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On a weighted two-phase boundary obstacle problem

In this work we consider an inhomogeneous two-phase obstacle-type problem driven by the fractional Laplacian. In particular, making use of the Caffarelli-Silvestre extension, Almgren and Monneau type monotonicity formulas and blow-up analysis, we provide a classification of the possible vanishing orders, which implies the strong unique continuation property. Moreover, we prove a stratification result for the nodal set, together with estimates on its Hausdorff dimensions, for both the regular and the singular part. The main tools come from geometric measure theory and amount to Whitney's Extension Theorem and Federer's Reduction Principle.


Introduction
In the present paper we investigate local properties of solutions of a stationary two-phase fractional problem, with particular emphasis on regularity aspects and on the structure of the nodal set. Our investigation focuses on the study of weak solutions of the following boundary value problem Here a ∈ (−1, 1), t + := max{t, 0} and t − := max{−t, 0} denote the positive and negative parts of t, λ − , λ + ≥ 0 are nonnegative constants, p ≥ 2 and, for r > 0, B + r := {z = (x, y) ∈ R N +1 + : |z| < r}, B ′ r := {x ∈ R N : |x| < r}, with for N ≥ 1. We will refer to the function h : B ′ 1 → R as the (thin) obstacle. Thanks to the extension procedure established in [10], one can see that solutions to (1.1) are closely related to solutions of an equation in N dimensions, driven by the fractional Laplacian where s = (1 − a)/2 ∈ (0, 1) and C(N, s) > 0 is a constant, while B ′ r (x 0 ) := {x ∈ R N : |x − x 0 | < r}. 1 Loosely speaking, it is shown in [10] that, in a suitable sense, for a certain (explicit) constant κ(N, a) > 0. Therefore, one can think of the trace on {y = 0} of u in (1.1) as a solution to Among many others, we refer for instance to [19] and [25] for a more thorough description of the operator (−∆) s , the fractional setting, and the extension procedure (see also [1,8,12]). Thanks to this connection, the problem we study can be read in two different ways. On the one hand, it can be seen as a two-phase penalized obstacle problem for the fractional Laplacian on R N , as in (1.2). On the other hand, in its extended formulation (1.1), one can understand it as a (weighted) two-phase boundary penalized obstacle problem associated to the operator − div(y a ∇(·)). Although we will keep in mind the parallel just outlined, in this work we will focus on the study of the extended problem (1.1). More precisely, our main objectives are to investigate the regularity of the solution and the structure of the free boundary ∂{u(·, 0) = 0} ∩ B ′ 1 . We now observe a couple of limit situations that may emerge from (1.1). If we consider both λ − = λ + = 0, then the boundary condition in (1.1) becomes of homogeneous Neumann type, while if we let λ + → +∞ and λ − = 0, then the condition becomes In the case s = 1/2, this is known as the thin obstacle problem (or even Signorini problem, see [35]). Boundary value problems such as (1.1) find many applications as physical models. We briefly describe an example. Let Ω ⊆ R N +1 be a continuum, conductive body, and let us consider the problem of regulating the temperature of its boundary ∂Ω. In particular, let us assume we have a reference temperature h : ∂Ω → R, and we want the temperature u(z, t), for (z, t) ∈ ∂Ω×(0, +∞), to stay as close as possible to h(z) at any time t > 0. The mathematical formulation of this phenomenon is given by the following boundary value problem (z, t) − ∆ z u(z, t) = 0, for (z, t) ∈ Ω × (0, +∞), ∂u ∂ν (z, t) = Φ(u(z, t)), for (z, t) ∈ ∂Ω × (0, +∞), where ν denotes the normal vector of ∂Ω and Φ should reflect the fact that, if u(z, t) is below the threshold h(z) then the temperature must be raised, while if u(z, t) is greater than h(z), it must be lowered. Moreover, the adjustment should be proportionally related to the difference |u(z, t) − h(z)|. Therefore a possible choice is
This corresponds to (1.1) in the case a = 0 and when a portion of ∂Ω lies on the thin space R N . In view of this physical interpretation, problem (1.1) can be said to possess a stable nature. We refer to [16,Section 2] and references therein for more details concerning physical motivations of the problem under investigation (see also [8] for the fractional framework). Free boundary problems associated to the fractional Laplace operator have been the object of intensive investigation in recent years. We refer the interested reader to [17], [31] for thorough monographs on the subject. For instance, in [5], [2] the authors studied the so-called lower dimensional two-phase membrane problem, while in [4], [3] and [16] the fractional version of a penalized obstacle problem is considered. A fairly complete picture for the regularity of the free boundary in the "classical" lower dimensional obstacle problem is given in [6], [7], [11], together with [26], [28]. In [37] the authors investigated geometric theoretical features of the nodal set of s-harmonic functions. We finally quote [39], [40], where the authors developed new techniques in order to study regularity properties of zero level sets of solutions to local equations with sublinear (and even singular) powers; [44] extends some of the results to the nonlocal framework.
In order to state our main results, we first introduce the appropriate functional setting. For any positive half ball B + r , we define the weighted Sobolev space H 1,a (B + r ) = H 1 (B + r ; y a dxdy) as the completion of C ∞ (B + r ) with respect to the norm . For a fixed function g ∈ H 1,a (B + 1 ) ∩ C(B + 1 ), we consider the problem In particular, we say that u ∈ H 1,a (B . The Dirichlet boundary datum on S + 1 in (1.3) appears (in contrast to (1.1)) in order to establish existence and uniqueness of a weak solution, see Proposition 2.2. Also, we explicitly note that, for the sake of simplicity, we have taken the thin obstacle h ≡ 0.
We carry out our analysis in the following steps. First of all, we establish the optimal regularity of the solution u in Hölder spaces, see Lemma 2.7. We point out that there is a substantial difference in this respect between the cases a = 0 and a = 0. While in the former the optimal Hölder space depends on the exponent p, this is no longer true in the latter case. This is a clear indication of the strong influence of the weight y a , which can be degenerate or singular when y → 0 + , and affects the behavior of the solution near the thin space ∂R N +1 + . Nevertheless, this weight belongs to the second Muckenhoupt class A 2 and therefore it enjoys nice properties, including e.g. Sobolev and trace embeddings, see for instance [20], [29] and also Lemma 3.7. At this point, we investigate the local behavior of u(x, y) when y → 0 + . In order to do so, we establish Almgren-and Monneau-type monotonicity formulas that, together with a blow up analysis for a proper rescaling of u, provide the asymptotic rate and shape of the solution near a free boundary point, in terms of a-harmonic polynomials (see Definition 2.3), which are even in the variable y and homogeneous of some degree. More precisely, for any integer k ≥ 0, let P a k denote the space of polynomials p : One can immediately notice that, given the other conditions, the first one is equivalent to Also, thanks to [38, Theorem 1.1], if the conditions above are satisfied, no other values of k apart from integer ones are allowed.
We are now able to state our first main result.
, then there exists an integer k ≥ 0 and p x 0 k ∈ P a k such that , as r → 0 + , for some α ∈ (0, 1).
Since the polynomial p x 0 k (which we refer to as blow-up limit of u at x 0 ) cannot vanish everywhere on the thin space R N × {0} (see Remark 4.3), the previous theorem readily implies the boundary strong unique continuation principle. Corollary 1.2 (Strong unique continuation). Let u ∈ H 1,a (B + 1 ) be the unique solution of (1.3) and assume that u(z) = o(|z| n ) as |z| → 0, z ∈ B + 1 for all n ∈ N. Then u ≡ 0 on B + 1 . This result tells us that the nodal set of u on the thin space Z(u) := ∂{x ∈ B ′ 1 : u(x, 0) = 0} has empty interior in the R N topology. Therefore, thanks to the continuity of u, it coincides with the free boundary that separates the two phases of the solution The last part of the present work is devoted to the analysis of the regularity and structural properties of the nodal set Z(u). We first remark that, when investigating the properties of the free boundary in low dimensions (N = 1, 2) some extra care is needed. Therefore, in order to avoid technicalities and to preserve the main idea, we hereafter assume N ≥ 2 or N ≥ 3 (we specify the choice in each statement). A fundamental tool is the notion of frequency. More precisely, we introduce, for any v ∈ H 1,a (B + 1 ), x 0 ∈ B ′ 1 and r > 0 , as a function of r, the frequency function of v at x 0 . In Proposition 3.8 we prove, if u the solution of (1.3), the existence of the limit In addition, we show in Theorem 4.1 that N x 0 (u, 0 + ) is a nonnegative integer, and it is the degree of the blow-up limit of u at x 0 . We refer to N x 0 (u, 0 + ) as the frequency of u at x 0 . To proceed, we split the nodal set Z(u) into two disjoint parts where, for any integer k ≥ 1, We call R(u) and S(u) the regular and singular part of the free boundary, respectively. At this point of our analysis, we are confronted again with the strong influence of the weight y a on the behavior of the solution u near the thin space. Indeed, we perform another classification of the zero points of u, based on whether the blow-up limit of the solution at x 0 ∈ Z(u) depends on the variable y or not. Namely, let us consider the following two spaces of polynomials P * k := {p ∈ P a k : p(x, y) = p(x) for all (x, y) ∈ R N +1 } and P y k := P k \ P * k . It is readily seen that the former can be characterized as Moreover we observe that P y 1 = ∅, and this leads to our first result about the free boundary, concerning its regular part.
On the other hand, since P y k = ∅ for k ≥ 2, we cannot expect the same regularity for the singular set. However, we are able to prove a stratification result which describes the structure of S(u). Once more, we must distinguish between points for which the blow-up limit depends on the variable y, and points for which this does not happen. If we define, for p x 0 k as in Theorem 1.1, , then the singular set naturally splits into the following two disjoint parts In order to prove their stratified structure, we need the notion of dimension of the singular set at one of its points.

