Hausdorff measure bounds for nodal sets of Steklov eigenfunctions

We study nodal sets of Steklov eigenfunctions in a bounded domain with $\mathcal{C}^2$ boundary. Our first result is a lower bound for the Hausdorff measure of the nodal set: we show that for $u_{\lambda}$ a Steklov eigenfunction, with eigenvalue $\lambda\neq 0$, $\mathcal{H}^{d-1}(\{u_{\lambda}=0\})\geq c_{\Omega}$, where $c_{\Omega}$ is independent of $\lambda$. We also prove an almost sharp upper bound, namely $\mathcal{H}^{d-1}(\{u_{\lambda}=0\})\leq C_{\Omega}\lambda\log(\lambda+e)$.


Introduction
Let a bounded domain in ‫ޒ‬ d , where d ≥ 2. A Steklov eigenfunction u λ ∈ H 1 ( ) is a solution of Here and throughout the paper we denote by ∂ ν the outward normal derivative.A number λ for which a solution to (1) exists is called a Steklov eigenvalue, and it is well known that Steklov eigenvalues form a discrete sequence accumulating to infinity.Moreover, Steklov eigenvalues coincide with the eigenvalues of the Dirichlet-to-Neumann operator, which is the operator that maps a function on ∂ to the normal derivative of its harmonic extension in , and a Steklov eigenfunction restricted to ∂ is an eigenfunction of the Dirichlet-to-Neumann operator.For a survey on the Steklov problem outlining many results and open questions see [Girouard and Polterovich 2017].
Inspired by a famous conjecture of Yau on the Hausdorff measure of nodal sets of Laplace eigenfunctions, an analogous question has been asked for nodal sets of Steklov eigenfunctions (it is stated explicitly in [Girouard and Polterovich 2017], for example); the conjecture can be formulated both for interior and boundary nodal sets.For the interior nodal set, the question is as follows: • Is it true that there exist positive constants c and C, depending only on , such that cλ ≤ H d−1 ({u λ = 0}) ≤ Cλ? (2) Similarly, for the boundary nodal set (which is the nodal set of an eigenfunction of the Dirichlet-to-Neumann operator) one can ask: 1238 STEFANO DECIO Here we do not deal with question (3) and just note that the upper bound was proved in [Zelditch 2015] when ∂ is real-analytic.About question (2), a polynomial upper bound was proved in [Georgiev and Roy-Fortin 2019], following the corresponding polynomial upper bound in the Laplace-Beltrami eigenfunction case proved in [Logunov 2018a].On real-analytic surfaces (that is, real-analytic metric in the interior and real-analytic boundary), the full conjecture (2) was established in [Polterovich et al. 2019].Again in the real-analytic category, the upper bound was recently obtained in any dimension in [Zhu 2020].Concerning lower bounds, as far as we know, the best result was contained in [Sogge et al. 2016], where the bound is obtained for a domain with C ∞ boundary (actually, a smooth Riemannian manifold with smooth boundary).The first contribution of the present article is an improvement on the lower bound; we show that the Hausdorff measure of the interior nodal set is bounded below by a constant independent of λ (so the result is really an improvement over [Sogge et al. 2016] if d ≥ 3).
Theorem 1.Let be a bounded domain in ‫ޒ‬ d with C 2 -smooth boundary, and let u λ be a solution of (1) in , λ ̸ = 0. Then there exists a constant c > 0 independent of λ such that In the previous work [Decio 2022] we established a density property of the zero set near the boundary, under weaker hypothesis on the boundary regularity: we transcribe the result below.
Theorem A [Decio 2022].Let be a Lipschitz domain in ‫ޒ‬ d , d ≥ 2, and let u λ be a solution of (1), where we assume λ ̸ = 0.There exists a constant C = C( ) such that for any ball B in ‫ޒ‬ d of radius C/λ centered at a point in ∂ .
The proof of Theorem 1 involves a combination of Theorem A and the recent breakthrough by Logunov [2018b] on Yau's conjecture.We cannot apply the results of [Logunov 2018b] directly and have to do some work to modify the necessary arguments.The fact that we are one power of λ away from the optimal result is a consequence of the deficiency of the density result, which we can only prove very close to the boundary, and not of the arguments in [Logunov 2018b].
Remark.It will be apparent from the proof that Theorem 1 extends without much difficulty to the case of manifolds equipped with a C 2 -smooth Riemannian metric and C 2 boundary.
The conjectured upper bound in (2) would be sharp, as the example of a ball shows; the second main contribution of this article is an almost sharp upper bound for Euclidean domains with C 2 boundary.
Theorem 2. Let be a bounded domain in ‫ޒ‬ d with C 2 -smooth boundary, and let u λ be a solution of (1) in .Then there is a constant C > 0 independent of λ such that the hyperplane lemma of [Logunov 2018a;Logunov et al. 2021].While, as remarked above, the proof of the lower bound can be extended almost verbatim to smooth Riemannian manifolds with boundary, for Theorem 2 we rely heavily on the fact that is a Euclidean domain, or at least we have to require that the metric inside is real analytic; this is because the results of [Donnelly and Fefferman 1988] require real analyticity.Our theorem lies in between previous results on the upper bound: the multiscale argument at the boundary allows for C 2 -regularity of the boundary only, as opposed to real analyticity as in the aforementioned paper [Zhu 2020]; on the other hand, if the metric inside is assumed to be only C 2 (or C ∞ ), the best result attainable with these methods is still the polynomial upper bound of [Georgiev and Roy-Fortin 2019].
