Domains without dense Steklov nodal sets

This article concerns the asymptotic geometric character of the nodal set of the eigenfunctions of the Steklov eigenvalue problem $$ -\Delta \phi_{\sigma_j}=0,\quad\text{ on }\Omega,\qquad\qquad \partial_\nu \phi_{\sigma_j}=\sigma_j \phi_{\sigma_j}\quad \text{ on }\partial\Omega $$ in two-dimensional domains $\Omega$. In particular, this paper presents a dense family $\mathcal{A}$ of simply-connected two-dimensional domains with analytic boundaries such that, for each $\Omega\in \mathcal{A}$, the nodal set of the eigenfunction $\phi_{\sigma_j}$"is $not$ dense at scale $\sigma_j^{-1}$". This result addresses a question put forth under"Open Problem 10"in Girouard and Polterovich, J. Spectr. Theory, 321-359 (2017). In fact, the results in the present paper establish that, for domains $\Omega\in \mathcal{A}$, the nodal sets of the eigenfunctions $\phi_{\sigma_j}$ associated with the eigenvalue $\sigma_j$ have starkly different character than anticipated: they are not dense at any shrinking scale. More precisely, for each $\Omega\in \mathcal{A}$ there is a value $r_1>0$ such that for each $j$ there is $x_j\in \Omega$ such that $\phi_{\sigma_j}$ does not vanish on the ball of radius $r_1$ around $x_j$.


Introduction
Let (M, g) be a compact Riemannian manifold with piecewise smooth boundary ∂M . The Steklov problem is given by There is a discrete sequence 0 = σ 0 < σ 1 ≤ σ 2 ≤ . . . of values of σ, with σ j → ∞ as j → ∞, for which non-trivial solutions satisfying (1.1) exist [HL01]. These are the Steklov eigenvalues and the corresponding functions φ σ j are the Steklov eigenfunctions. This paper studies the asymptotic character of the nodal set of the eigenfunctions of the Steklov eigenvalue problem in the case M equals a bounded open set Ω ∈ R 2 . In particular the results in this paper show that the nodal set of the eigenfunction φ σ j is not dense at scale σ −1 j for some such sets Ω-or, more precisely, that there is a dense family A of simply-connected two-dimensional domains with analytic boundaries such that, for each Ω ∈ A, the eigenfunction φ σ j in the domain Ω remains nonzero on a j-dependent ball of j-independent radius. This result addresses a question put forth under "Open Problem 10" in [GP17].
The behavior of both the Steklov eigenvalues (see e.g. [GP17,GPPS14,LPPS17]) and eigenfunctions (see e.g. [PST, GT19, BL15, Zhu16, Zel15, SWZ16, Sha71, HL01]) have been a topic of recent interest. When M has smooth boundary, the Steklov eigenfunctions φ σ j | ∂M behave much like high energy Laplace eigenfunctions with eigenvalue σ 2 j . In particular, they oscillate at frequency σ j . References [PST,BL15,Zhu16,Zel15,SWZ16,WZ15,GRF17,Zhu15] study the nodal sets of φ σ j | M , giving both upper and lower bounds on its Hausdorff measure similar to those for Laplace eigenfunctions. In fact, most results regarding Steklov eigenfunctions in the interior of M extract behavior similar to that of high energy Laplace eigenfunctions.
The purpose of this article is to show that, away from the boundary of M , Steklov eigenfunctions behave very differently than high energy Laplace eigenfunctions. Not only do they decay rapidly (see [GT19,HL01]) but, at least for a dense class of analytic domains, they oscillate slowly over 1 certain portions of the domain. Girouard-Polterovich [GP17, Open Problem 10(i)] raise the question of whether nodal sets of Steklov eigenfunctions are dense at scale σ −1 j in M . One consequence of the results in the present paper is a negative answer to this question. We show that arbitrarily close to any simply-connected domain with analytic boundary Ω 0 ⊂ R 2 , there is a domain Ω 1 for which the nodal sets are not σ −1 j dense and, indeed, that there is a region within Ω 1 where the nodal set density does not increase as σ j → ∞. Moreover, the Steklov eigenfunctions oscillate no faster than a fixed frequency in this region. These results are summarized in the following theorem.
