Recalling that u ∈ C 1,1 and that ∇u(x 1 ) = 0, we estimate the term in the middle with u x 1 ,r -u x 2 ,r L 1 (∂B 1 ) ≤ 2γ ≤ dist(Ω 0 , ∂Ω). By Theorem 3 we see that

u x 1 ,r -Q x 1 L 1 (∂B 1 ) + u x 2 ,r -Q x 2 L 1 (∂B 1 ) ≤ C(-log(r)) -1-γ 2γ = C(-log |x 1 -x 2 | - 1 -γ 2γ log(-log |x 1 -x 2 |)) -1-γ 2γ
(5.9)

Noticing that the inequality a-1-γ 2γ log a ≥ a/2 holds for a greater than a given a 0 > 0 (depending only on γ and therefore on d), we apply this inequality to a = -log |x 1 -x 2 | to get

u x 1 ,r -Q x 1 L 1 (∂B 1 ) + u x 2 ,r -Q x 2 L 1 (∂B 1 ) ≤ C(-log |x 1 -x 2 |) -1-γ 2γ .
Putting together the previous inequalities, we find (1.6). The conclusion follows then by standard arguments (see for instance [START_REF] Focardi | Monotonicity formulas for obstacle problems with Lipschitz coefficients[END_REF]) by applying the Whitney extension theorem (which holds also for C 2,log functions [START_REF] Fefferman | Extension of C m,ω -smooth functions by linear operators[END_REF]) and the implicit function theorem.

5.1. Proof of Theorem 5. We notice that if u ∈ H 1 (Ω) is a minimizer of E q or A q , then it is locally W 2,∞ by the results of [START_REF] Gerhardt | Global C 1,1 -regularity for solutions of quasilinear variational inequalities[END_REF] and moreover it is an almost-minimizer of the functional E with a constant C depending only on q C 0,γ (Ω) , c q and u W 2,∞ loc . We say that u ∈ H 1 (Ω) is an almost minimizer of E if there exists a constant C > 0 such that for every ball B r (x 0 ) ⊂ Ω and for every v ∈ H 1 (B r (x 0 )) which agrees with u on ∂B r (x 0 ) Br(x 0 ) |∇u| 2 + q(x 0 ) max{u, 0} dx ≤ (1 + Cr γ ) Br(x 0 ) |∇v| 2 + q(x 0 ) max{v, 0} dx . (5.10) In the following we show that the statement of Theorem 5, in particular the logarithmic estimate, holds true also if we drop the assumption that u ∈ W 2,∞ loc (Ω) is a minimizer of E q or A q and we only assume the almost minimality.

The main modifications with respect to the arguments of Section 5 appear in Proposition 5.1 and we outline them below.

Up to a rescaling, we may assume that q(x 0 ) = 1. Applying the epiperimetric inequality of Theorem 1 to u r,x 0 , we find that (5.2) has to be modified for almost monotonicity (5.10) to get d dr e(r) ≥ d + 2 r W (c r ) -Θ u (0) -e(r) + f (r) ≥ c 0 r e(r) 1+γ -c 1 r 1-α + 2f (r) (where e(r) := W (u r , 1) -Θ u (0) and the notation is the same as in Section 5) for some constants c 0 , c 1 > 0.

We define now ẽ(r) = e(r) + 2α -1 c 1 r α and we notice that from the previous inequality and since a 1+γ + b 1+γ For r sufficiently small, the previous inequality implies that ẽ (r) ≥ c 0 r ẽ(r) 1+γ + 2f (r)

From the previous inequality, we see that ẽ(r) satisfies the same inequality that e(r) solved in (5.2). Hence, with the same argument as in (5.3), we see that ẽ satisfies the same estimate as e in (5.4) e(r) + 2α -1 c 1 r α = ẽ(r) ≤ (-cγ log(r/r 0 ))

-1 γ .
This inequality implies that, up to a constant, also e(r) satisfies a logarithmic estimate and we can carry out the rest of the proof of Proposition 5.1 and of Theorems 3 and 4.

