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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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##### References:
 [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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