Singular perturbations of curved boundaries in dimension three. The spectrum of the Neumann Laplacian

Antoine Laurain (1) , Sergueï A. Nazarov (2) , Jan Sokolowski (3)

1 HU Berlin - Humboldt-Universität zu Berlin = Humboldt University of Berlin = Université Humboldt de Berlin
2 IPME - Institute of Mechanical Engineering Problems [St. Petersburg]
3 IECN - Institut Élie Cartan de Nancy

We calculate the main asymptotic terms for eigenvalues, both simple and multiple, and eigenfunctions of the Neumann Laplacian in a three-dimensional domain $\Om(h)$ perturbed by a small (with diameter $O(h)$) Lipschitz cavern $\overline{\om_h}$ in a smooth boundary $\partial\Om=\partial\Om(0)$. The case of the hole $\overline{\om_h}$ inside the domain but very close to the boundary $\partial\Om$ is under consideration as well. It is proven that the main correction term in the asymptotics of eigenvalues does not depend on the curvature of $\partial\Om$ while terms in the asymptotics of eigenfunctions do. The influence of the shape of the cavern to the eigenvalue asymptotics relies mainly upon a certain matrix integral characteristics like the tensor of virtual masses. Asymptotically exact estimates of the remainders are derived in weighted norms.

Mots clés: asymptotic analysis; singular perturbations; spectral problem; asymptotics of eigenfunctions and eigenvalue

Domaines: Equations aux dérivées partielles [math.AP]

Antoine Laurain, Sergueï A. Nazarov, Jan Sokolowski. Singular perturbations of curved boundaries in dimension three. The spectrum of the Neumann Laplacian. 2010. ⟨hal-00461233v2⟩