Bozhidar Velichkov: Laboratoire Jean Kuntzmann (LJK), Université Grenoble Alpes Bâtiment IMAG, 700 Avenue Centrale, 38401 Saint-Martin-d'Hères E-mail address: bozhidar.velichkov@univ-grenoble-alpes.fr

n→∞ ∇ψ n 2 L 2 (

 22 ∂B 1 ) = 0 andψ n (θ) ≥ θ 2 1 on the set {θ ∈ ∂B 1 : θ d = 0 , θ • ν n > 0}, which is in contradiction with the trace inequality

∂B 1 1 0 2 ( 5 . 8 )

 1258 |∇u(x 1 + rx + t(x 2 -x 1 ))||x 2 -x 1 | r 2 dt dH(x) ≤ C u C 1,1 (Br(x 0 )) (r + |x 2 -x 1 |) |x 2 -x 1 | r We choose r = |x 1 -x 2 |(-log |x 1 -x 2 |) -1-γ 2γand we assume that r 0 satisfies the inequality|r 0 |(-log |r 0 |) -1-γ

  ≥ 2 -γ (a + b) 1+γ for any a, b ≥ 0 ẽ (r) ≥ c 0 r e(r) 1+γ + c 1 r 1-α + 2f (r) ≥ c 0 r [e(r) + c 1 r α 1+γ ] 1+γ + 2f (r)


Ignacijo Biluš

Luka Lešnik

Matevž Dular
email: matevz.dular@fs.uni-lj.si

Numerical Simulation of Various Mechanisms of Cavitation Erosion

Keywords: Cavitation, Numerical simulation, Erosion

The results of numerical simulations of cavitation at Venturi section are presented in the contribution. A fully compressible, cavitating flow simulations was performed to resolve the formation of the shock waves at cloud collapse which are believed to be directly related to the formation of the damage. Good agreements were noticed between calculations and tests performed in previous work. The numerical simulations with short time step were performed to investigate the processes of cavitation collapse and shedding off in detail. The results indicate the importance of different individual cavitation structures collapse on erosion potential. Present analysis results shows a great potential for future development of techniques for accurate predictions of cavitation erosion by numerical means only.

INTRODUCTION

Since Rayleigh reported on the cavitation erosion issue on ship propellers in 1917, considerable progress has been made to discover its hydrodynamic mechanisms [START_REF] Kato | Possibility of Quantitative Prediction of Cavitation Erosion without Model Test[END_REF][START_REF] Berchiche | A Cavitation Erosion Model for Ductile Materials[END_REF][START_REF] Franc | Impact Load Measurements in an Erosive Cavitating Flow[END_REF][START_REF] Franc | Incubation Time and Cavitation Erosion Rate of Work-Hardening Materials[END_REF]. Two main mechanisms are usually discussed: the micro-jet and pressure shock wave. For the case of the micro-jet it is believed that the liquid jet penetrates the bubble as the surrounding pressure is imbalanced. The jet velocity can reach a magnitude order of 100 m/s, and when it impacts the solid wall enormous stresses that cause pit formation occur [START_REF] Franc | Material and Velocity Effects on Cavitation Erosion Pitting[END_REF]. On the other hand the pressure shock wave approach, considers the bubbles to remain spherical during the collapse what, at the final stage causes a shock wave generation with a magnitude order of several MPa [START_REF] Wang | Shock Wave Development in the Collapse of A Cloud of Bubbles[END_REF]. Fortes-Patella et al. [START_REF] Fortes-Patella | Numerical and Experimental Investigations on Cavitation Erosion[END_REF] studied the interaction between shock wave emitted by the implosions of a spherical bubble and material deformation. They concluded that the cavitation induced damage is directly related to the pressure shock wave and the characteristics of material. Subsequently, they proposed a so-called energy cascade theory. It postulated that potential energy contained in a macro cavitation cloud, considering the conservation of energy, would transfer into the radiation of acoustic pressure wave, which might be emitted either by spherical bubble or vortex collapse as well as micro jet. The rapidly development of computational power and numerical simulation technology, increase the potential and the accuracy of the methods for the prediction of cavitating flow and cavitation erosion power. Dular & Coutier-Delgosha [START_REF] Dular | Numerical Modelling of Cavitation Erosion[END_REF] presented a prediction method of cavitation damage based on the micro-jet theory, by coupling a computational fluid dynamics (CFD) and the proposed erosion model. In the present paper we are comparing results of numerical simulations against experimental data obtained by Petkovsek & Dular [START_REF] Petkovšek | Simultaneous observation of cavitation structures and cavitation erosion[END_REF][START_REF] Dular | On the Mechanisms of Cavitation Erosion -Coupling High Speed Videos to Damage Patterns[END_REF]. We performed fully compressible, cavitating flow simulations to resolve the formation of the shock waves during cavitation collapse phase which is directly related to the formation of the damage in the vicinity of the solid material surfaces. We have confirmed that cavitation cloud shedding causes different typical hydrodynamic mechanisms and results in extreme conditions which are connected with material erosion. The results are valuable for further numerical investigation and development of cavitation erosion prediction models.