Definition 1.4.
For any x 0 ∈ Z k (u), we define the dimension of Z k (u) at x 0 as We observe that, since We are now able to describe the structure of the singular set. Roughly speaking, we show and they are contained in a countable union of n-dimensional C 1 manifolds.
The last result of our paper yields estimates on the Hausdorff dimension of the nodal set and of its regular and singular part. Theorem 1.6. Let N ≥ 2. Let u ∈ H 1,a (B + 1 ) be the unique solution of (1.3) and assume Furthermore, in the latter case The proof relies on Theorem 1.1 and Federer's Reduction Principle. We observe that the previous result immediately implies the boundary unique continuation from sets of positive N -dimensional measure. Corollary 1.7. Let N ≥ 2. Let u ∈ H 1,a (B + 1 ) be the unique solution to (1.3). If |Z(u)| N > 0, then u ≡ 0 in B + 1 . We conclude the introduction with a couple of comments. We first emphasize that the assumption p ≥ 2 is made in view of the fact that the value 2 delineates a threshold, making the problem superlinear: indeed, in the sublinear regime p ∈ (1, 2), even though some of the results of the present paper still hold true, the structure of the of the nodal set might be qualitatively different and this is somehow suggested by the available results in the local case a = −1 (i.e. s = 1). For instance, already in dimension 1, the ODE −u ′′ = λ|u| p−2 u (corresponding to the case λ − = λ + = λ > 0) admits nontrivial solutions with arbitrary large nodal set. Therefore, the equation does not enjoy the unique continuation property when p ∈ (1, 2) (see e.g. [33]). We refer to [24] for the study of the free boundary ∂{u < 0} ∪ ∂{u > 0} of the problem above. In the nonlocal framework a similar situation is expected, even if we cannot exclude some peculiar features to emerge: anyway this requires further investigation, which may be interesting to pursue in the future.
We finally mention some further possible research topics, which are left open by our present work. In the same spirit as other classical problems, one might investigate higher regularity of the free boundary, concerning both the regular and the singular part. In addition to the classical works, starting points in this direction can be some recent outstanding approaches, such as the ones developed in [27,15,22] (see also [18,23,41]). Another interesting point one may investigate concerns the possible vanishing orders of solutions to (1.3). We proved in Theorem 1.1 that the vanishing order must be a nonnegative integer, but still it is not clear whether it could be any possible element of N or not. For instance there could be an upper bound (such as in the linear case, see [32]) or some integer numbers might be excluded.
1.1. Outline of the paper. The paper is organized as follows. In Section 2 we provide some basic properties of the solution, including the optimal regularity in Hölder spaces. Section 3 is devoted to the proof of monotonicity of a perturbed Almgren type frequency function and of a Monneau functional. These results are then applied in Section 4 to perform a blow-up analysis for a suitable scaling of the solution, which leads to the proof of Theorem 1.1. Finally, in Section 5 we prove the structure of the free boundary and Hausdorff dimension estimates.