Plan of the paper.We prove Theorem 1 in Sections 2 and 3; in Section 2 we discuss a procedure for extending a Steklov eigenfunction across the boundary, which gives rise to an auxiliary equation for which a statement very similar to Logunov's theorem [2018b] holds (see Theorem 3), and we use this together with Theorem A to prove the lower bound.Section 3 is quite long and contains the proof of Theorem 3, which requires us to review Logunov's argument carefully and use a combination of classical elliptic estimates and frequency function techniques.Section 4 is dedicated to the proof of Theorem 2.

Lower bound on nodal sets
Here we deduce Theorem 1 using Theorem A and ideas stemming from Logunov's solution [2018b] of a conjecture of Nadirashvili on nodal sets of harmonic functions.In order to do this, we transform a solution to (1) into a solution of an elliptic equation in the interior of a domain.To the best of our knowledge, this idea was introduced first in [Bellová and Lin 2015] and then also applied successfully in [Georgiev and Roy-Fortin 2019;Zhu 2015].
We now describe this extension procedure, which requires ∂ to be of class C 2 ; we follow [Bellová and Lin 2015] very closely.There is a δ > 0 such that the map ∂ × (−δ, δ) ∋ (y, t) → y + tν(y) is one-to-one onto a neighborhood of ∂ in ‫ޒ‬ d .We set d(x) = dist(x, ∂ ), and for ρ ≤ δ we define ρ = {x ∈ : d(x) < ρ} and ′ ρ = {x ∈ ‫ޒ‬ d : d(x) < ρ} \ .Let now u λ be a solution of (1), and for an easy computation shows that v satisfies where We want to study (8) at wavelength scale.In order to deal with its zero set we use the theorem below, which is just an extension to more general equations of the aforementioned theorem of Logunov on harmonic functions [2018b]; its proof, which merely consists of a tedious but necessary verification that Logunov's argument carries over in this slightly more general setting, is relegated to the next section.We warn the reader that below and in the rest of the paper we do not explicitly indicate dependence of the constants on the dimension.(i) A is a uniformly positive definite matrix; that is, A(x)ξ • ξ ≥ α|ξ | 2 for any ξ ∈ ‫ޒ‬ d .
(ii) A is Lipschitz; that is, i, j |a i j (x) − a i j (y)| ≤ γ |x − y|.
where ε 0 is a small enough constant depending on α, γ , K .
Then there exist r 0 = r 0 (α, γ , K ) < 1 and c 0 = c 0 (α, γ , K ) such that, for any solution u of (9) and any ball B(x, r ) ⊂ B(0, r 0 ) for which u(x) = 0, we have the lower measure bound Assume now that λ is large enough depending on and consider a ball B(x 0 , ε/λ) ⊂ D, where ε is a small enough constant, with smallness depending only on .We set v x 0 ,λ (x) = v(x 0 + εx/λ) for x ∈ B = B(0, 1); note that v x 0 ,λ satisfies the equation where the ellipticity constant of A x 0 ,λ is the same as that of A and the Lipschitz constant is the same if not better, and the coefficients satisfy Note that if λ is large enough then c x 0 ,λ ≥ 0. If we then take ε small enough, v x 0 ,λ satisfies (9) and assumptions (i)-(iv) with constants α, γ , K depending only on .By Theorem A, any ball centered at ∂ of radius C/λ contains a zero of the Steklov eigenfunction u λ and hence of v.We can reduce the radius of the balls and take a maximal disjoint subcollection of balls B(x i , C 1 /λ) ⊂ D, x i ∈ , such that v(x i ) = 0 and consider the corresponding rescaled functions v x i ,λ ; we can assume that C 1 < r 0 , so that by Theorem 3 we obtain (12) Note also that where C depends on only.Since there are ∼ λ d−1 such balls B(x i , C 1 /λ), we obtain , and Theorem 1 is proved.
Remark.If one could improve the result of Theorem A by showing that every ball of radius C/λ centered at any point in a corona of fixed (independent of λ) size around the boundary contains a zero of u λ , the optimal lower bound H d−1 ({u λ = 0}) ≳ λ would follow immediately by the preceding argument (actually more easily, since one could directly apply Logunov's result without the need to go through Theorem 3).

Proof of Theorem 3
This entire long section is dedicated to the proof of Theorem 3. We follow essentially the arguments of [Logunov 2018b], which carry through in this setting with few changes; the difference is that we have to use more general elliptic estimates, such as a weaker form of the maximum principle, and a frequency function that takes into account the lower-order terms in the equation.In Sections 3.1 and 3.2 we introduce the main tools we need in the proof, namely, classical elliptic estimates and the monotonicity of the frequency function.Section 3.3 will serve as a break from technicalities: here we try to convey an idea of the scheme of the proof to the reader.Sections 3.4-3.8contain the actual body of the proof with full details.
