Epiperimetric inequalities in the obstacle problem for the fractional Laplacian

Using epiperimetric inequalities approach, we study the obstacle problem \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\min \{(-\Delta )^su,u-\varphi \}=0,$$\end{document}min{(-Δ)su,u-φ}=0, for the fractional Laplacian \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(-\Delta )^s$$\end{document}(-Δ)s with obstacle \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varphi \in C^{k,\gamma }(\mathbb {R}^n)$$\end{document}φ∈Ck,γ(Rn), \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k\ge 2$$\end{document}k≥2 and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma \in (0,1)$$\end{document}γ∈(0,1). We prove an epiperimetric inequality for the Weiss’ energy \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$W_{1+s}$$\end{document}W1+s and a logarithmic epiperimetric inequality for the Weiss’ energy \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$W_{2m}$$\end{document}W2m. Moreover, we also prove two epiperimetric inequalities for negative energies \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$W_{1+s}$$\end{document}W1+s and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$W_{2m}$$\end{document}W2m. By these epiperimetric inequalities, we deduce a frequency gap and a characterization of the blow-ups for the frequencies \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda =1+s$$\end{document}λ=1+s and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda =2m$$\end{document}λ=2m. Finally, we give an alternative proof of the regularity of the points on the free boundary with frequency \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1+s$$\end{document}1+s and we describe the structure of the points on the free boundary with frequency 2m, with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m\in \mathbb {N}$$\end{document}m∈N and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$2\,m\le k.$$\end{document}2m≤k.

The aim of the paper is to established the optimal regularity of the solution and to describe the structure and the regularity of the free boundary Γ(u) := ∂Λ(u) where Λ(u) := {x ∈ R n : u(x) = ϕ(x)} is the contact set.
In particular, when s = 1 2 , i.e. a = 0 and L a = ∆, the problem (1.2) is the thin obstacle problem (also know as Signorini problem).
Localizing the problem in B 1 ⊂ R n+1 , the solution of (1.2) can be obtained by minimizing the functional among the admissible functions Here we denote by H 1 (Ω, a) := H 1 (Ω, |y| a ) the weighted Sobolev space.Similarly, L 2 (Ω, a) := L 2 (Ω, |y| a ) is the weighted Lebesgue space.
In the following, with a slight abuse of notation, we denote by u the L a −extension in R n+1 of u, and we suppose that u ∈ H 1 loc (R n+1 , a).Moreover, with a slight abuse of notation, we also denote by x 0 ∈ R n the point (x 0 , 0) ∈ R n × {0}.
1.3.Obstacle ϕ ≡ 0. We will say that u is a solution with 0 obstacle, if solves (1.2) with ϕ ≡ 0, i.e. if u satisfies (1.4)Moreover, we denote by x, −y)} the set of admissible function with ϕ ≡ 0, then u is a minimum of the functional (1.3) in the class K c , with c = u| ∂B 1 .
1.4.Reduction to 0 obstacle.Let u be a solution of (1.2) with obstacle ϕ ∈ C k,γ (R n ), k ≥ 2 and γ ∈ (0, 1).Let q x 0 k (x) be the k-th Taylor polynomial of ϕ at x 0 ∈ Γ(u) and q x 0 k (x, y) be a polynomial of degree k and the L a −harmonic extension of q x 0 k (x) (see Lemma A.3). Then q x 0 k (x, y) solves the following problem We can define u x 0 (x, y) = u(x, y) − q x 0 k (x, y) − (ϕ(x) − q x 0 k (x)), then u x 0 solves the following problem where h(x, y) = h x 0 (x, y) := ∆ x (ϕ(x) − q x 0 k (x)).Notice that starting from an obstacle problem with obstacle ϕ ≡ 0, we have reduced the problem to the case ϕ ≡ 0, where the right-hand side in the third and fourth line of (1.5) is not 0.However, the function h = h x 0 is very small near x 0 .Precisely, by the C k,γ −regularity of ϕ, (1.6) Notice that the function u x 0 inherits the local behavior of u.In what follows we will study the properties of the functions u x 0 at all the points x 0 on the boundary Γ(u) of the contact set Λ(u).
1.5.State of art.In this section we give a brief overview on the state of the art of the obstacle for the fractional Laplacian; for more details we refer to [DS18] for s ∈ (0, 1), and to [PSU12], [Fer22] for the case s = 1 2 .The obstacle problem for the fractional Laplacian was studied by Silvestre in [Sil07], where it was established the almost-optimal regularity C 1,α of the solution for all α ∈ (0, s), in the case ϕ ∈ C 2,1 (R n ).Moreover, in the same paper, it is proved that if ϕ ∈ C 1,β (R n ), then u ∈ C 1,α (R n ) for all α ∈ (0, min{s, β}).
