The local structure of the free boundary in the fractional obstacle problem

Building upon the recent results in \cite{FoSp17} we provide a thorough description of the free boundary for the fractional obstacle problem in $\mathbb{R}^{n+1}$ with obstacle function $\varphi$ (suitably smooth and decaying fast at infinity) up to sets of null $\mathcal{H}^{n-1}$ measure. In particular, if $\varphi$ is analytic, the problem reduces to the zero obstacle case dealt with in \cite{FoSp17} and therefore we retrieve the same results: (i) local finiteness of the $(n-1)$-dimensional Minkowski content of the free boundary (and thus of its Hausdorff measure), (ii) $\mathcal{H}^{n-1}$-rectifiability of the free boundary, (iii) classification of the frequencies up to a set of Hausdorff dimension at most $(n-2)$ and classification of the blow-ups at $\mathcal{H}^{n-1}$ almost every free boundary point. Instead, if $\varphi\in C^{k+1}(\mathbb{R}^n)$, $k\geq 2$, similar results hold only for a distinguished subset of points in the free boundary where the order of contact of the solution and the obstacle is less than $k+1$.


Introduction
Quasi-geostrophic flow models [9], anomalous diffusion in disordered media [4] and American options with jump processes [10] are some instances of constrained variational problems involving free boundaries for thin obstacle problems. In this paper we analyze the fractional obstacle problem with exponent s ∈ (0, 1), a problem that can be stated in several ways, each motivated by a different application and suited to be studied with different techniques. We follow here the variational approach: given ϕ : R n → R smooth and decaying sufficiently fast at infinity, one seeks for minimizers of the H s -seminorm s ∈ (0, 1), on the cone whereḢ s (R n ) is the homogeneous space defined as the closure in the H s seminorm of C ∞ c (R n ) functions. Existence and uniqueness of a minimizer w follow for all s ∈ (0, 1) if n ≥ 2 (the case n = 1 requires some care see [21] and [3]). In addition, defining the fractional laplacian as for v ∈Ḣ s (R n ), the Euler-Lagrange conditions characterize w as a distributional solution to the system of inequalities      w(x ′ ) ≥ ϕ(x ′ ) for x ′ ∈ R n , (−∆) s w(x ′ ) = 0 for w(x ′ ) > ϕ(x ′ ), (−∆) s w(x ′ ) ≥ 0 for x ′ ∈ R n . (1.1) The most challenging regularity issues are then that of w itself and that of its free boundary Γ ϕ (w) := ∂ x ′ ∈ R n : w(x ′ ) = ϕ(x ′ ) .
To investigate the fine properties of the solution w of (1.1) the groundbreaking paper by Caffarelli and Silvestre [7] introduces an equivalent local counterpart for the fractional obstacle problem in terms of the so called a-harmonic extension argument. Indeed, it is inspired by the case s = 1 /2, in which it is nothing but the harmonic extension problem. More precisely, setting a = 1 − 2s for s ∈ (0, 1) and m = |x n+1 | a L n+1 , it turns out that any function w satisfying (1.1) is the trace of a function u ∈ H 1 (R n+1 , dm) solving in the sense of distribution in R n+1 . (1.2) Note that u is unique minimizer of the Dirichlet energŷ R n+1 |∇ v| 2 |x n+1 | a dx on the class A := v ∈ H 1 (R n+1 , dm) : v(x ′ , 0) ≥ ϕ(x ′ ) . Viceversa, the trace u(x ′ , 0) on the hyperplane {x n+1 = 0} of a solution u to (1.2) is a solution w to (1.1). One then is interested into regularity issues for u and for the corresponding free boundary Γ ϕ (u) (with a slight abuse of notation we use the same symbol as for the analogous set for w): the topological