Throughout the section we consider the operator L defined by (9) satisfying conditions (i)-(iv).It will be convenient to denote by L 1 = L − cI the operator without the zeroth-order term.
3.1.Elliptic estimates.We first recall some standard elliptic estimates for L, paraphrasing the results in [Gilbarg and Trudinger 1983] in our notation.Note that whenever we consider a bounded domain we can assume for our purposes that it is contained in the unit ball, so we can ignore the dependency of the constants on the diameter of and on the radius of balls contained in .We start with the weak maximum principle.
Remark.In Theorem 9.20 in [Gilbarg and Trudinger 1983], the constants depend on R.However, they get worse as R increases and improve as R decreases; in this work we will only be concerned with small R, so that we can ignore the dependency on it.
We now come to the weak Harnack inequality and then the full Harnack inequality.
Theorem 7 [Gilbarg and Trudinger 1983, Theorem 9.22].Let Lu ≤ δ in , and suppose that u is nonnegative in a ball B(x, 2R) ⊂ .Then where p and C are positive numbers depending on α and K .
Proof.Call M = sup B(x 0 ,R) u and consider the function h = M − u, which is nonnegative in B(x 0 , R).
Note that Lh = cM, so that |Lh| ≤ εM.By applying to h Theorem 6 and then Theorem 7 with δ = εM, one gets that sup where the last inequality holds because u(x 0 ) ≥ 0. Hence we obtain sup B(x 0 ,2R/3) (−u) ≤ C M. Since clearly we have that sup B(x 0 ,2R/3) u ≤ M, the corollary is proved.□ 3.2.Frequency function and doubling index.The frequency function, which as far as we know was used first by Almgren and then subsequently developed in the works of Garofalo and Lin [1986;1987], is a powerful tool in the study of unique continuation and zero sets of elliptic PDEs.We are now going to define it for operators of the form ( 9) and state some of its properties, following mainly [Garofalo and Lin 1987;Han and Lin].
Let u ∈ W 1,2 loc (B) be a solution of (9).In [Garofalo and Lin 1987] and [Han and Lin] a metric g(x) = i, j g i j (x) d x i ⊗ d x j is introduced in the following way: let first where, as customary, a i j denote the entries of the matrix A −1 .To define g i j we assume here d ≥ 3; if d = 2, we can just add a "mute" variable.Next, one defines Finally, one sets Note that η is a positive Lipschitz function with Lipschitz constant depending on α, γ and K .Let G be the matrix (g i j ) and define |g| = det(G).We can now write (9) as Note that, since |g| −1/2 is a Lipschitz function bounded above and below by constants depending on α, γ and K only, b g and c g satisfy analogous bounds to b and c in (9).The following quantities are then introduced, where the integrals are with respect to the measure induced by the metric g: The frequency function is finally defined as Compared with the definition in [Garofalo and Lin 1987] and [Han and Lin] there is an extra factor of 2 for aesthetic reasons in later formulas.More often than not, we will forget about the point x and only write the dependance on the radius r .The key property of the frequency function is the following almost monotonicity: Theorem 10.There are constants r 0 , c 1 and c 2 depending on α, γ and K such that for r ∈ (0, r 0 ).Moreover, c 2 can be chosen to be 1 + ε for any ε > 0 if r 0 = r 0 (ε) is small enough.
Remark.The statement of Theorem 10 is implicit in [Garofalo and Lin 1987], and the proof is contained there; in [Han and Lin] the theorem is stated as it is here, and the proof given is essentially the one of [Garofalo and Lin 1987].The second assertion is not explicitly stated in [Garofalo and Lin 1987] or [Han and Lin] and needs some justification.In both papers, the strategy to prove the theorem is the following: one defines r 0 = {r ∈ (0, r 0 ) : β(r ) > max(1, β(r 0 ))} and proves that it is an open subset of ‫ޒ‬ and therefore it can be decomposed as r 0 = +∞ j=1 (a j , b j ) with a j and b j not belonging to r 0 ; it is then showed that β ′ (r )/β(r ) ≥ −C for any r ∈ r 0 .By integration, one has that β(r ) ≤ β(b j ) exp(C(b j − r )) for any r ∈ (a j , b j ).Since b j / ∈ r 0 , this implies that the constant c 2 can be chosen to be exp (Cr 0 ), which is close to 1 if r 0 is small.
In the course of the proof of Theorem 10 in [Garofalo and Lin 1987] and [Han and Lin] the differentiation formula is obtained; the formula can be rewritten as The next statement is an immediate consequence of this formula.
Proposition 11.There is a constant C depending on α, γ and K such that the function e Cr H (r )/r d−1 is increasing for r ∈ (0, r 0 ).
From (20) and almost monotonicity ( 19), by integration one obtains the following: holds, where again c 2 can be chosen to be 1 + ε if r 0 is small enough.