Later, in [CSS08], Caffarelli, Salsa and Silvestre proved the optimal regularity C 1,s of the solution for ϕ ∈ C 2,1 (R n ), thus generalizing the result previously obtained by Athanasopoulos and Caffarelli in [AC04] in the case s = 1 2 and ϕ ≡ 0. They use a modification of the following "pure" Almgren's frequency function which is monotone in r provided that ϕ ≡ 0. Precisely, setting v 2 |y| a dH n , (1.8) they introduce the following generalized Almgren's frequency function: Φ x 0 (r, v) = (r + Cr 1+p ) d dr log max{H x 0 (r, v), r n+a+2(k+γ−p) } with k = 2, γ = 1 and p = 1.This function is monotone in r also when ϕ ≡ 0, with ϕ ∈ C k,γ (R n ).Here we denoted by v = u x 0 , with x 0 ∈ Γ(u), a solution to a similar problem to (1.5).As a consequence of the monotonicity of Φ x 0 , they obtain that if x 0 ∈ Γ(u) is a free boundary point such that for some λ ∈ (0, 2), then the rescalings v r,x 0 (x, y) := v(x 0 + rx, ry) 1 r n+a ∂Br(x 0 ) u 2 |y| a dH n , converge, up to a subsequence, to a function v 0,x 0 which is a λ−homogeneous solution of (1.4) (with 0 obstacle) (see Lemma 6.1 and Lemma 6.2 in [CSS08]).
We will call Γ 1+s (u) the set of regular points and we will denote it by Reg(u) since it is known to be locally a C 1,α submanifold of dimension n − 1 (see [CSS08]).This is a generalization of the result of Athanasopoulos, Caffarelli and Salsa obtained in [ACS08] in the case s = 1 2 and ϕ ≡ 0. In the case ϕ ≡ 0, we can consider the "pure" Almgren's frequency function as in (1.7), and the free boundary can be decomposed as where Reg(u) = Γ 1+s (u) are the regular points, Sing(u) = m∈N Γ 2m are the so-called singular points, and Other(u) are all the remaining points in Γ(u).
For the singular points, in the case s = 1 2 , in [GP09], Garofalo and Petrosyan proved that Sing(u) is contained in the union of at most countably many submanifolds of class C 1 .In [GR20], using the monotonicity of the generalized Almgren's frequency for k ≥ 2, γ ∈ (0, 1) and p small enough, Garofalo and Ros-Oton extended the structure of singular set to any s ∈ (0, 1).Indeed in the case ϕ ∈ C k,γ (R n ), k ≥ 2 and γ ∈ (0, 1), they proved that is contained in the union of countably many submanifolds C 1 , where the bound 2m ≤ k is needed in order to assure the existence of blow-ups.Thus, they improved the result previously obtained in [GP09] (in the case s = 1 2 ), where more regularity of the obstacle ϕ was required.
Moreover, Focardi and Spadaro described the entire free boundary, up to sets of null H n−1 measure, in the case ϕ ≡ 0 in [FS18] and in the case ϕ ≡ 0 in [FS20].
An alternative proof of the regularity and structure of the free boundary uses epiperimetric inequalities approach for the Weiss' energy (1.9)In the case s = 1 2 and ϕ ≡ 0, Garofalo, Petrosyan and Smit Vega Garcia in [GPS16] and Focardi and Spadaro in [FS16] proved an epiperimetric inequality for W 1+s to deduce the regularity C 1,α of the regular points Reg(u).In the case s ∈ ( 1 2 , 1) and ϕ ≡ 0, using epiperimetric inequalities approach, the same regularity for Γ 1+s (u) was established by Garofalo, Petrosyan, Pop and Smit Vega Garcia in [GPPS17] (see also [Ger19] for ϕ ≡ 0 and s ∈ (0, 1)).
The regularity of the free boundary Γ 1+s (u) in the case of more general degenerate elliptic operators for variable coefficients was established recently by Banerjee, Buseghin and Garofalo in [BBG22], using again an epiperimetric inequality.
Following epiperimetric inequalities approach, Colombo, Spolaor and Velichkov in [CSV20] give an alternative proof of the structure of singular set Sing(u) in the case s = 1 2 and ϕ ≡ 0. They improved the regularity of the manifolds that contains the singular set up to C 1,log , due to the logarithmic epiperimetric inequality for W 2m .
1.6.Main results.The goal of this paper is to generalize the epiperimetric inequalities that we already know for the thin obstacle problem s = 1 2 , to any s ∈ (0, 1).With this generalization, we can deduce the previous results of regularity and structure of the free boundary, even for non-zero obstacles ϕ ≡ 0, with ϕ ∈ C k,γ (R n ), k ≥ 2 and γ ∈ (0, 1).In particular, we prove an epiperimetric inequality for W 1+s and a logarithmic epiperimetric inequality for W 2m , for each s ∈ (0, 1).
Before we state our main results (Theorem 1.1 and Theorem 1.2 below), we recall that ) is the set of admissible function, for each c ∈ H 1 (∂B 1 , a).We will also denote by W λ (u) the Weiss' energy W x 0 λ (r, u), when x 0 = 0 and r = 1.Theorem 1.1 (Epiperimetric inequality for W 1+s ).Let K c defined in (1.10) and z = r 1+s c(θ) ∈ K c be the (1 + s)−homogeneous extension in R n+1 of a function c ∈ H 1 (∂B 1 , a).Therefore there is ζ ∈ K c such that 2n+a+5 .Theorem 1.2 (Logarithmic epiperimetric inequality for W 2m ).Let K c defined in (1.10) and z = r 2m c(θ) ∈ K c be the 2m−homogeneous extension in R n+1 of a function c ∈ H 1 (∂B 1 , a).We also suppose that exists a constant Θ > 0 such that with β = n−1 n+1 and ε = ε(n, m, a) > 0 small enough.The first inequality was originally proved in [GPS16] and in [FS16] for s = 1 2 and generalized to any s ∈ (0, 1) in [GPPS17] and in [Ger19].For the proof, we use an alternative method, that follows the idea in [CSV20], decomposing the datum c ∈ H 1 (∂B 1 , a) in terms of eigenfunctions of L a restricted to ∂B 1 .
