Size of the zero set of solutions of elliptic PDEs near the boundary of Lipschitz domains with small Lipschitz constant

Let $\Omega \subset \mathbb R^d$ be a $C^1$ domain or, more generally, a Lipschitz domain with small Lipschitz constant and $A(x)$ be a $d \times d$ uniformly elliptic, symmetric matrix with Lipschitz coefficients. Assume $u$ is harmonic in $\Omega$, or with greater generality $u$ solves $\operatorname{div}(A(x)\nabla u)=0$ in $\Omega$, and $u$ vanishes on $\Sigma = \partial\Omega \cap B$ for some ball $B$. We study the dimension of the singular set of $u$ in $\Sigma$, in particular we show that there is a countable family of open balls $(B_i)_i$ such that $u|_{B_i \cap \Omega}$ does not change sign and $K \backslash \bigcup_i B_i$ has Minkowski dimension smaller than $d-1-\epsilon$ for any compact $K \subset \Sigma$. We also find upper bounds for the $(d-1)$-dimensional Hausdorff measure of the zero set of $u$ in balls intersecting $\Sigma$ in terms of the frequency. As a consequence, we prove a new unique continuation principle at the boundary for this class of functions and show that the order of vanishing at all points of $\Sigma$ is bounded except for a set of Hausdorff dimension at most $d-1-\epsilon$.


Introduction
In this paper, we study the size of the zero set of solutions u of a certain class of elliptic PDEs (see Section 2.1) near the boundary of a Lipschitz domain. Assume Ω ⊂ R d is a Lipschitz domain with small Lipschitz constant and Σ is an open set of the boundary ∂Ω where u vanishes. We investigate the dimension of the set S ′ Σ (u) = {x ∈ Σ | u −1 ({0})∩B(x, r)∩Ω = ∅, ∀r > 0}, the set where u changes sign in every neighborhood.
In a more regular setting, for example in the case Ω is a C 1,Dini domain (see [AE,DEK,KN,KZ1,KZ2] for the definition) and u is harmonic, S ′ Σ (u) coincides with the usual singular set at the boundary of u: S Σ (u) = {x ∈ Σ | |∇u(x)| = 0} (see Proposition 1.9). Note that all C 1,α domains (for any α > 0) are C 1,Dini and all C 1,Dini domains are C 1 , but the converse is not true. Nonetheless, in the case where Ω is a Lipschitz domain, ∇u(x) (or ∂ ν u) only exists in Σ in a weaker sense (see Appendix A) as far as we know, which anticipates that we will not be able to find fine estimates of the size and dimension of S Σ (u) (see Section 9). The situation is different for S ′ Σ (u), for which we prove the following Minkowski dimension estimate: Theorem 1.1. Let Ω ⊂ R d be a Lipschitz domain, and let A(x) be a uniformly elliptic symmetric matrix with Lipschitz coefficients defined on Ω. Let B be a ball centered in ∂Ω and suppose that Σ = B ∩ ∂Ω is a Lipschitz graph with slope τ < τ 0 , where τ 0 is some positive constant depending only on d and the ellipticity of A(x). Let u ≡ 0 be a solution of div(A(x)∇u(x)) = 0 in Ω, continuous in Ω that vanishes in Σ. Then there exists some small constant ǫ 1 (d) > 0 and a family of open balls (B i ) i , i ∈ N centered on Σ such that (1) u| B i ∩Ω is either strictly positive or negative, for all i ∈ N, (2) K\ i B i has Minkowski dimension at most d − 1 − ǫ 1 for any compact K ⊂ Σ. Moreover, in the planar setting (d = 2), the set K\ i B i is finite for any compact K ⊂ Σ.
Recall that the upper Minkowski dimension of a set E ⊂ R d−1 can be defined as ( 1.1) dim M E = lim sup j→∞ log(#{dyadic cubes Q of side length 2 −j that satisfy Q ∩ E = ∅}) j log 2 .
The previous result gives the following corollary: Corollary 1.2. Assume Ω ⊂ R d , Σ, A(x), τ < τ 0 , and u ≡ 0 as in the statement of Theorem 1.1. Then, there exists a constant ǫ 1 (d) > 0 such that Remark 1.3. We remark that Theorem 1.1 and Corollary 1.2 • are new even in the harmonic case as the set S ′ Σ (u) has not been well studied before (as far as I know), • are valid for harmonic functions in Riemannian manifolds with Lipschitz boundary (with small Lipschitz constant depending on the metric), • include the case when Ω is a C 1 domain, situation where not too much is known either, • give Hausdorff dimension estimates for the set S ′ Σ (u) by taking an exhaustion of Σ by compact sets. Note that Hausdorff dimension estimates are weaker than Minkowski dimension estimates but these were not known either.
Remark 1.4. Some of the results of the present paper suggest that the set S ′ Σ (u) might be a natural substitute of the usual singular set S Σ (u) in the case of Lipschitz domains (and rougher). These results are the fact that S Σ (u) = S ′ Σ (u) in the case Ω is a C 1,Dini domain (Proposition 1.9), the existence of an example of a Lipschitz domain where dim H S Σ (u) = d−1 and no better (see Section 9), and Corollary 1.2 showing that better dimension estimates are true for S ′ Σ (u). Moreover, we show an upper bound estimate on the size of the zero set of u on balls centered at Σ in terms of the frequency function N (x, r) (see Definition 3.3): Theorem 1.5. Assume Ω, Σ, A(x), τ < τ 0 , and u ≡ 0 as in the statement of Theorem 1.1. Let x ∈ Σ and 0 < r < r 0 with r 0 depending on dist(x, ∂Ω\Σ), the Lipschitz constant L A and the ellipticity Λ A of A(x), and d. There existsx ∈ Ω such that |x −x| ≈ Λ A dist(x, ∂Ω) ≈ Λ A r and H d−1 ({u = 0} ∩ B(x, r) ∩ Ω) ≤ Cr d−1 (N (x, Sr) + 1) α for some large S depending on L A , Λ A and d, and some α ≥ 1 depending on d.
The precisex appearing in the statement of Theorem 1.5 is the center of a certain dyadic cube related to a Whitney cube decomposition of Ω but we have freedom in choosing it. For more details see Section 4.1 and Remark 6.5. This result is analogous to Theorem 2 in [LMNN] for a more general class of functions but with a worse exponent. Further, we briefly discuss the application of this theorem to the study of the zero set of Dirichlet eigenfunctions of the operator div (A∇·) in Ω in Section 6.6. See [LMNN] for more background on this result and its applications in the harmonic case.
Let us give some historical background for the results of this paper. L. Bers asked the following question. Consider a harmonic function u in the upper half-plane R d + , C 1 up to the boundary such that there exists E ∈ ∂R d + = R d−1 where u = |∇u| = 0 on E. Does measure d−1 (E) > 0 imply u ≡ 0? This question has positive answer in the plane, thanks to the subharmonicity of log |∇u|. But in R d + , d ≥ 3, there are examples constructed by J. Bourgain and T. Wolff [BW] which give a negative response in general.
A related conjecture by F.-H. Lin [Lin] which is still open is the following: Conjecture.
Let Ω ⊂ R d be a Lipschitz domain and Σ = B ∩ ∂Ω for some ball B centered in ∂Ω. Let u be a harmonic function in Ω and continuous up to the boundary that vanishes in Σ. If the set where ∂ ν u = 0 in Σ has positive surface measure, then u must be identically zero.
This conjecture was proved in the C 1,1 case in [Lin], where it was also shown that S Σ (u) is a (d − 2)-dimensional set (see also [BZ] for more quantitative estimates). V. Adolfsson, L. Escauriaza and C. Kenig also gave a positive answer to the conjecture in [AEK] in the case Ω is a convex Lipschitz domain. Their work was followed by I. Kukavica and K. Nyström [KN], and V. Adolfsson and L. Escauriaza [AE] where the conjecture is solved in the case Ω is C 1,Dini . Moreover, in [AE] it is also proved that S Σ (u) has Hausdorff dimension at most d − 2.
In the interior of the domain, the singular and critical sets of u have been extensively studied too. In particular, the recent results from A. Naber and D. Valtorta [NV] are remarkable: they find (d − 2)-dimensional Minkowski content bounds and prove they are (d − 2)-rectifiable with the introduction of a new quantitative stratification. Some of their techniques have been adapted to study the Minkowski content of the singular set at the boundary in the convex Lipschitz case by S. McCurdy [Mcc] and in the C 1,Dini by C. Kenig and Z. Zhao [KZ1]. Unfortunately, the methods relying on pointwise monotonicity that have been commonly used to study this problem are no longer useful in general Lipschitz domains, and new ideas are needed in this case. The conjecture of F.-H. Lin saw no further progress until X. Tolsa proved the result for harmonic functions in Lipschitz domains with small Lipschitz constants in [To2]. His proof uses the new powerful methods developed by A. Logunov and E. Malinnikova (see [Lo1,Lo2,LM2]) to study zero sets of Laplace eigenfunctions in compact Riemannian manifolds. These techniques are used on the boundary of the domain Ω with the trade-off of restricting its Lipschitz constant. This idea from [To2] has also been successfully used by A. Logunov, E. Malinnikova, N. Nadirashvili and F. Nazarov in [LMNN] to study the zero set of Dirichlet Laplace eigenfunctions in Lipschitz domains with small Lipschitz constant. In [LMNN], the authors also develop novel methods to control the zero set near the boundary that will be relevant in the present paper.
The main tool used in this paper is Almgren's frequency function (see Definition 3.3), a quantity that controls the doubling properties of the L 2 averages of u in spheres. This function was also used in most of the works mentioned in this introduction. Our proof of Theorem 1.1 requires two technical lemmas. One is Lemma 4.1, an adaptation of Lemma 3.1 -Key lemma from [To2] and it controls the behavior of the frequency function at points in Ω near the boundary (see also Lemma 7 from [LMNN]). The other is Lemma 5.1, inspired by Lemma 8 -Second hyperplane lemma: cubes without zeros from [LMNN] and it controls the size of balls centered at the boundary that contain no zeros of u. Both results were originally proved only for harmonic functions, hence we first extend them for solutions of second order linear elliptic PDEs in divergence form. Afterwards, we combine both lemmas in a new combinatorial argument that controls the size of the zero set of u near the boundary. We remark that the extension of Lemma 8 from [LMNN] to the elliptic case presents many difficulties and it might be interesting on its own. Theorem 1.1 allows us to prove an analogous unique continuation at the boundary result to the one in [To2] for more general elliptic PDEs in divergence form: Corollary 1.6. Assume Ω, Σ, A(x), τ < τ 0 , and u ≡ 0 as in the statement of Theorem 1.1.
