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

1 LMPT - Laboratoire de Mathématiques et Physique Théorique

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.

Keywords : Eigenvalues Laplacian Dirichlet problem Domain deformation Heat trace

Author: Ahmad El Soufi - Saïd Ilias -

Source: https://hal.archives-ouvertes.fr/