Continuity of eigenvalues and shape optimisation for Laplace and Steklov problems

Alexandre Girouard,Mikhail Karpukhin,Jean Lagacé

doi:10.1007/s00039-021-00573-5

Abstract

We associate a sequence of variational eigenvalues to any Radon measure on a compact Riemannian manifold. For particular choices of measures, we recover the Laplace, Steklov and other classical eigenvalue problems. In the first part of the paper we study the properties of variational eigenvalues and establish a general continuity result, which shows for a sequence of measures converging in the dual of an appropriate Sobolev space, that the associated eigenvalues converge as well. The second part of the paper is devoted to various applications to shape optimization. The main theme is studying sharp isoperimetric inequalities for Steklov eigenvalues without any assumption on the number of connected components of the boundary. In particular, we solve the isoperimetric problem for each Steklov eigenvalue of planar domains: the best upper bound for the k-th perimeter-normalized Steklov eigenvalue is 8pi k, which is the best upper bound for the k^{text {th}} area-normalised eigenvalue of the Laplacian on the sphere. The proof involves realizing a weighted Neumann problem as a limit of Steklov problems on perforated domains. For k=1, the number of connected boundary components of a maximizing sequence must tend to infinity, and we provide a quantitative lower bound on the number of connected components. A surprising consequence of our analysis is that any maximizing sequence of planar domains with fixed perimeter must collapse to a point.

Highlights

For a compact, connected Riemannian manifold (M, g) of dimension d, with or without C1 boundary ∂M, the Laplace eigenvalue problem consists in determining all λ ∈ R for which the following eigenvalue problem admits a nontrivial solution: −Δgu = λu in M,∂nu = 0 on ∂M, when ∂M = ∅, where ∂nu is the outward normal derivative of u
In the first part of the paper we study the properties of variational eigenvalues and establish a general continuity result, which shows for a sequence of measures converging in the dual of an appropriate Sobolev space, that the associated eigenvalues converge as well
We solve the isoperimetric problem for each Steklov eigenvalue of planar domains: the best upper bound for the k-th perimeter-normalized Steklov eigenvalue is 8πk, which is the best upper bound for the kth area-normalised eigenvalue of the Laplacian on the sphere

Summary

Introduction

For a compact, connected Riemannian manifold (M, g) of dimension d, with or without C1 boundary ∂M , the Laplace eigenvalue problem consists in determining all λ ∈ R for which the following eigenvalue problem admits a nontrivial solution:. We use Theorem 1.11 to prove that there are domains with arbitrarily small area-normalised Neumann eigenvalues λk(Ω, g0), for which the Steklov eigenvalues are bounded away from zero. To Theorem 1.14, on any closed Riemannian surface (M, g) there exists a sequence of conformal metrics gn = efng and a sequence of domains Ωn ⊂ M such that σ1(Ωn, gn) −ε−→−→0 Σ1(M, g) while for each k ∈ N, the normalised Laplace eigenvalues satisfy λk(M, gn) −ε−→−→0 0. Let C ⊂ ∂M be the longest connected component of the boundary and fix y ∈ S2 It follows from the Koebe uniformization theorem that there exists a diffeomorphism Φ : M → Ω ⊂ S2y, conformal in the interior of M , sending C to the equator, i.e Φ(C ) = ∂S2y. Finding a better trial function would improve the bounds obtained in (2.6)

Admissible Measures and Associated Function Spaces

Variational Eigenvalues

First Examples of Spectrum Convergence

Homogenisation

Girouard