DOMAIN DEFORMATIONS AND EIGENVALUES OF THE DIRICHLET LAPLACIAN IN A RIEMANNIAN MANIFOLD

摘要

For any bounded regular domain Ω of a real analytic Rie-mannian manifold M we denote by λ_k(Ω) the k-th eigenvalue of the Dirichlet Laplacian of Ω. In this paper, we consider λ_k as a functional on 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 λ_k. These results rely on Hadamard type variational formulae that we establish in this general setting. As an application, we obtain a characterization of critical domains of the trace of the heat kernel under Dirichlet boundary conditions.