The second inequality was originally proved in [CSV20] for s =1 2 , but this is a new result for each s ∈ (0, 1).Moreover, we will give a proof of two epiperimetric inequalities for negative energies 1 for W 1+s and for W 2m (see Theorem 5.1 and Theorem 5.2), originally proved for s = 1 2 in [Car23] and in [CSV20] respectively.Using this two epiperimetric inequalities for negative energies, together with the first two epiperimetric inequalities in Theorem 1.1 and Theorem 1.2, we deduce a frequency gap.
Proposition 1.3 (Frequency gap).Let u be a solution of (1.2) and v = u x 0 be the solution of (1.5) with for some constants c ± m,a > 0, that depend only on n, m and a.In particular, if u is a λ−homogeneous solution of (1.4) (with 0 obstacle) with λ > 0, then the same conclusion hold for u.
) is a new result for any s ∈ (0, 1) and it is originally proved in [CSV20] for s = 1 2 and ϕ ≡ 0. Observe that the function −|y| 2s is a solution of (1.4) (with 0 obstacle), then we are able to prove the best frequency gap around 1 + s.Furthermore, we use the epiperimetric inequalities to deduce a characterization of the λ−homogeneous solutions of (1.4) (with 0 obstacle), in the case λ = 1 + s and λ = 2m, as described in the following proposition.
Finally we use the epiperimetric inequality for W 1+s in Theorem 1.1 and the logarithmic epiperimetric inequality for W 2m in Theorem 1.2 to get the regularity and the structure of the free boundary.In particular we prove the regularity of the points on the free boundary with frequency 1 + s , denoted by Γ 1+s (u), and we describe the structure of the points with frequency 2m, denoted by Γ 2m (u), with 2m ≤ k.See (2.6) below for the definition of Γ λ (u).
(1) Γ 1+s (u) is locally a C 1,α submanifold of dimension n − 1, for some α > 0, i.e. for all x 0 ∈ Γ 1+s (u), there is ρ > 0 and g : is contained in the union of at most countably many submanifolds of class C 1,log , for all 2m ≤ k.In particular, for such m, we have that for every j ∈ {0, . . .n − 1}, In particular, when ϕ ∈ C ∞ (R n ), the singular sets ∪ m∈N Γ j 2m (u) is contained in the union of countably many j-dimensional submanifolds of class C 1,log .Moreover, the singular set Sing(u) = ∪ m∈N Γ 2m (u) is contained in the union of countably many submanifolds of class C 1,log .
Here we have defined to be the dimension of the "tangent plane", where p x 0 2m is the unique homogeneous blow-up (see Proposition 1.4) of u at x 0 ∈ Γ 2m (u).
In particular, we improve the regularity of submanifolds that contain Γ j 2m (u) from C 1 (proved in [GR20]) to C 1,log .1.7.Structure of the paper.The paper is organized as follows.
• In Section 2, we recall the generalized Almgren's frequency function, the Weiss' energy W for 0 obstacle and the Weiss' energy W for nonzero obstacle.We also introduce the operator L S a , i.e.L a restricted to ∂B 1 , its eigenfunctions and its relation to the Weiss' energy W . Finally, we recall the properties of the function h s e defined in (1.12).• In Section 3, we prove Theorem 1.1, i.e. the epiperimetric inequality for W 1+s .• In Section 4, we prove Theorem 1.2, i.e. the logarithmic epiperimetric inequality for W 2m .• In Section 5, we state and prove two epiperimetric inequalities for negative energies W 1+s and W 2m (Theorem 5.1 and Theorem 5.2).• In Section 6, using the four epiperimetric inequalities proved in the previous sections, we establish a frequency gap in Proposition 1.3.• In Section 7, using Theorem 1.1 and Theorem 1.2, we prove Proposition 1.4, i.e. the characterization of the λ-homogeneous solutions of (1.4) (with 0 obstacle), in the case λ = 1 + s and λ = 2m.
• In Section 8, we use Theorem 1.1 and Theorem 1.2 to prove our main result on the regularity and the structure of the free boundary (Theorem 1.5).
Acknowledgment.I would like to thank Bozhidar Velichkov for all the useful discussion and encouragement.
The author was partially supported by the European Research Council (ERC), EU Horizon 2020 programme, through the project ERC VAREG -Variational approach to the regularity of the free boundaries (No. 853404).

Preliminaries
2.1.Generalized Almgren's frequency function.The original generalized Almgren's frequency function in [CSS08] must be modified in the case ϕ ∈ C k,γ (R n ), as in the following proposition.
We follow [GR20], but a similar generalized Almgren's frequency can be found in [CSS08] or in [BFR18].
