JournalsaihpcVol. 26, No. 2pp. 547–560

Singular asymptotic expansions for Dirichlet eigenvalues and eigenfunctions of the Laplacian on thin planar domains

  • Denis Borisov

    Faculty of Mathematics, Chemnitz University of Technology, D-09107 Chemnitz, Germany, Department of Physics and Mathematics, Bashkir State Pedagogical University, October rev. st., 3a, 450000 Ufa, Russia

  • Pedro Freitas

    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
Singular asymptotic expansions for Dirichlet eigenvalues and eigenfunctions of the Laplacian on thin planar domains cover
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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.

Cite this article

Denis Borisov, Pedro Freitas, Singular asymptotic expansions for Dirichlet eigenvalues and eigenfunctions of the Laplacian on thin planar domains. Ann. Inst. H. Poincaré Anal. Non Linéaire 26 (2009), no. 2, pp. 547–560

DOI 10.1016/J.ANIHPC.2007.12.001