The Dirichlet-to-Neumann map, the boundary Laplacian, and Hörmander's rediscovered manuscript
Alexandre Girouard, Mikhail Karpukhin, Michael Levitin, Iosif Polterovich
TThe Dirichlet-to-Neumann map, the boundary Laplacian, andHörmander’s rediscovered manuscript
Alexandre Girouard Mikhail Karpukhin Michael Levitin Iosif Polterovich11 February 2021
Abstract
How close is the Dirichlet-to-Neumann (DtN) map to the square root of the correspondingboundary Laplacian? This question has been actively investigated in recent years. Somewhatsurprisingly, a lot of techniques involved can be traced back to a newly rediscovered manuscriptof Hörmander from the 1950s. We present Hörmander’s approach and its applications, with anemphasis on eigenvalue estimates and spectral asymptotics. In particular, we obtain results forthe DtN maps on non-smooth boundaries in the Riemannian setting, the DtN operators for theHelmholtz equation and the DtN operators on differential forms.
Contents
MSC(2020):
Primary 58J50. Secondary 35P20
Keywords:
Dirichlet-to-Neumann map, Laplace–Beltrami operator, Dirichlet eigenvalues, Robin eigenvalues, eigen-value asymptotics.
A. G.:
Département de mathématiques et de statistique, Pavillon Alexandre-Vachon, Université Laval, Québec, QC,G1V 0A6, Canada; [email protected] ; http://archimede.mat.ulaval.ca/agirouard/ M. K.:
Mathematics 253-37, California Institute of Technology, Pasadena, CA 91125, USA; [email protected] ; http://sites.google.com/view/mkarpukh/home M. L.:
Department of Mathematics and Statistics, University of Reading, Pepper Lane, Whiteknights, Reading RG66AX, UK;
[email protected] ; I. P.:
Département de mathématiques et de statistique, Université de Montréal CP 6128 succ Centre-Ville, MontréalQC H3C 3J7, Canada; [email protected] ; a r X i v : . [ m a t h . SP ] F e b . Girouard, M. Karpukhin, M. Levitin, and I. Polterovich µ ≤ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134.3 DtN–Robin duality and domains with corners . . . . . . . . . . . . . . . . . . . . . 154.4 The case µ > . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 References 21 §1. Introduction and main results §1.1. The Steklov spectrum and the Dirichlet-to-Neumann map.
Let Ω be a bounded domain ina complete smooth Riemannian manifold X of dimension d ≥ and let ∆ be the (positive) Laplacianon Ω . Assume that the boundary ∂ Ω = M is Lipschitz. Consider the Steklov eigenvalue problem on Ω : (cid:40) ∆ U = in Ω ; ∂ U ∂ n = σ U on M . (1)We refer to [KK+14] for a historical discussion and to [GiPo17] for a survey on this eigenvalue problemand related questions in spectral geometry.The Steklov spectrum is discrete, and the eigenvalues form a sequence = σ < σ ≤ σ ≤ · · · (cid:37)+∞ . Alternatively, the Steklov eigenvalues can be viewed as the eigenvalues of the Dirichlet-to-Neumannmap , D : H ( M ) → H − ( M ), u (cid:55)→ ∂ n U , where ∂ n U : = ∂ U ∂ n = 〈 ( ∇ U ) | M , n 〉 , n is the unit outward normal vector field along M , and the solution U : = U u of ∆ U = in Ω , U | M = u , is the unique harmonic extension of the function u from theboundary into Ω (we refer to [MiTa99, Section 5] for uniqueness and existence results for the solutionsof the Dirichlet problem on Lipschitz domains in Riemannian manifolds). The eigenfunctions of D are the restrictions of the Steklov eigenfunctions to the boundary, and form an orthogonal basis of L ( M ) . §1.2. The Dirichlet-to-Neumann map and the boundary Laplacian. The goal of this paper is tooverview the links between the Dirichlet-to-Neumann map D and the boundary Laplacian ∆ M .If Ω has a smooth boundary M , D is a self-adjoint elliptic pseudodifferential operator of order oneon M with the same principal symbol as (cid:112) ∆ M , i.e. the square root of the (positive) boundary Lapla-cian. In this case, the following sharp Weyl’s law holds for the Steklov eigenvalues (see, for instance,[GPPS14]): N ( σ ) = σ k < σ ) = vol( (cid:66) d − ) vol( M )(2 π ) d − σ d − + O (cid:179) σ d − (cid:180) , (2)or, equivalently, σ k = π (cid:181) k vol( (cid:66) d − ) vol( M ) (cid:182) d − + O (1). (3)Page 2he Dirichlet-to-Neumann map and the boundary LaplacianLet us denote by = λ ≤ λ ≤ λ ≤ · · · (cid:37) +∞ the eigenvalues of the boundary Laplacian ∆ M .Comparing (3) with the sharp Weyl’s law for the boundary Laplacian (see [Shu01, Chapter III]), weobtain (cid:175)(cid:175)(cid:175) σ k − (cid:112) λ k (cid:175)(cid:175)(cid:175) < C , k ≥ (4)for some constant C > depending on Ω . Remark . As we show later, see Remarks 1.8 and 1.10, the constant C in (4) depends only on thegeometry of Ω in an arbitrary small neighbourhood of M . Moreover, our approach yields (4) undersignificantly weaker regularity assumptions on M , see Theorem 1.9.If d = , the asymptotic results above can be made much more precise. Theorem 1.2 ([GPPS14]) . For any smooth surface Ω with m boundary components of lengths l , . . . l m , σ k − (cid:112) λ k = o ( k −∞ ), k → +∞ , (5) where λ k is the k -th eigenvalue of the Laplacian on the disjoint union of circles of lengths l , . . . l m . For simply connected planar domains, this result was proved by Rozenblyum [Roz86] and redis-covered by Guillemin–Melrose, see [Edw93].Recall that the Steklov eigenvalues of the unit disk are
0, 1, 1, 2, 2, . . . , k , k , . . . , and hence σ k = (cid:112) λ k for all k ≥ . Moreover, the boundary Laplacian ∆ (cid:83) coincides with the square of the Dirichlet-to-Neumann map on the disk. Our first result shows that such an equality of operators occurs if andonly if the Euclidean domain is a disk. Theorem 1.3.
Let Ω ⊂ (cid:82) d be a smooth bounded Euclidean domain, d ≥ . If D = (cid:112) ∆ M then d = and Ω is a disk. Moreover, if M is connected and d ≥ , D commutes with ∆ M if and only if Ω is a ball. Note that the “if” part of the second statement follows from the well-known fact that the eigen-functions of the Dirichlet-to-Neumann map on the sphere are precisely the spherical harmonics. The-orem 1.3 is proved in §2 using a combination of symbolic calculus and some simple arguments fromdifferential geometry. §1.3. Hörmander’s identity and its applications.
The inequalities between the Steklov and Laplaceeigenvalues discussed in the previous subsection were obtained using pseudodifferential techniquesfor domains with smooth boundaries. It is more efficient to use other techniques in order to extendthese results to non-smooth domains, as well as to characterise the difference between the Dirichlet-to-Neumann operator and the boundary Laplacian in geometric terms. These questions have beenaddressed in a series of recent papers starting with the work of Provenzano–Stubbe [PrSt19, HaSi20,Xio18, CGH18].
Remark . The approach used in these papers is based on the so-called Pohozhaev’s identity [Poh65],which in turn is an application of the method of multipliers going back to Rellich (see [C-WGLS12, p.205] for a discussion). One of the objectives of the present paper is to show that, surprisingly enough,these results go back to an old unpublished work L. Hörmander [Hör18] that was originally writtenin 1950s (see also [Hör54] where an identity similar to Pohozhaev’s has been obtained).In what follows, we assume that Ω has a C boundary, i.e. at each point of the boundary thereexists a smooth coordinate chart on the ambient manifold X in which the image of ∂ Ω coincides with agraph of a C function. Note that in this case the outward unit normal vector field on ∂ Ω is Lipschitzcontinuous and the induced Riemannian metric on the boundary M is Lipschitz. We let ( , ) Ω ( , ) M , (cid:107) (cid:107) Ω , (cid:107) (cid:107) M denote the inner products and norms in L ( Ω ) and L ( M ) , respectively.Page 3. Girouard, M. Karpukhin, M. Levitin, and I. Polterovich Theorem 1.5.