boundary, in the relative topology of R n , of the coincidence set of a solution u The locality of the operator L a (v) := div |x n+1 | a ∇v(x) in (3.1) is the main advantage of the new formulation to perform the analysis of Γ ϕ (u). Indeed, being Γ ϕ (u) = Γ ϕ (w) it permits the use of monotonicity and almost monotonicity type formulas analogous to those introduced by Weiss and Monneau for the classical obstacle problem (cf. [5,6,22,18]). Optimal interior regularity for u has been established Caffarelli, Salsa and Silvestre in [8, Theoren 6.7 and Corollary 6.8] for any s ∈ (0, 1). The particular case s = 1 /2 had been previously addressed by Athanasopoulos, Caffarelli and Salsa in [1]. Instead, despite all the mentioned progresses, the current picture for free boundary regularity theory is still incomplete. In this paper we go further on in this direction and deal with the non-zero obstacle case following the recent achievements obtained in the zero-obstacle case in [13,14]. Drawing a parallel with the theory in the zero-obstacle case, the free boundary Γ ϕ (u) can be split as a pairwise disjoint union of sets: Γ ϕ (u) = Reg(u) ∪ Sing(u) ∪ Other(u), (1.3) termed in the existing literature as the subset of regular, singular and nonregular/nonsingular points, respectively. These sets are defined via the infinitesimal behaviour of the rescalings of the solution itself. More precisely, for x 0 ∈ Γ ϕ (u) a function ϕ x0 related to ϕ can be conveniently defined (cf. (3.46) and (3.48)) in a way that if u x0,r (y) := r n+a 2 u(x 0 + r y) − ϕ x0 (x 0 + r y) ´∂ Br (u − ϕ x0 ) 2 |x n+1 | a dH n 1 /2 , then the family of functions {u x0,r } r>0 is pre-compact in H 1 loc (R n+1 , dm) (see [8,Section 6]). The limits are called blowups of u at x 0 , they are homogeneous solutions of a fractional obstacle problem with zero obstacle. The set of all such functions is denoted by BU(x 0 ). Their homogeneity λ(x 0 ) depends only on the base point x 0 and not on the extracted subsequence, and it is called infinitesimal homogeneity or frequency of u at x 0 . It is indeed the limit value, as the radius vanishes, of an Almgren's type frequency function related to u which turns out to be non decreasing in the radius. Given this, one defines According to the regularity of ϕ different results are known in literature: (i) Regular points: in [8] for ϕ ∈ C 2,1 (R n ) optimal one-sided C 1,s regularity of solutions is established. Moreover, Reg(u) is shown to be locally a C 1,α submanifolf of codimension 2 in R n+1 (non-optimal regularity of the solution had been previously established in [21]); (ii) Singular points: for ϕ analytic and a = 0 it is proved in [15] that Sing(u) is (n − 1)rectifiable. The latter result has been very recently extended to the full range a ∈ (−1, 1) and to ϕ ∈ C k+1 (R n ), k ≥ 2, in [16]. Furthermore, fine properties of the singular set have been studied very recently by Fernández-Real and Jhaveri [11]. It is also worth mentioning the paper by Barrios,, in which the authors study the fractional obstacle problem (1.1) with non zero obstacle ϕ having compact support and satisfying suitable concavity assumptions. Under these assumptions, they are able to fully characterize the free boundary, showing that Other(u) = ∅ and that at every point of Sing(u) the blowup is quadratic, i.e. the only admissible value of m is 1. In addition, they are able to show that the singular set Sing(u) is locally contained in a single C 1 -regular submanifold.