Observe that the assumption that u vanishes continuously in Σ implies that ∇u exists σ-a.e. as a non-tangential limit in Σ, and ∇u = (∂ ν u)ν ∈ L 2 loc (σ). Here σ stands for the surface measure restricted to Σ and ν is the outer unit normal (see Remark 3.7 and Appendix A).
The proof of Corollary 1.6 uses that the elliptic measure ω A associated to the elliptic operator div(A∇·) is an A ∞ Muckenhoupt weight with respect to σ (see Definiton 7.1) and that, since |u| is locally comparable to a Green function near most points of Σ (thanks to not changing sign in a neighborhood), ∂ ν u is comparable to dω A /dσ. Theorem 1.1 also has a second corollary that controls the vanishing order of the zeros in the set where u does not change sign nearby.
Definition 1.7. The vanishing order of the zero at a point x ∈ Σ is defined as the supremum of the α > 0 such that there exist C α > 0 finite and r 0 > 0 satisfying B(x,r)∩Ω |u|dy ≤ C α r α , 0 < r ≤ r 0 .
Corollary 1.8. Assume Ω, Σ, A(x), τ < τ 0 , and u ≡ 0 as in the statement of Theorem 1.1. There exists some small constant ǫ 2 > 0 depending on d, the Lipschitz constant τ of Σ, and the ellipticity Λ A of the matrix A(x) such that for all x ∈ Σ outside a set of Hausdorff dimension d − 1 − ǫ 1 , the vanishing order of u at x is smaller than 1 + ǫ 2 . Moreover, for all x ∈ Σ, the vanishing order of u at x is greater than 1 − ǫ 2 . This corollary is proved by comparing u locally (in the neighborhoods where it does not change sign) with the Green function of a certain cone with angular opening related to the Lipschitz constant τ of Σ.
In Section 9 we provide an example showing that Corollary 1.6 cannot be improved in the sense of Hausdorff dimension estimates. This contrasts with the higher regularity case (C 1,Dini ) where the set S Σ (u) is known to be (d − 2)-rectifiable and, a fortiori, has Hausdorff dimension at most d−2 (see [KZ1]). Finally, in Section 10, we prove the following proposition relating S Σ (u) and S ′ Σ (u) in the smooth case.
Proposition 1.9. Let Ω ⊂ R d be a C 1,Dini domain, B be a ball centered in ∂Ω, and Σ = B ∩ ∂Ω. Let u be a harmonic function defined in Ω, continuous in Ω that vanishes in Σ.
The proof of the proposition follows from a local expansion of u as the sum of a homogeneous harmonic polynomial and an error term of higher degree proved in [KZ2].

Further questions.
We present some open questions related to the previous results:  [AE,KN,KZ1,KZ2] we know this cannot happen in the C 1,Dini case.
Outline of the paper. In Sections 2 and 3, we present some notation, tools and ideas that will be used throughout the paper (often without reference). The main aim of Section 4 is the proof of Lemma 4.1, although in Section 4.1 we construct a Whitney cube structure to Ω that will be used during the sections following after. Section 5 is devoted to Lemma 5.1. Both lemmas are then combined in a combinatorial argument in Section 6 to prove Theorems 1.1 and 1.5. In Section 6, we also briefly discuss the application of Theorem 1.5 to the study of the zero set of certain class of eigenfunctions. The rest of the paper is spent on the proofs of Corollary 1.6, Corollary 1.8, the example of a Lipschitz domain and harmonic function u with "large" S Σ (u), and the equality S Σ (u) = S ′ Σ (u) in the C 1,Dini case. This last part does not require the Whitney cube structure or Lemmas 4.1 and 5.1. In Appendix A, we discuss the existence of the non-tangential limit of ∇u in Σ.
Acknowledgments Part of this work was carried out while the author was visiting the Hausdorff Research Institute for Mathematics in Bonn during the research trimester Interactions between Geometric measure theory, Singular integrals, and PDE. The author thanks this institution and its staff for their hospitality. The author is also grateful to Xavier Tolsa for his guidance and advice, to Jaume de Dios for some useful discussions, and to the anonymous referee that helped improve the readability of the paper.

Lipschitz domains with small constant and some properties of elliptic PDEs
Notation: the letters C, c, c ′ ,c are used to denote positive constants that depend on the dimension d and whose values may change on different proofs. The constants c H and C N retain their values. The notation A B is equivalent to A ≤ CB, and A ∼ B is equivalent to A B A. Sometimes, we will also use the notation In the whole paper, we assume that Ω, Σ, and u ≡ 0 are as in Theorem 1.1. Moreover, we assume that Σ is a Lipschitz graph with Lipschitz constant τ with respect to the hyperplane H 0 := {x d = 0} and that locally Ω lies above Σ.
Remark 2.1. Note that a C 1 domain is a Lipschitz domain with local Lipschitz constant as small as we need. In particular, Theorem 1.1, Theorem 1.5, and its corollaries are valid for C 1 domains.
Remark 2.2. The Lipschitz constant of Ω is invariant by rescalings. If we consider a more general (anisotropic) scaling given by multiplication by a positive definite symmetric matrix with ellipticityΛ, then the Lipschitz constant of Σ changes by a factor of at mostΛ 2 .
2.1. Divergence form elliptic PDEs with Lipschitz coefficients. The function u we study solves div(A(x)∇u(x)) = 0 weakly in Ω where the matrix A(x) satisfies that By standard elliptic PDE theory (see [GT, FR] for example), we know that u ∈ C 1,α in any ball with closure inside Ω for any 0 < α < 1. Also u ∈ W 1,2 (Ω) and u ∈ W 2,2 (Ω ′ ) for any compactly embedded subdomain Ω ′ ⊂⊂ Ω. We extend the function u by 0 outside of Ω so that it is continuous through Σ. We also extend the matrix A(x) in a way that it preserves the ellipticity Λ A and Lipschitz L A constants up to a constant factor (the particular extension we choose will not matter). In particular, note that the absolute value of the extended function |u| is a subsolution in balls B such that B ∩ ∂Ω ⊂ Σ. This means that B (A∇|u|, ∇φ) dx ≤ 0, ∀φ ∈ C 1 c (B).

Modifying the domain and A(x).
Remark 2.3. Throughout this paper, we will require at different points the constants max(Λ A − 1, 1 − Λ −1 A ) and L A to be very small. We can obtain this by exploiting the fact that our considerations are local. Indeed, we can cover our initial domain Ω by a finite family of "small" domains and prove the results on the introduction on each one separately.
By "zooming" on a small domain, we decrease the Lipschitz constant L A of the matrix. By rescaling the domain by multiplication with an adequate matrix, we can force A(x) = I at a particular point x. This, in addition to the small Lipschitz constant L A of the new matrix, implies small ellipticity Λ A (we will show this below). Note, though, that this last operation changes the Lipschitz constant τ of Σ. For this reason, in the statement of Theorem 1.1, τ 0 depends on the ellipticity of A(x). Intuitively, we use the Lipschitz regularity of A(x) to exploit that, in small scales, A(x) is very close to a constant matrix.
Let us show the effects of "zooming" and rescaling the domain when the domain is a ball. Suppose u solves weakly div(A∇u) = 0 in a ball B R for some R < 1, that is, for all If we "zoom" in on the origin by considering the functionũ(y) = u(Ry) defined on B 1 , we can show it satisfiesˆB 1 (Ã(y)∇ũ(y), ∇φ(y)) dy = 0, ∀φ ∈ C 1 c (B 1 ), whereÃ(y) = A(Ry). Notice thatÃ(y) has improved Lipschitz constant R · L A < L A . Now suppose that A(0) = I and we want to change the function, the domain and the equation so that "A(0) = I" for the solution of the new equation.
Considerũ(x) = u(Sx) defined inS −1 B R whereS is the symmetric positive definite square root of A(0) and B R is a ball centered at the origin of radius R > 0. Then, for all Properties of the new matrix AS(x): • Observe that AS(0) =S −1 A(0)S −1 = I.
• The coefficients of AS(x) are also Lipschitz with constant L AS and the ratio between the Lipschitz constants of AS and A can be bounded above and below by positive constants depending only on d and Λ A . • AS(x) is uniformly elliptic with ellipticity constant bounded by min(Λ 2 A , L AS diam(S −1 B R )d). Indeed, the largest and smallest eigenvalues λ max and λ min of AS(z) at a point where · F is the Frobenius norm, · 2 is the spectral norm, and δ i,j is the Kronecker delta.

Frequency function for solutions of elliptic PDEs in divergence form
Let x ∈ Ω ∪ Σ such that A(x) = I. For r > 0 such that B r (x) ∩ ∂Ω ⊂ Σ, we denote µ x (y) := (A(y)(y − x), y − x)/|y − x| 2 and H(x, r) := r 1−dˆ∂ where dσ is the surface measure of the ball. Note that the quantity H(x, r) is nonnegative. To simplify the notation we will assume from now on that x is the origin and we will write H(r) := H(x, r) and µ(y) := µ x (y).
We also denote Note that I(x, r) is also nonnegative since A is positive definite. Moreover, I(x, r) is finite thanks to Caccioppoli's inequality supposing B r (0) ∩ ∂Ω ⊂ Σ. As before, we will write I(r) := I(x, r).
As a consequence there exists some constant c > 0 such that For the rest of this paper, we denote c H := cL A . In particular, H(r)e c H r is a nondecreasing function of r. Note that in the harmonic case, c H = 0 and H is nondecreasing (which is also a corollary of the subharmonicity of |u| 2 ).