Let X be a complete smooth Riemannian manifold, and let Ω ⊂ X be a bounded domainwith a C boundary. Let u ∈ H ( M ) , let U be the harmonic extension of u , and let F be a Lipschitzvector field on Ω , such that a restriction of F to M coincides with the outward unit normal n . Then ( D u , D u ) M − ( ∇ M u , ∇ M u ) M = (cid:90) Ω (cid:161) 〈∇ ∇ U F , ∇ U 〉 − |∇ U | div F (cid:162) d v Ω , (6) where ∇ ∇ U F denotes the covariant derivative of F in the direction of ∇ U and d v Ω is the Riemannianmeasure on Ω .Remark . Here and further on, we understand the covariant derivatives of a Lipschitz vector fieldas elements of L ∞ ( Ω ) . Remark . The integrand in the right-hand side of (6) can be expressed in terms of
Hörmander’s en-ergy tensor defined in [Hör54, Hör18]. If Ω is a Euclidean domain then 〈∇ ∇ U F , ∇ U 〉 = ( DF )[ ∇ U , ∇ U ] ,where DF is the Jacobian of F .The integrand in the right-hand side of (6) is a quadratic form in ∇ U with bounded coefficients,since F is Lipschitz. Hence, for some C > depending only on F , we obtain (see Corollary 3.2) | ( D u , D u ) M − ( ∇ M u , ∇ M u ) M | ≤ C (cid:90) Ω |∇ U | d v Ω . (7) Remark . Let d M ( · ) be the signed distance function to the boundary M defined on X , positiveinside Ω . Consider the vector field F : = ∇ (cid:161) d M χ (cid:162) , where χ ( · ) is a smooth cut-off function equal to onenear M and zero outside a small neighbourhood of M . Then F satisfies the assumptions of Theorem1.5, see [DeZo98, Section 3] and [PrSt19, Subsection 5.3]. Note that while [DeZo98, Theorem 3.1]is presented in the Euclidean setting, the required statement can be adapted to the Riemannian casein a straightforward way, cf. [CGH18, Section 2]. In particular, this implies that the constant C in(7) depends only on the geometry of Ω in an arbitrary small neighbourhood of M , cf. Remark 1.1.Moreover, if Ω is a smooth Riemannian manifold with boundary, it was shown in [CGH18] that theconstant C can be estimated in terms of the rolling radius of Ω , bounds on the sectional curvatures in atubular neighbourhood of M , and the principal curvatures of M . In fact, obtaining an explicit controlon C in terms of these geometric quantities was one of the main results of [PrSt19, Xio18, CGH18].Using the variational principle, one can deduce from (7) the following statement. Under C reg-ularity assumptions it was first proved in the Euclidean setting in [PrSt19, Theorem 1.7], and then inthe smooth Riemannian setting in [CGH18, Theorem 3]. Theorem 1.9.
Let X be a complete smooth Riemannian manifold, and let Ω ⊂ X be a bounded domainwith a C boundary. Then (cid:175)(cid:175)(cid:175) σ k − (cid:112) λ k (cid:175)(cid:175)(cid:175) ≤ C (8) holds for all k ∈ (cid:78) with the same constant C as in (7) . The proofs of Theorems 1.5 and 1.9 are presented in §3.
Remark . As shown in Remark 1.8, we can choose the constant C in the right-hand side of (8) todepend only on an arbitrarily small neighbourhood of the boundary.Theorem 1.9 together with the results of [Zie99, Ivr00] can be applied to extend Weyl’s asymptotics(3) to domains with non-smooth boundaries. Theorem 1.11.
Let X be a complete smooth Riemannian manifold of dimension d , and let Ω ⊂ X bea bounded domain with C α boundary for some α > . Then the sharp Weyl asymptotic formula (2) holds for the Steklov eigenvalues on Ω . Page 4he Dirichlet-to-Neumann map and the boundary LaplacianMoreover, if d = , the regularity assumption in Theorem 1.11 can be improved even further. Theorem 1.12.
Let X be a complete smooth Riemannian surface and let Ω ⊂ X be a bounded domainwith a C boundary. Then the Steklov eigenvalues σ k of Ω satisfy Weyl’s asymptotics σ k = π k | ∂ Ω | + O (1) (9) as k → ∞ . Theorems 1.11 and 1.12 are proved in §3.4. §1.4. Plan of the paper.
The paper is organised as follows. In §2 we prove Theorem 1.3 and discusssome related open questions. In §3 we obtain Proposition 3.3 which is an abstract form of Theorem 1.5,and then prove the main results of the present paper stated in §1.3. The rest of the paper is concernedwith extensions and applications of Theorem 1.5 to other settings. In §4 we consider the Dirichlet-to-Neumann operators associated with the Helmholtz equation ∆ U = µ U in Ω . In particular, we get ageneralisation of the bound (8) which is uniform for all µ ≤ , see Theorem 4.2. Interestingly enough,such a uniform estimate does not hold for domains with corners, see Proposition 4.6. In a way, thisobservation shows that the C regularity assumption which is needed for the proof of Theorem 4.2can not be relaxed by too much. Some difficulties arising in the case µ > are also discussed. In §5we extend the results of §1.3 to the Dirichlet-to-Neumann operator on differential forms as defined in[Kar19]. In particular, we prove an analogue of the estimate (8) comparing the eigenvalues of the DtNoperator on co-closed forms with the corresponding eigenvalues of the Hodge Laplacian. As a conse-quence, we obtain Weyl’s law for the DtN operator on forms which has not been known previously. Acknowledgements.
The authors are grateful to Asma Hassannezhad, Konstantin Pankrashkin,David Sher and Alexander Strohmaier for helpful discussions. A.G. and I.P. would also like to thankYakar Kannai for providing them with a copy of the original Hörmander’s manuscript before it waspublished as [Hör18]. The research of A.G. and I.P. is partially supported by NSERC, as well as byFRQNT team grant §2. Commutators and rigidity §2.1. Proof of Theorem 1.3.
We start by proving the second part of the theorem. Since the boundaryof Ω is smooth, the Dirichlet-to-Neumann operator is an elliptic pseudodifferential operator of orderone, and it is related to the boundary Laplacian by D = (cid:112) ∆ M + B , where B is a -th order pseudodifferential operator on M with the principal symbol (i.e. the subprin-cipal symbol of D ) β ( x , ξ ) = (cid:181) Q ( ξ , ξ ) | ξ | − H (cid:182) . We refer to [Tay96, Chapter 12, Proposition C1] and [PoSh15, formula (4.1.2)] for the derivation ofthis formula. Here, Q ( ξ , ξ ) is the second fundamental form of M in Ω and H is the mean curvatureof M , that is the trace of the second fundamental form. Consider the commutator T = [ ∆ M , D ] .Since the symbol | ξ | of the Laplacian commutes with the principal symbol | ξ | of D , the order twopart of the symbol of the commutator vanishes, and the operator T is of order one. Up to a constantmultiple, its principal symbol is given by the Poisson bracket { | ξ | , β ( x , ξ )} (see [Gu78, Appendix]).Page 5. Girouard, M. Karpukhin, M. Levitin, and I. PolterovichTo compute this expression, we use the boundary normal coordinates at a given point p ∈ M , so thatthe Riemannian metric satisfies g i j ( p ) = δ i j , and the first order derivatives of the metric tensor vanishat p . It follows that the Poisson bracket evaluated at x = p is given by { | ξ | , β ( x , ξ )} = (cid:88) k ∂∂ξ k g i j ξ i ξ j ∂∂ x k β ( x , ξ ) − (cid:88) k ∂∂ξ k β ( x , ξ ) ∂∂ x k g i j ξ i ξ j (cid:124) (cid:123)(cid:122) (cid:125) . The hypothesis that ∆ M and D commute therefore imply the following identity at x = p ∈ M : = (cid:88) k ξ k ∂∂ x k β ( x , ξ ) (cid:175)(cid:175) x = p = (cid:88) k ξ k (cid:181) ∂ Q i j ∂ x k ξ i ξ j | ξ | − ∂ H ∂ x k (cid:182) . (10)Our goal is to show that the mean curvature H is constant on M . We do this using different trialcovectors ξ ∈ T ∗ p M and substituting them into (10). Let us start, for each fixed i , by using ξ = d x i .This leads directly to (cid:88) j (cid:54)= i ∂∂ x i Q j j = for each i =
1, 2, · · · , d − (11)This means, in particular, that ∂∂ x i H = ∂∂ x i Q ii (12)Recall that the Codazzi equation for a submanifold M ⊂ Ω states that ( R ( X , Y ) Z ) ⊥ = ∇ X Q ( Y , Z ) − ∇ Y Q ( X , Z ), where R is the Riemannian curvature tensor of the ambient space and ⊥ denotes the projection on thenormal direction (see [Tay96, formula (4.1.6)]). Now, because the ambient space is (cid:82) d , the curvaturevanishes and this simply means that ∇ X Q ( Y , Z ) = ∇ Y Q ( X , Z ) . This has the useful consequence that ∂∂ x i Q ( e j , e k ) is symmetric in i , j , k . (13)Multiplying (10) by and setting ξ = d x i + d x j , i (cid:54)= j (we use here that d ≥ and hence dim M ≥ ),leads to = ∂∂ x j (cid:161) Q ii + Q i j + Q j j (cid:162) − ∂∂ x j H + ∂∂ x i (cid:161) Q ii + Q i j + Q j j (cid:162) − ∂∂ x i H = ∂∂ x j (cid:161) Q ii + Q i j + Q j j (cid:162) − ∂∂ x j Q j j + ∂∂ x i (cid:161) Q ii + Q i j + Q j j (cid:162) − ∂∂ x i Q ii = ∂∂ x j (cid:161) Q ii + Q i j − Q j j (cid:162) + ∂∂ x i (cid:161) − Q ii + Q i j + Q j j (cid:162) = ∂∂ x j (cid:161) Q ii − Q j j (cid:162) + ∂∂ x i (cid:161) − Q ii + Q j j (cid:162) . Here the second equality follows from (12), and the last equality is obtained by applying (13) to theterms containing Q i j and rearranging them afterwards.Finally, set ξ = d x i + d x j . Multiplying (10) by and using the same argument as above leads to = ∂∂ x i (cid:161) Q ii + Q i j + Q j j (cid:162) − ∂∂ x i Q ii + ∂∂ x j (cid:161) Q ii + Q i j + Q j j (cid:162) − ∂∂ x j Q j j = ∂∂ x i (cid:161) − Q ii + Q i j + Q j j (cid:162) + ∂∂ x j (cid:161) Q ii + Q i j − Q j j (cid:162) = ∂∂ x i (cid:161) − Q ii + Q j j (cid:162) + ∂∂ x j (cid:161) Q ii − Q j j (cid:162) Page 6he Dirichlet-to-Neumann map and the boundary LaplacianThe two previous computations lead to a simple linear system which implies, for i (cid:54)= j , ∂∂ x i Q ii = ∂∂ x i Q j j . Substitution in (11) leads to = d − ∂∂ x i Q ii . In particular, for d ≥ , this implies ∂∂ x i Q ii = for each i =
1, . . . , d − . Using (11) again, it followsthat ∂∂ x i H = ∂∂ x i Q ii = That is, the mean curvature H is constant on the hypersurface M = ∂ Ω ⊂ (cid:82) d , and it then follows fromAlexandrov’s “Soap Bubble” theorem [Ale58] that M must be a sphere.To prove the result in the opposite direction, we note that the eigenfunctions of the Dirichlet-to-Neumann map on a sphere are precisely the spherical harmonics. Therefore, since D and ∆ M share anorthogonal basis in L ( M ) , the two operators commute.For the first part of the theorem, notice first that the multiplicity of λ = is equal to the numberof connected components of M , while the multiplicity of σ = is equal to one since Ω is connected.Hence, M is connected. If d ≥ , since D = (cid:112) ∆ M , it follows from the second part of the theorem that M must be a sphere, say of radius R > . However, it is known that in that case, λ j = σ j + d − R σ j . Hence, we must have d = . Thus, Ω ⊂ (cid:82) is a bounded simply-connected domain. The length L ofits boundary is determined by λ = π L , and it follows that σ L = (cid:112) λ L = (cid:115) π L L = π . This is the equality case in the Weinstock inequality: Ω must be a disk (see [Wei54, GiPo10]). Thiscompletes the proof of the theorem. §2.2. Discussion and open problems. The proof of Theorem 1.3 uses the calculus of pseudodiffer-ential operators. This is the reason we have assumed that M is a smooth surface. It is quite likely thatthe result holds for surfaces of lower regularity. One possible approach to this problem is to expressthe Dirichlet-to-Neumann map using layer potentials. We note that Alexandrov’s theorem holds for C compact embedded surfaces, and it would be interesting to check whether Theorem 1.3 is true inthis case as well.It would be also interesting to understand whether the property of a ball described in Theorem1.3 is “stable”, i.e. if D and ∆ M almost commute (in some sense) then M is close to a sphere. In viewof stability results for Alexandrov’s theorem [MaPo19] it would be sufficient to show that the meancurvature of M is close to a constant in an appropriate norm.Recall that the proof of the second part of Theorem 1.3 relies on the condition d ≥ , since for d = the subprincipal symbol β of the Dirichlet-to-Neumann map is identically zero. Open Problem.