For ease of expositions we start with the simpler case in which the obstacle function ϕ is analytic, actually the slightly milder assumption (1.4) below suffices (see Section 3 for related results in the case ϕ ∈ C k+1 (R n )). Indeed, after a suitable transformation (see Section 2.1) such a framework reduces to the zero obstacle case since in this setting ϕ turns out to be exactly the a-harmonic extension of ϕ. Thus, in view of [13, Theorems 1.1-1.3] we may deduce the following result. Then, (i) the free boundary Γ ϕ (u) has finite (n − 1)-dimensional Minkowski content: more precisely, there exists a constant C > 0 such that where T r (Γ ϕ (u)) := {x ∈ R n+1 : dist(x, Γ(u)) < r}; (ii) the free boundary Γ ϕ (u) is (n − 1)-rectifiable, i.e. there exist at most countably many Moreover, there exists a subset Σ(u) ⊂ Γ ϕ (u) with Hausdorff dimension at most n − 2 such that for every The analysis is more involved in case ϕ is not analytic, since one cannot in principle avoid contact points of infinite order between the solution and the obstacle, and the free boundary can be locally an arbitrary compact set K ⊂ R n (explicit examples are provided in [12]). In view of this, we follow the existing literature and we consider only those points in the free boundary in which u has order of contact with ϕ less than k + 1: given u a solution to the fractional obstacle problem (1.2) and given a constant θ ∈ (0, 1) we set where H ux 0 is defined in (3.16) and it is related to the L 2 (∂B r , dm ′ ) norm of u x0,r (cf. Section 3 for more details). For this subset of points of the free boundary we can still prove some of the results stated in Theorem 1.1.

Analytic obstacles
In this section we deal with analytic obstacles. We report first on some results related to the Caffarelli-Silvestre a-harmonic extension argument that will be instrumental to reduce the analytic type fractional obstacle problem to the lower dimensional obstacle problem. We provide then the proof of Theorem 1.1.
2.1. Extension results. We start off stating a lemma in which it is proved that there exists a canonical a-harmonic extension of a polynomial which is a polynomial itself (see [16,Lemma 5.2]). We denote by P l (R n ) the finite dimensional vector space of homogeneous polynomials of degree l ∈ N.

C k+1 obstacles
In this section we deal with the more demanding case of C k+1 obstacles, k ≥ 2. It is convenient to reduce the analysis of (1.2) to that of the following localized problem In what follows, we shall assume that ϕ C k+1 (B ′ R ) ≤ 1. This assumption can be easily matched by a simple scaling argument (cf. the proof of Theorem 1.2).
For any x 0 ∈ B ′ 1 we denote by T k,x0 [ϕ] the Taylor polynomial of ϕ of order k at x 0 : 1 · · · x αn n , |α| := α 1 + . . .+ α n and α! := α 1 ! · · · α n !. In what follows we will repeatedly use that (recall where E l are the extension operators in Lemma 2.1. By the translation invariance of the operator, we point out that is the relative boundary in the hyperplane {x n+1 = 0}. We note that u x0 is not a solution of a fractional obstacle problem as in (3.1) with null obstacle, but rather of a related obstacle problem with drift as discussed in what follows (cf. (3.13)).
First, from the regularity assumption on ϕ and Lemma 2.1 we infer that L a (ϕ x0 ) is a function in L 1 (B 1 , dm). Indeed, for some C = C(n, a) > 0 and for all In turn, this yields that the distribution L a (u x0 ) is actually a function in for some constant C = C(n, a) > 0.
The following result resumes the regularity theory developed by Caffarelli, Salsa and Silvestre in [8,Proposition 4.3].
In particular, the function u is analytic in x n+1 > 0 (see, e.g., [17]) and the following boundary conditions holds: More precisely, the left hand side in (3.11) is a measure supported on B ′ 1 . Furthermore, for B r (x 0 ) ⊂ B 1 and x 0 ∈ B ′ 1 , an integration by parts implies that where in the second equality we have used that E [T k,x0 (ϕ)] is even with respect to the hyperplane {x n+1 = 0} to deduce that lim In particular, since the last addend in (3.12) only depends on the boundary values of u x0 , it follows that u x0 is a minimizer of the functional among all functions v ∈ u x0 + H 1 0 (B r (x 0 ), dm) and satisfying v(x ′ , 0) ≥ 0 on B ′ r (x 0 ). Equivalently, we will say that u x0 is a local minimizer of the functional in (3.13) subject to null obstacle conditions. Remark 3.2. We record here some bounds that shall be employed extensively in what follows. By using the linearity and continuity of the extension operator E k (cf. Remark 2.2), together with estimate (3.2) we get for all z ∈ B 1 and for some constant C = C(n, a, k) > 0.

