The Steklov Spectrum of Cuboids

Abstract

The paper is concerned with the Steklov eigenvalue problem on cuboids of arbitrary dimension. We prove a two-term asymptotic formula for the counting function of Steklov eigenvalues on cuboids in dimension d ≥ 3. Apart from the standard Weyl term, we calculate explicitly the second term in the asymptotics, capturing the contribution of the (d − 2)-dimensional facets of a cuboid. Our approach is based on lattice counting techniques. While this strategy is similar to the one used for the Dirichlet Laplacian, the Steklov case carries additional complications. In particular, it is not clear how to establish directly the completeness of the system of Steklov eigenfunctions admitting separation of variables. We prove this result using a family of auxiliary Robin boundary value problems. Moreover, the correspondence between the Steklov eigenvalues and lattice points is not exact, and hence more delicate analysis is required to obtain spectral asymptotics. Some other related results are presented, such as an isoperimetric inequality for the first Steklov eigenvalue, a concentration property of high frequency Steklov eigenfunctions and applications to spectral determination of cuboids. Lemma 2.1. Let M 1 and M 2 be smooth compact Riemannian manifolds with boundary. Let σ ≥ 0 be a Steklov eigenvalue of the product manifold M = M 1 × M 2 with the

The Robin Laplacian—Spectral conjectures, rectangular theorems

“…• minimality of the square for the first eigenvalue when α ∈ R (Theorem 3.2) • maximality of the square for the second eigenvalue when α ∈ [α − , α + ] (Theorem 3.6), where α − −9.4 and α + 33.2; outside that range the maximizer is the degenerate rectangle, • maximality of the square for the first nonzero Steklov eigenvalue (Corollary 3.7, which gives a new proof of a result by Girouard, Lagacé, Polterovich and Savo [23]) • maximality of the square for the spectral ratio λ 2 /λ 1 (Corollary 3.11) when α > 0.…”

“…Maximizing the first nonzero Steklov eigenvalue on rectangular boxes). The first nonzero Steklov eigenvalue σ 1 (B) is maximal for the cube and only the cube, among rectangular boxes B of given volume.The Steklov result in Corollary 3.5 was proved directly by Girouard et al[23, Theorem 1.6]. Indeed, they proved a stronger result, namely that the cube maximizes σ 1 among rectangular boxes of given surface area.…”

Uniqueness results in the inverse spectral Steklov problem

“…A much more refined asymptotic holds if M is a smooth surface, involving the lengths of the connected components of M , with a rate of decay of O(j −∞ ) (see [7]), but this formula fails for polygons ([8]). In some particular cases, when the boundary ∂M is just C 1 , it is also possible to find a one term asymptotic of the Steklov spectrum counted with multiplicity (see [1]).…”

“…In some particular cases, when the boundary ∂M is just C 1 , it is also possible to find a one term asymptotic of the Steklov spectrum counted with multiplicity (see [1]). For more specific domains with Lipschitz boundary, a recent paper ( [7]) proves a two-term asymptotic formula for cuboids, i.e domains defined, for n ≥ 3, as M = (−a 1 , a 1 ) × ... × (−a n , a n ) ∈ R n .…”

“…Moreover, it is well-known that some other geometrical quantities of the boundary of a Riemannian manifold surface are Steklov spectrum invariants. Let us mention for instance (see [7,8] for details and reference therein):…”