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 -> 0, establishing that the perturbation induced by the inclusion on the resolvent is not greater than O(vertical bar ln delta vertical bar-(gamma)) for some gamma > 0. We deduce convergence of the eigenvalues of the perturbed operator toward the limit operator.