3.1.
A frequency type function. Building upon the approach developed in [13] we consider a quantity strictly related to Almgren's frequency function and instrumental for developing the free boundary analysis in the subsequent sections. Let φ : [0, +∞) → [0, +∞) be defined by then given the solution u to (3.1), a point x 0 ∈ B ′ 1 and the corresponding function u x0 in (3.5), we define for all 0 < r < 1 − |x 0 | Hereφ indicates the derivative of φ. Clearly, I ux 0 (r) is well-defined as long as H ux 0 (r) > 0, in what follows when writing I ux 0 (r) we shall tacitly assume that the latter condition is satisfied.
For later convenience, we introduce also the notation and In particular, note that for all r > 0 by Cauchy-Schwarz inequality.
Remark 3.3. In case ϕ = 0, then u x0 = u for all x 0 ∈ B ′ 1 and G u = D u . Thus, I ux 0 boils down to the variant of Almgren's frequency function used in [13].
In particular, this shows that the frequency function is scaling invariant, in the sequel we will use this property repeatedly.

3.2.
Almost monotonicity of I ux 0 at distinguished points. In this subsection we prove the quais-monotonicity of I ux 0 for a suitable subset of points of the free boundary. We prove first some useful identities in a generic point x 0 of B ′ 1 . Lemma 3.5. Let u be a solution to the fractional obstacle problem (3.1) in B 1 . Then, for all (3.20) Remark 3.6. Note that the last addends in (3.18) and (3.20) are well-defined thanks to (3.7).
Proof. To show (3.18), (3.19) and (3.20), we assume without loss of generality that x 0 = 0. For (3.18) we consider the vector field V ( it suffices to take into account the one-sided C 1 regularity of u in Theorem 3.1 and the C k+1 regularity of ϕ to conclude. Instead, if (x ′ , 0) / ∈ Λ ϕ (u) we use (3.9) and that E [T k,x0 (ϕ)] is even with respect to the hyperplane {x n+1 = 0} to conclude. Thus the distributional divergence of V is the L 1 function given by ). Therefore, (3.18) follows from the divergence theorem by taking into account that V is compactly supported.
Next (3.19) is a consequence of (3.18) and the direct computation Finally, to prove (3.20) we consider the compactly supported vector field By Theorem 3.1 and Lemma 2.
Thus div W has no singular part in B ′ 1 , and we can compute pointwise the distributional divergence as follows for Therefore, we infer that and we conclude (3.20) by direct differentiation since As a consequence we derive a first monotonicity formula for H ux 0 in B 1 .
Corollary 3.7. Let u be a solution to the fractional obstacle problem (3.1). Then, for all Proof. The proof of (3.21) (and hence of (3.22) and (3.23)) follows from the differential equation in (3.19). The proof of (3.24) is a simple consequence of a dyadic integration argument: where in the last inequality we used that H ux 0 (s) ≤ H ux 0 (r) for s ≤ r by (3.23).
We establish next an auxiliary lemma containing useful bounds for some quantities related to the L 2 -norm of u x0 , for points x 0 in the contact set.
Proof. By the co-area formula for Lipschitz functions we check that and Therefore, an integration by parts gives Analogously, we get with C = C(n, a) > 0. Integrating the latter inequality we find (3.25). Instead, by first multiplying formula (3.31) by t and then integrating over ( r /2, r), we concludê Thus, (3.26) and (3.27) follow directly.
Next we show an explicit expression for the radial derivative of I ux 0 at all points x 0 ∈ B ′ 1 . We follow here [13, Proposition 2.7]. Proposition 3.9. Let u be a solution to the fractional obstacle problem and C 3.9 = C 3.9 (n, a) > 0.