Proof.The proof is technical and we skip it, since it is standard and proved in many variations.We refer to Proposition 6.1 in [GR20] for the complete proof.
By the previous proposition, if v = u x 0 , it is well-defined In the case of obstacle ϕ ≡ 0, if the "pure" Almgren's frequency N x 0 (0 + , u) = λ, with N x 0 as in (1.7), then we can consider a blow-up of u at x 0 , that is a λ−homogeneous solution (see [ACS08]).
We want a similar result for ϕ ∈ C k,γ (R n ).Precisely, we will show that if Φ x 0 (0 + , v) = n + a + 2λ, then the blow-up of v at x 0 is a λ−homogeneous solution.But this is only true if λ < k + γ, and this is the motivation for defining a new generalized Almgren's frequency, which was originally introduced in [CSS08].
To prove this result, we need the following two lemmas from [GR20].
Proof.The proof of the first part follows by the monotonicity of generalized Almgren's frequency and an integration from r to r 0 .The second part is similar.See Lemma 6.4 in [GR20] for more details.
Now we proceed with the second lemma.
Lemma 2.3.Let u be a solution of (1.2) and v = u x 0 be the solution of (1.5) with Proof.By (1.6), we obtain where in the last inequality we used (2.1).Now we are ready to prove the existence of a λ−homogeneous blow-up, when λ < k + γ.We denote by where for all R > 0 to a blow-up v 0,x 0 , as r → 0 + , which is a solution of (1.4) (with 0 obstacle) and it is λ−homogeneous.
Proof.We can proceed as in the proof of Proposition 6.6 in [GR20].
In particular, the monotonicity of the function r → r(1 Notice that the blow-up v 0,x 0 is not a priori unique.Still, all the blow-ups at x 0 have the same homogeneity.Therefore, for λ < k + γ, the following set is well-defined: (2.6) Remark 2.5.As in the proof of the last proposition, we get vh|y| a dX , (2.7) for r small enough.Then, by (2.2) and (2.3), we obtain 2.2.Properties of the Weiss' energy W .In the case ϕ ≡ 0, we can consider the following Weiss' energy, which is a small modification of the Weiss' energy W for the obstacle ϕ ≡ 0, defined in (1.9).Precisely, we define vh|y| a dX and we drop the dependence on x 0 if x 0 = 0.
Lemma 2.6.Let u be a solution of (1.2) and v = u x 0 be the solution of (1.5) with and vh|y| a dH n . (2.9) Proof.We give only the idea of the proof; for more details we refer to [GPPS17].First we can change the variables in the expression of H(r) and with a standard computation we get (2.8).Now, if X = (x, y), then using and an integration by parts, we get (2.9).
We next show that also the Weiss' energy W satisfies a monotonicity formula.
Since we have , by (2.8) and (2.9), we obtain which proves the claim.
Proposition 2.8.Let u be a solution of (1.2) and v = u x 0 be the solution of (1.5) with In particular, using (2.10), we get for all r ∈ (0, r 0 ).
Proof.We can write Hence, using (2.2) and Remark 2.5, we get lim which concludes the proof.

Homogeneous rescalings and homogeneous blow-up.
The sequence (2.5) has good rescaling properties with respect to the Almgren's frequency function.Now we consider another sequence of rescalings, the homogeneous rescalings, which have good rescaling properties with respect to the Weiss' energy.
Proposition 2.9.Let u be a solution of (1.2) and v = u x 0 be the solution of (1.5) with r,x 0 be the homogeneous rescalings of v at x 0 ∈ Γ(u), defined as Then, up to a subsequence, the homogeneous rescalings converge in C 1,α a (B + R ) for all R > 0 (defined in (2.4)), as r → 0 + , to a blow-up v (λ) 0,x 0 , which is λ−homogeneous and is a solution to (1.4) (with 0 obstacle).
Proof.By the Poincaré inequality (A.8), in order to show the boundness of v r in H 1 (B R , a), it is sufficient to prove the boundness of v r in L 2 (∂B R , a) and the boundness of |∇v r | in L 2 (B R , a).The first bound follows by (2.1).In fact . Also the second bound follows from (2.1): where in the last inequality we used (2.2), (2.3) and the monotonicity of the function r → Φ x 0 (r, v).Thus, for r small enough, as in the proof of Proposition 2.4.Hence, up to subsequences, v r converges to some v 0 weakly in H 1 (B R , a) and (2.14) by (2.3), with an equality in R n+1 \ ({v r = 0} ∩ {y = 0}).Therefore, we can use the estimate in [CSS08] (Proposition 4.3 and Lemma 4.4) to get the convergence in C 1,α a , since v r are solutions of (1.5), where the right-hand sides in the third and the fourth lines are as in (2.14).Moreover, v 0 is a solution of (1.4) (with 0 obstacle), since we can send r → 0 + in (2.14).
Finally, we show that v 0 is homogeneous.Indeed, by (2.10), we have that for all 0 where sending r k → 0 + the left-hand side vanishes, by (2.11).Thus, by arbitrariness of R 1 and R 2 , we obtain that v 0 is homogeneous, since ∇v 0 •x = λv 0 on ∂B r for all r ∈ (0, r 0 ).
