International Journal of Mathematics and Mathematical Sciences
Volume 2005 (2005), Issue 8, Pages 1201-1220
The integral equation methods for the perturbed Helmholtz eigenvalue problems
Centre de Mathématiques Appliquées (CMAP), École Polytechnique, Palaiseau cedex 91128, France
Received 10 May 2004; Revised 31 January 2005
Copyright © 2005 Abdessatar Khelifi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
It is well known that the main difficulty in solving eigenvalue problems under shape deformation relates to the continuation of multiple eigenvalues of the unperturbed configuration. These
eigenvalues may evolve, under shape deformation, as separated, distinct eigenvalues. In this paper, we address the integral equation method in the evaluation of eigenfunctions and the corresponding eigenvalues of the two-dimensional Laplacian operator under boundary variations of the domain. Using surface
potentials, we show that the eigenvalues are the characteristic values of meromorphic operator-valued functions.