Proof. It is not restrictive to assume x 0 = 0. We use the identities in (3.18), (3.19) and (3.20) to compute (the lenghty details are left to the reader) From this we conclude (3.32) straightforwardly. For (3.33), we estimate separately each term appearing in the integral defining R ux 0 (t). We start with with C 3.34 = C 3.34 (n, a) > 0. Arguing similarly we infer  Estimate (3.33) turns out to be useful to analyze the subsets of points Γ ϕ,θ (u) of Γ ϕ (u), for every θ ∈ (0, 1) (cf. (1.7)). With fixed θ ∈ (0, 1), we then look at points of the free boundary in the subset where δ > 0 is any.
Hence, in what follows it is enough to consider the values of δ small enough.
We derive next an additive quasi-monotonicity formula for the frequency.
Proof. Under the standing assumptions, the quasi-monotonicity of I ux 0 and (3.42) yield that d dr for r sufficiently small. Hence, we conclude (3.44) at once by integration.

3.3.
Lower bound on the frequency and compactness. We first show that the frequency of a solution u to (3.1) at points in Z ϕ,θ,δ (u) is bounded from below by a universal constant.
For the free boundary analysis developed in [13] it is mandatory to consider the nodal set of a solution. In the current framework, the natural subsitute for the nodal set is given by the first inclusion is a consequence of (3.9)).
We can then give the following compactness result. For u : B 1 → R solution of (3.1) and x 0 ∈ B ′ 1 we introduce the rescalings among all functions v ∈ u x0,r + H 1 0 (B 1 , dm) satisfying v(x ′ , 0) ≥ 0 on B ′ 1 . Corollary 3.15. Let δ > 0 be given. Let (u l ) l∈N be a sequence of solutions to the fractional obstacle problem (3.1) in B 1 with obstacle functions ϕ l equi-bounded in C k+1 (B 1 ), and let x l ∈ Z ϕ l ,θ,δ (u l ) be such that sup l I (u l )x l (̺ l ) < +∞, for some ̺ l ↓ 0.
Then, there exist a subsequence l j ↑ ∞ and a solution v ∞ to the fractional obstacle problem (3.1) in B 1 with null obstacle function, such that on setting v j := (u lj ) x l j ,̺ l j we have Proof. By taking into account inequality (3.43) in Proposition 3.11 we get for l large In particular, we infer that sup l D (u l )x l ,̺ l (1) < ∞. Thus, a subsequence v j := (u lj ) x l j ,̺ l j converges weakly H 1 (B 1 , dm) to some function v ∞ . Moreover, v j is a local minimizer of (3.46)-(3.47)). By taking into account that x lj ∈ Z ϕ l j ,θ,δ (u lj ), inequality (3.6) implies that for some constant C = C(n, a, θ) > 0 and for all y Therefore, one can easily show that the sequence (F j ) j Γ(L 2 (B 1 , dm))-converges to the functional Items

Main estimates on the frequency
In this section we prove the principal estimates on the frequency that we are going to exploit in the sequel. We start with an elementary lemma. Recall that all obstacles functions ϕ are assumed to satisfy the normalization condition ϕ C k+1 (B ′ 1 ) ≤ 1.
Remark 4.2. Note that as a byproduct of the first estimate in (4.1) in Lemma 4.1 frequencies at the scale ̺ are well-defined at every point x ∈ B ′ ̺ /2 , recalling that 0 ∈ Z ϕ,θ,δ (u). Proof. In order to prove (4.1), we argue by contradiction: we can assume that there exist A, δ > 0 and solutions u j to the fractional obstacle problem with obstacles ϕ j , ϕ j C k+1 (B ′ 1 ) ≤ 1, with 0 ∈ Z ϕj ,θ,δ (u j ), such that I (uj )0 (2̺ j ) ≤ A, for some ̺ j ↓ 0, and there exist points x j ∈ B ′ ̺ j/2 contradicting one of the sets of inequalities in (4.1).