From now on we denote with letters c, C, c 1 , . . .constants which may vary from line to line and that depend only on α, γ and K without explicitly saying so every time.Additional dependencies will be indicated.We now define a quantity related to the frequency function: the doubling index.
Definition 13.For B(x, 2r ) ⊂ B, the doubling index N (x, r ) is defined by The doubling index and the frequency function are comparable in the following sense: Lemma 14.Let ε > 0 be sufficiently small, and let r 0 be small enough that the constant c 2 in (21) is 1 + ε; then, for 4r < r 0 , The proof of Lemma 14 is an easy computation using the elliptic estimate ( 14), Proposition 11 and inequality (21); in fact, by ( 14), sup and further by integration and Proposition 11.From here on the computation is identical to the one in [Logunov 2018a, Lemma 7.1].Using this, one can derive a scaling property for the doubling index; see [Logunov 2018a, Lemmas 7.2 and 7.3] for details of the computation.

3.
3. An informal outline of the proof.We include here a brief discussion of the scheme of the proof avoiding details and technicalities; the latter are all included in the next subsections.Let us first note that in dimension 2 Theorem 3 is an easy consequence of the weak maximum principle (Corollary 5): if u vanishes at the center of a ball, the weak maximum principle tells us that there can be no small loops of zeros containing the center and therefore the nodal component containing the center must exit the ball, implying that its length must be greater than the diameter of the ball.In higher dimensions, this simple argument does not give any lower measure bound because a priori the nodal set could be a very thin tube crossing the ball.However, a slightly more sophisticated argument, still using essentially only the maximum principle, does give a nonoptimal lower bound: we prove in Proposition 16 that if u(x) = 0, where N is an upper bound for the doubling index N x, 1 2 r .Note that when d = 2 this is already optimal, as it should be.If d ≥ 3, this naive lower bound gets worse as the doubling index gets larger.This however contradicts intuition, since we are dealing with solutions of elliptic PDEs: if the doubling index is large, meaning that there is strong growth of u, then there should be many zeros.This suggests that one could use induction on N to promote the naive lower bound to the optimal one.The key to achieving this is Proposition 23, which shows that if the doubling index is comparable to N ≫ 1 in balls of radii 1 4 r to r (we call this "stable growth", see Definition 22), there are many zeros in the ball of radius r ; more precisely, there are at least [ such that u vanishes at the center.The fact that f (N ) grows with N essentially shows that indeed there are more zeros as the doubling index increases, and it is needed to close the induction in Section 3.8.The proof of Proposition 23 uses crucially Theorem 19, which tells us that if a cube is partitioned into some large number B d of subcubes, the number of subcubes which have doubling indices dropping by an amount increasing with B compared to the doubling index of the original cube form the vast majority of the subcubes.The argument goes as follows: since the doubling index is comparable to N on scales 1 4 r to r , we can assume that in the ball of radius 1 4 r , |u| ≤ 1, while in the ball of radius 1 2 r , |u| ≥ 2 cN .We then connect points where u is small to points where u is large by many chains of cubes (called "tunnels" later): since there is considerable growth of u from one endpoint of the tunnel to the other, the Harnack inequality tells us that there must be zeros and the growth happens in the cubes with zeros; an application of Theorem 19 gives us that most of the cubes in the tunnel have doubling index much smaller than N, so that the growth from one endpoint to the other cannot be realized in very few cubes, and hence each tunnel must have many cubes with zeros.The formal proof is a matter of quantifying what "small", "large", "few" and "many" mean.
The only issue remaining is ensuring that there are balls of stable growth: this is done in Claim 3, and the proof uses the estimates in Section 3.6 which are consequences of the almost monotonicity of the frequency function.
Let us emphasize once again that the proof scheme described above is due to Aleksandr Logunov, and it appeared first in [Logunov 2018b].In our case we have to adapt it to elliptic equations with lower-order terms, but the more general estimates that we need are collected above in Sections 3.1 and 3.2, and using those estimates the proof runs in the same way as for harmonic functions.
3.4.Local asymmetry.We now derive a lower estimate for the relative volume of the set {u > 0} in balls centered at zeros of u, and consequently a nonoptimal lower estimate for the measure of the zero set.The estimate and the proof are analogous to the Laplace-Beltrami eigenfunctions case, for which see, for example, [Logunov and Malinnikova 2018;Mangoubi 2008].For the reader's convenience, we reproduce here essentially the same proof as [Logunov and Malinnikova 2018].
Proposition 16.Let B(x, r ) ⊂ B and u be a solution of (9) such that u(x) = 0. Suppose that N x, 1 2 r ≤ N, where N is a positive integer.Then the lower measure bound holds for some c > 0.