2.4.The operator L S a .The strategy to prove the epiperimetric inequalities is to decompose a trace c ∈ H 1 (∂B 1 , a) in terms of the eigenfunctions of the operator L a restricted to ∂B 1 .The restriction L S a is defined for any function where Φ(x) = φ x |x| .Remark 2.10.Since in spherical coordinates we have By Liouville Theorem A.5, if we suppose that φ is even in the y direction, then Using the theory of compact operators, we can prove that there exist an increasing sequence of eigenvalues {λ a k } k∈N ⊂ R ≥0 and a sequence of eigenfunctions The (normalized) eigenspace corresponding to eigenvalue λ a is . for all λ a ⊂ {λ a k } k∈N .We denote by α k ∈ N the grade of the polynomial that corresponds to the eigenvalue λ a k , i.e. the only natural number such that λ a (α k ) = λ a k .In particular λ a 1 = λ a (0) = 0 and E(λ a 1 ) is the space of constant functions, while λ a 2 = . . .= λ a n+2 = λ a (1) = n + a and E(λ a 2 ) is the space of linear functions.Finally λ a k ≥ λ a (2) for k ≥ n + 3. 2.5.The Weiss' energy W and eigenfunctions of −L s a .The following lemma is a generalization of Lemma 2.3 and Lemma 2.4 in [CSV20], for the Weiss' energy W with weight.It will be used in several proof later.
where φ k normalized eigenfunctions of −L S a as above, and let r µ φ(θ) be the µ−homogeneous extension, then (2.17) Moreover, if (2.20) and Proof.The proof is very similar to the one in [CSV20], but we briefly recall it for the sake of completeness.By (A.3) we get from where (2.17) and (2.19) follow.Now if r µ+t c is a solution, then W µ+t (r µ+t c) = 0, by (A.6).Therefore (2.20) holds.
2.6.Properties of h s e .Finally we recall the properties of the function h s e defined in (1.12), which is the only (1 + s)−homogeneous solution of (1.4) (with 0 obstacle).The latter follows from Proposition 1.4, but it was originally proved in [CSS08].
Proposition 2.12.Let e ∈ ∂B ′ 1 and h s e as in (1.12), then h s e is a (1 + s)−homogeneous solution of (1.4) (with 0 obstacle), h s e = 0 on B ′ 1 ∩{x•e ≤ 0} and it holds lim Proof.The proof is a simple computation, hence it will be omitted.

Epiperimetric inequality for W 1+s
The proof of the epiperimetric inequality for W 1+s follows the ideas of the proof from [CSV20] in the case s = 1 2 , i.e. we decompose the trace c ∈ H 1 (∂B 1 , a) in terms of eigenfunction of L S a .3.1.Decomposition of c.Let c ∈ H 1 (∂B 1 , a) even in the y direction and such that c ≥ 0 in ∂B ′ 1 .We decompose c using eigenfunctions of the operator −L S a defined in (2.15).The projection on linear functions E(λ a 2 ) of c has the form c 1 (x • e) for some e ∈ ∂B ′ 1 , then the projection of h s e on E(λ a 2 ) has the same form C(x • e) for C > 0. Thus we can choose C > 0 such that c and Ch s e has the same projection on E(λ a 2 ).Notice that the function u 0 (x, y) = |y| 1+s restricted on ∂B 1 has 0 projection on E(λ a 2 ).Therefore we can choose c 0 ∈ R such that the projections of c − Ch s e and u 0 on the constant functions E(λ a 1 ) are the same.Then where Proof.By Proposition 2.12 and (A.6), we obtain Additionally, since u 0 is (1 + s)−homogeneous and we have that Finally, we notice that 2) which, together with the previous identities, gives the claim.

Logarithmic epiperimetric inequality for W 2m
The proof of the logarithmic epiperimetric inequality for W 2m follows the ideas of the proof from [CSV20] in the case s = 1 2 .The strategy is the same as the one of the proof of Theorem 1.1.

4.1.
Construction of h 2m .For the proof of the logarithmic epiperimetric inequality, we need to build an eigenfunction of −L S a as follows.Remark 4.1.There is a 2m−homogeneous L a −harmonic polynomial h 2m such that h 2m ≡ 1 on ∂B ′ 1 .The polynomial is given by where the constants C k are yet to be chosen.Notice that Thus, h 2m is L a −harmonic if and only if Therefore, we can choose for k ∈ {1, . . .m} and C 0 = 1, which concludes the construction of h 2m .with and Let z = r 2m c be the 2m−homogeneous extension of c and let h 2m as in Remark 4.1, therefore where M = max{P − (θ ′ ) : θ ′ ∈ ∂B ′ 1 } and P − (θ) is the negative part of P (θ).We choose a competitor ζ extending with homogeneity α > 2m the high modes on the sphere and leaving the rest unchanged, i.e.
Defining κ a α,2m as in (2.18), we will choose ε = ε(n, m) > 0 small enough and 2m < α < 2m + 1 2 such that for Θ > 0 and β = n−1 n+1 .Notice that to be able to choose such α, we must have an estimate of the type for some C n,m > 0, that should depend only on n, m and a, since we want that α depends only on n, m and a.For this reason we ask for the bounds c 2 L 2 (∂B 1 ),a ≤ Θ and |W 2m (z)| ≤ Θ.