Theorem 1. Let Ω 0 ⊂ R 2 be a bounded simply-connected domain with analytic boundary, and let k > 0 and ε > 0 be given. Then there exist a set Ω 1 ⊂ R 2 with analytic boundary given by (where ν denotes the outward unit normal to ∂Ω 0 and where g is an analytic function defined on ∂Ω 0 ), a point x 0 ∈ Ω 1 and numbers 0 < r 1 < r 0 , (B(x 0 , r 0 ) ⊂ Ω 1 ) such that: for each Steklov eigenvalue σ for the domain Ω 1 there exists a point x σ ∈ B(x 0 , r 0 ) such that B(x σ , r 1 ) ⊂ B(x 0 , r 0 ) and each Steklov eigenfunction φ σ of eigenvalue σ for the domain Ω 1 satisfies Additionally, "φ σ has bounded frequency on B(x 0 , r 0 )" (a precise statement follows in Theorem 2).
Figure 1. Fixed-sign sets for Steklov eigenfunctions over the elliptical domain Ω = x 2 + y 2 1.01 2 = 1. The yellow and blue regions indicate the subsets over which the eigenfunctions are positive and negative, respectively. The left and right images correspond to the eigenvalues σ 20 = 9.9502 and σ 30 = 14.9253, respectively. For a circle the nodal lines coincide with a set of j uniformly arranged radial lines from the center to the boundary: they are dense at scale σ −1 j = j −1 over the complete domain, including the origin. Under the barely-visible perturbation of the unit disc into the slightly elliptical domain Ω, regions of asymptotically fixed size on which the eigenfunction does not change sign open-up within Ω. Indeed, the nodal set corresponding to σ 30 (right image) shows such an opening, whereas the nodal set corresponding to σ 20 (left image) does not; cf. also Remark 1.2.
Theorem 1 is a consequence of the more precise results presented in Theorems 2 and 3 and Corollary 2.2. In particular, these results establish that, for each domain Ω in a dense class A of two-dimensional domains, an estimate holds for the truncation error in certain "mapped Fourier expansions" of the eigenfunctions φ σ (i.e., Fourier expansions of φ σ under a change of variables).
This estimate is uniformly valid over a subdomain of Ω for all eigenfunctions φ σ with σ large enough. To state these results we first introduce certain conventions and notations, and we review known facts and results from complex analysis.
In what follows, and throughout the reminder of this article, R 2 is identified with the complex plane C, Ω ⊂ C denotes a bounded, simply-connected open set with analytic boundary, and D := {z ∈ C | |z| < 1} denotes the open unit disc in the complex plane. Under these assumptions it follows from the Riemann mapping theorem [BK87] that there is a smooth map f : D → C such that f | D : D → Ω is a biholomorphism and |∂ z f | > 0 on D-that is to say, f | D : D → Ω is a biholomorphic conformal mapping of Ω up to and including ∂Ω. We call such a function f a mapping function for Ω. Note that, denoting by ∂ r and ∂ ν the radial derivative on the boundary of D and the normal derivative on the boundary of Ω, respectively, we have ∂ r = |∂ z f |∂ ν and |∂ z f | > 0. Thus, for z ∈ ∂D the function and, hence, the generalized Steklov eigenvalue problem Finally we introduce notation for the relevant Fourier analysis. For v ∈ C(D) we let denote the "boundary Fourier coefficients", namely, the Fourier coefficients of the restriction v| ∂D of v to ∂D. Where notationally useful, we write F[v] =v.
Definition 1.1. We say that the Steklov problem on Ω satisfies the tunneling condition if there is m 0 > 0 and a mapping function f for Ω, such that for all K > 0 there is C 0 > 0 satisfying for any m Lemma 4.1 shows that any tunneling Steklov problem there exist σ 0 > 0 so that for each m ∈ Z there is a constant C > 0 such that for σ > σ 0 , This estimate and its connections with similar results in quantum mechanics motivate the "tunneling" terminology introduced in Definition 1.1. To explain this, recall that u is an eigenfunction of the Dirichlet to Neumann map which is a pseudodifferential operator on ∂Ω with symbol |ξ| g where g is the metric on ∂Ω [Tay11, Sec. 7.11, Vol 2]. Therefore, the classical problem corresponding to the Steklov problem is the Hamiltonian flow for the Hamiltonian |ξ| g on T * ∂Ω at energy |ξ| g = σ-which describes the motion of a free particle on ∂Ω. The allowable energies for this classical problem are given by {|ξ| g = σ} which, in the Fourier series representation correspond to σ = |ξ| g ∼ |k|. Thus, the classically forbidden region is σ −1 |k| − 1 > c > 0. Equation (1.6) tells us that, in cases for which the Steklov problem on Ω is tunneling, Steklov eigenfunctions carry positive energy even in the classically forbidden region σ −1 |k| 1, with an energy value that is no smaller than exponentially decaying in σ. (Using the estimates of [GT19] one can also see that Steklov eigenfunctions carry at most exponentially small energy in the forbidden region.) Theorem 2. Assume that the Steklov problem on Ω is tunneling and let σ denote a Steklov eigenvalue for the set Ω. Let Then, there exist a constant c > 0 such that, for each integer N > 0, there are constants C N , σ 0 , δ 0 , and m 0 > 0 so that for all 0 < δ < δ 0 , m > m 0 , and σ j > σ 0 the inequality holds.