For planar domains, is it true that the Dirichlet-to-Neumann map and the boundaryLaplacian commute if and only if the domain is a disk or a rotationally symmetric annulus?
Page 7. Girouard, M. Karpukhin, M. Levitin, and I. PolterovichFinally, let us note that it would be interesting to find a geometric characterisation of Rieman-nian manifolds with (possibly disconnected) boundary, where the Dirichlet-to-Neumann map andthe boundary Laplacian commute. The examples of such manifolds include balls in space forms (see[BiSa14]) and cylinders over closed manifolds (see [GiPo17, Example 1.3.3]). Note that in this setting thesymbol calculus can not possibly yield a complete solution. Indeed, any symbolic computation onlycaptures the information in an arbitrary small neighbourhood of the boundary, whereas the Dirichlet-to-Neumann map depends on the interior of the manifold as well. §3. The proofs of Hörmander’s identity and its corollaries §3.1. Pohozhaev’s and Hörmander’s identities.
Let us start with a useful Pohozhaev-type identity(as discussed in Remark 1.4) which has various applications, see [CGH18, Lemma 20].
Theorem 3.1 (Generalised Pohozhaev’s identity) . Let X be a complete smooth Riemannian manifold,and let Ω ⊂ X be a bounded domain with a C boundary. Let F be a Lipschitz vector field on Ω , let u ∈ H ( M ) , and let U be the unique harmonic extension of u into Ω . Then (cid:90) M 〈 F , ∇ U 〉 ∂ n U d v M − (cid:90) M |∇ U | 〈 F , n 〉 d v M + (cid:90) Ω |∇ U | div F d v Ω − (cid:90) Ω 〈∇ ∇ U F , ∇ U 〉 d v Ω = (14) Proof.
We follow the argument in [CGH18]. Since ∆ U = div ∇ U = in Ω , using the standard identi-ties for the divergence of a product and for the gradient of a scalar product, we obtain div ( 〈 F , ∇ U 〉 ∇ U ) = 〈∇ 〈 F , ∇ U 〉 , ∇ U 〉 = 〈∇ ∇ U F , ∇ U 〉 + ∇ U [ F , ∇ U ] , where the last term in the right-hand side is understood as the application of the bilinear form given bythe Hessian ∇ U to the vectors F and ∇ U (note that the Hessian is well-defined since U is harmonic).At the same time, div (cid:161) |∇ U | F (cid:162) = ∇ U [ F , ∇ U ] + |∇ U | div F . Subtracting the first equality from the second, we get div (cid:181) 〈 F , ∇ U 〉 ∇ U − |∇ U | F (cid:182) = 〈∇ ∇ U F , ∇ U 〉 − |∇ U | div F . Integrating this identity over Ω and using the divergence theorem completes the proof of Theorem3.1. The original Pohozhaev’s identity [Poh65, Lemma 2] was proved in a different setting. As wasmentioned in Remark 1.4, the results of this kind are also often referred to as Rellich’s identities, see[HaSi20, Theorem 3.1]. Proof of Theorem 1.5.
Setting F = n on M in (14), we obtain (cid:90) M ( ∂ n U ) d v M − (cid:90) M |∇ u | d v M = (cid:90) Ω 〈∇ ∇ U F , ∇ U 〉 d v Ω − (cid:90) Ω |∇ U | div F d v Ω . Note that on M we have |∇ U | = |∇ M u | + ( ∂ n U ) . Therefore, with account of D u = ∂ n U , ( D u , D u ) M − (cid:90) M (cid:161) |∇ M u | + ( ∂ n U ) (cid:162) d v M = (cid:90) Ω 〈∇ ∇ U F , ∇ U 〉 d v Ω − (cid:90) Ω |∇ U | div F d v Ω . Page 8he Dirichlet-to-Neumann map and the boundary LaplacianMultiplying by and re-arranging, we obtain ( D u , D u ) M − ( ∇ M u , ∇ M u ) M = (cid:90) Ω (cid:161) 〈∇ ∇ U F , ∇ U 〉 d v Ω − |∇ U | div F (cid:162) d v Ω , which completes the proof of the theorem. Corollary 3.2 ([Hör18]) . There exists a constant C > depending only on the geometry of Ω in anarbitrary small neighbourhood of M such that for any u ∈ H ( M ) inequality (7) holds, i.e. | ( D u , D u ) M − ( ∇ M u , ∇ M u ) M | ≤ C ( D u , u ) M . Proof.
Note that the integrand on the right-hand side of (6) is a quadratic form in ∇ U with boundedcoefficients, since the vector field F is Lipschitz continuous, see Remark 1.6. Therefore, there exists aconstant C > such that (cid:175)(cid:175)(cid:175)(cid:175)(cid:90) Ω (cid:161) 〈∇ ∇ U F , ∇ U 〉 d v Ω − |∇ U | div F (cid:162) d v Ω (cid:175)(cid:175)(cid:175)(cid:175) ≤ C ( ∇ U , ∇ U ) Ω = C ( D u , u ) M , where the last equality follows from the divergence theorem. Moreover, C can be chosen only depend-ing on the geometry of Ω in an arbitrary small neighbourhood of M , see Remark 1.8. This completesthe proof of the corollary. §3.2. An abstract eigenvalue estimate. Before proceeding to the proof of Theorem 1.9, we statethe following abstract result generalising the idea of Hörmander [Hör18].
Proposition 3.3.