Optimal regularity and main properties of the solution
In this section we higlight some important features of the solution and we establish its optimal regularity.
In order to prove existence and uniqueness of a solution to problem (1.3), it is useful to set up a variational framework. Hence, we introduce, for any v ∈ H 1,a (B + 1 ) ∩ L p (B ′ 1 ), the following energy functional We will need the following Poincaré inequality.

Lemma 2.1. For all r > 0 and for all
Proof. Just by integration of the identity div(y a u 2 z) = 2y a u∇u · z + (N + a + 1)y a u 2 over B + r and Young's inequality (with z = (x, y)). This variational setting allows us to prove the following.

Proposition 2.2. There exists a unique solution to problem
In particular, the solution u is the unique minimum of the functional J under the constraint Θ. Here Θ and J are as in (2.1) and (2.2).
Proof. Thanks to the form of the functional J, the proof of existence is standard and is based on Lemma 2.1, weak lower-semicontinuity of the H 1,a -norm, compactness of the trace embedding H 1,a (B + 1 ) ֒→ L 2 (B ′ 1 ), and Fatou's Lemma. Uniqueness follows from convexity of the functions t → (t ± ) p .
The first notable property of the solution is that its positive and negative part are subharmonic functions, with respect to the operator − div(y a ∇(·)), in B + 1 ∪ B ′ 1 . We first give the precise definition.

