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Zeitschrift für Analysis und ihre Anwendungen


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Volume 30, Issue 2, 2011, pp. 145–180
DOI: 10.4171/ZAA/1429

Published online: 2011-04-03

Singular Perturbations of Curved Boundaries in Three Dimensions. The Spectrum of the Neumann Laplacian

Antoine Laurain[1], Sergei A. Nazarov[2] and Jan Sokolowski[3]

(1) TU Berlin, Germany
(2) Institute for Problems in Mechanical Engineering RAS, St. Petersburg, Russian Federation
(3) Université Henri Poincaré, Vandoeuvre les Nancy, France

We calculate the main asymptotic terms for eigenvalues, both simple and multiple, and eigenfunctions of the Neumann Laplacian in a three-dimensional domain $\Omega(h)$ perturbed by a small (with diameter $O(h)$) Lipschitz cavern $\overline{\omega_h}$ in a smooth boundary $\partial\Omega=\partial\Omega(0)$. The case of the hole $\overline{\omega_h}$ inside the domain but very close to the boundary $\partial\Omega$ 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 $\partial\Omega$ 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.

Keywords: Asymptotic analysis, singular perturbations, spectral problem, asymptotics of eigenfunctions and eigenvalues

Laurain Antoine, Nazarov Sergei, Sokolowski Jan: Singular Perturbations of Curved Boundaries in Three Dimensions. The Spectrum of the Neumann Laplacian. Z. Anal. Anwend. 30 (2011), 145-180. doi: 10.4171/ZAA/1429