JournalszaaVol. 30, No. 2pp. 145–180

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

  • Sergei A. Nazarov

    Institute for Problems in Mechanical Engineering RAS, St. Petersburg, Russian Federation
  • Antoine Laurain

    TU Berlin, Germany
  • Jan Sokolowski

    Université Henri Poincaré, Vandoeuvre les Nancy, France
Singular Perturbations of Curved Boundaries in Three Dimensions. The Spectrum of the Neumann Laplacian cover
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Abstract

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

Cite this article

Sergei A. Nazarov, Antoine Laurain, Jan Sokolowski, Singular Perturbations of Curved Boundaries in Three Dimensions. The Spectrum of the Neumann Laplacian. Z. Anal. Anwend. 30 (2011), no. 2, pp. 145–180

DOI 10.4171/ZAA/1429