Letting {φ σ j } ∞ j=1 denote an orthonormal basis of Steklov eigenfunctions and calling u σ j = φ σ j •f , Theorem 2 shows in particular that In other words, for r small, u σ j is well approximated by a function with finitely many Fourier modes. If there is c > 0 such that then we obtain u σ j =û σ j (0) + O((r + e −cσ j )|û σ j (0)|) and u σ j is nearly constant on small balls centered around 0. In general, however, finitely many Fourier modes are necessary to capture the lowest-order asymptotics, as indicated in equation (1.9).
One of the main components of the proof of Theorem 1, in addition to Theorem 2, is the construction of a large class of domains Ω for which the Steklov problem is tunneling. To this end, we introduce some additional definitions. A function v ∈ C(D) will be said to be boundary-bandlimited providedv(k) = 0 except for a finite number of values of k ∈ Z. We say that a mapping function f is boundary band limited conformal (BBLC) if |∂ z f | is boundary band-limited. If in addition, |∂ z f || ∂D is non-constant, we will write that Ω is BBLCN. Finally, we say the domain Ω is BBLC (BBLCN) if and only if a BBLC (BBLCN) mapping function, f : D → Ω exists. We now present the main theorem of this paper.
Theorem 3. Assume Ω is BBLCN. Then the Steklov problem on Ω is tunneling.
Remark 1.2. It is not clear whether the elliptical and kite-shaped domains (equations (6.1) and (6.2)) considered in Figures 1, 4 and 5 satisfy the BBLCN condition or, more generally, whether they have tunneling Steklov problems (we have not as yet been able to establish that the tunneling condition holds for domains that are not BBLCN). However, domain-opening observations such as those displayed in Figure 1 and Section 6, suggest that these domains may nevertheless be tunneling. This and other domain-opening observations provide support for Conjecture 1.3 below.
(Steklov eigenfunctions on a domain which satisfies the BBLCN condition, and, therefore, in view of Theorem 3, is known to be tunneling, are displayed in Figure 2.) In view of Remark 1.2 we conjecture that every Steklov problem on an analytic domain is tunneling unless the Steklov domain Ω is a disc: Let Ω ⊂ R 2 be a bounded, simply-connected domain with real analytic boundary that is not equal to B(x, r) for any x ∈ R 2 , r > 0. Then the Steklov problem on Ω is tunneling.
Outline of the paper. This paper is organized as follows. Section 2 shows that arbitrary analytic, bounded, simply-connected domains can be approximated arbitrarily closely by BBLCN domains. Then, Sections 3 and 4 provide proofs for Theorems 3 and 2, respectively. The numerical methods used in this paper to produce accurate Steklov eigenvalues, eigenfunctions, and associated nodal sets are presented in Section 5. Section 6, finally, illustrates the methods with numerical results for elliptical and kite-shaped domains.
Remark 1.4. Throughout this article we abuse notation slightly by allowing C to denote a positive constant that may change from line to line but does not depend on any of the parameters in the problem. In addition C N is a positive constant that may change from line to line and depends only on the parameter N .

Approximation by tunneling domains
This section shows that any analytic domain can be approximated arbitrarily closely (in a sense made precise in Corollary 2.2) by a BBLCN domain. To do this, first let M ≥ 0, α i ∈ C \ D for i = 1, . . . , N , and let N i ≥ 1, i = 1, . . . M , and let us seek approximating BBLCN domains whose mappings f : D → C take the form In words: f is the integral of the square of a polynomial with roots outside D. It follows that In particular, We next show that an arbitrary non-vanishing analytic function on D can be approximated by the square of a polynomial.
Lemma 2.1. Let g : D → C smooth with g| D analytic and |g| > 0 on D. Then, for any ε 0 > 0 and Then, since U is simply-connected and |g| > 0 on D, h is analytic in D with smooth extension to D. In addition, is an analytic function on D such that w 2 (z) = g(z) and w extends smoothly to D. Then, for all ε > 0, there is a polynomial p ε such that In particular, since |g| > c > 0 on D, for 0 < ε small enough, p ε has no zeros in D. Hence, , 1) proves the result with α 0 = β 2 0 and α i = β i .