Proof.For notational simplicity we assume x = 0 and write B r = B(0, r ).We can also safely assume that N ≥ 4, say.Note that by ( 17) and ( 13) we have sup Let now r j = r 3 4 + j/(4N ) for j = 0, 1, . . ., N, and consider the concentric spheres S j = {|x| = r j }.Write m + j = max S j u and m − j = min S j u.From the weak maximum principle (13) (applied to u as well as −u), it follows that so at most 1 4 N , say, of the τ + j are greater than some C independent of N. The same holds for the τ − j , so that for at least |u| for y = x 0 and s = r/(16N ) to get, for x ∈ B(x 0 , θr/N ) with θ a sufficiently small number, We thus found a ball centered on S k of radius θr/N where u is positive, call it b + .Replace now u with −u, which is also a solution of (9): repeating the argument above with m − k and τ − k instead of m + k and τ + k gives us a ball centered on S k of radius θr/N where u is negative, call it b − .Now consider the sections of the two balls with hyperplanes through the origin that contain the center of the balls: any path within the annulus {r k−1 < |x| < r k+1 } that connects these two sections contains a zero of u, since u is positive on b + and negative on b − .This implies that the measure of the zero set is greater than the measure of the section of the balls, that is to say, .
The above holds for all indices k for which m + k+1 ≤ Cm + k and |m − k+1 | ≤ C|m − k |, and recall that there are at least 1 2 N such indices.Summing the inequality above over those indices, we see that (24) holds.□ Remark.Note that the argument above also shows that 3.5.Counting doubling indices.We now recall some very useful results from [Logunov 2018a;2018b;Logunov and Malinnikova 2018] that allow us to find many small cubes with better doubling index than the original ball (or cube).The proofs are combinatorial in nature.First we define a version of the doubling index for cubes, which are more suitable for partitioning than balls.Given a cube Q and a solution u of (9), we define the doubling index N (Q) as The constant 10d is there for technical reasons and the reader should not worry about it.It is clear that with this definition Proposition 17.Let D be a bounded domain with C 2 boundary, and let B be a ball of radius ρ < 1.
Let u be a solution of (9  Remark. In [Logunov 2018a;Georgiev and Roy-Fortin 2019], Proposition 17 is applied when ∂ D is flat; this is sufficient to prove the theorem below.We will use the proposition in the nonflat case later in Section 4, to prove a version of the hyperplane lemma. Theorem 18 [Logunov 2018a, Theorem 5.1;Georgiev and Roy-Fortin 2019, Theorem 4.1].There exist a constant c > 0 and an integer A > 1 depending on the dimension only, and positive numbers N 0 = N 0 (α, γ , K ) and R 0 = R 0 (α, γ , K ) such that for any cube Q ⊂ B(0, R 0 ) the following holds: if Q is partitioned into A d equal subcubes, then the number of subcubes with doubling index greater than Theorem 18, in [Logunov 2018b] an iterated version is proved, which is the one decisively used in the proof of the lower bound on zero sets.We state it below and refer to [Logunov 2018b] for the proof.
Theorem 19 [Logunov 2018b, Theorem 5.3].There exist positive constants c 1 , c 2 , C and an integer B 0 > 1 depending on the dimension only, and positive numbers N 0 = N 0 (α, γ , K ) and R 0 = R 0 (α, γ , K ) such that for any cube Q ⊂ B(0, R 0 ) the following holds: if Q is partitioned into B d equal subcubes, where B > B 0 is an integer, then the number of subcubes with doubling index greater than 3.6.Estimates in a spherical shell.In the following we always indicate by u a solution of (9); the frequency function and doubling index are relative to u.Consider a ball B( p, s) ⊂ B 0, 1 4 r 0 ; we are going to establish some estimates for the growth of u near a point of maximum.Let x ∈ ∂ B( p, s) be a point where the maximum of |u| on B( p, s) is almost attained, in the sense that sup B( p,s) |u| ≤ 2|u(x)|; the existence of such an x is guaranteed by Corollary 5. Write M = |u(x)|.In the next two lemmas we will assume that there is a large enough number N and Lemma 20 (variation on [Logunov 2018b, Lemma 4.1]).Let (25) be satisfied.There exist positive constants C and c such that sup Proof.Let us prove (26) only.By ( 21) and ( 25), we have that for t 1 < t 2 ∈ I, where we assume that r 0 is small enough to take c 2 = 2 in (21).We estimate , where the first inequality is just the estimate of the L 2 -norm by the L ∞ -norm and the second inequality comes from (28).By integration and Proposition 11 we have |u| 2 .

Let now
|u| 2 ; moreover, by ( 14) we have Combining the estimates we obtain Since log 1+ 1 2 δ ≥ 1 4 δ, it follows easily from the above and δ ≳ 1/ log 100 N that M 2 ≥ C 6 exp 1 100 N δ M 2 , from which one obtains (26) recalling the definitions of M and M. □ Using the properties of the doubling index, we now derive some estimates on small balls close to x; we keep on denoting by x the point on ∂ B( p, s) where the maximum of |u| on B( p, s) is almost attained.
