Justification of asymptotic expansions of the eigenvalues of nonselfadjoint singularly perturbed elliptic boundary value problems. (English. Russian original) Zbl 0618.35005

Nazarov, S. A.

Math. USSR, Sb. 57, 317-349 (1987); translation from Mat. Sb., Nov. Ser. 129(171), No. 3, 307-337 (1986).

General boundary value problems for scalar elliptic PDE’s of the order 2m, with a small parameter \(\epsilon\) with higher derivatives (singular perturbed boundary value problem) and degenerating for \(\epsilon\to 0\) to a not selfadjoint boundary value problem of the order 2n, \(m>n>0\), are examined. Asymptotic distributions and their substantiations for eigenvalues, eigenvectors, and solutions are presented. Substantiations of the asymptotic expansions are based on a norm estimation of an inversion operator for the operator of the singular perturbed problem.

Reviewer: J.Smid

MSC:

35B25	Singular perturbations in context of PDEs
35P20	Asymptotic distributions of eigenvalues in context of PDEs
35J40	Boundary value problems for higher-order elliptic equations

Keywords:

small parameter; Asymptotic distributions; asymptotic expansions