Note that the proof of Proposition 3.2 is quite simple in the case B(0, r) ∩ ∂Ω = ∅. But in our setting we require some extra considerations to apply the divergence theorem in balls touching the boundary since u only belongs in W 1,2 (B(0, r) ∩ Ω). To address this problem, we use Appendix A and follow the ideas of [To2] in the harmonic case.
Proof. First, using that u ∈ W 1,2 (B(0, r)), we apply the divergence theorem to obtain Using Remark 3.1, we expand the term inside the integral as We compute (formally) the derivative of H: Note that the only problematic term is r −d´∂ Br 2u(∇u, A(x)x)dσ(x). Observe though, that ∂Br |∇u|dσ exists and is finite for almost every r since´b 0´∂Br |∇u| dσ dr =´B r |∇u|dx is finite. Nonetheless, using the divergence theorem, we show that for all [a, b] where A(0, a, b) is the annulus B(0, b)\B(0, a) for 0 < a < b. Thus, we have the identity for H ′ (r) for almost every r.
Now we want to prove that We denote Σ ǫ = Σ + ǫe d and Ω ǫ = Ω + ǫe d for small ǫ > 0 where e d = (0, . . . , 0, 1) and we have from the fact that u vanishes continuously in Σ and ∇u converges as a non-tangential limit in L 2 loc (Σ), as shown in Appendix A. Now, since u ∈ W 2,2 (B r ∩ Ω ǫ ), we may use divergence's theorem and that u solves div(A∇u) = 0 to obtain Finally, the limit as ǫ → 0 of the last term exists because u ∈ W 1,2 (B r ).
Summing up, we have with the second term being O(L A H(r)).
We will assume, as before, that x is the origin and we will write N (r) := N (x, r).
The following geometric lemma is essential to ensure good behavior for N (x, r) in balls intersecting Σ.
Proof. Without loss of generality, we assume x is the origin, denote T = dist(0, Σ) and suppose T ≤ r. Otherwise, Σ ∩ B r = ∅.
First, note that A(y) − I 2 ≤ CL A r for all y ∈ B r ∩ ∂Ω by the Lipschitz continuity of A. Using this, we obtain (ν(y), A(y)y) = (ν(y), y) + (ν(y), (A − I)y) where α is the angle between y and ν(y).
If the domain were flat (τ = 0 and Σ were an open subset of a hyperplane) then all the points would have the same fixed normal vectorν and we would have cos α ≥ T r . Since the domain is not flat (τ = 0), we have that the normal vector ν makes at most an angle of arctan(τ ) withν. Using this we can bound from above α by arccos T r + arctan τ . Now, assuming r is small enough so that T r > CL A r and τ is also small enough so that cos arccos T r + arctan τ ≥ CL A r, we obtain the desired inequality.
Definition 3.5. We will say that the origin 0 ∈ Ω ∪ Σ (assuming A(0) = I) and a radius r are admissible if they satisfy the assumptions of the previous lemma. If A(0) = I, we will say that 0 ∈ Ω ∪ Σ and r are admissible if 0 and Λ

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A r are admissible for the transformed domainS −1 Ω and the matrix AS (see Remark 2.3). We can extend this definition to a point x = 0 by translating the domain.
(1) If dist(x, Σ) > r, we have that B r ∩ ∂Ω = ∅ and the integral is 0. In this case we also say that x and r are admissible.
(2) If x and r ′ are admissible, then x and r are admissible for all 0 ≤ r < r ′ .
(3) In the case A(x) = I, we can ensure admissibility if we impose The reason is that, in the transformed domain, the Lipschitz constant τ increases at most by a factor Λ A and the distance from the point to Σ decreases at most by a factor Λ Remark 3.7. Since the outer unit normal vector ν is only defined σ-a.e. the derivative ∂ ν u may not exist everywhere. Its existence, the non-tangential convergence, and the fact that ∇u = (∇u, ν)ν in L 2 (σ) are proven in Appendix A.
We prove an almost monotonicity property for N (r) which was first observed in [GL] in the interior of the domain.
Proposition 3.8. Assume 0 and r ′ are admissible, A(0) = I, and the ellipticity constant Λ A of A(x) is smaller than 2. Then there exists C N > 0 depending on L A such that e rC N N (r) is nondecreasing in the interval (0, r ′ ).
For the proof of this proposition we will only keep track of the relation between the constant C N and the Lipschitz constant L A . C N also depends on the ellipticity constant Λ A but, by assuming (for example) Λ A < 2, we can omit it. We may do so thanks to Remark 2.3. Notice also that C N ≡ 0 in the harmonic case.
Our proof is an adaptation of the one in [LM1] but with the inclusion of the case B r ∩∂Ω = ∅. We need special care when B r ∩ ∂Ω = ∅ as with the proof of Proposition 3.2. Again, we use Appendix A and follow the ideas of [To2] in the harmonic case to circumvent these problems.
Proof. Fix a compact interval I ⊂ (0, r ′ ). We will show that the derivative of N (r) is positive a.e. r ∈ I. Since I(r) = r 1−dˆr 0ˆ∂Bs (A∇u, ∇u) dσds and r is bounded away from 0, we have that I is absolutely continuous. Also note that we are only considering the case 0 ∈ Ω (since we ask for admissibility). For this reason, H is of class C 1 and bounded away from 0 and N is also absolutely continuous. The derivative is Let's compute The previous identity is true for a.e. r ∈ I.
Let w(y) := µ(y) −1 A(y)y be a vector field in Ω. Observe that (w(y), y) = |y| 2 and that y/r = ν, the normal vector in ∂B r . We can rewrite A as To study C we can use that B r ∩ ∂Ω ⊂ Σ and, thus, ∇u = (∇u, ν)ν on Σ (in the sense of non-tangential limits, see Appendix A). Then, we have that This term is a bit problematic and we will treat it later with the help of Lemma 3.4. Let's use the divergence theorem on B (we do this in B r ∩ Ω ǫ and then let ǫ → 0, as in the proof of Proposition 3.2), obtaining We have that div(w(x)(A∇u, ∇u)) = div(w)(A∇u, ∇u) + (w, ∇(A∇u, ∇u)) = div(w)(A∇u, ∇u) + 2(w, Hess(u)(A∇u)) + (A D,w ∇u, ∇u) Let's compute the integrals with all these terms in (3.1) one by one. First, we obtain Using that u satisfies div(A∇u) = 0 in Ω, we get We can rewrite the previous equation using the divergence theorem: where the last term on the right hand side coincides with C defined previously.
One of the terms we missed in (3.1) behaves aŝ and the other asˆB Summing everything up, and we obtain Since C ≥ 0 thanks to the fact that 0 and r ′ are admissible, and Lemma 3.4, we can use Cauchy-Schwarz inequality to show that the whole first term is positive. Indeed, note that from the proof of Proposition 3.2, we have Thus, using Cauchy-Schwarz, we obtain Therefore, there exists C ≥ 0 such that and we denote C N := CL A .
3.1. Frequency function centered at arbitrary points. We have only considered H(x, r) and N (x, r) centered at points x ∈ Ω ∪ Σ where A(x) = I. We can treat general points by making a change of variables such as the one in Section 2.2. Assume A(0) = I. LetS be the symmetric positive definite square root of A(0),ũ(x) = u(Sx), and AS =S −1 A(Sx)S −1 . For the transformed equation div(AS∇ũ) = 0 with AS(0) = I, we can compute After some computations and a change of variables, we can check that it is equal (in the original domain) to Remark 3.9. Note that detS, (A(y)ν(y), ν(y)) and r −1 |A(0) −1 y| can be upper and lower bounded by a constant depending only on the ellipticity constant Λ A .
In particular, by assuming Λ A bounded, we have On the other hand, we have which, in the original domain, is equal to This allows us to compute N (x, r) for general points x. Beware that AS may have different Lipschitz and ellipticity constants but this is not a problem since the change can be controlled as discussed in Section 2.2.
3.2. Auxiliar lemmas on the behavior of H(r) and N (r). First, we will present a lemma that controls the growth of H(r) using N (r).
Proof. Using polar coordinates, writê and use that e c H r H(r) is nondecreasing (Proposition 3.2), and |A(0, r, r In a similar fashion, if A(x 0 ) = I and we assume Λ A bounded we obtain the following result.
Proof. Make a change of variables so that A(x 0 ) = I. The ball B(x 0 , r) is sent to an ellipsoide contained in the ball B(x 0 , Λ

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A r). Proceed as in the previous lemma by using that µ ≈ 1. The next lemma is a perturbation result for H(z, r): it shows that we can bound H(0, r) by CH(z, r ′ ) if 0 and z are close compared to r. Moreover, it does not assume that A(z) = I.
We omit the dependence of C on Λ A in this lemma.

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A , 1 + O(L A γr)) which will be useful in the proof of the next lemma.

Now we can bound
We make the change of variables x =Sy, dx = (detS)dy = (1+ O(L A γr)) dy and integrate in polar coordinates to get Finally, we use that e c H r H(r) is increasing to see that Summing up, we obtain Finally, we prove a perturbation result for the frequency function N .
Lemma 3.14. Let r > 0 and z ∈ Ω with |z| ≤ γr for γ > 0 small enough. Assume L A r is small enough, 0 ∈ Ω, A(0) = I, the ellipticity Λ A of A(x) is small enough, and the point z and distance 4r are admissible. Then we have the following bound If γ merely satisfies 0 < γ < (Λ A + 1) −1 , we obtain for some constant C > 0.
Proof. Let δ ∈ (0, 1) to be chosen later. Using Lemma 3.11 we get the following upper and lower bounds By Lemma 3.10, we have We aim to upper bound this quantity by something of the form log(H(z, r 1 )/H(z, r 2 )). To do this we will proceed as in Lemma 3.13. Since A(0) = I, we have λ max (A(z)) = 1 + O(L A γr) and λ min (A(z)) = 1 − O(L A γr) (maximum and minimum eigenvalues of A(z)). LetS z be the positive definite symmetric square root of A(z). From now on, we will denote λ max (S z ) =: λ max and λ min (S z ) =: λ min , both depending on z.