In particular, by almost monotonicity of the frequency function (cf. Proposition 3.11) and the lower bound on the frequency (cf. Corollary 3.16) we infer that 1+s ≤ I (uj )0 (t) ≤ A e C 3.11 (2̺j ) θ ≤ A e C 3.11 =: A ′ for all t ∈ (0, 2̺ j ]. By Corollary 3.15, up to a subsequence, v j := (u j ) 0,̺j converges strongly in H 1 (B 2 , dm) to a function u ∞ solution of the fractional obstacle problem in B 2 with zero obstacle function. We assume in addition that ̺ −1 j x j → x ∞ ∈B ′ 1 /2 . To prove the first set of inequalities in (4.1), we compute Moreover, by estimate (3.14) in Remark 3.2 we get for Therefore, recalling that 0 ∈ Z ϕj,θ,δ (u j ), from (4.4) we infer Since ̺ j ↓ 0, by contradiction lim j H (u j ) 0 (̺j ) ∈ {0, ∞}. Moreover, by (4.3) and (4.5), by the strong L 2 (B 2 , dm) and local uniform convergence on B 2 of v j → v ∞ we conclude that, Being the left hand side finite, necessarily , and thus v ∞ ≡ 0 on the whole of B 1 by analiticity. A contradiction to H v∞ (1) = 1 that follows from strong L 2 (B 1 , dm) convergence and the equality H vj (1) = 1 for all j.
The second set of inequalities in (4.1) is proven by the same argument. Indeed, assuming that and since by (3.15) in Remark 3.2 and by (3.39) By the strong convergence of v j to v ∞ in H 1 (B 2 , dm), we infer that the left hand side is finite and then actually 0, so thatˆB Thus, by analiticity v ∞ is constant on B 1 , and we may conclude that The latter equality contradictŝ that follows from strong H 1 (B 2 , dm) convergence and recalling that H vj (1) = 1 and 1 + s ≤ Finally, (4.2) follows straightforwardly from (4.1): We introduce next the following notation for the radial variation of (modified) frequency at a point x ∈ Z ϕ,θ,δ (u) of a solution u in B 1 : given 0 < r 0 < r 1 < 1 − |x|, we set , if r 1 is sufficiently small and if Λ ≥ A C 3.13 (cf. Corollary 3.13). We do not indicate the dependence of ∆ r1 r0 on Λ since such a parameter will be fixed appropriately in the next result. 3 > 0 such that, if x 0 ∈ Z ϕ,θ,δ (u), and I ux 0 (r 1 ) ≤ A, with 2r 1 ≤ ̺ 4.3 , then for every r 0 ∈ ( r1 /8, r 1 ) we havê Proof. We start off with the following computation: We now use the following integral estimate (whose elementary proof is left to the readers)

4.1.
Oscillation estimate of the frequency. The following lemma shows how the spatial oscillation of the frequency in two nearby points at a given scale is in turn controlled by the radial variations at comparable scales.
for every x 1 , x 2 ∈ B ′ τ ∩ Z ϕ,θ,δ (u). Proof. We start off noting that by Remark 4.2 and the choice 144τ < ̺ 4.4 , if the constant ̺ 4.4 is suitably chosen, a simple scaling argument yields that I ux (10τ ) is well-defined for every x ∈ B ′ 77τ /4 . To ease the readability of the proof we divide it in several substeps. 1], and consider the map t → I ux t (10τ ). The differentiability of the functions x → H ux (10τ ) and x → D ux (10τ ) yields that Set e := x 1 − x 2 , then e · e n+1 = 0; and set for all y ∈ R n+1 δ t (y) := d dt u xt (x t + y) .
