Domain Deformations and Eigenvalues of the Dirichlet Laplacian in a Riemannian Manifold
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چکیده
Abstract. For any bounded regular domain Ω of a real analytic Riemannian manifold M , we denote by λk(Ω) the k-th eigenvalue of the Dirichlet Laplacian of Ω. In this paper, we consider λk 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 λk. 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.
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تاریخ انتشار 2007