Definition 2.3. We say that a function
. We now prove a-subharmonicity of the positive and negative part of solutions to (1.3).
Proof. Let ϕ ∈ H 1,a 0,S + 1 (B + 1 ) such that ϕ ≥ 0 a.e. and, for ǫ > 0 small, let u ǫ := (u − ǫϕ) + − u − . One can immediately notice that Since u ǫ = u on S + 1 , by minimality and (2.3) we have This yields By expanding the last term we obtain Since u + and u − are a-subharmonic, the weak maximum principle (together with boundedness of g on S + 1 and a reflection argument) immediately implies the following, see [20, Theorem 2.2.2] and [29, Theorem 6.7].
. We also recall the validity of mean value inequalities for a-subharmonic functions. The following result immediately follows from [45, Lemma A.1] and an even-in-y reflection.
). Remark 2.8. We point out that the result is optimal when a = 0. Indeed, let Ψ(x) be harmonic in R N . Then, the function u(x, y) = (y 1−a + 1 − a)Ψ(x) is a solution of (1.3) with λ + = λ − = 1 and p = 2, and so Lemma 2.7 is sharp. On the contrary, when a = 0 the regularity is actually higher, depending on the exponent p (we refer to [16] for a detailed exposition).
Even though the global regularity cannot exceed the one just stated, if we look separately at the derivatives in x and the weighted derivative in y, we are able to prove something more.
, for a certain C > 0 depending on N, a and b L ∞ (B ′ 1 ) . Remark 2.10. By inspecting the proof of the previous lemma, which is based on an iterative Moser-type argument, one can find a more explicit expression for the constant C and emphasize the dependence on b L ∞ (B ′ 1 ) . More precisely where the c i 's are positive constants depending on N and a. Lemma 2.11. Let u ∈ H 1,a (B + 1 ) be the unique solution to problem (1.3). Then there exists α ∈ (0, 1) ∩ (0, 1 − a), R 0 ∈ (0, 1/4) and C > 0, possibly depending on N, a, p, g L ∞ (S + 1 ) , such that Proof. For i = 1, . . . , N , let e i be the i-th vector of the standard basis of R N +1 and, for h ∈ R sufficiently small and z = (x, y) ∈ B + 1/4 , let This function weakly satisfies From Lemma 2.9 we know that u h ∈ L ∞ (B + r 1 ), for some r 1 ∈ (0, 1/4) and that . Thanks to the Lipschitz continuity of the function t → (t ± ) p−1 on real compact intervals and to Lemma 2.5, we have that V h ∈ L ∞ (B ′ 1/4 ) and its L ∞ -norm is bounded by a constant that depends on p, λ − , λ + and u L ∞ (B + 1 ) , uniformly with respect to h. More precisely, a direct study of the function Combining this fact with (2.6), in view of Remark 2.10 and Lemma 2.5, we have that , withC 1 > 0 depending on N, a, λ + , λ − , p and g L ∞ (S + 1 ) . From [38, Theorem 1.5] we know that, for any α ∈ (0, 1) ∩ (0, 1 − a) there exists r 2 ∈ (0, 1/4) and C 2 > 0, depending on N, a, α such that .

Monotonicity Formulas
In this section we introduce a perturbed Almgren type frequency function, suitable for our problem, and we prove its monotonicity. As a consequence we deduce the existence of the limit of the standard Almgren frequency function of the solution, as the radius of the underlying ball vanishes. Finally, we introduce a Monneau type functional and we show it is nondecreasing with respect to the radius. For y a v 2 dS and the classical frequency function Moreover we let Then we introduce the perturbed frequency functioñ The first goal of the section is to prove monotonicity ofÑ x 0 (u, r) with respect to r ∈ (0, R 0 ), where u is the unique solution to problem (1.3). Hereafter, when we compute the functionals above at the origin x 0 = 0 and for v = u, we may drop this dependence in our notation and simply writeÑ (r). Without loss of generality, in this section we can assume x 0 = 0.
There holds the following formula, for z ∈ B + 1 , Proof. Since the proof is essentially just computations, we only sketch it. We first have that (3.4) div(y a |∇v| 2 z − 2y a (∇v · z)∇v) Moreover it is easy to check that the following identity holds By plugging this into (3.4) we obtain the desired conclusion.

Lemma 3.2.
For a.e. r ∈ (0, R 0 ) we have that Proof. Since similar computations can be found in literature, we give a sketch of the proof and we refer to, among others, [21,Theorem 3.7] for a precise justification. In order to prove (3.5) we first integrate (3.3) over B + r , which, taking into account the equation (1.3) satisfied by u, gives Now, integrating by parts in B ′ r and using the identity we obtain the following expression for the last term in (3.7) where F is as in (3.2). Combining the previous identity with (3.7) and reorganizing the terms yields (3.5). Finally, (3.6) comes from multiplying (1.3) by u and integrating over B + r . Lemma 3.3. We have that H(r) > 0 for all r ∈ (0, R 0 ).
In the following lemma we state the regularity of the function H and we compute its derivative.
Proof. Let us fix r 0 ∈ (0, R 0 ) and let us examine the limit In order to pass to the limit under the integral sign, we make use of Lebesgue Dominated Convergence Theorem. In fact, from Lagrange Theorem and the regularity of the solution (see Lemma 2.11) we derive that, for all z ∈ S + 1 and r ∈ (r 0 /2, R 0 ), with C > 0 depending on R 0 and a. We also recall that the weight y a is integrable, which allows us to conclude the proof of (3.8). Finally, (3.9) is a consequence of (3.8), (3.6) and the definition ofD (3.1). The fact that H ′ (r) ≥ 0 follows from the fact that D and G are nonnegative. The proof is thereby complete.
Proof. For any r ∈ (0, R 0 ) we let First of all, it is easy to verify that E ∈ L 1 (0, R 0 ). Secondly, we compute its distributional derivative. By the coarea formula we have where dσ denotes the surface N − 1 dimensional measure of R N . Again, it is easy to see that E ′ ∈ L 1 (0, R 0 ), thus implying that E ∈ W 1,1 (0, R 0 ). Now, by definition we have that This, together with identity (3.5), yields the thesis.

Proof. Combining Lemma 3.4 and Lemma 3.5 we have that
where the last inequality follows from Cauchy-Schwartz inequality, Lemma 3.4 and the assumption a ∈ (−1, 1).
We now state a trace inequality on half-balls. The proof is rather standard, but we include it for the sake of completeness.