Let H be a Hilbert space with an inner product ( · , · ) H . Let A , B be two non-negativeself-adjoint operators in H with discrete spectra Spec( A ) = { α ≤ α ≤ . . . } and Spec( B ) = { β ≤ β ≤ . . . } and the corresponding orthonormal bases of eigenfunctions { a k } , { b k } . Assume additionallythat a k ∈ Dom( B ) and b k ∈ Dom( A ) , k ∈ (cid:78) , where the domains are understood in the sense ofquadratic forms. Suppose that for some C > | ( A u , A u ) H − ( B u , u ) H | ≤ C ( A u , u ) H for all u ∈ D : = Dom( B ) ∩ Dom( A ). (15) Then (cid:175)(cid:175) α k − β k (cid:175)(cid:175) ≤ C α k (16) and consequently (cid:175)(cid:175)(cid:175) α k − (cid:113) β k (cid:175)(cid:175)(cid:175) ≤ C (17) for all k ∈ (cid:78) , with the same constant C as in (15) .Proof. We note that (15) is equivalent to (cid:40) ( B u , u ) H ≤ ( A u , A u ) H + C ( A u , u ) H ,( A u , A u ) H − C ( A u , u ) H ≤ ( B u , u ) H , (18)and (16) is equivalent to (cid:40) β k ≤ α k + C α k , β k ≥ α k − C α k . (19)From the variational principle for the eigenvalues of B and the first inequality in (18) we have β k ≤ sup (cid:54)= u ∈ V k ⊂ Dom( B ) ( B u , u ) H ( u , u ) H ≤ sup (cid:54)= u ∈ V k ⊂ Dom( B ) ( A u , A u ) H + C ( A u , u ) H ( u , u ) H (20)Page 9. Girouard, M. Karpukhin, M. Levitin, and I. Polterovichfor any subspace V k with dim V k = k . Take V k = Span{ a , . . . , a k } . As for any u = c a +· · ·+ c k a k ∈ V k with | c | + · · · + | c k | = we have due to orthogonality ( A u , A u ) H + C ( A u , u ) H ( u , u ) H = k (cid:88) j = | c j | ( α j + C α j ) ≤ α k + C α k , the first inequality (19) follows immediately from (20).We now prove the second inequality (19). Let K : = max{ k ∈ (cid:78) : α k ≤ C } . We note that for k ≤ K the second inequality (19) is automatically satisfied since in this case β k ≥ ≥ α k ( α k − C ) , so we needto consider only k > K . We re-write the second inequality (18) as (cid:161) ˜ A u , ˜ A u (cid:162) H ≤ ( B u , u ) H + C u , u ) H , where ˜ A : = A − C . Let ˜ α k denote the eigenvalues of ˜ A enumerated in non-decreasing order. Wenote that ˜ α k = (cid:161) α k − C (cid:162) for k > K (this may not be the case for k ≤ K but as mentioned abovewe can ignore these values of k ). Writing down the variational principle for ˜ α k similarly to (20) andchoosing a test subspace V k = Span{ b , . . . , b k } leads in a similar manner to ˜ α k = (cid:181) α k − C (cid:182) ≤ β k + C which gives the second inequality (19) after a simplification.Finally, we note that (16) implies, for α k β k (cid:54)= , (cid:175)(cid:175)(cid:175) α k − (cid:113) β k (cid:175)(cid:175)(cid:175) ≤ C α k α k + (cid:112) β k ≤ C , yielding (17). Note that α k = implies β k = by (19). §3.3. Proof of Theorem 1.9. We apply Lemma 3.3 with A = D , B = ∆ M , and therefore α k = σ k and β k = λ k , taking into account Corollary 3.2, the obvious ( ∆ M u , u ) M = ( ∇ M u , ∇ M u ) M , and thefact that, as follows from [MiTa99, Proposition 7.4], the eigenfunctions of D belong to H ( M ) ; notethat since M is not smooth, this is not a priori evident. Thus we can take D = H ( M ) , and (8) followsimmediately from (17). §3.4. Applications to spectral asymptotics. Theorem 1.9 allows us to prove results on the asymp-totic distribution of Steklov eigenvalues using similar results for the Laplacian.
Proof of Theorem 1.11.
Since the boundary of Ω has regularity C α for some α > , the normal vectorto the boundary has regularity C α . Indeed, the boundary is locally given by a graph of a C α func-tion, and the normal vector is calculated in terms of its first derivatives. Hence the induced Rieman-nian metric on M = ∂ Ω has C α coefficients. At the same time, it was shown in [Ivr00, Theorem 3.1](see also [Zie99] for a similar result under slightly stronger regularity assumptions) that sharp Weyl’slaw holds for the Laplace eigenvalues on manifolds with a Riemannian metric having coefficients ofregularity C α for some α > . In other words, the asymptotic formulas (2) and (3) hold on M with σ k replaced by (cid:112) λ k . Therefore, in view of (8) they holds for σ k as well, and this completes the proofof the theorem. Remark . To our knowledge, in dimension d > , the sharp asymptotic formula (3) was previouslyavailable in the literature only for domains with smooth boundaries.Page 10he Dirichlet-to-Neumann map and the boundary Laplacian Proof of Theorem 1.12.
Since Ω is two-dimensional, its boundary M has dimension one and is thereforeisometric to a union of circles. Hence, the Laplace eigenvalues of M are explicitly known (recall thatthe unit circle has the Laplace spectrum given by
0, 1, 1, 4, 4, . . . , k , k , . . . ) and satisfy the sharp Weyl’slaw. Therefore, by (8) the sharp Weyl’s law (3) holds for the Steklov eigenvalues σ k which yields (9)since d = . This completes the proof of the Theorem. Remark . One expects sharper results to hold for domains in two dimensions. In particular, fordomains with C r boundaries, r ≥ , it is likely that analogue of (5) holds, with the right-hand sidedecaying polynomially in k , with the order of decay depending on r . Some results in this directionhave been obtained in [CaLa21]. Remark . It would be interesting to understand how much one can relax the C regularity as-sumption so that the asymptotic formula (9) remains true. For instance, it is known to hold for planarcurvilinear polygons with sides that are C regular, see [LPPS19]. Moreover, for a large class of curvi-linear polygons, (cid:175)(cid:175)(cid:175) σ k − (cid:112) λ k (cid:175)(cid:175)(cid:175) = o (1) , provided the boundary Laplacian is defined as a certain quantumgraph Laplacian on the circular graph modelled by the boundary, with the matching conditions at thevertices determined by the corresponding angles.We say that the eigenvalue asymptotics satisfies a rough Weyl’s law if formula (3) holds with theerror term o (cid:161) σ d − (cid:162) instead of O (cid:161) σ d − (cid:162) . For Euclidean domains with C boundary, a rough Weyl’slaw for Steklov eigenvalues was first obtained by L. Sandgren in in [San55]. Using heavier machinery,a similar result can be also proved for Euclidean domains with piecewise C boundaries [Agr06]. Letus conclude this section by a challenging open problem going back to M. Agranovich in 2000s. Open Problem.
Show that a rough Weyl’s law holds for the Steklov problem on any bounded Lipschitzdomain in a smooth Riemannian manifold. §4. Dirichlet-to-Neumann map for the Helmholtz equation §4.1. Parameter-dependent Dirichlet-to-Neumann map.
Let, as before, Ω be a bounded domainin a complete smooth Riemannian manifold X of dimension d ≥ ; we assume that the boundary M = ∂ Ω is Lipschitz. Consider the standard Dirichlet and Neumann Laplacians ∆ D = ∆ D Ω and ∆ N = ∆ N Ω acting on Ω . Their spectra are discrete, and we will denote their eigenvalues by < µ D1 < µ D2 ≤ · · · (cid:37) +∞ , and = µ N1 < µ N2 ≤ · · · (cid:37) +∞ , respectively.Let µ ∈ (cid:82) \ Spec( ∆ D Ω ) . Then the boundary value problem (cid:40) ∆ U = µ U in Ω , U = u on M (21)has a unique solution U : = U µ : = U µ , u ∈ H ( Ω ) for every u ∈ H ( M ) . We will call U the µ - Helmholtz extension of u ; for µ = it is just a harmonic extension. The parameter-dependent operator D µ : H ( M ) → H − ( M ), u (cid:55)→ ∂ n U µ , is called the Dirichlet-to-Neumann map for the Helmholtz equation ∆ U = µ U . The Dirichlet-to-Neumann map for the Laplacian, D , considered in §§1–3, is just the special case D = D . The DtNPage 11. Girouard, M. Karpukhin, M. Levitin, and I. Polterovichmap D µ can be also defined for µ coinciding with an eigenvalue µ D k of the Dirichlet Laplacian if itsdomain is restricted to the orthogonal complement, in L ( M ) , to the span of the normal derivativesof the corresponding Dirichlet eigenfunctions.For every µ ∈ (cid:82) \ Spec (cid:161) ∆ D (cid:162) , the DtN map D µ is a self-adjoint operator in L ( M ) with a discretespectrum; we enumerate its eigenvalues with account of multiplicities as σ µ ,1 < σ µ ,2 ≤ · · · (cid:37) +∞ . Bythe variational principle and integration by parts, σ µ , k = inf V k ⊂ H ( M )dim V k = k sup (cid:54)= u ∈ V k (cid:161) D µ u , u (cid:162) M ( u , u ) M = inf W k ⊂ H ( Ω )dim W k = k sup (cid:54)= U ∈ W k ( ∇ U , ∇ U ) Ω − µ ( U , U ) Ω ( U , U ) M . (22)Moreover, if M is smooth, then D µ is an elliptic pseudodifferential operator of order one with thesame principal symbol as D = D and therefore as (cid:112) ∆ M , with the same eigenvalue asymptotics (2),(3). In the remarkable paper [Fri91], L. Friedlander investigated the dependence of the eigenvalues ofoperator D µ upon the parameter µ in the Euclidean setting, and used them to prove the inequalities µ N k + ≤ µ D k , k ∈ (cid:78) , (23)between the Neumann and Dirichlet eigenvalues for any bounded domain Ω ⊂ (cid:82) d with smoothboundary M (this was later extended to non-smooth boundaries by N. Filonov [Fil05] using a dif-ferent approach). Friedlander’s results were based on the following main observations:– The eigenvalues σ µ , k are monotone decreasing continuous functions of µ on each interval ofthe real line not containing points of Spec( ∆ D ) (this can be easily seen from the variationalprinciple (22) since the corresponding quadratic form ( ∇ U , ∇ U ) Ω − µ ( U , U ) Ω is monotonedecreasing in µ for every non-trivial U ∈ H ( Ω ) ).– At each Neumann eigenvalue µ N ∈ Spec( ∆ N ) of multiplicity m µ N , exactly m µ N eigenvaluecurves σ µ , k (as functions of µ ) cross the axis σ = from the upper half-plane into the lowerone.– At each Dirichlet eigenvalue µ D ∈ Spec( ∆ D ) of multiplicity m µ D , exactly m µ D eigenvalue curves σ µ , k (as functions of µ ) blow down to −∞ as µ → µ D − and blow up to +∞ as µ → µ D + .– Therefore the eigenvalue counting functions of the Dirichlet problem N D ( µ ) : = µ D k < µ } , ofthe Neumann problem N N ( µ ) : = µ N k < µ } , and of the DtN operator N D µ ( σ ) : = σ µ , k < σ } ,are related, for any µ ∈ (cid:82) , by the relation N N ( µ ) − N D ( µ ) = N D µ (0). Friedlander then demonstrated that N D µ (0) , that is, the number of negative eigenvalues of D µ , is atleast one for any µ > for a domain Ω in a Euclidean space, thus implying (23) (this need not be true fordomains on a Riemannian manifold, see [Maz91]). We also refer to [ArMa12] for extensions of Fried-lander’s approach to a Lipschitz case and a comprehensive discussion of various other generalisationsand alternative approaches, and to [Saf08] for an abstract scheme encompassing the above.A typical behaviour of eigenvalues σ µ , k as functions of µ is illustrated by Figure 1, which showssome of the eigenvalues for a unit disk for which the spectrum of the Dirichlet-to-Neumann map D µ is given by the multisets Spec( D µ ) = (cid:110) (cid:112) µ J (cid:48) n ( (cid:112) µ ) J n ( (cid:112) µ ) , n ∈ (cid:78) ∪ {0} (cid:111) if µ ≥ (cid:110) (cid:112)− µ I (cid:48) n ( (cid:112)− µ ) I n ( (cid:112)− µ ) , n ∈ (cid:78) ∪ {0} (cid:111) if µ < (24)Page 12he Dirichlet-to-Neumann map and the boundary Laplacianwith J n and I n being the Bessel functions and the modified Bessel functions, respectively, and eigen-values with n > should be taken with multiplicity two.Figure 1: Some eigenvalues σ µ , k of D µ for a unit disk plotted as functions of µ . The solid curvescorrespond to n > in (24) and are double; the dashed curve corresponds to n = and is single. Thedotted vertical lines indicate the positions of the Dirichlet eigenvalues (points from Spec( ∆ D ) ), andthe intersections of the curves with the axis σ = are at the Neumann eigenvalues from Spec( ∆ N ) .We will also make use of the following generalised Pohozhaev’s identity which extends Theorem3.1 to solutions of the Helmholtz equation. Theorem 4.1 (Generalised Pohozhaev’s identity [HaSi20, Theorem 3.1]) . Let X be a complete smoothRiemannian manifold, and let Ω ⊂ X be a bounded domain with a C boundary. Let F be a Lipschitzvector field on Ω , let u ∈ H ( M ) , and let U = U µ , u be the µ -Helmholtz extension of u into Ω . Then (cid:90) M ∂ u ∂ n ( F , ∇ u ) d v M − (cid:90) M |∇ u | ( F , n ) d v M + µ (cid:90) M u ( F , n ) d v M + (cid:90) Ω |∇ U | div F d v Ω − (cid:90) Ω ( ∇ ∇ U F , ∇ u ) d v Ω − µ (cid:90) Ω U div F d v Ω = (25) §4.2. The case µ ≤ . We aim to prove the following
Theorem 4.2.