4.3.Proof of Theorem 1.2.First we want to compute the term W 2m (ζ)− (1−κ a α,2m )W 2m (z), and we see that for α near 2m and α > 2m, it is negative.This is contained in the following lemmas.Lemma 4.2.In the hypotheses of Theorem 1.2, we have 3) for some C 1 , C 2 > 0 depending only on n, m and a.
Proof.We introduce the following functions: Then, we can write and in H 1 (B 1 , a) to ϕ µ for µ = α, 2m.Additionally, H 2m is L a −harmonic and 2m−homogeneous, therefore using (A.6), we get ).Notice now that, since ψ has only frequencies lower than 2m, we have W 2m (ψ) < 0. Thus, using (2.17), we get where in the second equality we used (2.19) with C 1 and C depending only on n, m and a. Observe that (4.4) where we used and we have chosen .
Hence, we conclude by choosing C 2 = CC 2 and C 1 as above.
Lemma 4.3.In the same hypotheses of the Theorem 1.2, we have for some C 3 > 0 that depends only on n, m and a.
Proof.Since φ k are C 1 on ∂B 1 and only a finite number of k is such that ≤ L m for all k such that α k ≤ 2m and for some L m > 0.
Moreover, the coefficients c k corresponding to P are bounded by Θ by Proposition A.6, and since φ contains only eigenfunctions corresponding to eigenvalue λ a k > λ a (2m).Finally we claim that in fact the norm in L 2 of P 2 − (θ) is controlled by the volume of an (n − 1)dimensional cone with height M 2 and radius of the base M L , since the graph of P − must be above this cone, by the Lipschitz continuity of P − .Thus, since 1 , we obtain which is precisely (4.5).
Proof of Theorem 1.2.Notice that we can suppose W 2m (z) > 0, otherwise we have done.
First, as already observed, we must show the estimate (4.2).In fact, as in (4.4) with α = 2m, we have where in the last equality we used (2.17).
Using the orthogonality of P and φ and again (2.17), we obtain where we used that c 2 L 2 (∂B 1 ) ≤ Θ and |W 2m (z)| ≤ Θ, that concludes the estimate.
If we choose ε < C 2 C 1 C 3 , where C 1 , C 2 , C 3 are the constant in (4.3) and (4.5) and if we choose κ α,2m as in (4.1), we deduce that that conclude the proof.
Remark 4.4.We have proved a stronger version of the logarithmic epiperimetric inequality, that is

Epiperimetric inequalities for negative energies
In this section we prove two epiperimetric inequalities for negative energies 2 W 1+s and W 2m .
These epiperimetric inequalities are a generalization for the case s = 1 2 and they allow us to prove the backward frequency gap in Proposition 1.3.
5.1.Epiperimetric inequality for negative energies W 1+s .In the case s = 1 2 , the epiperimetric inequality for negative energies W 1+s was proved in [Car23].We follow the same idea.
For I, we notice that the function −r 2s u 0 (θ) = −|y| 2s is a solution of (1.4) (with 0 obstacle), then using (2.20), we obtain and using (2.21), we get , with a simple calculation.For J, using (2.17), we deduce that For K, we integrate by parts Now, by (2.16), we get where in the last equality we used (A.2).Hence, we obtain , with a simple calculation.For L, by Proposition (2.12), we have where we used that h s e (θ) = u 0 (θ) = 0 for θ ∈ B ′ 1 ∩ {x • e < 0} and that φ = c ≥ 0 in B ′ 1 ∩ {x • e < 0}.Finally, since I, J, K, L ≤ 0, we conclude by using (5.1).5.2.Epiperimetric inequality for negative energies W 2m .In the case s = 1 2 , the epiperimetric inequality for negative energies W 2m was proved in [CSV20].We follow the same idea.
Theorem 5.2 (Epiperimetric inequality for negative energies W 2m ).Let K c be defined as in (1.10) and let z = r 2m c(θ Proof.Notice that we can suppose W 2m (z) < 0, otherwise we have done.
The explicit competitor is where we notice that As above, we can define and we write z = ψ 2m + H 2m + ϕ and ζ = ψ α + H 2m + ϕ.
Moreover, since the functions ψ µ are orthogonal to ϕ in L 2 (B 1 , a) and in H 1 (B 1 , a), since H 2m is L a −harmonic and 2m−homogeneous, and since W 2m (ϕ) > 0 by (2.17), we have that where in the second equality we used (2.19).Furthermore, if where the last inequality follows by , where we used the Cauchy-Schwartz inequality and that all norms in a finite dimensional space are equivalent.
We deduce that where we have chosen ε = ε(n, m, a) > 0 small enough.

Frequency gap
Once we have proved the epiperimetric inequalities in Theorems 1.1, 1.2, 5.1 and 5.2, the proof of the frequency gap is standard, as done in [CSV20].

Proof of Proposition 1.3.
Step 1.To prove the frequency gap around 1 + s, it is sufficient to check that if λ = 1 + s + t with t > 0, then t ≥ 1 − s, while if t < 0, then t ≤ −(1 − s).
Hence, since we have a negative energy, we get Step 2. Let c ∈ H 1 (∂B 1 , a) be a trace of a (2m + t)−homogeneous solution, say by (2.20).