Lemma 21 (variation on [Logunov 2018b, Lemma 4.2]).Let (25) be satisfied.There exists C > 0 such that and, for any Proof.Note that since B(x, δs) ⊂ B( p, s(1 + δ)), the first estimate ( 29) is an immediate consequence of ( 27).By definition of doubling index and (29) we have that and ( 30) is proved.Now recall the scaling properties (23); by those and (30) we obtain sup B( x,δs/4) |u| where the last inequality holds because δ ≳ 1/ log 100 N .Since, by the distance condition, sup B( x,δs/4) |u| ≥ |u(x)| = M, (31) follows.□ 3.7.Finding many balls around the zero set.We follow the arguments in Section 6 of [Logunov 2018b], in the reformulation contained in [Logunov and Malinnikova 2020]; the estimates in the spherical shell will be used together with the combinatorial results on doubling indices.We use the notion of "stable growth", which is taken from [Logunov and Malinnikova 2020] and was not present in [Logunov 2018b].
Definition 22.We say that u has a stable growth of order N in a ball B(y, s) if N y, 1 4 s ≥ N and N (y, s) ≤ 1000N .
The number 1000 does not have any special meaning, it is just a large enough numerical constant.The following result is the key to the proof of the lower bound.
Proposition 23 (variation on [Logunov 2018b, Proposition 6.1]).Let B( p, 2r ) ⊂ B(0, r 0 ).There exists a number N 0 > 0 large enough such that, for N > N 0 and any solution u of (9) that has stable growth of order N in B( p, r ), the following holds: there exist at least [ Proof.Assume without loss of generality that sup B( p,r/4) |u| = 1.The stable growth assumption then implies that sup |u| ≥ 2 N and sup We denote by x the point on ∂ B p, 1 2 r where the maximum over B p, 1 2 r is almost attained, so that by the above |u(x)| ≥ 2 N −1 .We now divide the ball B( p, 2r ) into cubes q i of side length cr/

√
N and organize these cubes into tunnels in the following way: the centers of the cubes in each tunnel lie on a line parallel to the segment that connects p and x.A tunnel contains at most C √ N cubes.Let us call a cube q i "good" if for some constant c.We will call a tunnel "good" if it contains only good cubes; by Theorem 19, most of the cubes are good and most of the tunnels are good.Another application of Theorem 19 gives the following: Claim 1.The number of good tunnels containing at least one cube with distance from x less than r/ log 2 N is greater than c( The proof of the proposition is then completed with the help of the next claim.
Claim 2. Any good tunnel that contains at least one cube with distance from x less than r/ log 2 N also contains at least 2 c 2 log N / log log N cubes with zeros of u.
Proof.Take one such tunnel T. Note that T contains at least one cube q a ⊂ B p, 1 4 r , so that sup q a |u| ≤ 1.Call q b a cube in T with distance from x less than r/ log 2 N ; we want to show that the supremum of |u| over q b is large.To this end, we apply Lemma 21 with δ ∼ 1/ log 2 N and x being the center x b of the cube q b .By the stable growth assumption and the comparability of the doubling index and frequency function (Lemma 14), inequality (25) is satisfied for N large enough.Then (31) gives us sup We now follow T from q a to q b and find many zeros.The proof is at this point identical to the one given in [Logunov 2018b]; for completeness we provide the details.We enumerate the cubes q i from q a to q b such that q a is the first and q b is the last.Since T is a good tunnel, by (32) we have that for any two adjacent cubes log sup q i+1 /2 |u| We split the set of indices S into two sets S 1 and S 2 , where S 1 is the set of i such that u does not change sign in q i ∪ q i+1 and S 2 = S \ S 1 .The advantage of this is the possibility to use the Harnack inequality on S 1 ; the aim is to get a lower bound on the cardinality of S 2 .In fact, for i ∈ S 1 , by ( 16) we have that log sup q i+1 /2 |u| We then estimate log sup q b /2 |u| sup q a /2 |u| = S 1 log sup q i+1 /2 |u| on the other hand, recall that log sup q b /2 |u| sup q a /2 |u| ≥ cN.
Combining the two estimates one obtains The last quantity is larger than 2 c 2 log N / log log N if N is large enough, and the claim is proved.□ It is now a straightforward matter to finish the proof of Proposition 23: by Claim 1 there are at least c( √ N / log 2 N ) d−1 tunnels satisfying the hypothesis of Claim 2, and hence there are at least c( √ N / log 2 N ) d−1 2 c 2 log N / log log N cubes that contain zeros of u; the last quantity can be made larger than ( √ N ) d−1 2 c 1 log N / log log N , and then one replaces cubes by balls.□ 3.8.Proof of the lower bound.We take r 0 small enough that ( 19), ( 21), Lemma 14 and (23) hold.Writing N (0, r 0 ) = sup {B(x,r )⊂B(0,r 0 )} N (x, r ), we define where the infimum is taken over all balls B(x, ρ) ⊂ B(0, r 0 ) and all solutions u of ( 9) such that u(x) = 0 and N (0, r 0 ) ≤ N. Theorem 3 then follows immediately from the following: Theorem 24.F(N ) ≥ c, where c is independent of N.