Let X be a complete smooth Riemannian manifold, and let Ω ⊂ X be a bounded domainwith a C boundary. Let µ ≤ . Then, with some constant C > , the bounds (cid:175)(cid:175)(cid:175) σ µ , k − (cid:113) λ k − µ (cid:175)(cid:175)(cid:175) < C . (26) hold uniformly over µ ∈ ( −∞ , 0] and k ∈ (cid:78) . Before proceeding to the actual proof of Theorem 4.2 we require a Helmholtz analogue of Theo-rem 1.5. Repeating literally the proof of Theorem 1.5, with account of extra µ -dependent terms in (25)compared to (14), we arrive at Page 13. Girouard, M. Karpukhin, M. Levitin, and I. Polterovich Theorem 4.3.
Let X be a complete smooth Riemannian manifold, and let Ω ⊂ X be a bounded domainwith a C boundary. Let F be a Lipschitz vector field on Ω such that F | M = n , let u ∈ H ( M ) , and let U = U µ , u be the µ -Helmholtz extension of u into Ω .. Then (cid:161) D µ u , D µ u (cid:162) M − ( ∇ M u , ∇ M u ) M + µ ( u , u ) M = (cid:90) Ω (cid:161) 〈∇ ∇ U F , ∇ U 〉 − |∇ U | div F + µ U div F (cid:162) d v Ω . (27)Theorem 4.3 leads to the crucial Helmholtz analogue of Corollary 3.2: Corollary 4.4.
Under conditions of Theorem 4.3, there exists a constant C > depending only on thegeometry of Ω in an arbitrary small neighbourhood of M such that for any u ∈ H ( M ) and any µ ≤ (cid:175)(cid:175)(cid:161) D µ u , D µ u (cid:162) M − (cid:161) ( ∆ M − µ ) u , u (cid:162) M (cid:175)(cid:175) ≤ C (cid:161) D µ u , u (cid:162) M . (28) Proof of Corollary 4.4.
Take the absolute values in both sides of equality (27). Then the left-hand sidebecomes the left-hand side of (28) after an integration by parts on M . The first two terms in the right-hand side can be estimated above by C ( ∇ U , ∇ U ) Ω by the same argument as in Corollary 3.2, and thelast term by C | µ | ( U , U ) Ω (possibly with a different constant but also depending on F only). Since fornon-positive µ we have | µ | = − µ , the bound in the right-hand side becomes C (cid:161) ( ∇ U , ∇ U ) Ω − µ ( U , U ) Ω (cid:162) = C (cid:161) D µ u , u (cid:162) M , and the result follows.Theorem 4.2 now follows immediately from Corollary 4.4 and Lemma 3.3 by taking in the latter A = D µ and B = ∆ M − µ (which are both non-negative for µ ≤ ). Remark . The remarkable feature of Theorem 4.2 is that the constant appearing in the right-handside of the bound is in fact independent of both the eigenvalue’s number k and the parameter µ (aslong as µ is non-positive). As we will see shortly, such uniform bounds are impossible if the boundary M has corners.We illustrate Theorem 4.2 by plotting, in Figure 2, some eigenvalues of D µ for a unit disk and,for comparison, the values of (cid:112) λ k − µ , in two regimes: firstly, for negative µ close to zero, and loweigenvalues of D µ , and secondly, for very large negative µ , and relatively high eigenvalues of D µ .Figure 2: Some eigenvalues σ µ , k of D µ for a unit disk plotted as functions of µ (solid curves), and,for comparison, the plots of (cid:112) λ k − µ (dashed curves). In the left figure, µ ∈ [ −
20, 0] , and k is chosenin the set {1, 3, 5, 7, 9} . In the right figure, µ ∈ [ − × , − × + ] , and k is chosen in the set {100, 102, 104, 106, 108} . Page 14he Dirichlet-to-Neumann map and the boundary Laplacian §4.3. DtN–Robin duality and domains with corners. Consider the two-parametric problem (cid:40) ∆ v = µ v in Ω , ∂ n v = σ v on M . (29)There are two ways in which we can treat (29) as a spectral problem. Firstly, as we have already donebefore, we can treat σ as a spectral parameter, and µ ∈ (cid:82) as a given parameter; then for every µ ∈ (cid:78) the eigenvalues σ µ , k (that is, the values of σ for which there exists a non-trivial solution v ∈ H ( Ω ) of (29)) are exactly the eigenvalues of the Dirichlet-to-Neumann map D µ . Reversely, we can treat µ as a spectral parameter and σ ∈ (cid:82) as a given parameter. The corresponding eigenvalues µ R − σ , k are thenexactly the eigenvalues of the Robin Laplacian ∆ R, γ with γ = − σ , that is of the Laplacian in Ω subjectto the boundary condition (cid:181) ∂ v ∂ n + γ (cid:182) v = on M , with the quadratic form (cid:161) ∆ R, γ v , v (cid:162) Ω = ( ∇ v , ∇ v ) Ω + γ ( v , v ) M , v ∈ H ( Ω ). (Note that there is no uniform convention on the choice of sign in the Robin condition, thereforesome care should be exercised when comparing results in the literature.) It is immediately clear that µ ∈ Spec (cid:161) ∆ R, − σ (cid:162) if and only if σ ∈ Spec (cid:161) D µ (cid:162) , and it is easy to check that in this case the dimensions of thecorresponding eigenspaces of ∆ R, − σ and D µ coincide, see [ArMa12]. Moreover, due to monotonicityof eigenvalues σ µ , k of D µ in the parameter µ ∈ ( −∞ , µ D1 ) , the functions µ (cid:55)→ σ µ , k on this interval arejust the inverse functions of σ (cid:55)→ µ R − σ , k .The study of the Robin eigenvalues µ R k , γ , in particular in the physically important regime γ →−∞ , has grown significantly in the last two decades, starting with some acute observations in [LOS98],their rigorous justification in [LePa08], and most recently mostly due to K. Pankrashkin, M. Khalile,N. Popoff and collaborators, see in particular [Kha18, KhPa18, KO-BP18, Pan20, Pop20] and referencestherein. Without going into the full details of these works, we mention only that, as it turns out, theasymptotics of the Robin eigenvalues in Ω as γ → −∞ depends dramatically on the smoothness of M = ∂ Ω . Specifically, as shown in the above-cited references, if Ω is a curvilinear polygon in (cid:82) withat least one angle less than π , then the following dichotomy is observed: there exists a number K ∈ (cid:78) such that µ R k , γ = (cid:40) − C k γ + o (cid:161) γ (cid:162) for k =
1, . . . K , with C ≥ · · · ≥ C K > − γ + o (cid:161) γ (cid:162) for k > K , as γ → −∞ (30)(for a smooth boundary one should take K = ; in many cases the remainder estimates can be vastlyimproved). Based on this, we make the following observation showing that our uniform bounds ofTheorem 4.2 cannot be extended in the same uniform manner to domains with corners, whateverboundary Laplacian we choose (see Remark 3.6). Proposition 4.6.