We want to find an ellipsoidal annulus (with shape given byS z ) centered at z that contains A 2 0 := A(0, 2r, r(2 + δ)) and another one that is contained in If we take an annulus A(0, r 1 , r 2 ) and deform it byS z , we get an "ellipsoidal annulus" S z A(0, r 1 , r 2 ) such that Using this, we can choose the following annuli and this other one (recall |z| ≤ γr) For this last annulus to be well defined, we need We also require γ ≤ 1 λmax λ min + 1 < 1 2 so that δ can satisfy δ ≤ 1. Now we can proceed in the exact same way as in Lemma 3.13 to get In an analogous way, we can lower bound Putting both expressions together log . (3.3) Using Lemma 3.10 again, we can upper bound log H(z,r 1 ) H(z,r 2 ) in terms of N as Now we need to choose δ. Remember that δ has to satisfy We will choose δ equal to the geometric mean of the left hand side and the right hand side of inequality (3.5), that is On the other hand, γ has to satisfy Notation: From now on, for this proof, we will write 1 − ǫ : for ǫ small enough (L A r small enough). Now let's bound every term that has appeared before on Equations (3.3) and (3.4). First we bound Let's write the first order expansion of the terms in δ: Now we can use this in the first term As for the other term, we proceed similarly Using all the bounds obtained in the last paragraphs together with Equations (3.3) and (3.4), we get for L A r and γ small enough. Finally, we can bound λ −1 min (2 + γ + δ)r ≤ 4r and use that e C N r N (r) is increasing to get a simpler expression.
Remark 3.15. To prove the second part of the lemma (γ not necessarily small), we just need to choose δ as the arithmetic mean of the left hand side and the right hand side of inequality (3.5). The rest of the proof is straightforward.

Behavior of the frequency function on cubes near the boundary
The aim of this section is to prove the first technical lemma concerning the behavior of N near the boundary. For an analogous proof in the harmonic case, see Section 3 of [To2] or Sections 4.1 and 4.2 of [LMNN]. 4.1. Whitney cube structure on Ω. We will consider the same Whitney cube structure in Ω as [To2].
Let H 0 be the horizontal hyperplane through the origin, and B 0 be a ball centered in Σ such that M B 0 ∩ ∂Ω ⊂ Σ for some very large M . We also assume that M B 0 ∩ ∂Ω is a Lipschitz graph with slope τ small enough with respect to H 0 .
We consider the following Whitney decomposition of Ω: a family W of dyadic cubes in R d with disjoint interiors and constants W > 20 and D 0 ≥ 1 such that . We will denote by ℓ(Q) the side length of Q and by x Q the center of the cube Q. From these properties it is clear that dist(Q, ∂Ω) ≈ ℓ(Q). Also we consider the cubes small enough so that diam(Q) < 1 20 dist(Q, ∂Ω). Now we will introduce a "tree" structure of parents, children and generations to this Whitney cube decomposition.
Let Π denote the orthogonal projection on H 0 and choose R 0 ∈ W such that R 0 ⊂ M 2 B 0 . It will be the root of the tree and we define D 0 . By the properties of the Whitney cubes, we can observe that Further, for any Q ∈ W, we denote its center by x Q , its associated cylinder by [To2] one can find more details about the construction of this Whitney cube structure and its projections. 4.2. Lemma on the behavior of the frequency in the Whitney tree. Now we can present the first main lemma required in the proofs of Theorem 1.1 and Theorem 1.5. This lemma controls probabilistically the behavior of the frequency function in the tree of Whitney cubes defined in the last section. See [To2,Lemma 3.1] for a version of this lemma for harmonic functions. Our proof is very similar. The reader only needs to consider that the properties of the frequency function for elliptic PDEs are slightly worse than those of the frequency function for harmonic functions, and that A(x) is a perturbation of the identity matrix. Also this Lemma should be compared with the (interior) Hyperplane lemma of [Lo1] and [LMNN,Lemma 7]. Note that, in what follows, we refer to the frequency function N of a solution of div(A∇u) = 0 in Ω as in the statement of Theorem 1.1.
Lemma 4.1. Let N 0 > 1 be big enough. There exists some absolute constant δ 0 > 0 such that for all S ≫ 1 big enough the following holds, assuming also the Lipschitz constant τ of Σ is small enough. Let R be a cube in D W (R 0 ) with ℓ(R) small enough depending on S and L A that satisfies N (x R , Sℓ(R)) ≥ N 0 . Then, there exists some positive integer K = K(S) big enough such that if we let Note that (2) It is important that δ 0 does not depend on S. Other constants such as M , K, and the upper bound on τ do depend on S. Finally, N 0 only depends on the dimension d.
and suppose u is C 1 smooth up to the boundary and A(x) is as discussed in Section 2.1. Let Suppose that u L 2 (B + ) ≤ 1 and u W 1,∞ (Γ) ≤ ǫ for some ǫ ∈ (0, 1). Then This result is proved in great generality in [ARRV,Theorem 1.7]. It will also be useful for the proof of Lemma 5.1. Before starting the proof of Lemma 4.1, we note that we will require both Λ A − 1 and L A very small in what follows (see again Remark 2.3 to see why we can do so).
Proof of Lemma 4.1. Let S ≫ 1 and then choose R ∈ D W (R 0 ) with ℓ(R) small enough depending on S and L A . For some j ≫ 1 independent of S that will be fixed below, consider the hyperplane L parallel to H 0 (and above H 0 ) such that dist(L, Σ ∩ C(R)) = 2 −j ℓ(R).
From now on, we will denote by J the family of cubes from W that intersect L ∩ C( 1 2 R). By our construction of the Whitney cubes, we have ℓ(Q) ≈ 2 −j ℓ(R) and Π(Q) ⊂ Π(R) for all Q ∈ J. Notice that if τ is small enough (depeding on j), then Denote by Adm(2W Q) the set of points x ∈ Ω∩2W Q such that the interval (0, diam(25W Q)) is admissible for x. We assume that the Lipschitz constant of the domain τ is small enough so that 3Q ⊂ Adm(2W Q) (using Remark 4.2). Then by Lemma 3.14, for Q small enough hence it satisfies the conditions required by Lemma 3.14 (we use that the ellipticity constant is small enough).
Claim. There exists some Q ∈ J such that 4C 0 if j is big enough (but independent of S) and we assume that τ 0 is small enough depending on j, and also N 0 is big enough.
Proof of the claim. From now on, we denote N = N (x R , Sℓ(R)). Our aim is to prove the claim using Theorem 4.3 in a small half-ball centered at z R , the projection of x R onto the hyperplane L. Set Also, letz R := z R + (0, . . . , 0, ℓ(R)/8) ∈ B + . Note that, rescaling B + ,z R corresponds to the point (0, . . . , 0, 1/2) in the statement of Theorem 4.3.
We aim for a contradiction, so we assume that N ( |u| 2 dx by standard properties of solutions of elliptic PDEs. Then, by Lemma 3.12, we obtain where c H is the constant from Lemma 3.10. Note that e c H diam(20W Q) = 1 + O(L A ℓ(Q)), omiting the dependence on Λ A (which we may assume is very close to 1). Using Lemma 3.10 we obtain the following bound using the frequency function: 2N (x Q ,diam(20W Q))(1+O(L A ℓ(Q))) (1+O(L A ℓ(Q))).
Note that for the previous step we need τ small enough depending on j. At other points of the proof we will require τ small enough but without further reference. Now we estimate H(x Q , ℓ(R)) as follows where we have used Remark 3.9, and that for some fixed ). Finally, using Lemma 3.11 we can bound Moreover, using again Lemma 3.10, we can further bound as follows and recalling that B(z R , ℓ(R)/8) ⊂ B + even after considering the reescaling by A(z R ) 1/2 , we obtain H(z R , C 1 ℓ(R)) (1 + O(L A ℓ(R))) |B + |ˆB + |u| 2 dy · (16C 1 ) 2Ñ (z R ,C 1 ℓ(R))(1+O(L A ℓ(R))) . Now, using that L A ℓ(R) is very small, we can bound all the terms O(L A ℓ(R)) by 1, for example. Thus, summing up all the computations we have done, we can write By Lemma 3.14, we have for some positive constants C ′′ and C ′′′ . We have used that diam(20W Q) < ℓ(R) by choosing j large enough.
Using interior estimates for solutions of elliptic PDEs From the last two estimates we deduce that if j is big enough (depending on the absolute constants C ′′ and C ′′′ ) and N 0 (and thus also N ) is big enough too, then there exists some c ′ > 0 such that Since the cubes 3 2 Q with Q ∈ J cover the flat part of the boundary of B + , we can apply a rescaled version of Theorem 4.3 to B + to get Using these bounds in (4.5) we obtain H(z R , ℓ(R)/16) 2 −jc ′ N α H(z R , ℓ(R)).
By Lemma 3.10, this implies for some fixed c ′ > 0. But for j big enough this contradicts the fact that N (z R , ℓ(R)) N by (4.4). Observe though that j "big enough" does not depend on the election of S. Now we may introduce the set G K (R). Fix Q 0 ∈ J such that (4.2) holds for Q 0 . Notice that, by (4.1) since N ≥ N 0 and we assume N 0 big enough. The precise value of the constant 10 11 2 is not important. Finally, we can define The property (1) follows from (4.6). Indeed if P ∈ G K (R), then taking into account that x P ∈ Adm(2W Q 0 ) for τ small enough (depending on S) and using Proposition 3.8 we get N (x P , Sℓ(P )) ≤ 11 10 N (x P , ℓ(Q 0 )) ≤ 11 10 2 N (x P , W ℓ(Q 0 )) ≤ N 2 where we have bounded the terms 1 + O(L A ℓ(Q 0 )) by 11/10. Notice also that and recall that j is independent of S. So (1) holds with δ 0 ≈ 2 −j(d−1) . The property (2) is a consequence of Lemma 3.14. Indeed for any P ∈ D K W (R), since |x P − x R | ℓ(R), taking γ ≈ S −1 in the Lemma 3.14, we deduce Assuming that ℓ(R) S −2 and N 0 large enough, we obtain for certain constant C.