Recalling the very definition of u xt in (3.5), it turns out that because by linearity (the details of the elementary computations are left to the readers) Thus from (4.11) by direct calculation it follows that Hence we conclude that 12) and moreover that ∇δ t (y) = d dt ∇u xt (x t + y) . 2. Thanks to the previous formulas, for all λ ∈ R we infer (4.14) In addition, integrating by parts gives δ t (y) − λu xt (x t + y) ∇u xt (x t + y) · y |yn+1| a |y| dy + 2λG ux t (10τ ) − 2ˆφ |y| 10τ δ t (y) L a (u xt (x t + y)) dy . Then, by formula (3.18) together with (4.14) and (4.15), we have that t .
In what follows we estimate separately the two terms J 3. We start off with J t . With this aim, first note that I ux 0 (10τ ) ≤ A e 72 θ C 3.11 by Proposition 3.11 since 144τ < ̺ 4.4 , provided the latter is small enough. In turn, as x t ∈ B ′ τ , by (4.2) in Lemma 4.1 we infer that I ux t (10τ ) ≤ C 4.1 + A e 72 θ C 3.11 .
We estimate separately the factors of the integrand defining J (setting x t + y = z). We start off with the first one as follows with C = C(n, k) > 0. Moreover, by choosing λ := I ux 2 (10τ ) − I ux 1 (10τ ), we infer Using inequalities (3.14)- (3.15) in Remark 3.2, we estimate the last two addends as follows for some constant C > 0, for i = 1, 2. In the last inequality we have also used that |z − x i | ≤ 12τ , being z ∈ B 10τ (x t ). Therefore, we have (4.16) For the second factor, we note that for i = 1, 2 To estimate the last three addends we use the very definition of u xt in (3.5), formula (4.2) and inequalities (3.14)-(3.15) in Remark 3.2, to conclude that for i = 1, 2 In the last inequality we have used again that |z − x i | ≤ 12τ . Therefore, we get By collecting (4.16) and (4.17), using Hölder inequality we conclude that there exists C > 0 Clearly, we have that In the second inequality, we have used that 1], and that |z −x i | ≤ 2|z −x t | as z ∈ B 10τ (x t )\B 5τ (x t ), i = 1, 2. Moreover, in the third inequality we have applied estimate (4.6) in Lemma 4.3 to x 1 , x 2 ∈ B ′ τ ∩ Z ϕ,θ,δ (u), with r 1 = 24τ and r 0 = 3τ . Furthermore, thanks to Corollary 3.7, we concludê In addition, thanks to (3.24) and |z − x t | ≥ 5τ we get (4.20) By collecting (4.18)-(4.20) we conclude that for some C > 0 where, in the last inequality, we have used Lemma 4.1 and that x 1 , x 2 ∈ B ′ τ ∩ Z ϕ,θ,δ (u). Finally, in view of the very definition of the spatial oscillation of the frequency and Corollary 3.13, we deduce that 4. We estimate next J (2) t . We start off noting that d dt Then, arguing as in (4.4), thanks to estimate (3.14) in Remark 3.2, we get as k ≥ 2 ˆφ |y| 10τ In addition, (3.7) and (4.12) yield ˆφ |y| 10τ δ t (y) L a (u xt (x t + y)) dy ux t (10τ ). Therefore, by applying repeatedly Lemma 4.1, we infer The conclusion in (4.10) follows at once from estimates (4.21) and (4.23).

Mean-flatness.
Here we show a control of the Jones' β-number by the oscillation of the frequency. Given a Radon measure µ in R n+1 , for every x 0 ∈ R n and for every r > 0, we set where the infimum is taken among all affine (n − 1)-dimensional planes L ⊂ R n+1 . If x 0 ∈ R n+1 and r > 0 is such that µ(B r (x 0 )) > 0, setx x0,r the barycenter of µ in B r (x 0 ), i.e.
From the very definitions of B p and of its barycenter we deduce where α := 1 µ(B r )ˆB r (p) I ux (9r)dµ(x).
