Singular asymptotic expansions for Dirichlet eigenvalues and eigenfunctions of the Laplacian on thin planar domains |
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| Authors: | Denis Borisov Pedro Freitas |
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| Institution: | 1. Faculty of Mathematics, Chemnitz University of Technology, D-09107 Chemnitz, Germany;2. Department of Physics and Mathematics, Bashkir State Pedagogical University, October rev. st., 3a, 450000 Ufa, Russia;3. Department of Mathematics, Faculdade de Motricidade Humana (TU Lisbon) and Group of Mathematical Physics of the University of Lisbon, Complexo Interdisciplinar, Av. Prof. Gama Pinto 2, P-1649-003 Lisboa, Portugal |
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| Abstract: | We consider the Laplace operator with Dirichlet boundary conditions on a planar domain and study the effect that performing a scaling in one direction has on the spectrum. We derive the asymptotic expansion for the eigenvalues and corresponding eigenfunctions as a function of the scaling parameter around zero. This method allows us, for instance, to obtain an approximation for the first Dirichlet eigenvalue for a large class of planar domains, under very mild assumptions. |
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| Keywords: | primary 35P15 secondary 35J05 |
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