Balls without zeros near the boundary
In this section we will prove the second main lemma concerning the behavior of N near the boundary. This lemma shows that if we have a ball near the boundary with bounded frequency, then we can find a smaller ball centered at the boundary where u does not change sign. The following lemma should be compared with [LMNN,Lemma 8] that treats the harmonic case. Note that, in what follows, we refer to the frequency function N (x, r) of a solution of div(A∇u) = 0 in Ω as in the statement of Theorem 1.1. Moreover, we consider Ω with the Whitney structure defined in Section 4.1.
Lemma 5.1. For any N > 0 and S ≫ 1 large enough there exist positive constants τ 0 (N, S) and ρ(N, S) such that the following statement holds. Suppose the Lipschitz constant τ of Σ is smaller than τ 0 and Q is a cube in D W (R) such that N (x Q , Sℓ(Q)) ≤ N . Then there exists a ball B centered in Σ ∩ C(Q) with radius ρℓ(Q) such that u does not vanish in B ∩ Ω.
First, we will prove a "toy" version of this lemma on the half ball B + for harmonic functions. The following lemma is essentially [LMNN,Lemma 9] but formulated using the frequency function instead of the doubling index (a closely related quantity used in [LMNN]).
Lemma 5.2. Let B be the unit ball in R d and let B + be the half ball, Let u be a function harmonic in B + such that u ∈ C(B + ), u = 0 on Γ := ∂B + ∩ {y ′′ = 0}, and sup 1 4 B + |u| = 1.
The notation 1 n B + used in the previous statement stands for {y ∈ R d | ny ∈ B + }. Proof. Let B − be the reflection of the half-ball B + with respect to Γ = {y ′′ = 0}∩∂B + . Since u vanishes on Γ, u can be extended to a harmonic function in B by the Schwarz reflection principle. We also denote this extension by u.
Using Cauchy estimates we can uniformly bound every partial derivative of u inside B(0, 1/8) obtaining sup for some positive C and γ ∈ (0, 1). Then ffl ∂B(0,r 0 /4) u 2 dσ ≤ Cδ 2γ and by subharmonicity ffl ∂B(0,1/2) u 2 dσ ≥ c sup B(0,1/4) u 2 = c. By the monotonicity of N in the harmonic case, we also have We can conclude that Clearly, at this point, |∇u(x ′ * , 0)| = |∂ d u(x ′ * , 0)| (the derivative in the direction normal to Γ). Without loss of generality, assume that ∂ d u(x ′ * , 0) = δ. Observe that the second derivatives of u are uniformly bounded in B(0, 1/8) using Cauchy estimates (as we have done before). Thus, we have ∂ d u(y) > δ/2 when dist(y, (x ′ * , 0)) < ρ = min{c 0 δ, r 0 } where c 0 only depends on the bound on the second derivatives, and thus is an absolute constant. Using this, we finally get Now we will prove an elliptic extension of the previous lemma. Unfortunately, the proof presents some complications since we do not have an adequate substitute to Schwarz's reflection principle. To overcome this, we assume that our function u is a solution of div(A(x)∇u) = 0 where A(x) is a small (Lipschitz) perturbation of the identity matrix, and we show that there is a harmonic function v very close to u in C 1 norm for which the previous lemma holds. Then, we obtain that there is a smaller ball where u does not vanish either. where 1 4 B + = 1 4 B + . For any N > 0 and 0 < r 0 < 1/32, there exist ρ = ρ(N, r 0 ) > 0 and c 0 = c 0 (N, r 0 ) > 0 such that if N (0, 1/2) ≤ N , then there is x ′ ∈ R d−1 with |x ′ | < r 0 such that |u(y)| ≥ c 0 y ′′ , for any y = y ′ , y ′′ ∈ B x ′ , 0 , ρ ∩ B + assuming that L A and Λ A − 1 are small enough depending on N and r 0 (where L A and Λ A are the Lipschitz and ellipticity constants of A(x), respectively). In particular, u does not have zeros in B ((x ′ , 0) , ρ) ∩ B + .
Proof. Let v be the harmonic extension of u| ∂( 1 2 B + ) defined in 1 2 B + . We intend to use Lemma 5.2 to find a ball B such that |v(y ′ , y ′′ )| y ′′ for (y ′ , y ′′ ) ∈ B ∩ Ω. Afterwards, we will see that if L A and Λ A − 1 are small enough, then the difference v − u is arbitrarily small in W 1,∞ (B ∩ Ω) which will prove the lemma (for a smaller concentric ball).
First, we bound the frequency (associated to ∆) of v as follows 2 B + u 2 dσ using that v is the minimizer of the Dirichlet energy for the boundary condition u| ∂(B + /2) . We also have that obtaining an upper bound for N v (0, 1/2). Analogously, we may also obtain a lower bound for N v (0, 1/2) in terms of N u (0, 1/2) using that u minimizes a weighted Dirichlet energy.
We aim to bound h(x) and ∇h(x) in 1 16 B + in terms of N, L A , and Λ A . Using the Green's function G A (x, y) for the elliptic operator div(A(x)∇·) in 1 2 B + , we can represent h(x) as In both integrals we bound |A − I| by Λ A − 1. Also, in the first integral, we are going to bound |∇v(y)| 2 cN for some c > 0 using Cauchy estimates. To this end, we consider v extended to 1 2 B using Schwarz reflection principle, and then, for y ∈ 1 8 B, we obtain where we have also used H u (0, 1/4) sup 1 4 B + |u| 2 = 1. Remember that H v (x, r) = r 1−d´∂ Br(x) |u(z)| 2 dσ(z), as v is harmonic.
Then, using that ∇ y G A (x, ·) has weak L d d−1 norm bounded by a constant depending only on Λ A and d (see [GW,estimate (1.6)] together with the symmetry of A), we get Thus, we can bound the first integral bŷ For the other integral, we use Cauchy-Schwarz to obtain Invoking [GW,Theorem 3.3], we have that |∇ y G A (x, y)| ≤ C|x − y| 1−d 1 for x ∈ 1 16 B + and y ∈ 1 2 B + \ 1 8 B + , which allows us to bound A . We estimate B as follows: where we have used that H v (0, 1/2) 2 cN , as shown before. Summing up all the previous estimates, we have obtained |h(x)| C(N, Λ A )(Λ A − 1) for x ∈ 1 16 B + . Now, [GT,Corollary 8.36] gives us h C 1,α ( 1 for some α ∈ (0, 1). Using the product rule for derivatives, we get

Finally, using interior Cauchy estimates to estimate
To end the proof, we apply Lemma 5.2 to v to get a ball where |v(y)| ≥ c 0 y ′′ , for any y = y ′ , y ′′ ∈ B x ′ , 0 , ρ ∩ B + and we make L A and Λ A − 1 small so that ∂ d h(x) ≤ c 0 /2 in 1 32 B + . This implies |h(y ′ , y ′′ )| ≤ c 0 2 y ′′ and, since |u| ≥ |v| − |h|, it finishes the proof. We need to state a last lemma before proving Lemma 5.1.

The proof of this lemma is
Step 1 of the proof of [DS, Theorem 1.1]. For a simpler proof in the harmonic case (in Lipschitz domains with small Lipschitz constant), see Appendix B in [FR].
In the following proof, we will assume again that L A and Λ A −1 are very small (see Remark 2.3).
Proof of Lemma 5.1. Let S ≫ 1 and Q be a cube of our Whitney cube structure such that x Q (the center of Q) and Sℓ(Q) are admissible and N (x Q , Sℓ(Q)) ≤ N . Note that, to attain this, we need the Lipschitz constant of the domain τ small enough depending on S. Further, we assume that Sℓ(Q) = 8 by rescaling the domain and the Whitney cube structure. This rescaling changes the Lipschitz constant L A of the matrix A(x) corresponding to the elliptic operator. But if the cube Q is small enough, the rescaling improves it, that is, makes L A smaller.
Letx be the projection (in the direction e d ) of x on Σ. Then, if S is big enough, we have by Lemma 3.10 and Lemma 3.13 (since we do not assume A(x Q ) = I or A(x) = I, we use first Remark 3.9). We need S large enough so that B(x, Λ A Sℓ(Q)/2 k ) ⊂ B(x Q , Sℓ(Q)/2 k−1 ) and B(x, Sℓ(Q)/2 k+1 ) ⊃ B(x Q , Λ A Sℓ(Q)/2 k+2 ) (we are using that Λ A is very close to 1) for k = 1, . . . , 5. From now on, S is fixed. We also fix the following normalization for u: 2) and B k,+ the upper half of B k , k = 1, 2 (the half of B k that intersects Ω).
Let g 0 be the solution of div(A∇g 0 ) = 0 on B 2,+ such that g 0 ≡ 1 on ∂B 2,+ \Γ 0 and g 0 ≡ 0 on Γ 0 . By the maximum principle g 0 ≥ 0 on B 2,+ and g 0 ≥ |u| on Ω ∩ B 2 ⊂ B 2,+ because of the normalization (5.2) of u. Notice that, moreover, we have the bound 1 ) for x ∈ B 1,+ which gives us a bound for |u| in Ω ∩ B 1 . In the case A ≡ I, this follows from reflection and interior Cauchy estimates for ∇u. In the general case, we may use that g 0 (x) is comparable to the Green function G A (x, y) of the domain B 2,+ (with pole y = (0, . . . , 0, 1.5) for example) by the boundary Harnack inequality inside B 1,+ . By [GW,Theorem 3.3], since B 2,+ satisfies an exterior sphere condition, we have that G(x, y) dist(x, ∂B 2,+ )|x − y| 1−d ≈ dist(x, ∂B 2,+ ). Further, for x ∈ B 1,+ we have that dist(x, ∂B 2,+ ) = x ′′ − x ′′ 1 which gives us the desired bound.
6. Proof of Theorems 1.1 and 1.5 We will combine Lemmas 4.1 and 5.1 to prove Theorems 1.1 and 1.5. Note that, in the present section, we refer to the frequency function N of a solution of div(A∇u) = 0 in Ω as in the statement of Theorem 1.1. Without further mention, we will use that the constants L A and Λ A − 1 from the matrix A(x) are very small thanks to Remark 2.3. Moreover, we consider Ω with the Whitney structure defined in Section 4.1.