Next we estimate the two sides of (5.4).

5.2.
Rigidity of homogeneous solutions. In this section we extend the results on the rigidity of almost homogeneous solutions established in [13].
We denote by H λ the space of all non-zero λ-homogeneous solutions to the thin obstacle problem As observed in [13], the maximal dimension of the spine of a function in H is at most n − 1 and we set u ∈ H top if u ∈ H and dim S(u) = n − 1, and H low := H \ H top . All functions in H top are classified in [13,Lemma 5.3]. Note also that by Caffarelli, Salsa and Silvestre [8] H 1+s ⊆ H top . (5.10) We next introduce the notion of almost homogeneous solutions. Given δ > 0 and x 0 ∈ Z ϕ,θ,δ we set J ux 0 (t) := e C 3.11 t θ I ux 0 (t) ∀ t ∈ (0, ̺ 3.11 ] . Definition 5.2. Let η > 0 and let u : B 1 → R be a solution to thin obstacle problem (3.1) with obstacle ϕ (as usual ϕ C k+1 (B ′ 1 ) ≤ 1). Assume that 0 ∈ Z ϕ,θ,δ and ̺ ≤ ̺ 3.11 , u is called η-almost homogeneous in B ̺ if J u0 ( ̺ /2) − J u0 ( ̺ /4) ≤ η.
The following lemma justifies this terminology and it is the analog of [13, Lemma 5.5]. for some homogeneous solution w ∈ H.
Proof. The proof follows by a contradiction argument similar to [13,Lemma 5.5]. Assume that for some ε, A > 0 we could find sequences of numbers δ l , ̺ l and of 1 /l-almost homogeneous solutions u l of (3.1) in B ̺ l , with ̺ l arbitrarily small, such that 0 ∈ Z ϕ,θ,δ (u l ) and inf l inf w∈H (u l ) 0,̺ l − w H 1 (B 1/4 ,dm) ≥ ε , (5.12) and satisfying the bounds I (u l )0 (̺ l ) ≤ A.
Consider v l := (u l ) 0,̺ l , then by Corollary 3.15 applied to v l there would be a subsequence, not relabeled, converging in H 1 (B 1 , dm) to a solution v ∞ of the thin obstacle problem with zero obstacle. By Proposition 3.11 there is some A ′ independent of l such that I (u l )0 (t) ≤ A ′ for all being T k, z /R [ ϕ](·) = T k,z [ϕ](R·). Thus, we get that z ∈ Z R ϕ,θ, 1 /j (u) if and only if z /R ∈ B ′ 1 /2 ∩ Z ϕ,θ,R 2(k+1−θ) /j ( u). In addition, it is easy to check that . We choose R > 0 sufficiently small so that ϕ C k+1 (B ′ R ) ≤ 1 and the smallness conditions on the radii in all the statements of Sections 3-5 are satisfied.
In such a case the proof, of the main results can be obtained by following verbatim [13, Sections [6][7][8]. Indeed, [13,Proposition 6.1], that leads both to the local finiteness of the Minkowskii content of Z ϕ,θ,δ ( u) and to its (n − 1)-rectifiability, is based on a covering argument that exploits the lower bound on the frequency in Corollary 3.16, the control of the mean oscillation via the frequency in Proposition 5.1, the rigidity of almost homogeneous solutions in Proposition 5.4, the discrete Reifenberg theorem by Naber and Valtorta [19,Theorem 3.4,Remark 3.9], and the rectifiability criterion either by Azzam and Tolsa [2] or by Naber and Valtorta [19,20]. Therefore, the only extra-care needed in the current setting is to start the covering argument from a scale which is small enough to validate the conclusions of the lemmas and propositions of the previous sections.
Finally, the classification of blow-up limits is exactly that stated in [13,Theorem 1.3], and proved in [13,Section 8], in view of Lemma 3.14 and Corollary 3.15.