Using the logarithmic epiperimetric inequality for W 2m , i.e.Theorem 1.2, and (6.1), we obtain where in the last equality we used (2.21).Therefore, since we have a negative energy, we have which gives that t ≥ c + m , for some explicit constant c + m > 0.
(2) If t < 0 then W 2m (r 2m+t c) = t < 0 by (2.20), therefore using the epiperimetric inequality for negative energies W 2m , i.e.Theorem 5.2, we obtain where in the last equality we used (2.21).Thus we get

Characterization of blow-ups
The epiperimetric inequality approach allows us to give an alternative proof of the characterization of blow-ups, in the spirit of [CSV20].For the original proof we refer to [CSS08] and [GR20].
Step 2. Suppose that z is a solution of (1.4) (with 0 obstacle) and that z is 2m−homogeneous.Then we claim that z = p 2m for some polynomial p 2m which is L a −harmonic.
Let c ∈ H 1 (∂B 1 , a) be the trace of z and let ζ be the competitor in the logarithmic epiperimetric inequality for W 2m .Without loss of generality, we can suppose c L 2 (∂B 1 ,a) = 1.Hence, by the strong version of the logepiperimetric inequality 4.7, we have Thus φ ≡ 0 and c contains only eigenfunctions corresponding to eigenvalues λ a k ≤ λ a (2m), i.e. c(θ) Using (2.17), we obtain i.e. the frequencies α k < 2m must vanish.Therefore c is an eigenfunction corresponding to eigenvalue λ a (2m) and it follows that the homogeneous extension z is a 2m−homogeneous L a −harmonic polynomial.
8. Regularity of Γ 1+s (u) and structure of Γ 2m (u) We conclude this paper with the most important application of the epiperimetric inequalities in Theorem 1.1 and Theorem 1.2, i.e. the proof of Theorem 1.5.The proof following a standard argument, for instance see [GPPS17] for the case s ∈ (0, 1) or [FS16], [GPS16], [CSV20], for the case s = 1 2 .In this last section, we recall the results from [GPPS17], where the claim (1) was proved in the case s ∈ ( 1 2 , 1).The regularity assumption for ϕ, that is ϕ ∈ C k,γ (R n ), allows us to generalize the result to any s ∈ (0, 1) by using the same argument.Indeed, in [GPPS17] it was proved that the function r → W x 0 λ (r, v) + Cr 2s−1 is increasing (with a slight difference in the definition of v), where 2s − 1 > 0. We notice that in the case ϕ ∈ C k,γ (R n ), we have that the function r → W x 0 λ (r, v) + Cr k+γ−λ is increasing, with k + γ − λ > 0, by (2.10), with v = u x 0 the solution of (1.5).
For the case (2), we give the complete proof, which is based on similar arguments and uses ideas from [CSV20] and [GP09].Notice that in the case λ = 2m, the condition λ < k + γ become λ = 2m ≤ k since 2m and k are integers.
Proof.Using (2.8), we obtain where in the last inequality we used (2.3).By the definition of the homogeneous rescalings v r from (2.13), we have where c r is the λ−homogeneous extension of v r | ∂B 1 and where in the last equality we used (A.3).Hence we have the following inequality is the function such that the homogeneous rescaling of ζ at x 0 is ζ r , then v is the minimizer of wh dX ), then we can use (2.1) to deduce the estimate (2.3) for Br ζh dX.Now we separate the case λ = 1 + s and λ = 2m.
Remark 8.2.In the Step 2 of the proof of Proposition 8.1, we used the logarithmic epiperimetric inequality for the rescaled c r , but to use Theorem 1.2, we have to check that the conditions (1.11) hold with Θ > 0 that does not depend on r.
Moreover, by (A.4) where in the equality we used (A.3) and in the last inequality we used the boundness of v r in H 1 (B 1 , a), as in the proof of Proposition (2.9).8.2.Decay of homogeneous rescalings.The decay of the Weiss' energy allows us to prove a decay of the norm in L 1 (∂B 1 , a) of the homogeneous rescalings.As a consequence, we get the uniqueness of the homogeneous blow-up.Proposition 8.3 (Decay of homogeneous rescalings).Let u be a solution of (1.2) and v = u x 0 be the solution of (1.5) with r,x 0 be the homogeneous rescalings from (2.13), and K ⊂ Γ λ (u) ∩ R n+1 be a compact set.
Proof.Dropping the dependence on x and λ, for 0 < r ′ < r < r 0 , where r 0 is as in the previous Proposition, using (2.10), we get where we used that W x 0 λ (r ′ , v) + C(r ′ ) k+γ−λ ≥ 0, by (2.10) and (2.12).The conclusion follows by a dyadic decomposition as in [FS16], [GPS16] or [CSV20], and by using (8.1) for λ = 1 + s, and (8.2) for λ = 2m.8.3.Non-degeneracy of homogeneous blow-up.Another consequence of the decay of the Weiss' energy is the non-degeneracy of the homogeneous blow-ups, i.e. the homogeneous blow-ups cannot vanish identically.Proposition 8.4 (Non-degeneracy of homogeneous blow-up).Let u be a solution of (1.2) and v = u x 0 be the solution of (1.5) with 0 r n+a+2λ for all x 0 ∈ Γ λ (u) and for all r ∈ (0, r 0 ), for some r 0 > 0, where In particular the homogeneous blow-up v Proof.Note that the inequality follows by the proof of (2.1), in fact we can obtain that the function r → H x 0 (r) r n+a+2λ is increasing for r small enough.Hence it is sufficient to prove that H x 0 0 > 0. Let v r be the homogeneous rescalings as in (2.13) of v at x 0 , then v r converge in C 1,α a to some v 0 , λ−homogeneous solution, as r → 0 + , up to subsequences.
