Splitting of resonant and scattering frequencies under shape deformation |
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Authors: | Habib Ammari Faouzi Triki |
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Institution: | Centre de Mathématiques Appliquées, CNRS UMR 7641, Ecole Polytechnique, 91128 Palaiseau, Cedex, France |
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Abstract: | 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, and the splitting may only become apparent at high orders in their Taylor expansion. In this paper, we address the splitting problem in the evaluation of resonant and scattering frequencies of the two-dimensional Laplacian operator under boundary variations of the domain. By using surface potentials we show that the eigenvalues are the characteristic values of meromorphic operator-valued functions that are of Fredholm type with index 0. We then proceed from the generalized Rouché's theorem to investigate the splitting problem. |
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Keywords: | 35J05 35C15 35P05 |
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