First, we give a corollary to Lemma 5.1 in a language closer to that of Lemma 4.1.
Corollary 6.1. For any N > 0 and S ≫ 1 large enough there exist positive constants τ 0 (N, S) and K(N, S) such that the following statement holds. Suppose Ω ⊂ R d has Lipschitz constant τ < τ 0 and Q is a cube in D W (R) such that N (x Q , Sℓ(Q)) ≤ N . Then, for all These cubes Q ′′ j are the cubes that are contained in the ball B given by Lemma 5.1 and are vertical translation of cubes in DK W (Q). From now on, given a cube Q ∈ D W (R), we denote by t(Q) the unique cube Q ′ such that its center lies on Σ and Q ′ is a vertical translation of Q.
Next, we present a modified frequency function for which we prove good behavior as a consequence of Lemmas 4.1 and 5.1. 6.1. Modified frequency function. Let R be a cube in W such that it satisfies the conditions of Lemma 4.1 with S = S 1 ≫ 1 large enough so that CS −1/2 1 (also from the statement of Lemma 4.1) is small enough (we will specify the precise relation later). The use of this lemma gives us constants K 1 := K(S 1 ) and δ 1 := δ 0 (that does not depend on S 1 ).
Consider also Corollary 6.1 with fixed N = 2N 0 +1 (where N 0 is the constant of Lemma 4.1) and S = S 2 large enough but smaller than S 1 (note that both constants are independent). The use of this corollary gives us constants K 2 := K(N, S 2 ) and δ 2 := δ 0 (N, S 2 ). In particular, we may assume that K 2 is smaller or equal than K 1 and that CS −1/2 1 is small enough depending on min(δ 1 , δ 2 ).
We remark that the use of Lemma 4.1 and Corollary 6.1 with constants S 1 and S 2 respectively requires that the domain has small enough Lipschitz constant τ . For the rest of this section, we will denote ǫ := CS −1/2 1 , K := K 1 , and δ 0 = min(δ 1 , δ 2 ) (in particular, δ 0 and S 1 are independent constants). Define for any Q ∈ D jK W (R) for j ≥ 0. Notice that 1+2N (Q) ≥ N (x Q , S 2 ℓ(Q)) thanks to Proposition 3.8 (we assume x Q and S 1 ℓ(Q) are admissible for all children Q of R in the Whitney tree structure, thanks to τ being small enough).
We define the modified frequency function N ′ (Q) for Q ∈ D jK W (R), j ≥ 0, inductively. For j = 0, we define N ′ (R) = max(N (R), N 0 /2). Assume we have N ′ (P ) defined for all cubes P ∈ D iK W (R) for 0 ≤ i < j. Fix Q ∈ D (j−1)K W (R) and consider its vertical translation t( Q) centered on Σ. Then: (a) if u restricted to t( Q)∩Ω has no zeros, define N ′ (Q) = N ′ ( Q)/2 for ⌈δ 0 ·card{D K W ( Q)}⌉ of the cubes Q in D K W ( Q) and N ′ (Q) = (1 + ǫ)N ′ ( Q) for the rest of cubes in D K W ( Q) (the particular choice is irrelevant), (b) if u restricted to t( Q) ∩ Ω has zeros, choose Q ∈ D K W ( Q), and 1. if its vertical translation t(Q) satisfies that u restricted to t(Q) ∩ Ω has no zeros, Note that if a cube Q satisfies that u restricted to t(Q) ∩ Ω has no zeros, then all its descendants in the Whitney cube structure will satisfy the same property and (a) applies to them. Alternatively, if a cube Q satisfies that u restricted to t(Q) ∩ Ω changes sign, then all its predecessors in the Whitney cube structure will satisfy the same and (b2) applies to them. Now, a combination of Lemma 4.1 and Corollary 6.1 yields the following behavior for N ′ (Q) for Q ∈ D jK W (R), j ≥ 0. Consider a cube Q and its vertical translation t( Q). Then: • If u restricted to t( Q) ∩ Ω has zeros and N ( Q) ≥ N 0 , then Lemma 4.1 tells us that at least ⌈δ 0 · card{D K W ( Q)}⌉ cubes in D K W ( Q) satisfy N (Q) ≤ N ( Q)/2. Moreover, in this case, N ′ (Q) = max(N (Q), N 0 /2) ≤ N ( Q)/2 = N ′ ( Q)/2 where we have used that N ( Q) > N 0 and that (b) applies. For the rest of the cubes Q in D K W ( Q), we have where we have used again that N ( Q) > N 0 and (b) applies. • If u restricted to t( Q) ∩ Ω has zeros and N ( Q) < N 0 , then Corollary 6.1 enters in play and it tells us that at least ⌈δ 0 · card{D K W ( Q)}⌉ cubes Q in D K W ( Q) satisfy that t(Q) ∩ Ω does not contain zeros of u. For these cubes, N ′ (Q) = N ′ ( Q)/2 = max(N ( Q), N 0 /2)/2 < N 0 /2. For the rest of the cubes Q in D K This is similar to the behavior of the frequency function N given by Lemma 4.1 but without the restriction N ′ ( Q) ≥ N 0 .
Let's summarize the dependence of the constants that have appeared in this section (omitting its dependence on the dimension d and on L A and Λ A by assuming we are in a setting like the one described in Remark 2.3). On one hand, the constants K 2 and δ 2 given by Corollary 6.1 are absolute since they depend on S 2 and N = 2N 0 + 1 which also are absolute constants. We do require τ small enough to use Corollary 6.1. On the other hand, we have K 1 depending on S 1 , ǫ = CS −1/2 1 , and an absolute constant δ 1 given by Lemma 4.1. To use Lemma 4.1, we require τ small enough depending on S 1 . For the arguments that follow, we need ǫ small enough depending on δ 0 = min(δ 1 , δ 2 ). Though, since both constants (ǫ and δ 0 ) are independent, we can choose S 1 large enough and, thus, we require τ small enough (depending on S 1 ) to use Lemma 4.1.
6.2. Proof of Theorem 1.1. The idea behind the proof is that Lemma 4.1 allows us to use that most cubes in any generation of the Whitney tree satisfy N (x Q , Sℓ(Q)) ≤ N 0 . Then, we can apply Lemma 5.1 to these cubes, thus covering most of Σ by balls where u does not change sign.
Proof of Theorem 1.1. We will prove the result for the projection of a single cube Π(R). Afterwards, we can cover any compact in Σ by a finite union of such cubes which leaves stable the Minkowski dimension estimate.
For x ∈ Π(R), we denote by Q j (x) the unique cube Q j ∈ D jK W (R) such that x ∈ Π(Q j ) for some integer K large enough that will be fixed later. We will say that Q j (x) is a good cube if N ′ (Q j ) ≤ 1 2 N ′ (Q j−1 ) and that it is bad otherwise. Remark 6.2. Note that, with the previous definitions, N ′ (Q) < N 0 /2 implies that for all x ∈ t(Q)∩Σ there is a neighborhood where u does not vanish in Ω. Thus we only need to study the Minkowski dimension of the set of points x ∈ Π(R) such that they are not in Π(Q) for some Q ∈ D W (R) with N ′ (Q) < N 0 /2. Also, notice that the map Π : Σ∩Π −1 (Π(R)) → Π(R) is biLipschitz and thus it preserves Minkowski dimensions.
We define the goodness frequency of a point x ∈ Π(R) as (in particular, α < δ 0 ) and ǫ 0 (α) > 0 such that Note that for any 0 < ǫ < ǫ 0 , we have For all j > 1, define Claim. For all j > 1, the following holds Proof of claim. We have that by (6.1), and the definition of F j (x) and µ j . Now, thanks to the previous claim and Remark 6.2, we can reduce the problem to studying the Minkowski dimension of the set of points If we consider a random sequence of cubes (Q j ) j with Q 0 = R and Q j ∈ D K W (Q j−1 ), and let x ∈ j≥0 Π(Q j ), the probability that F j (x) ≤ β j for j ∈ N and β j ∈ (0, 1) is bounded above by Note that choosing randomly such a sequence is equivalent to choosing a random x ∈ Π(R) uniformly. In what follows we will assume that β j satisfy 2 < δ 0 1−δ 0 1−β j β j < 4 for all j > 0, in particular β < δ 0 . Let's find an upper bound for the previous quantity for very large j: Observe that for β j < 1/2 we have This is because Iterating this inequality, we obtain Using this observation we can bound A j by We use Stirling's formula to approximate We also estimate Observe that the comparability constants in the previous approximations depend on the upper and lower bounds of β j but not on j for j large enough depending on δ 0 . Summing everything up, we obtain for j large enough. Observe that z(β j ) < 1 for β j < δ 0 . Now, we will choose a suitable covering of the set E by projections of cubes in D jK W (R). First, pick ǫ < ǫ 0 (equivalently S 1 large enough, recall the discussion in Section 6.1). Observe that by choosing ǫ we also fix K. For j ≥ 1, set so that E = j E j . Let's upper bound the (K-adic) Minkowski dimension of E by finding a certain cover of E j by projections of cubes in D jK W (R) (note that there are M = 2 (d−1)K cubes in D K W (R)).
Using the previous asymptotics (setting β j = α + µ j ), we can cover E j (for j large enough) with projections of cubes in D jK W (R) and each of those cubes has side length M −j/(d−1) . Now we are ready to upper bound the Minkowski dimension of the set E. We will use the following definition of upper Minkowski dimension which is equivalent to the dyadic one in (1.1). By covering a single set E j ⊃ E and making j → ∞, we obtain the following upper Minkowski dimension estimate 6.3. Planar case of Theorem 1.1. In the planar case, Theorem 1.1 asserts that we can cover any compact K ⊂ Σ by balls where u does not change sign inside apart from a finite set of points. Moreover, this is valid for general Lipschitz domains (and domains with worse boundary regularity).
Proof of Theorem 1.1 in the planar case. Without loss of generality, suppose that Ω is simply connected and bounded. By [AIM,Theorem 16.1.4], there exists a K-quasiconformal map φ : C → C such that u = w • φ with w harmonic in φ(Ω). Since φ is biHölder continuous it is enough to prove the desired result for harmonic functions in Jordan domains. Now, consider a conformal mapping between φ(Ω) and the disk D which extends to a homeomorphism up the boundary, and let v be the induced harmonic function in the disk.