Let (v r ) ρ be the rescalings as in (2.5) of v r at 0, then (v r ) ρ converge in C 1,α a to some v r , λ−homogeneous solution, as ρ → 0 + , up to subsequences.Arguing by contradiction, we suppose that H x 0 0 = 0. Then Therefore, using (8.6) and (8.7), together with the regularity of the solution, we obtain Finally, it is sufficient to choose ρ > 0 small enough and choose a corresponding r = r(ρ) > 0 small enough to obtain a contradiction.8.4.Regularity of blow-ups.Roughly speaking, we prove the regularity of the map that to any point x ∈ Γ λ (u) ∩ K associates the homogeneous blow-up of v = u x at x, where λ = 1 + s or λ = 2m.We are able to prove the regularity with an explicit modulus of continuity that depend on the right-hand side of (8.6) and (8.7).
Proposition 8.5.Let u be a solution of (1.2) and v = u x be the solution of (1.5) with (1) If λ = 1 + s and v (1+s) 0,x = λ x h s e(x) , with λ x > 0, e(x) ∈ ∂B ′ 1 and h s e as in (1.12), then up to decreasing α in Proposition 8.3, it holds 0,x = λ x p x with λ x > 0 and p x a 2m−homogeneous polynomial such that p x L 2 (∂B 1 ,a) = 1, then Proof.First note that the characterization of blow-up and λ x > 0 follows by Proposition 1.4 and Proposition 8.4.Secondly, we can use (2.8) to get Then, for r small enough, integrating the last equality, we obtain where we denote by the modulus of continuity.We indicate explicitly the dependence on α since it must be reduced to obtain the final claim.Thus, since x i c 0 , for some dimensional constant c 0 > 0, we obtain c 0 H x i (r) r n+a+2λ − λ 2 x i ≤ Cω α (r).(8.12) Let now x 1 , x 2 ∈ Γ λ (u) ∩ K and r = |x 1 − x 2 | σ with σ ∈ (0, 1) to choose.Let w r be the homogeneous rescalings in (2.13) of w = u − ϕ.Using the regularity of the solution and ∇w(x 2 , 0) = 0, we deduce that (8.13) where we can choose, for example, σ = 1 2m for λ = 2m.Now, recalling Q x i as in Lemma (A.7), we have where in the last inequality we have used Lemma A.7 for the first term, and the same computation in (8.13) for the second term.Therefore, recalling the definition of u x i that is a solution of (1.4), we deduce that if σ ∈ (0, 1) such that (k + γ − λ)σ = 1 − σ in the case λ = 1 + s.
By the estimates (8.12) and (8.14) (with the regularity of the solution), we obtain Since the function x → λ 2 x is ω−continuous and K is a compact set, we get that (8.8) and (8.10) hold.
Hence, by the homogeneity of u 0,x 1 − u 0,x 2 and by Proposition A.6, it follows that where in the last inequality we used (8.15) (with the regularity of the solution) and (8.16).
Using (8.8), (8.10) and (8.17), we get where we used the ω−continuity of the function x → λ x for x ∈ K to estimate from below λ x 1 .Now, by (8.18) and the definition of h s e in (1.12), for λ = 1 + s we have which implies (8.9).Finally, for λ = 2m, since all norms in a finite dimensional space are equivalent, we get By a similar computation as in (8.18), we obtain where we used (8.15), then we conclude (8.11) 8.5.Proof of Theorem 1.5.For the regularity of Γ 1+s (u) and the structure of Γ 2m (u) for 2m ≤ k we can proceed with a standard argument, as in [FS16], [GPS16], [CSV20], [GPPS17], [GP09].We briefly recall the proof.
Step 2. Now we prove part (2), i.e. the structure of Γ 2m (u) for 2m ≤ k, in particular we prove that Γ j 2m (u), defined in Theorem 1.5, is locally contained in a C 1,log submanifold of dimension j, with Γ j 2m (u) := {x 0 ∈ Γ 2m : d x 0 2m = j}, where d x 0 2m is as in (1.13), according to Proposition 8.4.Let v 0,x 0 = λ x 0 p x 0 be the only homogeneous blow-up, as in Proposition 8.5, with x 0 ∈ Γ j 2m (u) ∩ K where K ⊂ Γ j 2m (u) ∩ R n+1 is a compact set.The function q x 0 (x, y) = p x 0 (x − x 0 , y) 3 It is sufficient to use the frequency gap in Proposition 1.3 and the upper semicontinuity of the function x0 → Φ x 0 (0 + , u), since is an infimum of continuous functions.
Finally, the L 2 (∂B 1 , a) projection on linear functions of h s e has the form C(x • e) for some C > 0.