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

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.

To appear in Illinois J. Math