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

Xavier Claeys (1, 2)

1 ALPINES - Algorithms and parallel tools for integrated numerical simulations
2 LJLL - Laboratoire Jacques-Louis Lions

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. We prove convergence in operator norm of the resolvent of this problem as the thickness goes to 0, establishing that the perturbation on the resolvent induced by the inclusion is not greater than some (negative) power of the logarithm of the thickness. We deduce convergence of the eigenvalues of the perturbed operator toward the limit operator.

Mots clés: Laplace; Dirichlet; asymptotics; thin wire; eigenvalues

Domaines: Physique mathématique [math-ph]

Xavier Claeys. Asymptotics of the eigenvalues of the Dirichlet-Laplace problem in a domain with thin tube excluded. Quarterly of Applied Mathematics, 2016, 74 (4), pp.595-605. ⟨hal-01120422v2⟩