Perturbations of eigenvalues of the Neumann problem due to variations of a domain’s boundary. (English. Russian original) Zbl 0827.35086

Nazarov, S. A.; Olyushin, M. V.

St. Petersb. Math. J. 5, No. 2, 371-387 (1994); translation from Algebra Anal. 5, No. 2, 169-188 (1993).

Summary: We construct the leading terms of asymptotic series expansions of solutions to the spectral Neumann problem in a plane domain \(\Omega_\varepsilon\) depending on a small parameter \(\varepsilon\). The perturbation of the boundary is a small nonuniform translation boundary \(\partial \Omega_0\) of the initial domain.
It is shown that the asymptotic formulas for the eigenvalues \(\lambda (\varepsilon)\) in the cases of regular (smooth) and nonregular (having steps) perturbation are of different form. The method of matched asymptotic expansions is used, boundary layers near steps are found, and the following asymptotic terms are calculated: \[ \lambda (\varepsilon) \sim \lambda_0 + \varepsilon \lambda_1 + \varepsilon^2 (\lambda^1_2 \ln \varepsilon + \lambda^0_2). \] Asymptotic formulas are also given for a three-dimensional problem.

MSC:

35P10	Completeness of eigenfunctions and eigenfunction expansions in context of PDEs
35B25	Singular perturbations in context of PDEs

Keywords:

plane domain; perturbation of the boundary; asymptotic formulas for the eigenvalues