Proof.Let u be a solution of (9) in competition for the infimum in the definition of F(N ); let F(N ) be almost attained on u, in the sense that for some B(x, r ) ⊂ B(0, r 0 ) with u(x) = 0. Recall the easy bound (24): Estimate (34) already finishes the proof if N x, 1 4 r is bounded uniformly in N ; let us then argue by contradiction and assume that N x, 1 4 r is large enough.Denote N = N x, 1 4 r and suppose first that u Claim 3 gives an order of stable growth that is again large enough to get a contradiction with (33) if N x, 1 4 r and hence N 1 is large enough.This means that N x, 1 4 r is bounded from above by some N 0 independently of N , and therefore by ( 33) and ( 34) we obtain which concludes the proof of the theorem.□

Upper bound
Here we give the proof of Theorem 2. Throughout this section ∂ is assumed to be of class C 2 .As remarked in the introduction, the proof uses the Donnelly-Fefferman bound [1988] in the interior of the domain and a multiscale induction argument at the boundary.As will be apparent from the proof, the result with a C ∞ -metric inside would follow from an upper bound for zero sets of elliptic PDEs with smooth coefficients that is linear in the frequency; the best we have thus far is polynomial in the frequency [Logunov 2018a].
We introduce now a version of the doubling index that takes into account the boundary.Namely, for x ∈ and u ∈ C( ) a harmonic function, we let Note that if v is the extension across the boundary of the Steklov eigenfunction u λ as in Section 2 and dist(x, ∂ ) ≲ 1/λ, r ≲ 1/λ, we have that N * u λ (x, r ) ∼ N v (x, r ), where N v (x, r ) is defined as in ( 22); this will allow us to use the almost monotonicity property (23).It was proved in [Zhu 2015] (using the extension v) that for any r < r 0 ( ) mirroring a corresponding statement for Laplace eigenfunctions proved by Donnelly and Fefferman.It will once again be convenient to define a maximal version of the doubling index for cubes; for where s(Q) is the side length of Q and c 1 and c 2 are positive dimensional constants.
With this notation, we state the following important result of [Donnelly and Fefferman 1988].
Theorem 26.Let u be a harmonic function in .Then there is C > 0, independent of u, such that for any Whitney cube Q.
From now on, we will denote by u a Steklov eigenfunction with eigenvalue λ.We will first use the theorem above to bound the measure of the zero set of u in the interior, up to a distance from the boundary comparable to 1/λ.We will assume λ > λ 0 .As in the previous section, write d(x) = dist(x, ∂ ); Let c 0 be a small constant depending only on .We write the decomposition with C depending on only.The next lemma estimates the contribution of the nodal set in Mid.
Lemma 27.There is C > 0 depending only on such that Proof.We set M k = {x ∈ : c 0 2 k−1 /λ < d(x) < c 0 2 k /λ}, and we have We perform a decomposition of into Whitney cubes with disjoint interior (the statement that this is possible is usually called the Whitney covering lemma).Define In the following lines we will denote by | • | both the cardinality of a discrete collection and the Lebesgue measure of cubes; it should cause no confusion.Note that if We can then estimate, using Theorem 26 and (37), To prove Theorem 2 the only thing left is to estimate H d−1 (Z u ∩ Bd).We cover Bd with ∼ λ d−1 cubes q λ centered at ∂ of side length s(q λ ) = 4c 0 /λ; then Theorem 2 follows from (37) and the following proposition: Proposition 28.Let q λ be one of the cubes above, and suppose N * u (4q λ ) ≤ N. Then Remark.In the following we will rescale so that q λ becomes a cube Q of side length s < 1, where s is small enough depending on but independent of λ, and h satisfies h = 0 in 10Q ∩ and ∂ ν h = h on ∂ ∩ 10Q.Note that the doubling index is unchanged under this rescaling.Proposition 28 will follow from The main ingredient in the proof of Proposition 28 is a version of the hyperplane lemma of [Logunov 2018a] with cubes touching the boundary, the proof of which uses quantitative Cauchy uniqueness as stated in Proposition 17.The proof is very similar to the one contained in [Logunov et al. 2021], we reproduce it here for the reader's convenience.
Lemma 29.Let h be as in (42) and Q a cube of side length s as in the remark above.There exist k and N 0 large enough depending on s and such that if Q ∩ ∂ is covered by 2 k(d−1) cubes q j with disjoint interior centered at ∂ of side length 2 −k s, and N * h (Q) = N > N 0 , then there exists q j 0 such that N * h (q j 0 ) ≤ 1 2 N. Proof.We note first that since ∂ is of class C 2 , h is harmonic in 10Q ∩ and ∂ ν h = h on ∂ ∩ 10Q, we can use the extension-across-the-boundary trick described in Section 4, namely, consider v(x) = e d(x) h(x); recall that the coefficients of the second-order term in the equation satisfied by v are at least Lipschitz.This gives us access to elliptic estimates that hold up to the boundary for h.In particular, we will use the gradient estimate where the implied constant depends on s and .Denote now by x Q ∈ ∂ the center of the cube Q.Consider a ball B centered at x Q such that 2Q ⊂ B, and let M = sup B∩ |h|.By contradiction, suppose that N * h (q j ) > 1 2 N for any j; by definition, this implies that for any j there is x j ∈ q j ∩ and r j ≤ 2 −k √ ds =: r 0 such that N * h (x j , r j ) > 1 2 N. Assuming N large enough, we use (23) to get sup We are now ready to prove Proposition 28, or actually (43).The argument is an iteration at the boundary; it originates in [Logunov et al. 2021].