EXPERIMENT

Numerical simulation was performed on the case of Venturi geometry installed into a small cavitation tunnel at the Laboratory for Water and Turbine Machines, University of Ljubljana. The experimental analysis of cavitation erosion process was performed in the past and extensively reported [START_REF] Petkovšek | Simultaneous observation of cavitation structures and cavitation erosion[END_REF][START_REF] Dular | On the Mechanisms of Cavitation Erosion -Coupling High Speed Videos to Damage Patterns[END_REF]. According to this, only a brief description with key features of experimental set-up is given in the following. The Venturi geometry (Fig. 1a), which was 10 mm wide with a converging angle of 18º and diverging angle of 8º. The throat dimensions were 10×10mm 2 . The test section was made out of transparent plexi glass so that observation from all directions was made possible. The section was mounted into a cavitation tunnel (Fig. 1b) with a closed loop circle what enables to vary both the flow rate and the system pressure. This way a wide range of operating points can be achieved. The idea was to obtain sufficient damage in a very short period of time (in about 1 second) experimentaly. To set the operating conditions, first the test rig pressure was set to a desired value (490000 Pa, absolute pressure), then the valve upstream of the test section was closed and the pump was switched on to a determined rotating frequency. As the upstream valve was rapidly opened the velocity increased from 0 to 27.4 m/s in about 0.05 s achieving cavitation number σ = 1.3.

The idea of the experiment was to simultaneously record images of cavitation structures and cavitation erosion and by doing this to determine the mechanisms that lead to formation of pits. For this purpose several high speed cameras were used to simultaneously record the sequences of the cavitation dynamics and the damage sustained on the surface of the solid body.

The upper side of the foil is covered by vapour structures that obstruct the view, hence one needs to look at the foil from the bottom side to see the damage. Consequently the whole test section had to be made of transparent material and, equally important, the foil had to be thin enough so that the cavitation damage, which occurs on the side exposed to cavitation was also visible on the other side.

For the present experiment high speed cameras were synchronized and recorded at 30000fps at a reduced (608×86 pixels) resolution. One camera was used for observation of the aluminium foil while the other one captured images of cavitation structures. Erosion was evaluated in image pairs: image at time t+∆t was subtracted from image at time t. This way the change between times t and t+∆t could be detected and recognized as damage [START_REF] Petkovšek | Simultaneous observation of cavitation structures and cavitation erosion[END_REF].

Results -cavitation erosion mechanisms

The general behavior of the developed cavitating flow follows a distinctive pattern where cavitation structures of different shapes and sizes are shed from the attached cavity (Fig. 2). Not less than 5 different mechanisms that lead to the formation of the pit were determined (Fig. 2), although it is possible that the physical background of some is more or less the same.

Figure 3 shows the actual instants when the damage occured during the measurements -the damage is enlarged (3 times) for better representation and appears smaller in reality (its position and shape are not altered). The observed mechanisms were spherical cavitation cloud collapse (H), horseshoe cavitation cloud collapse (L), the "twister" cavitation cloud collapse (O) and in addition it was found that pits also appear at the moment of cavitation cloud separation (E) and near the stagnation point at the closure of the attached cavity (C).

