Asymptotic expansions of the eigenvalues of boundary value problems for the Laplace operator in domains with small holes. (English. Russian original) Zbl 0566.35031

Maz’ya, V. G.; Nazarov, S. A.; Plamenevskij, B. A.

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