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 $\delta$. We prove convergence in operator norm of the resolvent of this problem as $\delta \to 0$, establishing that the perturbation induced by the inclusion on the resolvent is not greater than $O(\vert \ln \delta \vert ^{-\gamma })$ for some $\gamma >0$. We deduce convergence of the eigenvalues of the perturbed operator toward the limit operator.