On the Hadamard Formula for Second Order Systems in Non-Smooth Domains

Abstract

Perturbations of eigenvalues of the Dirichlet problem for a second order elliptic system in a bounded domain Ω in ℝ n are studied under variations of the domain Ω. We investigate the case when the perturbed domain is located in a d-neighborhood of the reference Lipschitz domain. A new asymptotic formula is derived; it contains terms that are absent in the classical formula of Hadamard. The latter is valid only for smooth domains and smooth perturbations. We give conditions that guarantee the validity of the Hadamard formula. The general asymptotic formula is applied when Ω is perturbed by small curvilinear and circular cuts. Most of the results are new even for the Laplace operator.