(This polynomial was obtained as the N -th order Taylor polynomial of √ ∂ z f .) In this case, according to Theorems 1 and 2, the Steklov eigenfunctions should be slowly oscillating in a σ independent neighborhood of z 0 . Figure 2 displays corresponding Steklov eignfunction or various orders as well as a typical eigenfunction for the exact disc. Note the dramatic change that arises in the Steklov eigenfunctions from a barely visible boundary perturbation of the disc.

BBLCN domains and tunneling Steklov problems
This section presents a proof of Theorem 3. In preparation for that proof, let Ω ⊂ C be a BBLCN domain, and denote by f the corresponding mapping function. Define  (bottom left). Note that, according to Corollary 2.2 the set Ω is a BBLCN approximation to the disk. As predicted by Theorem 2, oscillations avoid a region around z 0 for high σ. The bottom-right image displays a typical eigenfunction on the exact disc. Note the dramatic change that arises in the Steklov eigenfunctions from a barely visible boundary perturbation of the disc.
Since Ω is a BBLCN domain, the function ∂ z f | ∂D is band limited and ∂ z f | ∂D is not identically constant. It follows that m 0 := sup{|n| : |a n | = 0} satisfies 1 ≤ m 0 < ∞. Denoting byû(n) the boundary Fourier coefficients of an eigenfunction u, the corresponding boundary Fourier coefficients of ∂ r u are given by |n|û(n). Thus, a solution to (1.4) is uniquely determined as an 2 solution to the equation In what follows we may, and do, assume that solutionsû have 2 -norm equal to one.
Proof of Theorem 3. Since it follows that (3.1) can be re-expressed in the form and, then, for all |n| ≤ Kσ, The second inequality follows from the fact that a n ≡ 0 for |n| ≥ m 0 , while the third one results from the relation a 0 > 0 and the positivity, σ > 0, of all nontrivial eigenvalues σ, which imply that ||n| − σa 0 | ≤ max(|n|, σ|a 0 |)≤ σ(max(K, a m ∞ )).
Making an identical argument, but solving forû(n − m 0 ), and using that |a m 0 | = |a −m 0 | = 0, we have for all |n| ≤ Kσ, We now use equation (3.3) to prove the first half of our tunneling estimate. Then, there exists C 0 > 0 so that for all σ > 0 and for −Kσ ≤ n + m ≤ Kσ we have An almost identical argument gives the −Kσ − m ≤ n ≤ 0 case.

Analysis of Tunneling Steklov Problems
The proof of Theorem 2 now follows in two steps. First, we show that, for eigenfunctions of any tunneling Steklov problem, the boundary Fourier coefficients of low frequency contain a mass no smaller than exponential in σ. To finish the proof, we use the fact that the harmonic extension of e inθ decays exactly as r |n| . Examining the solution on the ball of radius δ > 0 for some δ small enough, it will be shown that the low frequencies dominate the behavior of u.
We can now present a proof of Theorem 1.
Since the derivative of f never vanishes, for δ < δ 1 and for a certain E > 0 there is a ball B of radius Er m,δ such that φ σ does not vanish on B. The proof is now complete. Unfortunately, however, the single layer operator S on the right side of this equation is not always invertible. In order to avoid singular right-hand sides and the associated potential sensitivity to round-off errors, in what follows we utilize the Kress potential (where φ denotes the average of φ over ∂Ω), which leads to the modified eigenvalue equation [Akh16] (5.4) ( The right-hand operator in this equation is invertible [Kre14,Thm. 7.41], as desired. For either formulation, the evaluation of a given eigenfunction u requires evaluation of the SLP, in accordance with either (5.1) or (5.3), for the solution φ of the corresponding generalized eigenvalue problem (5.2) or (5.4), respectively, at all required points x ∈ Ω.
Remark 5.1. Note that for a given harmonic function u in Ω, φ in (5.2) and that in (5.4) are not the same.

5.2.
Fourier expansion and exponential decay. In terms of a given 2π-periodic parametrization C(t) of ∂Ω, the Steklov eigenfunction u corresponding to a given solution (φ, σ) of the regularized eigenvalue problem (5.4), which is given by the single layer expression (5.3), can be expressed, for a given point x = (x 1 , x 2 ) ∈ Ω, where C(t) = (C 1 (t), C 2 (t)) and where φ denotes the average of φ over the curve ∂Ω. Unfortunately, a direct use of this expression does not capture important elements in the eigenfunction within Ω, such as the nodal sets, since, for analytic domains, the eigenfunctions decay exponentially fast within Ω as the frequency increases [PST,GT19]. In regions where the actual values of the eigenfunction may be significantly below machine precision the expression (5.5) must be inaccurate: this expression can only achieve the exponentially small values via the cancellations that occur as the the solution φ becomes more and more oscillatory. But such cancellations cannot take place numerically below the level of machine precision. In order to capture the decay explicitly within the numerical algorithm we proceed in a manner related to the construction used in [PST].