Lemma 3.7. There exists a constant
and for all r > 0.
Proof. We provide the proof only for v ∈ C ∞ (B + r ) and conclude by standard density arguments. From the embedding H 1,a (B + 1 ) ֒→ L 2 (B ′ 1 ) (see [30, Theorem 2.3]) we know that there exists a constant c = c(N, a) > 0 such that We now prove that (3.11) (N + a) By integration of the identity div(y a v 2 z) = (N + a + 1)y a v 2 + 2y a v∇v · z , y), for some small δ > 0, we obtain Letting δ → 0 + , this implies Now, applying Young's inequality to the last term, using the fact that |z| ≤ 1 and rearranging the terms, we reach (3.11). At this point, a combination of (3.10) and (3.11) yields for somec =c(N, a) > 0. By a scaling argument we may easily conclude the proof.
We obtain as a consequence that the "unperturbed" frequency function N has a limit as r → 0 + . Proposition 3.8. Let k ∈ [0, ∞) be as in Theorem 3.6. Then there exists the limit of N (r) as r → 0 + and lim r→0 + N (r) = k.
Proof. Let r > 0 be sufficiently small. Trivially N (r) ≤Ñ (r). On the other hand, thanks to the boundedness of u, we have that Therefore, applying Lemma 3.7, we deduce
Proof. From Lemma 3.4 and Theorem 3.6 we know that By integrating the inequality above in (r, R 0 ), thanks to the boundedness of u, we conclude the proof of point (i). On the other hand, we recall that Keeping (3.12) in mind, we obtain Hence, since lim r→0 + N (r) = k, for any δ > 0 there exists r 0 = r 0 (δ) > 0 such that, for all r ∈ (0, r 0 ) Integrating the inequality above and taking into account the boundedness of u, we obtain that H(r) ≥ C 2 r 2k+δ for all r ∈ (0, r 0 ).
Thanks to continuity and positiveness of H we may conclude the proof of (ii).
The previous result readily implies an estimate on the growth rate of the solution u near the origin. Corollary 3.10. Let x 0 ∈ B ′ R 0 /2 and let k = lim r→0 +Ñ x 0 (u, r). Then there existsC > 0 depending on N , a, g L ∞ (S + 1 ) and k (independent of x 0 ) such that Proof. Let x ∈ B ′ R 0 /2 and let r := |x|. Thanks to Lemma 2.4 and Lemma 2.6 we have that From the Cauchy-Schwartz inequality we deduce that , for some C 1 > 0 depending on N, a and independent of x 0 . Gathering in chain all the previous inequalities we obtain that with C 2 > 0 again depending only on N, a. Now, Corollary 3.9, (i), implies that where C 3 > 0 depends on N, a, g L ∞ (S + 1 ) , k. In fact, the proof of Corollary 3.9, (i), can be trivially adapted to the function H x 0 (u, r) with the same constant K 1 . The proof may now be concluded by combining the inequality above with (3.13).
We now introduce the following functional of Monneau type , r ∈ (0, 1) and γ ≥ 0, with H x 0 as in (3.1). As indicated before, we will drop the index x 0 in the notation for M x 0 γ when x 0 = 0. In the following proposition we prove the monotonicity of this functional with respect to r ∈ (0, R 0 /2) when the solution u and a polynomial in P a k are "compared". Proposition 3.11. Let k = lim r→0 +Ñ (r) and let p k ∈ P a k be an a-harmonic homogeneous polynomial of degree k, even in y, i.e. respectively satisfying (1.6), (1.5), (1.4).
for someĈ > 0 depending on N , a, g L ∞ (S + 1 ) and k. (3.15) we thus need to compute the latter. By integrating the identity div(y a u∇p k ) = y a ∇u · ∇p k + u div(y a ∇p k ) in B + r we obtain since p k is a-harmonic and y a ∂p k ∂y = 0 on B ′ r . Moreover, on S + r , we have On the other hand, we have that the function u − p k weakly satisfies that, multiplied by u − p k and integrated gives that Let us denote Reasoning as in the proof of Lemma 3.4, we have that d dr Therefore, combining it with (3.18) and (3.17) we deduce that d dr In addition, since p k is homogeneous, then D(p k , r) = kH(p k , r); hence (3.19) d dr Putting together (3.19) and (3.20) with (3.15) we obtain that M k (u, p k , ·) ∈ C 1 (0, R 0 /2) and d dr From Corollary 3.10 we have that while, being p k an homogeneous polynomial of degree k, for someĈ > 0 depending on N , a, g L ∞ (S + 1 ) and k. In view of the inequality above and (3.21), together with the fact thatÑ (r) ≥ k and that p ≥ 2, the proof is complete.
Proof. From Proposition 3.11 we can derive that Hence the limit exists and it is finite. Since a ∈ (0, 1) and p ≥ 2, we know that r k(p−2)+1−a vanishes as r → 0 + , and therefore In addition M k (u, p k , r) ≥ 0 for all r ∈ (0, R 0 /2) and so M k (u, p k , 0 + ) ≥ 0 as well. Finally, by integration in (0, r) of (3.23) we obtain (3.22) and we conclude the proof.

