Domain deformations and eigenvalues of the Dirichlet Laplacian in a Riemannian manifold

TL;DR

This paper studies how the eigenvalues of the Dirichlet Laplacian change under domain deformations in a Riemannian manifold, introducing critical domains and deriving conditions for extremality using variational formulas.

Contribution

It introduces a notion of critical domain for eigenvalues of the Dirichlet Laplacian in Riemannian manifolds and provides necessary and sufficient conditions for extremality.

Findings

Characterization of critical domains for eigenvalues

Necessary and sufficient conditions for local minima and maxima

Derivation of Hadamard type variational formulas

Abstract

For any bounded regular domain $\Omega$ of a real analytic Riemannian manifold $M$, we denote by $\lambda_{k}(\Omega)$ the $k$-th eigenvalue of the Dirichlet Laplacian of $\Omega$. In this paper, we consider $\lambda_k$ and as a functional upon the set of domains of fixed volume in $M$. We introduce and investigate a natural notion of critical domain for this functional. In particular, we obtain necessary and sufficient conditions for a domain to be critical, locally minimizing or locally maximizing for $\lambda_k$. These results rely on Hadamard type variational formulae that we establish in this general setting.