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

Ahmad El Soufi (1) , Saïd Ilias (1)

1 LMPT - Laboratoire de Mathématiques et Physique Théorique

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.

Mots clés: Eigenvalues; Laplacian; Dirichlet problem; Domain deformation; Heat trace

Domaines: Géométrie métrique [math.MG]; Théorie spectrale [math.SP]

Ahmad El Soufi, Saïd Ilias. Domain deformations and eigenvalues of the Dirichlet Laplacian in a Riemannian manifold. Illinois Journal of Mathematics, 2007, 51, pp.645--666. ⟨hal-00145250⟩