Blow-up analysis
This section is devoted the blow-up analysis of a particular normalization of the solution, which will lead to the proof of Theorem 1.1.
Hereafter, we fixR ∈ (0, R 0 /2). In this section we denote, for z ∈ B + 1 and r ∈ (0,R), The first step is the blow-up analysis for the function u r as in (4.1).
Theorem 4.1. Let k = lim r→0 + N (r) and u r as in (4.1). Then k ∈ N ∪ {0} and for any sequence r n → 0 + there exists a subsequence r n i → 0 + and a functionũ ∈ P a k ,ũ ≡ 0, such that Proof. Let r n → 0 + . First, we have that u rn ∈ H 1,a (B + 1 ) and S + 1 y a u 2 rn dS = H(u rn , 1) = 1.
Therefore, thanks to the Poincaré-type inequality Lemma 2.1, we know that {u rn } n is bounded in H 1,a (B + 1 ). Thus there exists a subsequence r n i → 0 + and a functionũ ∈ H 1,a (B + 1 ) such that u rn i ⇀ũ weakly in H 1,a (B + 1 ) as i → ∞. Due to the compact embedding, u rn i →ũ strongly in L 2,a (S + 1 ), which implies that ũ L 2,a (S + 1 ) = 1 and thereforeũ ≡ 0. Next, we have that u r ∈ H 1,a (B + 1 ) satisfies . We now observe that, thanks to Corollary 3.10, for another constant C > 0. Combining this with (4.4) we obtain that B + 1 y a ∇u rn i · ∇ϕ dxdy ≤ Cr k(p−1)+1−a n i H(r n i ) .
up to renaming the constant C. Now, by choosing δ = 1 − a in Corollary 3.9, (ii), and plugging the estimate on H in the previous equation, we derive that possibly renaming again the constant C. Thus, passing to the limit for i → ∞ yields thatũ is y a -harmonic in B + 1 ∪ B ′ 1 . Thanks to the regularity of u r in Hölder spaces (see Lemma 2.7 and Lemma 2.11) we can easily deduce that An integration by parts implies that u rn i H 1,a (B + 1 ) → ũ H 1,a (B + 1 ) . Hence (4.2) and (4.3) are proved. Now we can pass to the limit in the Almgren frequency function and obtain that, for any R ∈ (0, 1) which implies thatũ is homogeneous of degree k (for the proof of this fact see e.g. [37,Proposition 4.6]). Therefore, the even reflection ofũ through {y = 0} is an a-harmonic homogeneous polynomial of degree k. But such a polynomial must be smooth (see e.g. [38, Theorem 1.1]) and so we know that k must be a nonnegative integer and thatũ ∈ P a k .
Next, we prove a nondegeneracy result of the solution at any boundary point.

Proposition 4.2.
There exists two constants C 1 , C 2 > 0 such that the following estimates from above and below hold: Proof. In order to prove (4.5), let us assume by contradiction that there exists a sequence r n → 0 + such that sup Together with Corollary 3.9 and a change of variable this means that But in view of Theorem 4.1 this gives rise to a contradiction for n sufficiently large. Now let us prove (4.6). We observe that (4.6) is equivalent to By explicit computation it is possible to check that for someC > 0 (depending on N and a), thus implying that So let us contradict (4.7) and assume that, up to a subsequence and so that Letũ ∈ H 1,a (B + 1 ) be, as in Theorem 4.1, the blow-up limit of u r (z) = [H(u, r)] −1/2 u(rz), possibly passing to another subsequence as r → 0 + . By definition of the Monneau-type functional, we have Thanks to homogeneity ofũ, we notice that (4.10) H(ũ, r) = r 2k H(ũ, 1).
On the other hand, by Hölder's inequality, (4.8) and (4.10) as r → 0 + . Inserting this, together with (4.8) and (4.10), in (4.9) we obtain as r → 0 + . Then, by monotonicity of M k (u,ũ, r), proved in Proposition 3.11, we have that We infer for the chosen sequence r → 0 + . But, after rescaling and dividing by r 2k H(u, r), thanks again to the homogeneity ofũ, (4.11) becomes Now, let us apply Corollary 3.9, (ii), with δ = 1 − a to find that there exists K 2 > 0 such that H(u, r) ≥ K 2 r 2k+ 1−a 2 . Continuing with the inequality above we get Passing to the limit as r → 0 + we obtain −2H(ũ, 1) = −2 S + 1 y aũ2 dS ≥ 0, thus giving rise to a contradiction since H(ũ, 1) > 0. If not, in fact, the unique continuation principle would be violated, as H(ũ, 1) = r −2k H(ũ, r) for all r ∈ (0, 1). Then there exists a constant C > 0 such that for all r ∈ (0, 1) and this proves (4.6). The second part of the thesis comes from the inequality above and Corollary 3.9, (i).
Next, we use once more the monotonicity of the Monneau functional (3.14) to uniquely identify the blow-up limit and to prove Theorem 1.1.
Proof of Theorem 1.1. Let r n → 0 + . By Theorem 4.1 there exists a subsequence r n i → 0 + and an a-harmonic homogeneous polynomialũ, even in y, such that u rn i →ũ in H 1,a (B + 1 ), C 0,α (B + 1 ) and C 1,α (B ′ 1 ) as i → ∞. Therefore, if we let γ := lim r→0 + r −2k H(u, r) ∈ (0, +∞), whose existence is ensured by Proposition 4.2, we have that u(rz) r k . By Corollary 3.12 the functional M k (u, u 0 , r) admits a limit as r → 0 + , and therefore Now we prove that the limit does not depend on the choice of the subsequence. Let r n i → 0 + another subsequence andũ 0 another possbile limit. Again,ũ 0 will be an a-harmonic homogeneous polynomial of degree k and even in y. Hence, analogously as r → 0 + and so u 0 =ũ 0 . The proof is concluded. and since lim y→0 + y a ∂u 0 ∂y = 0, then u 0 cannot vanish everywhere on the thin space {y = 0}, because otherwise its trivial extension to the whole R N +1 would violate the unique continuation principle for A 2 -weighted elliptic equations, see [43,Corollary 1.4] (see also [25,Lemma 5.2]).