Let Ω be a curvilinear polygon in (cid:82) with at least one angle less than π . Then thebounds (26) cannot hold uniformly over µ ∈ ( −∞ , 0] and k ∈ (cid:78) for any choice of a boundary Laplacian ∆ M with eigenvalues λ k .Proof. Suppose, for contradiction, that the bounds (26) hold uniformly. Passing, for a fixed k , in (26)to the asymptotics as µ → −∞ with account of (cid:113) λ k − µ = (cid:113) − µ + O (cid:179) ( − µ ) − (cid:180) , Page 15. Girouard, M. Karpukhin, M. Levitin, and I. Polterovichand using the DtN–Robin duality, we deduce that all the Robin eigenvalues should then satisfy µ R γ , k = − γ + o (cid:161) γ (cid:162) as γ → −∞ , thus contradicting the condition C , . . . , C K > in (30). §4.4. The case µ > . For simplicity, in this subsection we assume that Ω ⊂ (cid:82) d with a smoothboundary M . It is immediately clear, for example from Figure 1, that there is little hope of extend-ing the simple bound (26) of Theorem 4.2 to the case µ > as the first m eigenvalues of the DtN mapblow up to −∞ as µ approaches the Dirichlet eigenvalues µ D of multiplicity m from below. Indeed,using the results of [Fil15] or [BBBT18] for the asymptotics of low eigenvalues of the Robin problemwith parameter γ → +∞ , and the Robin–DtN duality, one can easily see that as µ → µ D − , the first m eigenvalues of D µ behave asymptotically as σ µ , k = O (cid:181) µ D − µ (cid:182) , k =
1, . . . , m . Therefore, any conceivable generalisation of Theorem 4.2 to the case µ > should take into accountthe distance d D ( µ ) : = dist (cid:161) µ , Spec (cid:161) ∆ D (cid:162)(cid:162) between µ and the Dirichlet spectrum.Taking Theorem 4.3 as a starting point, we in fact obtain Theorem 4.7.
Let Ω ⊂ (cid:82) d with a smooth boundary M . Let µ > . Then there exist positive constants C and C depending only on the geometry of Ω and a positive constant C = C ( µ ) which additionallydepends on µ such that − C σ µ , k − C (cid:161) d D ( µ ) (cid:162) ≤ λ k − σ µ , k ≤ C σ µ , k + C µ (cid:195) + µ (cid:161) d D ( µ ) (cid:162) (cid:33) for all k ∈ (cid:78) . (31) The first inequality in (31) holds uniformly over all µ ∈ [0, µ ] , and the second one uniformly over all µ ≥ . We outline a sketch of the proof of Theorem 4.7. The main problem is that for µ ≥ we can nolonger deduce an analogue of Corollary 4.4 from (27) and then apply Proposition 3.3, for a coupleof reasons: firstly, because the operators D µ and ∆ M − µ are no longer non-negative, and secondly,because for positive µ the bound on the right-hand side of (27) becomes C (cid:161) ( ∇ U , ∇ U ) Ω + µ ( U , U ) Ω (cid:162) = C (cid:161)(cid:161) D µ u , u (cid:162) M + µ ( U , U ) Ω (cid:162) , introducing an extra µ -dependent term.To bypass these difficulties, we first write the µ -Helmholtz extension U µ , u of u ∈ H ( M ) (that is,the solution of (21)) in terms of the harmonic extension U u of u and the resolvent of the DirichletLaplacian as U µ , u = (cid:179) + µ (cid:161) ∆ D − µ (cid:162) − (cid:180) U u . We further use the standard resolvent norm bound in terms of the distance to the spectrum, (cid:176)(cid:176)(cid:176)(cid:161) ∆ D − µ (cid:162) − U (cid:176)(cid:176)(cid:176) Ω ≤ d D ( µ ) (cid:107) U (cid:107) Ω , Page 16he Dirichlet-to-Neumann map and the boundary Laplaciantogether with the trace bound (cid:107) U u (cid:107) Ω ≤ const ·(cid:107) u (cid:107) M (see e.g. [JeKe95]) and a bound on the first Robin eigenvalue [Fil05, formula (1.7)] which after usingthe DtN–Robin duality becomes σ µ ,1 ≥ − const · max (cid:189)
1, 1 d D + ( µ ) (cid:190) with some constant independent of µ and with d D + ( µ ) : = dist (cid:161) µ , Spec( ∆ D ) ∩ [ µ , +∞ ) (cid:162) ≥ d D ( µ ). Theorem 4.7 then follows by using an extended version of Proposition 3.3 which takes care of the extraterms. We leave out the details. §5. Dirichlet-to-Neumann operator on forms and the boundary Hodge Laplacian §5.1. Notation.
Given a m -dimensional Riemannian manifold Y with or without boundary (in ourcase either Y = Ω or Y = M ), we denote by Λ p ( Y ) the space of smooth differential p -forms on Y , ≤ p ≤ m . Throughout this section Ω is a compact smooth Riemannian manifold with boundary, ∂ Ω = M . We assume in addition that Ω is orientable, so that we can use the standard Hodge theory.Denote by i : Ω → M the embedding of the boundary. Given a p -form ω ∈ Λ p ( Ω ) , the p -form i ∗ ω ∈ Λ p ( M ) is a part of the Dirichlet data for ω . Let d : Λ p ( Ω ) → Λ p + ( Ω ) be the differential , i be the interior product and n , as before, be the outward normal vector to the boundary. Then i n d ω ∈ Λ p ( M ) is a part of the Neumann data of ω (in a slight abuse of notation, d ω is understoodhere as the restriction of this form to the boundary).Let δ : Λ p ( Y ) → Λ p − ( Y ) be the codifferential on the space of p -forms and (cid:63) : Λ p ( Y ) → Λ m − p ( Y ) be the Hodge star operator, We recall the standard relations (cid:63)(cid:63) = ( − p ( m − p ) and (cid:63) δ = ( − p d (cid:63) ,The operator ∆ = d δ + δ d : Λ p ( Y ) → Λ p ( Y ) is the Hodge Laplacian .Finally, recall that a form ω ∈ Λ p ( Y ) is called closed if d ω = , co-closed if δω = , exact if ω = d α , α ∈ Λ p − ( Y ) and harmonic if ∆ ω = . If Y has no boundary then a form is harmonic if and only if itis both closed and co-closed.Let c C p ( M ) and E p ( M ) denote the spaces of co-closed and exact forms on M , respectively. Thenit follows from the Hodge decomposition that Λ p ( M ) = c C p ( M ) ⊕ E p ( M ) . §5.2. The Dirichlet-to-Neumann map on differential forms. We first recall the definition ofthe Dirichlet-to-Neumann operator D ( p ) on p -forms defined in [Kar19], see also Remark 5.7. Given φ ∈ Λ p ( M ) , one can show that there exists ω ∈ Λ p ( Ω ) such that ∆ ω = in Ω ; δω = in Ω ; i ∗ ω = φ on M . (32)One then sets D ( p ) ( φ ) = i n d ω ∈ Λ p ( M ) . In [Kar19] the following properties of D ( p ) are proved. Theorem 5.1.
The operator D ( p ) : Λ p ( M ) → Λ p ( M ) is well-defined and self-adjoint. Furthermore, onehas (a) E p ( M ) ⊂ ker D ( p ) ; Page 17. Girouard, M. Karpukhin, M. Levitin, and I. Polterovich(b)
The restriction on the space of co-closed forms D ( p ) : c C p ( M ) → c C p ( M ) is an operator withcompact resolvent. The eigenvalues of the restriction form a sequence ≤ σ ( p )1 ( Ω ) ≤ σ ( p )2 ( Ω ) ≤ . . . (cid:37) ∞ , with the account of multiplicities. (c) The eigenvalues satisfy the following variational principle, σ ( p ) k = inf E k sup ω ∈ E k \{0} (cid:107) d ω (cid:107) Ω (cid:107) i ∗ ω (cid:107) M , where E k ranges over k -dimensional subspaces in Λ p ( Ω ) satisfying i ∗ E k ⊂ c C p ( M ) . From now on, we will consider the Dirichlet-to-Neumann map D ( p ) as an operator on c C p ( M ) . §5.3. Pohozhaev and Hörmander type identities for differential forms. Let us first prove aPohozhaev-type identity for differential forms (cf. Theorem 3.1).
Theorem 5.2.
Let Ω be a compact smooth orientable manifold with boundary ∂ Ω = M . Let F be aLipschitz vector field on Ω , and let ω ∈ Λ p ( Ω ) be a differential form satisfying δ d ω = in Ω . Then (cid:90) M (cid:173) i ∗ ( i F d ω ), i n d ω (cid:174) d v M − (cid:90) M 〈 F , n 〉 | d ω | d v M + (cid:90) Ω | d ω | div F d v Ω + (cid:90) Ω ( L F g )[ d ω , d ω ] d v Ω = where L F is a Lie derivative.Remark . To simplify notation, we denote the bilinear form induced on the space of differentialforms by the Riemannian metric by the same letter g as the original metric on the manifold. Theintegrand ( L F g )[ d ω , d ω ] in the last term should be understood as follows: we take the Lie derivativeof the Riemannian metric g on Λ p + ( Ω ) in the direction of F and evaluate the resulting bilinear format the pair [ d ω , d ω ] . Proof.
