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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.

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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[1] Antunes, P.; Freitas, P., A numerical study of the spectral gap, J. phys. A, 41, (2008) · Zbl 1142.35054
[2] Berezin, F.A.; Shubin, M.A., The Schrödinger equation, (1991), Kluwer Dordrecht · Zbl 0749.35001
[3] Freitas, P., Precise bounds and asymptotics for the first Dirichlet eigenvalue of triangles and rhombi, J. funct. anal., 251, 376-398, (2007) · Zbl 1137.35049
[4] P. Freitas, D. Krejčiřík, A sharp upper bound for the first Dirichlet eigenvalue and the growth of the isoperimetric constant of convex domains, Proc. Amer. Math. Soc., in press · Zbl 1147.58030
[5] L. Friedlander, M. Solomyak, On the spectrum of the Dirichlet Laplacian in a narrow strip, preprint, 2007 · Zbl 1170.35487
[6] Joseph, D.D., Parameter and domain dependence of eigenvalues of elliptic partial differential equations, Arch. ration. mech. anal., 24, 325-351, (1967) · Zbl 0152.11302
[7] Ladyzhenskaya, O.A.; Uraltseva, N.N., Linear and quasilinear elliptic equations, (1968), Academic Press New York-London · Zbl 0164.13002
[8] Olejnik, O.A.; Shamaev, A.S.; Yosifyan, G.A., Mathematical problems in elasticity and homogenization, Studies in mathematics and its applications, vol. 26, (1992), North-Holland Amsterdam · Zbl 0629.35093
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