MATHEMATICAL MODEL

The governing equations and homogenous flow model

The applied governing equations were based on the conservation form of the Reynolds averaged Navier-Stokes equations, including mass continuity (1), momentum equation ( 2) and energy equation ( 3):

( ) 0 m m i i u t x ρ ρ ∂ ∂ + = ∂ ∂ (1) (2)
( )

[ ( )] ( ) m m eff u t E E p k T ρ ρ ∂ + ∇ ⋅ = ∂ + ∇ ⋅ ∇ v (3)
The liquid phase and vapor phase are treated as a homogeneous mixture based on the volume of fraction. The mixture density and viscosity are defined as a function of vapor volume fraction:

(1 )

m v v v l ρ ρ α ρ α = + - (4)
(1 )

m v v v l µ µ α µ α = + - ( 5
)
where p is the pressure, ρ is the density, u is the velocity, μ and μt stand for the laminar viscosity and turbulent viscosity, δij is the Kronecker delta function,

2 2 m E h p u ρ = - +
, h is the entropy, keff is the effective conductivity, α is the volume fraction. The subscripts m,l,v indicate the mixture, liquid and vapor, respectively.

Turbulence model

As known, the turbulence model plays a significant role in the prediction of cavitating flow. Since the standard k-ε model is over-estimating the eddy viscosity in the mixture region, it cannot effectively resolve the detachment of the cavity from solid surface and excessively attenuates the cavitation instability. However, the shedding motion and subsequent collapse are the primary reason causing cavitation erosion. Therefore, a modified Re-normalized group (RNG) k-ε model, proposed by Coutier-Delgosha et al. [START_REF] Coutier-Delgosha | Evaluation of the Turbulence Model Influence on the Numerical Simulations of Unsteady Cavitation[END_REF], was employed in this work. It can successfully reduce the eddy viscosity by defining the turbulent viscosity as:

2 ( ) t m k f C µ µ ρ ε = (6) ( ) ( ) 1 (
)

m n m v v n v l f ρ ρ ρ ρ ρ ρ - - = + - (7)
where the coefficient Cμ=0.09, identical with k-ε model, and the exponent n=10, recommended by Coutier-Delgosha et al. [START_REF] Coutier-Delgosha | Evaluation of the Turbulence Model Influence on the Numerical Simulations of Unsteady Cavitation[END_REF].

Cavitation model

The vapor generation and disappearance are controlled by a mass transport equation model (TEM) based on the vapor volume fraction, expressed as:

( )

v j v j u m m t x α α + - ∂ ∂ + = + ∂ ∂ & & (8) The source term m +
& and m - & represent the mass rates of liquid evaporation and vapor condensation. In this paper, the Zwart-Gerber-Belamri model [START_REF] Zwart | A Two-Phase Model for Predicting Cavitation Dynamics[END_REF], deduced from the Rayleigh-Plesset equation, was applied, since it has, based on our previous experience, a precise cavitating prediction performance and a good convergence behavior. It is defined as:

3 (1 ) 2 3 nuc v v v vap B l p r p F R m α ρ ρ + - - = & if p<pv (9) 3 2 3 v v v cond B l p p F R m α ρ ρ - - = & if p>pv ( 10
)
where Fvap and Fcond are the empirical calibration coefficients of evaporation and condensation, respectively, rnuc stands for the nucleation site volume fraction, RB is the nucleation site radius. Vaporization is initiated at nucleation sites, which can be regarded as the non-condensible gases. pv represents the water vaporization pressure. The recommended values of these coefficients are: Fvap=50, Fcond=0.01, rnuc=5×10-4, RB=2×10 -6 m and pv=3574Pa.

As compressible approach was adopted the vapour obeyed the ideal gas law and the liquid density variation was described via Tait equation:

n l ref ref p B p B ρ ρ + = + ( 11
)
where ρref and pref denote the reference liquid density and pressure 200mm upstream of the Venturi section. As for constant B and n, they are 300MPa and 7 for water, respectively.

NUMERICAL MODEL AND SIMULATION SETUP

To get a better accuracy and convergence behavior, the structured hexahedral grid was generated to model the fluid computational domain. The grid independence test was conducted on the basis of three kinds of mesh density.

Considering the calculation time and accuracy, the grid with the total number of the elements about 0.5 million was applied.

