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Asymptotic expansions of the eigenvalues of boundary value problems for the Laplace operator in domains with small holes. (English. Russian original) Zbl 0566.35031
Math. USSR, Izv. 24, 321-345 (1985); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 48, No. 2, 347-371 (1984).
Let $$\Omega$$ be a domain in $$R^ 2$$ or in $$R^ 3$$, $$\Omega_{\epsilon}=\Omega \setminus \{\omega \epsilon \},$$ where $$\epsilon >0$$ is a small parameter. The authors consider the eigenvalue problem $$\Delta u=\lambda u,$$ $$x\in \Omega_{\epsilon}$$, $$\partial u/\partial n|_{\partial \Omega}=0$$, $$u|_{\omega \epsilon}=0.$$ The authors determine the asymptotic behavior of the first eigenvalues when $$\epsilon \to +0$$.
Reviewer: M.V.Fedoryuk

##### MSC:
 35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation 35P15 Estimates of eigenvalues in context of PDEs
##### Keywords:
small parameter; asymptotic behavior; first eigenvalues
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