Proof of (43).First, we consider again v(x) = e d(x) h(x) and its even extension across the boundary (which we still call v).Recall from Section 2 that v satisfies an elliptic PDE with Lipschitz second-order coefficients and bounded lower-order coefficients.The results of [Hardt and Simon 1989] then apply to this situation.Let Q be any cube with s(Q) < s 0 small enough.By [Hardt and Simon 1989, Theorem 1.7], we have that for any ball B(x, ρ) ⊂ Q where v(x) = 0 and ρ < ρ 0 (N v (Q)).Covering Z h ∩ Q with balls of such small radius and summing the estimate above over all those balls, it follows that there is a function Ã : Let now Q be as above a cube centered at ∂ of side s, with s small enough depending on .Fix a large number N 0 ; if N * h (Q) < N 0 , (45) already implies the result.Otherwise, cover Q ∩ with smaller cubes of side length 2 −k s, where k = k( ) is given by Lemma 29, in the following way: first Q ∩ ∂ is covered by cubes q ∈ B centered at ∂ with disjoint interior, and then the rest of Q ∩ is covered by cubes q ∈ I with dist(q, ∂ ) > cs(q) for some constant c > 0 independent of k.Cubes in B will be called boundary cubes and cubes in I will be called inner cubes; inner cubes are allowed to overlap, while boundary cubes are not.Write N * h (Q) = N.By (38) and almost monotonicity, By Lemma 29, there is a boundary cube, call it q 0 , such that N * h (q 0 ) < 1 2 N. The other cubes in B will be enumerated from 1 to 2 k(d−1) −1.We have that We define where the supremum is taken over all harmonic functions h in 2Q with ∂ ν h = h on ∂ ∩ 2Q, N * h (Q) ≤ N and all cubes q ⊂ Q.By (45), A(N ) < +∞.From the inequality above, we get − 1)A(N )2 −k(d−1) , from which A(N ) < C(k)N + A 1 2 N .(Beware that C(k) changes value from line to line and depends also on ).Iterating the last inequality until 1 2 N < N 0 , we obtain A(N ) < C(k)N + A(N 0 ) < C(k)(N + 1), which concludes the proof.□ Theorem 2 now follows by combining ( 39), ( 40), ( 41) and ( 37).We believe that the extra log λ factor is not necessary and is an artificial feature of the proof; it appears in the proof of (40) and it is due to the necessity of getting to cubes of side length ∼ λ −1 .
where A = I , b = −2λ∇d and c = λ 2 − λ d.Consider now the reflection map : δ → ′ δ given by (y + tν(y)) = y − tν(y), where y ∈ ∂ ; since v satisfies a Neumann boundary condition on ∂ , we can extend it "evenly" across the boundary, i.e., set v( (x)) = v(x) for x ∈ δ .Write (x) = x ′ .Another easy computation shows that on ′ δ the extended function (which we still call v) satisfies the equation div( Ã∇v) + b • ∇v + cv = 0, we abuse notation and denote by A, b and c the functions that are equal to the previous A, b and c in δ and equal to Ã, b and c in ′ δ .In [Bellová and Lin 2015] it is shown that A is Lipschitz across ∂ with Lipschitz constant depending only on , and A is uniformly positive definite, again with constant depending only on .Pasting together the pieces, one obtains that v is a strong solution of the uniformly elliptic equation div

Theorem 3 .
Consider a strong solution of the equation Lu = div(A∇u) + b • ∇u + cu = 0 (9) in B = B(0, 1) ⊂ ‫ޒ‬ d , with the following assumptions on the coefficients: which is analogous to the best known lower bound (when d ≥ 3) for the local asymmetry of Laplace eigenfunctions [Mangoubi 2008].

B
(x j ,2r 0 )∩ |h| ≤ (C2 −k ) N /10 sup B∩ |h| ≤ Me −cN k if k is large enough.Using (44)Me −cN k ,with the implied constant depending on s and .Note that since q j ⊂ B(x j , r 0 ) the two estimates above give bounds for the Cauchy data of h on ∂ ∩ Q.On the other hand, if B ′ is the ball centered at x Q such that 4B ′ ⊂ Q we have sup 2B ′ ∩ |h| ≤ M and sup 2B ′ ∩ |∇h| ≲ M/s.Recalling that r 0 = 2 −k √ ds, we can then apply Proposition 17 with η = 2 k e −cN k to get supB ′ ∩ |h| ≤ C(s, )2 βkc d e −cβ N k M.But then N * h (x Q , √ ds) ≥ C d log sup B∩ |h| sup B ′ ∩ |h| ≥ C d (cβ N k − c d βk − C),and the rightmost expression is larger than N if k and N are large enough depending on s and ; this is a contradiction with N * h (Q) = N. □ Consider each such k and let x 0 ∈ S k be such that u(x 0 ) = m + k .Denote by b the ball centered at x 0 of radius r/(8N ); then by (13) and the choice of k, sup b u ≤ sup B(y,s)