[13]. The commercial CFD code "ANSYS-CFX" was used to solve the URANS equations summarized above. A static pressure boundary conditions are imposed on the inlet and outlet respectively. Meanwhile, the turbulent intensity at the inlet is set as zero gradient. A no-slip wall is applied on the Venturi surface with scalable wall function. The convergence criteria are all set to 1×10 -4 . On the premise of ensuring the temporal accuracy, all the simulations were initiated by running the calculation under upwind scheme based on COUPLED algorithm for four shedding periods until a timeperiodic solution has been reached, with a time step of 1.6×10 -4 s. Afterwards, the algorithm was changed to the second-order scheme and additionally, the time step was reduced to only 1.6×10 -6 s to obtain a more precise resolution concerning to the cavity shedding off and collapse.

RESULTS

The numerical simulation results are presented in Figures 567.

The left columns show isosurfaces of 10% vapor volume fractions while absolute pressure ratio p/pmax is shown in right columns. The value of pmax present the maximum pressure in each sequence analysed. The typical cavitation structure cycles are presented in time series with ∆t=1.6×10 -6 s. Figure 5 show the mechanism of stagnation point formation at closure of attached cavity. It is evident, that cavitation structure (left column) remains relatively stable. From pressure field (right column) it is evident, that elevated pressure region match the stagnation point created by flow, directed towards solid face. Following the fact that surface area where pressure peaks are present is relatively small, it is assumed that small scale cavitation structures (bubbles) collapse in this region. According to stable nature of attached cavity, it is expected the damage on surface to occur gradually. On the other hand, the damage progress is limited by re-entrant jet development since it blocks the shedding of small scale structures.

The sequence shown on Figure 6

CONCLUSION

In this paper, fully compressible cavitating flow simulations were performed to resolve the formation of the shock waves during cavitation collapse phase in a Venturi section. At the beginning, the results were validated against visualization experiments performed on the cavitation tunnel [START_REF] Petkovšek | Simultaneous observation of cavitation structures and cavitation erosion[END_REF]. The simulation results, obtained with a time step of 1.6×10 -4 s, are in a good agreement with the visualization experiments. The analysis confirmed that cavitation cloud shedding causes extreme conditions which are connected to different typical hydrodynamic mechanisms causing pressure waves in analyzed domain. On the other hand, the simulation results obtained with a time step of 1.6×10 -6 s showed the periodic nature of pressure instabilities during the collapse of four different cavitation structures types. The pressure peak areas are predicted by homogeneous model which corresponds to that by collapse of tiny bubbles in larger vapour structures and which considerably contributes to the erosive energy potential of cavitating flow in the pressure recovery region of Venturi section. The numerical model used and results obtained will be useful for further numerical investigation and development of cavitation erosion prediction models.

Figure 1 .

 1 Figure 1. The Venturi geometry (left) and the experimental setup (right): (1) pump, (2) upstream tank, (3) test section, (4) downstream tank, (5 & 6) valves, (7) electromagnetic flow meter, (8) thermocouple, (9) pressure sensor, (10) compressor and (11) vacuum pump.

Figure 2 .

 2 Figure 2. Schematic representation of observed mechanisms that lead to occurrence of cavitation erosion. [10]. The pressure difference between the outer flow and the inside of the attached cavity, forces the streamlines to curve towards the cavity and the surface beneath it. This causes attached cavity to close and the formation of a stagnation point at which the flow is split into outer flow which reattaches to the wall and the re-entrant jet which travels upstream, carrying a small quantity of the liquid to the inside the cavity. As the reentrant jet travels upstream it loses momentum due to shear forces, turns upwards and "cuts" the attached cavity, causing cavitation cloud separation (shedding). The cloud is then entrained downstream by the main flow and can violently collapse in a region of pressure recovery. During the separation, circulation around the structure appears. It causes the reshape of separated cloud and formation of horseshoe cavitation structure which results in damage in the region where vortex leg touches solid surface during break up phase. Meanwhile the attached cavity begins to grow and the process is periodically repeated [14]. The instants (C, E, H, L and O) when the damage occurs are highlighted by the darker colour.

Figure 3 .

 3 Figure 3. Cavitation structures at the moments at which the damage appeared on the foil. The position and the size of the damage were obtained from the images of the foil and were superimposed to the images of the structures.

Figure 4 .

 4 Figure 4. Computational mesh near the throat of the Venturi.

Figure 5 .Figure 6 .Figure 7 .

 567 Figure 5. Numerical simulation results -closure of attached cavity.