Consider the ( d − -form α : = i F d ω ∧ (cid:63) d ω . Then by Cartan’s identity L F = d i F + i F d andsince δ d ω = implies d (cid:63) d ω = , one has d α = ( d i F d ω ) ∧ (cid:63) d ω + ( − p i F d ω ∧ ( d (cid:63) d ω ) = ( L F d ω ) ∧ (cid:63) d ω = 〈 L F d ω , d ω 〉 d v Ω = (cid:161) ∇ F | d ω | − ( L F g )[ d ω , d ω ] (cid:162) d v Ω , where the last equality follows the well-known formula for the Lie derivative of a (0, 2) -tensor (see[Pet06, Appendix, Theorem 50]). Since div (cid:161) | d ω | F (cid:162) = ∇ F | d ω | + | d ω | div F , one has div (cid:161) | d ω | F (cid:162) d v Ω − d α = | d ω | div F + ( L F g )[ d ω , d ω ] d v Ω . Using the Stokes and divergence theorems we obtain (cid:90) Ω div (cid:161) | d ω | F (cid:162) d v Ω − d α = (cid:90) M | d ω | 〈 F , n 〉 d v M − (cid:90) M i ∗ ( i F d u ) ∧ i ∗ ( ∗ d u ) = (cid:90) M | d ω | 〈 F , n 〉 − (cid:173) i ∗ ( i F d ω ), i n d ω (cid:174) d v M . Rearranging the terms completes the proof the theorem.Page 18he Dirichlet-to-Neumann map and the boundary LaplacianNow we can prove a Hörmander-type identity for differential forms.
Theorem 5.4.
Let Ω be a compact smooth orientable manifold with boundary ∂ Ω = M . Let φ ∈ c C p ( M ) and let F be a Lipschitz vector field on Ω , such that F | M = n . If ω ∈ Λ p ( Ω ) is such that i ∗ ω = φ and δ d ω = , then (cid:161) D ( p ) φ , D ( p ) φ (cid:162) M − (cid:161) d M φ , d M φ (cid:162) M = (cid:90) Ω (cid:161) | d ω | div F − ( L F g )[ d ω , d ω ] (cid:162) d v Ω , where d M is the differential acting on Λ p ( M ) .Proof. The result follows from Theorem 5.2 by noting that | d ω | = | D ( p ) ( ω ) | + | d M φ | on M .The following analogue of Corollary 3.2 holds. Corollary 5.5.
There exists a constant C > , depending only on the geometry of Ω in an arbitrarilysmall neighbourhood of M , such that for any φ ∈ Λ p ( M ) one has (cid:175)(cid:175)(cid:161) D ( p ) φ , D ( p ) φ (cid:162) M − (cid:161) d M φ , d M φ (cid:162) M (cid:175)(cid:175) ≤ C (cid:161) D ( p ) φ , φ (cid:162) M . (33) Proof.
Let ω be a solution of (32). Then = ∆ ω = ( d δ + δ d ) ω = δ d ω . Thus, one can apply Theo-rem 5.4. Since F is Lipschitz one has (cid:175)(cid:175)(cid:161) D ( p ) φ , D ( p ) φ (cid:162) M − (cid:161) d M φ , d M φ (cid:162) M (cid:175)(cid:175) ≤ C ( d ω , d ω ) Ω . Since δ d ω = , if follows from Green’s formula for differential forms (see [Kar19, formula (2)]) that ( d ω , d ω ) Ω = (cid:161) D ( p ) φ , φ (cid:162) M . This completes the proof of the corollary. §5.4. The Hodge Laplacian and Weyl’s law for the Dirichlet-to-Neumann map.
Let ∆ M de-note the Hodge Laplacian on M . Then [ d M , ∆ M ] = [ δ M , ∆ M ] = . Thus, E p ( M ) and c C p ( M ) areinvariant subspaces and, in particular, the restriction ∆ M : c C p ( M ) → c C p ( M ) is a non-negative self-adjoint elliptic operator with eigenvalues ≤ (cid:101) λ ( p )1 ( M ) ≤ (cid:101) λ ( p )2 ( M ) ≤ . . . (cid:37) ∞ . The eigenvalues (cid:101) λ ( p ) k satisfy the variational principle (cid:101) λ ( p ) k = inf F k ⊂ c C p ( M ) sup φ ∈ F k \{0} (cid:107) d M φ (cid:107) M (cid:107) φ (cid:107) M , where F k ranges over k -dimensional subspaces of c C p ( M ) . Theorem 5.6.
Let Ω be a compact smooth orientable Riemannian manifold with boundary M = ∂ Ω .Then (cid:175)(cid:175)(cid:175)(cid:175) σ ( p ) k − (cid:113)(cid:101) λ ( p ) k (cid:175)(cid:175)(cid:175)(cid:175) ≤ C holds with the same constant as in (33) .Proof. The result follows from Corollary 5.5 in the same way as Theorem 1.9 follows from Corol-lary 3.2. Page 19. Girouard, M. Karpukhin, M. Levitin, and I. Polterovich
Remark . The restriction of the Hodge Laplacian to co-closed forms is an operator that has beeninvestigated in other contexts (see [JaSt07]) and has applications to physics, in particular, to the studyof Maxwell equations, see [BeSh08, KKL10] and references therein. The definition of the Dirichlet-to-Neumann map on differential forms given in [Kar19] which is used in the present paper is inspiredby the one introduced in [BeSh08] (see [RaSa12, JoLi05] for other definitions) and is also motivatedin part by the connection to Maxwell equations. Theorem 5.6 indicates that this definition of theDirichlet-to-Neumann map is natural from the viewpoint of comparison with the boundary HodgeLaplacian.Similarly to the proof of Theorem 1.11, one can use Theorem 5.6 to obtain Weyl’s law for σ ( p ) k fromthe spectral asymptotics for ˜ λ ( p ) k . Theorem 5.8.
Let Ω be a compact smooth orientable manifold of dimension d (cid:62) with boundary M = ∂ Ω . Then the eigenvalue counting function for the Dirichlet-to-Neumann map satisfies the asymp-totic relation N ( p ) ( σ ) : = (cid:179) σ ( p ) k < σ (cid:180) = (cid:195) d − p (cid:33) vol( (cid:66) d − ) vol( M )(2 π ) d − σ d − + o (cid:179) σ d − (cid:180) . (34) Proof.
The theorem follows immediately from the fact that (34) holds with σ ( p ) k replaced by (cid:113)(cid:101) λ ( p ) k ( M ) .As was explained to the authors by A. Strohmaier [Str21], this result is essentially contained in [JaSt07,LiSt16]. Indeed, combining the standard Karamata Tauberian argument with [LiSt16, formula (1.22)]giving the heat trace asymptotics, we obtain the asymptotic formula for the counting function. Hereone takes P to be the Hodge Laplacian and A to be the pseudodifferential projection onto the spaceof co-closed forms. In order to calculate the leading term, let us apply [LiSt16, formula (1.23)]. Let S ∗ M be the cosphere bundle and let σ A ( ξ ) ∈ End( Λ p ( M )) , ξ ∈ S ∗ M , be the principal symbol of A .As computed in [JaSt07, formula (29)], σ A ( ξ )[ ω ] = i ξ ( ξ ∧ ω ), where ξ ∈ SM is the image of ξ under the musical isomorphism. For a fixed x ∈ M and ξ ∈ S ∗ x M , weidentify Λ px ( M ) with Λ p (cid:161) (cid:82) d − (cid:162) and set (cid:82) d − = ξ ⊕ (cid:82) d − . This induces the decomposition Λ px ( M ) ∼= (cid:179) ξ ∧ Λ p − (cid:179) (cid:82) d − (cid:180)(cid:180) ⊕ Λ p (cid:179) (cid:82) d − (cid:180) . It is easy to see that σ A ( ξ ) is the projection on the second summand and, thus, tr( σ A ( ξ )) = dim Λ p (cid:179) (cid:82) d − (cid:180) = (cid:195) d − p (cid:33) . Integrating the trace over ξ ∈ S ∗ M completes the proof. Remark . It is quite likely that the error estimate in (34) can be improved to the bound O ( σ d − ) .This amounts to proving the sharp Weyl’s law for (cid:101) λ ( p ) k , which should be possible by further developingthe techniques of [LiSt16].Another way to prove (34) would be to show that D ( p ) is an elliptic pseudodifferential operator oforder one, and apply the methods of microlocal analysis directly to this operator. However, unlike theDirichlet-to-Neumann map defined in [RaSa12], the fact that the operator D ( p ) is pseudodifferentialis not yet available in the literature (see [Kar19, Remark 2.4]). Remark . One can check directly the validity of formula (34) for specific values of p and d for M = (cid:83) d − . In this case the eigenvalues of D ( p ) are known explicitly (see [Kar19, Theorem 8.1]) andtheir multiplicities coincide with the multiplicities of the corresponding eigenvalues of the HodgeLaplacian that can be found in [Ik00, formula (17)]. It is then easy to calculate the leading term inWeyl’s asymptotics using the heat trace expansion.Page 20he Dirichlet-to-Neumann map and the boundary Laplacian References [Agr06] M. S. Agranovich,
On a mixed Poincare-Steklov type spectral problem in a Lipschitz domain ,Russ. J. Math. Phys. (3) (2006), 281–290.[Ale58] A. D. Aleksandrov, Uniqueness theorems for surfaces in the large. V , Vestnik Leningrad. Univ. , no. 19 (1958), 5–8 (in Russian); English translation: Amer. Math. Soc. Transl. (2) (1962),412–416.[ArMa12] W. Arendt and R. Mazzeo, Friedlander’s eigenvalue inequalities and the Dirichlet-to-Neumannsemigroup , Comm. on Pure and Appl. Analysis (6) (2012), 2201–2212.[BBBT18] F. Belgacem, H. BelHadjAli, A. BenAmor, and A. Thabet, Robin Laplacian in the Large cou-pling limit: Convergence and spectral asymptotic , Ann. Scuola Norm. Superiore Pisa