Denote byΣ the open set in ∂D where v vanishes and byẼ the set of points inΣ such that for every neighborhood v changes sign. ThenẼ must be a discrete set. This is because it coincides with the zero set of ∇v which is holomorphic (we can locally extend v to D c by reflection using the Kelvin transform nearΣ). Thus, since it is a discrete set, it is countable and finite inside any compact. Note that all maps we have considered are homeomorphisms, thus the set where v changes sign in every neighborhood is transported to another countable discrete set in Σ ⊂ Ω.
6.4. Estimates on the measure of nodal sets in the interior of the domain. Theorem 6.1 in [Lo1] estimates the (d − 1)-dimensional Hausdorff measure of the nodal set in a cube in terms of its doubling index, which is a quantity intimately related to the frequency function used in this paper. We present the following reformulation of Logunov's theorem avoiding the use of doubling indices. Theorem 6.3. There exist positive constants r, R, C depending on the Lipschitz constant L A of A(x), ellipticity constant Λ A of A(x), and dimension d such that the following statement holds. Let u be a solution of div(A∇u) = 0 on B(0, R) ⊂ R d . Then, for any cube Q ⊂ B(0, r), we have Remark 6.4. If we assume N (x Q , 16 diam(Q)) > N 0 for some N 0 positive, we can rewrite the previous theorem as where now C depends also on N 0 and α.
6.5. Proof of Theorem 1.5. To prove Theorem 1.5 we will follow the ideas of [LMNN] but exchanging the use of the Donnelly-Fefferman estimate for the size of nodal sets (see [DF]) by Logunov's estimate (Theorem 6.3). This gives rise to a worse estimate (polynomial in the frequency) than the one of [LMNN] (which is linear in the frequency but only valid for harmonic functions).
Proof of Theorem 1.5. Let Q be a small enough cube of the Whitney structure (so that we can use the modified frequency function defined in Section 6.1). Again, we will use the notation t(Q) for the unique cube Q ′ with center on Σ such that Q ′ is a vertical translation of Q.
Claim. The following equation holds for any K compact inside Ω with C 0 and α independent of K.
Proof of claim. Note that Equation (6.2) holds for all cubes Q small enough, since t(Q)∩K = ∅ if Q is small enough. This is because dist(K, Σ) > 0. We will proceed to prove this estimate by induction going from small cubes to large cubes. Assume it holds for all small cubes with ℓ(Q) < s . Now choose a larger cube P in the Whitney cube structure with ℓ(P ) < 2 K ℓ(Q) (with K given by Lemma 4.1 as discussed in Section 6.1). Given such a cube P , we can cover t(P ) ∩ Ω with small cubes t(Q i ) (intersecting the boundary) where Q i ∈ D K W (P ) and with small cubes Q ′ i far from the boundary (small enough so that we can apply Theorem 6.3 on them).
Using that the Lipschitz constant of ∂Ω is small, we can bound the number of small cubes Q ′ i necessary. Moreover, using Lemma 3.14, we can bound N (x Q ′ , 16 diam(Q ′ )) < 2N (P ) + 1. Note that N (P ) ≤ N ′ (P ) in the case that t(P ) ∩ Ω ∩ {u = 0} = ∅. This allows us to bound the size of the nodal set on Q ′ i ∩ Ω using Theorem 6.3 by We still need to bound the size of the nodal set in the boundary cubes t(Q i ), which satisfy N ′ (Q i ) ≤ (1 + ǫ)N ′ (P ). Moreover, we know that at least for ⌈δ 0 card{D K W (P )}⌉ cubes Q i we have N ′ (Q i ) ≤ N ′ (P ) 2 . Now we can use the induction hypothesis We choose ǫ small enough (by increasing S 1 ) so that (1 + ǫ) α (1 − δ 0 ) + δ 0 2 α < 1 noting that δ 0 does not depend on S 1 or ǫ. Finally, we choose C 0 large enough so that it absorbs all terms, that is, Note that the previous estimates do not depend on the compact K chosen and we prove the claim.
To treat a general (small) ball B centered at Σ, we first cover it by a comparable cube Q centered at Σ. Now, we choose a translate and dilation of the dyadic cube structure of R d such that Q = t(P ) for some P in the Whitney cube structure and we apply the previous claim.
Remark 6.5. In the statement of Theorem 1.5, the pointx that appears is the center of a particular Whitney cube appearing in the proof. Nonetheless, Lemma 3.14 (and that the Lipschitz constant τ of the boundary is small to preserve admissibility) gives us a lot of freedom to choosex. 6.6. (d − 1)-dimensional Hausdorff measure of Dirichlet eigenfunctions. Theorem 1.5 allows us to study the zero set of solutions of the Dirichlet eigenvalue problem div(A∇u λ ) = −λu λ , in Ω, u λ = 0, on ∂Ω.
In fact, one can show that for bounded domains Ω with local Lipschitz constant small enough for some α = α(d) > 1. For a detailed account of the proof in the harmonic case (and a sharper result), see Section 6 in [LMNN]. We will only briefly sketch the main ideas behind its proof. Also note that this problem is intimately related to Yau's conjecture on nodal sets of Laplace eigenfunctions in manifolds (see [LM2,Lo1,Lo2,DF]). The first step consists in passing from eigenfunctions to solutions of div(A∇u) = 0. Consider the function u(x, t) = u λ (x)e √ λt in the cylinder domain Ω × R ⊂ R d+1 . Let Clearly, we have {u(x, t) = 0} = {u λ (x) = 0} × R. Thus, we can restrict us to the study of the nodal set of u(x, t). The next necessary step is the Donnelly-Fefferman frequency estimate [DF]. For small balls B contained in the domain, it is shown in [LM1,DF] The previous result is also true for balls intersecting the boundary (see Lemma 10 in [LMNN] for a proof in the harmonic case).
Finally, using the previous estimate together with Theorems 1.5 and 6.3, one obtains the result in (6.3).
7. Proof of Corollary 1.6 Theorem 1.1 tells us that we can decompose Σ in its intersection with a countable family of balls (B i ) i and a set of Hausdorff dimension smaller than d − 1 by taking an exhaustion of Σ by compacts. Thanks to countable additivity, we only need to prove that for any ball B ∈ (B i ) i given by the decomposition of Theorem 1.1.
Before starting the proof, we define the concept of A ∞ weight.
Proof of Corollary 1.6. Consider B centered on Σ such that u| B∩Ω does not change sign. Without loss of generality, we assume that u is positive. By Dahlberg's theorem [Dah], harmonic measure for the domain B ∩ Ω is an A ∞ weight with respect to surface measure. By [FKP], since the matrix A(x) is uniformly elliptic and has Lipschitz coefficients, its associated elliptic measure ω A is another A ∞ weight. In particular, it is well known that this implies that the density dω A dσ can only vanish in a set of zero surface measure.
On the other hand, the density of elliptic measure is comparable with (A∇g, ν) at the boundary (where g is the Green function with pole outside 2B). By the boundary Harnack inequality (see [DS] for example), since u is positive in B ∩ Ω, we have that A∇u on Σ ∩ B is comparable to A∇g. This finishes the proof.
Remark 7.2. There is also a different approach to proving Corollary 1.6, which is the one adopted in [To2] for harmonic functions. The ingredients are Lemma 4.1, [AE,Lemma 0.2] (which is also valid for the type of PDEs we consider, see the paragraph below the proof of [AE,Lemma 2.2], also [To2,Lemma 4.3]), and a modification of [To2,Lemma 4.1]. In particular, the tools of Sections 5 and 6 are not indispensable for this result.
8. Proof of Corollary 1.8 Definition 8.1. We define a non-truncated coneC τ with aperture τ ∈ R and vertex at 0 as Notice that when τ = 0, thenC 0 is a half space and when τ < 0,C τ is a non-convex cone. A truncated cone C τ,s is a non-truncated coneC τ intersected with the ball B(0, s).
First we will prove the result for harmonic functions.
Proof of Corollary 1.8 in the harmonic case. Let x ∈ B ∩ Σ for some ball B where u does not vanish given by Theorem 1.1. Assume without loss of generality that u| B∩Ω > 0.
Since Σ is Lipschitz with Lipschitz constant τ , we can find a small truncated cone C τ,s of aperture τ with vertex at x and contained in Ω.
Let g be the Green function (for the Laplacian) with pole at infinity of the non-truncated coneC τ . Since u| Cτ,s is nonnegative up to the boundary, we can lower bound it by some adequate multiple of g (that is u g in C τ,s ) by boundary Harnack inequality. Notice that g is of the form g(x) = g r (|x|)g θ ( x |x| ) where g θ is the first Dirichlet eigenfunction of the Laplace-Beltrami operator ∆ in the domainC τ ∩ ∂B(0, 1) with eigenvalue λ τ and g r (|x|) = (see [Anc,Theorem 1.1]). Thus g r gives an upper bound on the order of vanishing at the origin. Also notice that λ 0 = 1 − d (whenC 0 is a half space) and thus g r (|x|) = |x|. Since the Dirichlet eigenvalues λ τ ofC τ ∩ ∂B(0, 1) for the Laplace-Beltrami operator vary continuously with the aperture τ ofC τ , we can ensure that the order of vanishing of g is close enough to 1 by making τ small enough. The continuity of the variation of the eigenvalues with the aperture can be easily shown using the Rayleigh quotient.
For the lower bound on the order of the vanishing at the origin, consider a coneC −τ of aperture −τ (concave cone) and its Green function g with pole at the infinity. Now g| Ω∩C u and we can follow the same argument.
Next, we deal with the elliptic case, but first, we need several lemmas.
Lemma 8.2. Let u be a solution of div(A∇u) = 0 in Ω and x 0 ∈ Ω ∪ Σ. Then the vanishing order α of u at x 0 satisfies H(x 0 , r) when the limit exists.