To accurately obtain the exponentially decaying values of the Steklov eigenfunction we proceed as follows. We first consider the Fourier expansion A n e int .
of the product φ(C(t)) − φ Ċ (t) ; note that, as is easily checked, the n = 0 term in the Fourier expansion (5.6) is indeed equal to zero. Inserting this expansion in (5.5) we obtain Then, assuming an analytic boundary, as is relevant in the context of this paper, and further assuming, for simplicity, that C(t) is in fact an entire function of t (as are, for example, all parametrizations C(t) given by vector Fourier series containing finitely many terms), we introduce, for x = (x 1 , x 2 ) ∈ Ω, the quantities log (x 1 − C 1 (t + is sgn(ns))) 2 + (x 2 − C 2 (t + is sgn(ns))) 2 e int dt.
A proof of Lemma 5.2 is given in Appendix B. It follows from Lemma 5.2 that equation (5.8) optimally captures the exponential decay of the B n terms as σ → ∞. Note that this setup does not capture the exponential decay of the coefficients A n below machine precision away from |n| ∼ σ, and, therefore, the accuracy of the resulting interior eigenfunction reconstructions does not exceed that accuracy level. But the function λ(x 1 , x 2 ) does capture the exponential decay and the geometrical character of the eigenfunction as long as the (spatially constant) coefficients A n for low n remain above machine precision.
For general curves C(t) no closed form expressions exist for the function λ(x), and a numerical algorithm must be used for the evaluation of this quantity, as part of a numerical implementation of the eigenfunction expression (5.9). In our implementation the function λ was evaluated via an application of Newton's method to the nonlinear equation Explicit expressions can be obtained for circles and ellipses, however: (1) For a circle of radius 1: (2) For an ellipse of semiaxes a > b: The derivation of the expression (5.11) is outlined in Appendix A.  5.3. Exponential decay and verification of Cauchy's theorem. Tables 1 and 2 demonstrate the validity of equation (5.8) (since in both cases the results in the second and third columns closely agree with each other for n ≤ 50), as well as the exponential decay of the exact coefficients B 0 n -as born by the results in the third column of these tables. The disagreement observed for n > 50 is caused by the lack of precision of the results in the second column beyond machine accuracy, a problem that is eliminated in the third column via an application of the relation (5.8).  Table 2. Same as Figure (1) but for the kite-shaped domain Ω bounded by the curve (6.2).
Appendix B. Proof of Lemma 5.2 Then, for |Im z| < λ(x 1 , x 2 ), the expression defines the principal branch of log h(z)-which is, then, an analytic function in the strip |Im z| < λ.
Lemma B.1. Let h(z) denote an analytic function defined on an open neighborhood of the set {z : |Im z| ≤ λ} which does not vanish for |Im z| < λ, but which vanishes to order k at z 0 = t 0 + iλ. Then, lim Similarly, if h vanishes to order k at z 0 = t 0 − iλ, Proof. Note that for ε > 0 small enough {h(z) = 0} ∩ {|z − z 0 | < ε} = z 0 . Therefore where Γ is any contour starting at z 0 − ε 2 , ending at z 0 + ε 1 , and lying in In particular, let Letting ε 1 and ε 2 tend to zero completes the proof for the case z 0 = t 0 +iλ. The proof for z 0 = t 0 −iλ follows by substituting z by −z.
Since |t − t 0 | −k |h(t + iλ)| is smooth and bounded away from zero on the support of χ, the second term in ( Then, using (B.3) together with the fact that log 1 = 0 we may approximate the first term on the right-hand side of (B.2) by χ(t) log |h(t + iλ)|e int dt = −πke int 0 1 |n| + O(n −2 ), Let us now estimate the second term on the right-hand side of (B.1). We have χ(t)iIm log[h(t + iλ))]e int dt = t 0 0 iχ(t)Im log[h(t + iλ))]e int dt + where in the last equality Lemma B.1 was used.
We may now complete the proof of Lemma 5.2. Let 0 ≤ t 1 < t 2 < · · · < t M < 2π denote the zeroes of h(t + iλ) as a function of t, and let k j (0 ≤ j ≤ M ) denote the vanishing order at t = t j .
Then, by Lemma B.2, for χ j supported close enough to t j with χ j ≡ 1 near t j , and n > 0, χ j (t) log h(t + iλ)e int dt = − 2πk j e int j |n| + O(n −2 ).