Asymptotics of the eigenvalues of the Dirichlet-Laplace problem in a domain with thin tube excluded

We consider a Laplace problem with Dirichlet boundary condition in a three dimensional domain containing an inclusion taking the form of a thin tube with small thickness δ. We prove convergence in operator norm of the resolvent of this problem as δ → 0, establishing that the perturbation induced by the inclusion on the resolvent is not greater than O(| ln δ|^-γ) for some γ > 0. We deduce convergence of the eigenvalues of the perturbed operator toward the limit operator.