Proof. The proof has two parts. First, we see that Notice that for r small enough, we have By Remark 3 (page 953) in [AE], we obtain the doubling propertŷ uniformly for all r small enough. The remark is stated for convex domains, but the condition needed is that the domain is star-shaped with respect to 0. In the case of cones, this is trivially true. Using Lemma 3.12, we obtain that Summing up, we get for all r small enough.
The following lemma shows that the blow-up of positive solutions in cones converges to the Green function for the Laplacian in the domainC τ with pole at ∞ (see the proof of Corollary 1.8 in the harmonic case).
Lemma 8.4. Let u be a positive solution of div(A∇u) = 0 in a truncated cone C τ,s that vanishes on ∂C τ,s ∩ B(0, s) and assume that A(0) = I. Consider any sequence of radii r k ↓ 0 (such that r 1 < |p|). Let Then u k converges in W 1,2 loc (C τ ) and in C 1,α loc C τ \{0} to a multiple of the Green function g with pole at ∞ for the Laplacian inC τ for some exponent 0 < α < 1. In particular, u and g have the same vanishing order at 0.
Proof. Without loss of generality assume that s > 2 and r 1 = 1. Clearly, u ∈ W 1,2 (C τ,1 ) and u k ∈ W 1,2 (C τ,1/r k ) for all k > 0. Also notice that N u (0, r k ) = N u k (0, 1) where N u k is the frequency for the new PDE. Moreover, each u k satisfies 1 ≈ H u k (0, 1) and ∇u k 2 L 2 (C τ,1 ) ≈ N u k (0, 1). Observe that we can also bound ∇u k L 2 (C τ,1/r j ) and H u k (0, r j ) for any j ≤ k (with bound depending on r j ).
Thanks to Lemma 8.3, we have that the sequence (u k ) k is bounded in Sobolev norm in any bounded set. Moreover, by boundary Schauder estimates (see Lemma 6.18 in [GT]), u k ∈ C 1,α (K) for any compact K ⊂C τ \{0} (for k large enough depending on K). By Arzelà-Ascoli, there is a subsequence (u k j ) j that converges in norm C 1 toũ in C τ,1 \B 1/8 . Moreover, we can also assume thatũ is a weak limit in W 1,2 loc (C τ ). Thus,ũ is harmonic iñ C τ . To see this, fix any compact K ⊂⊂C τ . Then, for all φ ∈ C 1 C (K), we havê K (A(r k x)∇u k , ∇φ)dx = 0, ∀k > 0.
On the other hand, sinceũ is the weak limit of u k , we havê The term A is zero by the definition of weak solution and we can estimate B aŝ Summing up, we get thatũ is harmonic in K.
Claim. The only functions which are positive and harmonic inC τ are the multiples of the Green function g for the Laplacian inC τ with pole at ∞.
For the proof of the claim see [KT,Lemma 3.7].
Since we have C 1 convergence away from the pole and´∂ B 1 u 2 k dσ = 1 for all k > 0, the multiple of the Green function in the claim is fixed. Thus, the convergence does not depend on the sequence r k chosen and is not up to subsequences. For this reason, we assume without loss of generality that r k = 2 −k+1 .
The general elliptic case is a direct consequence of the previous lemma following the same proof as in the harmonic case.
The following example by Xavier Tolsa shows that in a Lipschitz domain Ω (even with small Lipschitz constant) there is no hope for a (non-trivial) Hausdorff dimension bound of the set of points of Σ where ∂ ν u(x) = 0.
Remark 9.1. We are interested in the normal derivative, but it may not exist at every point x ∈ Σ. Nonetheless, since we are in a Lipschitz domain Ω for every point x ∈ Σ we can consider a non-tangential cone C τ (x) contained in the domain. Thus, we can consider a non-tangential approach to the normal derivative. That is, for x ∈ Σ, we define = 0. This is a consequence of [Dah, Lemma 1] which shows for some C depending on the dimension and the Lipschitz constant of the domain. Now we can start setting up an appropriate domain. Consider, for λ ∈ (0, 1), the λ-Cantor set defined as We will refer to the intervals in [0, 1]\E λ k by gaps. Remark 9.3. The set C λ has Hausdorff dimension between 0 and 1 depending on λ, but by making λ small enough we can obtain a set with dimension arbitrarily close to 1. From now, on we will denote it by s = s(λ) = dim H C λ .
Remark 9.4. If λ = 1/(2k + 1) for some k ∈ N, then the set C λ coincides with the set of real numbers in [0, 1] such that its decimal expansion in basis 2k + 1 does not contain the digit k. Instead of choosing λ, we will choose k large enough so that the Hausdorff dimension s of C λ is as close as we want to 1.
The proof of the theorem is analogous to the proof of Borel's normal number theorem. Also, we will say that the points x ∈ C λ that satisfy this are C λ -normal.
Fix a ∈ (0, 1) and let where C a (p) are vertical open cones of aperture a with vertex at p. Thus, Ω λ is the union of all the open cones with aperture a centered at a point of the Cantor set C λ . The domain Ω λ clearly has Lipschitz boundary and its Lipschitz constant can be made arbitrarily small as it depends only on the aperture of the cones a.
10. S Σ (u) = S ′ Σ (u) in the C 1,Dini case Recall that for a harmonic function u in a Lipschitz domain that vanishes in a relatively open subset of the boundary Σ ⊂ ∂Ω, we define ∩ Ω = ∅, ∀r > 0}. Remark 10.1. Note that S Σ (u) is well defined in the C 1,Dini case since u ∈ C 1 (Ω ∪ Σ) thanks to the results of [DEK].
The proof of Proposition 1.9 follows from a local expansion of u as the sum of a homogeneous harmonic polynomial and an error term of higher degree (see [KZ2,Theorem 1.1]).
Proof of Proposition 1.9. By [KZ2, Theorem 1.1], for every x ∈ Σ there exists a positive radius R = R(x) and a positive integer N = N (x) such that where P N is a non-trivial homogeneous harmonic polynomial of degree N and the error term ψ satisfies lim y→0 |ψ(y)||y| −N = 0 and lim y→0 |∇ψ(y)||y| −N +1 = 0.
Appendix A. Existence of non-tangential limits for ∇u We will closely follow the ideas of Appendix A of [To2] in order to prove the L 2 convergence of the non-tangential limits of ∇u in Σ.
Let Ω ⊂ R d be a Lipschitz domain, B a ball centered on ∂Ω, Σ = B ∩ ∂Ω, and σ denote the surface measure on Σ. Without loss of generality, we assume Ω is locally above Σ in the direction of e d = (0, . . . , 0, 1). For σ-a.e. x ∈ Σ, the outer unit normal vector ν(x) is well defined. For a parameter a ∈ (0, 1) and x ∈ Σ, we consider the inner cone and outer cone X + a (x) = {y ∈ R d |(x − y, ν(x)) > a|y − x|}, X − a (x) = {y ∈ R d | − (x − y, ν(x)) > a|y − x|}, respectively. For a function f defined on R d \Σ, we define the non-tangential limits We prove the following theorem about the convergence of the non-tangential limits of the gradient of the solution of an elliptic PDE (of the type of Section 2.1).
Claim. In the sense of distributions, div(A∇u) = div(A∇u + ) − div(A∇u − ) restricted to B is a signed Radon measure supported on Σ.
Using the boundary Harnack inequality (see [DS], for example), we can bound supp ψ δ ∩Ω |u|dx ˆs upp ψ δ ∩Ω g dx δ 2 where g is the Green function with fixed pole far from B. To prove the second inequality, we cover supp ψ δ by a finite family of cubes Q i with side length ℓ(Q) ≈ δ. We can do this with approximately r(B) δ d−1 cubes. By standard estimates for elliptic measure, we have where ω is the elliptic measure associated to div(A∇·) for Ω with respect to a fixed pole p ∈ Ω\B. Finally, we havê where we have used the doubling properties of elliptic measure. Summing up, we take the limit as δ → 0 in equation (A.1) to obtain where we have used that the supp ψ δ → Σ as δ → 0, Thus div(A∇u)| B is a Radon measure supported on Σ.
Moreover, this Radon measure is absolutely continuous with respect to elliptic measure.
Claim. In the sense of distributions, we have div(A∇u)| B = ρω| Σ , where ρ ∈ L ∞ loc (Σ) and ω is the elliptic measure associated to div(A∇·) for Ω with respect to a fixed pole p ∈ Ω\B.
We can assume that B is small enough so that Ω\2B = ∅ and p ∈ Ω \ 2B.
Proof of the claim. To prove the claim, let B ′ be an open ball concentric with B such that B ′ ⊂ B. We will show that that there exists some constant C depending on B ′ and p such that for any compact set K ⊂ Σ ∩ B ′ , it holds (A.2) (div(A∇u), χ K ) ≤ Cω(K).
By duality, this implies the claim. Given ǫ ∈ 0, 1 2 dist(K, R d \B ′ ) , let {Q i } i∈I be a lattice of cubes covering R d such that each Q i has diameter ǫ/2. Let {φ i } i∈I be a partition of unity of R n , so that each φ i is supported in 2Q i and satisfies ∇ j φ i ∞ ℓ(Q i ) −j , for j = 0, 1, 2. Then, we have (div(A∇u), χ K ) = (div(A∇u), where I ′ is the collection of indices i ∈ I that satisfy 2Q i ∩ K = ∅. By the regularity properties of Radon measures, we obtain |(div(A∇u), i∈I ′ φ i − χ K )| ≤ | div(A∇u)|, χ Uǫ(K)\K → 0 as ǫ → 0, where U ǫ (K) is the ǫ-neighborhood of K. For the other term, we have using that A is elliptic and Lipschitz. Since |u| is a continuous subsolution in B that vanishes in B\Ω, by the boundary Harnack inequality, we have again |u(x)| g(x) for all x ∈ B ′ ∩ Ω, where g is the Green function of Ω for div(A∇·) with pole in Ω\B. The constant C depends on u, p, Λ A (the ellipticity constant of A), and B ′ , but not on K. Thus, proceeding as in the proof of the previous claim, we obtain |(div(A∇u), Letting ǫ → 0, we have ω(U 4ǫ (K)) → ω(K) from which equation (A.2) follows.