The Steklov and Laplacian spectra of Riemannian manifolds with boundary

Abstract

Given two compact Riemannian manifolds M 1 and M 2 such that their respective boundaries Σ 1 and Σ 2 admit neighbourhoods Ω 1 and Ω 2 which are isometric, we prove the existence of a constant C such that |σ k (M 1 ) − σ k (M 2 )| ≤ C for each k ∈ N. The constant C depends only on the geometry of Ω 1 ∼ = Ω 2 . This follows from a quantitative relationship between the Steklov eigenvalues σ k of a compact Riemannian manifold M and the eigenvalues λ k of the Laplacian on its boundary. Our main result states that the difference |σ k − √ λ k | is bounded above by a constant which depends on the geometry of M only in a neighbourhood of its boundary. The proofs are based on a Pohozaev identity and on comparison geometry for principal curvatures of parallel hypersurfaces. In several situations, the constant C is given explicitly in terms of bounds on the geometry of Ω 1 ∼ = Ω 2 .1991 Mathematics Subject Classification. 35P15 (primary), 58C40, 35P20 (secondary).1 The notation O(k −∞ ) designates a quantity which tends to zero faster than any power of k.

Compact manifolds with fixed boundary and large Steklov eigenvalues

“…These results indicate a strong link between the Steklov eigenvalues of a manifold and the geometry of its boundary. See also [4,15] for recent similar results on Riemannian manifolds. In fact, on smooth surfaces the spectral asymptotics is completely determined by the geometry of the boundary [8].…”

Uniqueness results in the inverse spectral Steklov problem

“…where u is the unique weak solution of the Dirichlet problem (1), and where v is any element of H 1 (M ) such that v |∂M = φ. When ψ is sufficiently smooth, this definition coincides with the usual one in local coordinates, that is (2) ∂ ν u = ν i ∂ i u.…”

“…(0) f only depends on the metric on the boundary component (K, f (0)g K ). This fact can be compared to a recent result (see [2]) where it is proved that the previous bound C Ω can be chosen uniformly with respect to a class of manifolds M satisfying some geometrical conditions only in a neighborhood of the boundary (Theorem 3, p.3).…”

“…If λ / ∈ σ(−∆ g ), the Dirichlet problem (1) has a unique solution u ∈ H 1 (M ), and we can define the so-called Dirichlet-to-Neumann (DN) operator as the map Λ g (λ) : H 1/2 (∂M ) → H −1/2 (∂M ) ψ → ∂u ∂ν ∂M , where ∂ ν is the unit normal derivative with respect to the unit outer normal vector on ∂M . This normal derivative has to be understood in the weak sense by ∀(ψ, φ) ∈ H 1/2 (∂M ) 2…”

Hypersurfaces with Prescribed Boundary and Small Steklov Eigenvalues

“…One of our main interests in recent years has been to understand the particular role that the boundary Σ plays with respect to Steklov eigenvalues. Some papers studying this question are [6,15,4,2,17,11,7,5,16]. In particular, we have considered the effect of various geometric constraints on individual eigenvalues σ k .…”