Remark 4.4. We recall that
Thanks to Theorem 1.1, for any x 0 ∈ Z(u) there exists a unique integer k ≥ 0 and a unique a-harmonic polynomial p x 0 k ∈ P a k , homogeneous of degree k such that . We refer to this polynomial as the blow-up limit of u at the point x 0 . In particular, the fact that u(x 0 , 0) = 0 readily implies that k ≥ 1.
We conclude the section by proving the continuous dependence of the blow-up limits on the nodal points. This result plays a pivotal role in the analysis of the structure of the free boundary.
and p x 0 k as in Remark 4.4. Moreover for any compact → 0 as r → 0 + , uniformly for x 0 ∈ K.
In particular there exists a modulus of continuity σ K (t) ≥ 0, σ K (0 + ) = 0 such that Proof. First we observe that P a k is a subset of finite dimensional space, namely the space of homogeneous polynomial of degree k, and so all the norms are equivalent: we choose to endow this space with the L 2,a (S + 1 )-norm. Thanks to Theorem 1.1, for any x 0 ∈ Z k (u) and any ǫ > 0 there exists r ǫ = r ǫ (x 0 ) ∈ (0, ǫ) such that for all r < r ǫ . Moreover, due to α-Hölder continuity of u and the previous inequality, there exists δ ǫ = δ ǫ (x 0 , k, α) > 0 such that for anyx ∈ Z k (u) satisfying |x − x 0 | < δ ǫ and for any r < r ǫ . Then, the almost monotonicity of Mx k (u, p x 0 k , r) (see Corollary 3.12) yields (4.13) ≤ Mx k (u, p x 0 k , r) +Ĉ 1 r k(p−2)+1−a < 3ǫ and for anyx ∈ Z k (u) such that |x − x 0 | < δ ǫ and for any r < r ǫ (up to taking a smaller r ǫ , chosen independently ofx). This proves the first part of the statement. Now, let us fix a compact K ⊆ Z k (u) and let x 0 ∈ K. From now until the end of the proof, we denote by C a positive constant, whose value may change from line to line, that is independent of ǫ and of the choice ofx. From (4.12) we deduce that which, integrated in r and up to a change of variable, yields, for anyx ∈ Z k (u) such that |x − x 0 | < δ ǫ , and for any r < r ǫ . By a change of variable in (4.13) we obtain that Again, integrating in r and with a backwards change of variable, we can infer ≤ Cǫ for anyx ∈ Z k (u) such that |x − x 0 | < δ ǫ , and for any r < r ǫ . Combining the previous estimate with (4.14) we have that for anyx ∈ Z k (u) such that |x − x 0 | < δ ǫ , and for any r < r ǫ . We now observe that the function w k,x r := u k,x r − px k ∈ H 1,a (B + 1 ) weakly solves Therefore, from Lemma 2.9, by virtue of the previous inequality and (4.15), we can say that for anyx ∈ Z k (u) such that |x − x 0 | < δ ǫ , and for any r < r ǫ . In particular we can choose r ǫ sufficiently small (independently ofx) in such a way that The conclusion of the proof follows by a covering argument of the compact set K.

Regularity of the free boundary
The goal of this last section is to prove the results regarding the structure of the free boundary, more precisely its rectifiability and estimates on the Hausdorff dimension of its parts. A first step is the proof of the regularity of the regular part R(u). Proof. The proof of the first claim immediately follows from the upper-semicontinuity of the map x 0 → N (x 0 , u, 0 + ), combined with the fact that the frequency function takes no values in the range (1,2). In view of Proposition 4.2, the proof of the second claim is the same as [26, Lemma 1.5.3], so we omit it.
Proof of Proposition 1.3. First of all, we recall that any a-harmonic, 1-homogeneous polynomial p 1 (x, y) must be smooth (see e.g. [38]) and so it must depend only on the variable x and be harmonic in x. In particular p 1 (x, y) = x · ν for some ν ∈ S N −1 = S N ∩ {y = 0}. Then, from Theorem 1.1 we know that, for any x 0 ∈ R(u) there exists ν x 0 ∈ S N −1 such that as x → x 0 . Moreover, we deduce from Theorem 4.5 that the map x 0 → ν x 0 is continuous on R(u). Since u(·, 0) ∈ C 1,α (B ′ 1 ) we can compute the directional derivative and observe that ∇ x u(x 0 , 0) = ν x 0 , thus proving that R(u) ⊆ {x 0 ∈ Z(u) : |∇ x u(x 0 , 0)| = 0}. The opposite inclusion trivially comes from Theorem 1.1. Finally, the implicit function theorem allows us to conclude the proof.
The following result is a key step in the proof of Theorem 1.5, and we refer to [37,Theorem 7.7] for the proof, which basically relies on Theorem 4.5, Proposition 5.1, Whitney's extension theorem (as stated in [46]) and the implicit function theorem.

