Spectrum Stability of an Elliptic Operator to Domain Perturbations

This paper raises the question of the stability of the full spectrum of an elliptic operator in divergence form (with homogeneous Dirichlet boundary conditions) with respect to domain perturbations which modify continuously a limited number of “small” eigenvalues. If the perturbations are semi-compact and modify the measure continuously, we prove that the stability of the first eigenvalue implies the stability of the full spectrum, under the hypothesis that the perturbed domain is connected.