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Singular asymptotic expansions for Dirichlet eigenvalues and eigenfunctions of the Laplacian on thin planar domains. (English) Zbl 1168.35401
Summary: We consider the Laplace operator with Dirichlet boundary conditions on a planar domain and study the effect that performing a scaling in one direction 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.

MSC:
35P10 Completeness of eigenfunctions and eigenfunction expansions in context of PDEs
35P15 Estimates of eigenvalues in context of PDEs
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
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