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

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

2007-05-09

arxiv.org/abs/0705.1263v1

Illinois Journal of Mathematics 51 (2007) 645--666

Mathematics

Metric Geometry

To appear in Illinois J. Math

Scientific paper

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.

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