Spectra of domains with partial degeneration

J. Math. Sci. Univ. Tokyo
Vol. 16 (2009), No. 3, Page 269--414.

JIMBO, Shuichi; KOSUGI, Satoshi
Spectra of domains with partial degeneration
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We consider the eigenvalue problem of the Laplacian (Neumann B.C.) in the domain which has a very thin subregion. We give an elaborate characterization of the asymptotics of theeigenvalues and the eigenfunctions. The perturbation formulas take various forms depending on the type of the eigenvalue and geometric situations (dimension, shape).

Keywords: Singularly perturbed domain, Laplacian, eigenvalues, asymptotic behavior

Mathematics Subject Classification (2000): Primary 35P15, Secondary 35J20
Mathematical Reviews Number: MR2597393

Received: 2007-11-12