Lemma 5.2.
For any k ≥ 2 and for any n = 0, . . . , N − 1, the sets Z n k (u) are contained in a countable union of n-dimensional C 1 manifolds.
As a consequence, we are now able to prove Theorem 1.5.
Proof of Theorem 1.5. For any fixed n ≥ 0, the sets Z y k 1 (u) ∩ Z n k 1 (u) and Z y k 2 (u) ∩ Z n k 2 (u) are disjoint for k 1 = k 2 ; therefore S y n (u) = S y (u) ∩ k≥2 Z n k (u) and so it is contained in a countable union of n-dimensional C 1 manifold, thanks to Lemma 5.2. Then, taking the union for n = 0, . . . , N − 1, we obtain the desired result for S y (u). The same applies for the stratum S * (u), but in this case the dimension at any point cannot exceed the threshold N − 2, since the polynomials in P * k are harmonic in R N . The last part of this section is dedicated to the proof of Theorem 1.6, regarding the Hausdorff dimension of the nodal set Z(u), and of its regular and singular part. The main tool we exploit is a generalization of the Federer's Reduction Principle. The basic idea is the following. Consider a family of functions F defined on R N invariant under rescalings and translations, and a map Σ which associates to every function in F a subset of R N . This principles establishes that, under suitable conditions on F and Σ, in order to control the Hausdorff dimension of Σ(f ) for every f ∈ F, you just need to control the Hausdorff dimension of Σ(f ) for the functions f that are homogeneous of some degree. To the best of our knowledge, this principle (originally proved by Federer) appears for the first time in the form we need in [36, Appendix A]. We make use of the following version of the principle, which is a particular case of the generalization made by Chen, see [14,Theorem 8.5] and [13,Proposition 4.5].

Theorem 5.3 (Federer's reduction principle).
Let F ⊆ C 1,α loc (R N ) and let, for any u ∈ F, x 0 ∈ R N and r > 0 u x 0 r (x) := u(x 0 + rx). We say that u n → u in F as n → ∞ if u n → u in C 1,α loc (R N ). Let us assume that the family F satisfies the following conditions: (F1) (closure under appropriate translations and rescalings) for any r, ρ ∈ (0, 1), x 0 ∈ B ′ 1−r and u ∈ F we have that ρu x 0 r ∈ F; (F2) (existence of a homogeneous blow-up limit) for any x 0 ∈ B ′ 1 , r n → 0 + and u ∈ F, there exists a sequence ρ n > 0, a real number α ≥ 0 and an α-homogeneous function u ∈ F such that, up to a subsequence ρ n u x 0 rn →û in F; (F3) (singular set hypotheses) there exists a map Σ : F → C, where C := {A ⊆ R N : A ∩ B ′ 1 is relatively closed in B ′ 1 }, such that (i) for any r, ρ ∈ (0, 1), x 0 ∈ B ′ 1−r and u ∈ F we have that Σ(ρu x 0 r ) = Σ(u) − x 0 r ; (ii) for any x 0 ∈ B ′ 1 , r n → 0 + , u,û ∈ F such that there exists ρ n > 0 for which, up to a subsequence, ρ n u x 0 rn →û in F, the following holds true: for any ǫ > 0 there exists n ǫ > 0 such that Σ(ρ n u x 0 rn ) ⊆ {x ∈ B ′ 1 : dist(x, Σ(û)) ≤ ǫ} for all n ≥ n ǫ . Then either Σ(u) = ∅ for every u ∈ F or dim H (Σ(u)) ≤ d for every u ∈ F, where d := max{dim V : V is a subspace of R N and there exists µ ≥ 0 and u ∈ F such that Σ(u) = ∅ and u x 0 r = r µ u for all x 0 ∈ V, r > 0}, Furthermore, in the latter case there exists a function ψ ∈ F, a d-dimensional subspace V ⊆ R N and a real number µ ≥ 0 such that ψ b r = r µ ψ for all b ∈ V, r > 0 and Σ(ψ) = V ∩ B 1 . Finally, if d = 0 the set Σ(u) ∩ B ′ r is discrete for every u ∈ F and r ∈ (0, 1). Let us consider the following class of problems, to be intended in a weak sense , ∇ x u ∈ C 0,α loc (R N ), u(·, 0) ≡ 0 in R N and solves (5.1) for some R ≥ 1 and λ ≥ 0 .

Remark 5.4.
We observe that if u ∈ F is homogeneous of degree µ > 0, then λ = 0, because the function t → λ − (t − ) p−1 − λ + (t + ) p−1 is homogeneous of degree p − 1 with p ≥ 2. This means that the even reflection of such u is harmonic in B 1 .
Now we are ready to prove the last main result of our paper.
Proof of Theorem 1.6. Let F be as in (5.2). It is straightforward to check that assumptions (F1) and (F2) in Theorem 5.3 are satisfied by F, in view also of Theorem 1.1. Then, we choose the map Σ depending on the claim. Hausdorff dimension of Z(u). In order to prove (1.7), we let Σ(u) := Z(u). It's easy to prove that the singular set hypotheses in (F3) are fulfilled by this choice of the map Σ. Therefore Theorem 5.3 applies and we have that either Z(u) = ∅ or dim H (Z(u)) ≤ d. Assume