XVIII ,issue 2 (2018), 565–591.[BeSh08] M. Belishev and V. Sharafutdinov,
Dirichlet to Neumann operator on differential forms , Bull.Sci. Math. , no. 2 (2008), 128–145.[BiSa14] Binoy and G. Santhanam,
Sharp upper bound and a comparison theorem for the first nonzeroSteklov eigenvalue , J. Raman. Math. Soc. (2014), 133–154.[CaLa21] B. Causley and J. Lagacé, Private communication , 2021.[C-WGLS12] S. Chandler-Wilde, I. Graham, E. Langdon, and E. Spence,
Numerical-asymptotic boundary in-tegral methods in high-frequency acoustic scattering , Acta Numerica (2012), 89–305.[CGH18] B. Colbois, A. Girouard, and A. Hassannezhad, The Steklov and Laplacian spectra of Rieman-nian manifolds with boundary , J. Funct. Anal. , no. 6 (2020), 108409.[DeZo98] M. C. Delfour and J.-P. Zolesio,
Shape analysis via distance functions: Local theory , in: M. C.Delfour (Ed.),
Boundaries. Interfaces, and Transitions , CRM Proceedings and Lecture NotesVol. 13, AMS, Providence, RI, 1998, 91–124.[Edw93] J. Edward,
An inverse spectral result for the Neumann operator on planar domains , J. Func. Anal. (1993), 312–322.[Fil15] A. Filinovsky,
On the asymptotic behavior of the first eigenvalue of Robin problem with large pa-rameter . J. Elliptic and Parabolic Eqns. (2015), 123–135.[Fil05] N. Filonov, On an inequality between Dirichlet and Neumann eigenvalues for the Laplace oper-ator , St. Petersburg Math. J. (2005), 413–416.[Fri91] L. Friedlander, Some inequalities between Dirichlet and Neumann eigenvalues , Arch. RationalMech. Anal. (1991), 153–160.[GPPS14] A. Girouard, L. Parnovski, I. Polterovich, and D. A. Sher,
The Steklov spectrum of surfaces:asymptotics and invariants , Math. Proc. Cambridge Philos. Soc. (3) (2014), 379–389.[GiPo10] A. Girouard and I. Polterovich,
Shape optimization for low Neumann and Steklov eigenvalues ,Math. Methods Appl. Sci. (4) (2010), 501–516.[GiPo17] A. Girouard and I. Polterovich, Spectral geometry of the Steklov problem , J. Spectral Theory ,no. 2 (2017), 321–359.[Gu78] V. Guillemin, Some spectral results for the Laplace operator with potential on the n -sphere , Adv.Math. (1978), 273–286.[HaSi20] A. Hassannezhad and A. Siffert, A note on Kuttler–Sigillito’s inequalities , Ann. Math. Qué. ,no. 1 (2020), 125–147. Page 21. Girouard, M. Karpukhin, M. Levitin, and I. Polterovich [Hör54] L. Hörmander,
Uniqueness theorems and estimates for normally hyperbolic partial differentialequations of the second order , in:
Tolfte Skandinaviska Matematikerkongressen , Lunds Univer-sitets Matematiska Institution, 1954, 105–115.[Hör18] L. Hörmander,
Inequalities between normal and tangential derivatives of harmonic functions ,in: L. Hörmander,
Unpublished manuscripts , Springer, 2018, 37–41.[Ik00] A. Ikeda,
Spectral zeta functions for compact symmetric spaces of rank one , Kodai Math. J. (2000), 345–357.[Ivr00] V. Ivrii, Sharp spectral asymptotics for operators with irregular coefficients , Int. Math. Res. Not. , no. 22 (2000), 1155–1166.[JaSt07] D. Jakobson and A. Strohmaier,
High energy limits of Laplace-type and Dirac-type eigenfunctionsand frame flows , Comm. Math. Physics, , no. 3 (2007), 813–833.[JeKe95] D. Jerison and C. E. Kenig,
The inhomogeneous Dirichlet Problem in Lipschitz domains , J. Funct.Anal. (1995), 161–219.[JoLi05] M. S. Joshi and W. R. B. Lionheart,
An inverse boundary value problem for harmonic differentialforms , Asymptot. Anal. , no. 2 (2005), 93–106.[Kar19] M. Karpukhin, The Steklov problem on differential forms , Canadian J. Math. , no. 2, (2019),417–435.[Kha18] M. Khalile, Spectral asymptotics for Robin Laplacians on polygonal domains , J. Math. Anal. Appl. (2) (2018), 1498–1543.[KO-BP18] M. Khalile, T. Ourmières-Bonafos, and K. Pankrashkin,
Effective operator for Robin eigenvaluesin domains with corners , arXiv:1809.04998 (2018), to appear in Ann. Inst. Fourier.[KhPa18] M. Khalile and K. Pankrashkin,
Eigenvalues of Robin Laplacians in infinite sectors , Math. Nachr. (5–6) (2018), 928–965.[KKL10] K. Krupchyk, Ya. Kurylev, and M. Lassas,
Reconstruction of Betti numbers of manifolds foranisotropic Maxwell and Dirac systems , Comm. Anal. Geom. , no. 5 (2010), 963–985.[KK+14] N. Kuznetsov, T. Kulczycki, M. Kwaśnicki, A. Nazarov, S. Poborchi, I. Polterovich, and B.Siudeja, The legacy of Vladimir Andreevich Steklov , Notices Amer. Math. Soc. , no. 1 (2014),9–22.[LOS98] A. A. Lacey, J. R. Ockendon, and J. Sabina, Multidimensional reaction diffusion equations withnonlinear boundary conditions , SIAM J. Appl. Math. , no. 5 (1998), 1622–1647.[LePa08] M. Levitin and L. Parnovski, On the principal eigenvalue of a Robin problem with a large param-eter , Math. Nachr. , no. 2 (2008), 272–281.[LPPS17] M. Levitin, L. Parnovski, I. Polterovich, and D. A. Sher,
Sloshing, Steklov and corners: Asymp-totics of sloshing eigenvalues , arXiv:1709.01891, to appear in J. d’Anal. Math.[LPPS19] M. Levitin, L. Parnovski, I. Polterovich, and D. A. Sher,
Sloshing, Steklov and corners: Asymp-totics of Steklov eigenvalues for curvilinear polygons , arXiv:1908.06455, 1–106.[LiSt16] L. Li and A. Strohmaier,
The local counting function of operators of Dirac and Laplace type , Jour-nal of Geometry and Physics (2016), 204–228.[MaPo19] R. Magnanini and R. Poggesi,
On the stability for Alexandrov’s soap bubble theorem , J. d’Anal.Math (2019), 179–205.[Maz91] R. Mazzeo,
Remarks on a paper of L. Friedlander concerning inequalities between Neumann andDirichlet eigenvalues , Int. Math. Res. Not. , no. 4 (1991), 41–48.
Page 22he Dirichlet-to-Neumann map and the boundary Laplacian [MiTa99] M. Mitrea and M. Taylor,
Boundary layer methods for Lipschitz domains in Riemannian man-ifolds , J. Funct. Anal. (1999), 181–251.[Pan20] K. Pankrashkin,
An eigenvalue estimate for a Robin p -Laplacian in C domains , Proc. Amer.Math. Soc. (2020), 4471–4477.[Pet06] P. Petersen, Riemannian geometry , Graduate texts in mathematics 171, Springer, 2006.[Poh65] S. Pohožaev,
Eigenfunctions of the equation ∆ u + λ f ( u ) = , Soviet Math.Dokl (1965), 1408–1411.[PoSh15] I. Polterovich and D. A. Sher, Heat invariants of the Steklov problem , J. Geom. Anal., , no. 2(2015), 924–950.[Pop20] N. Popoff, The negative spectrum of the Robin Laplacian , in: P. Miranda, N. Popoff, G.Raikov (Eds.),
Spectral Theory and Mathematical Physics , Latin American Mathematics Series,Springer, Cham, 2020, 229–242[PrSt19] L. Provenzano and J. Stubbe,
Weyl-type bounds for Steklov eigenvalues , J. Spectr. Theory , no.1 (2019), 349–377.[RaSa12] S. Raulot and A. Savo, On the first eigenvalue of the Dirichlet-to-Neumann operator on forms , J.Funct. Anal. , no. 3 (2012), 889–914.[Roz86] G. V. Rozenblyum,
On the asymptotics of the eigenvalues of certain two-dimensional spectral prob-lems , Sel. Math. Sov. (1986), 233–244.[San55] L. Sandgren, A vibration problem , Comm. Sém. Math. Univ. Lund (1955), 1–84.[Saf08] Yu. Safarov, On the comparison of the Dirichlet and Neumann counting functions , in: T. Suslina,D. Yafaev (Eds.), “Spectral Theory of Differential Operators: M. Sh. Birman 80th AnniversaryCollection”, Amer. Math. Soc. Transl. Ser. 2, vol. 225, Providence, RI (2008), 191–204.[Shu01] M. Shubin,
Pseudodifferential operators and spectral theory , Springer-Verlag, 2001.[Str21] A. Strohmaier,
Private communication , 2021.[Tay96] M. Taylor,
Partial differential equations II: Qualitative studies of linear equations , AppliedMath. Sciences , Springer, 1996.[Wei54] R. Weinstock,
Inequalities for a classical eigenvalue problem , J. Rational Mech. Anal. (1954),745–753.[Xio18] C. Xiong, Comparison of Steklov eigenvalues on a domain and Laplacian eigenvalues on itsboundary in Riemannian manifolds , J. Funct. Anal. , no. 12 (2018), 3245–3258.[Zie99] L. Zielinski,
Sharp spectral asymptotics and Weyl formula for elliptic operators with non-smoothcoefficients , Math. Phys. Anal. Geom. (1999), 291–321.(1999), 291–321.