On the spectrum of the Neumann problem for Laplace equation in a domain with a narrow slit

Article type: Research Article

Authors: Gadyl'shin, Rustem R. | Il'in, Arlen M.^;

Affiliations: The Bashkir State Pedagogical University, Ufa, Russia. E-mail: gadylshin@yandex.ru | Chelyabinsk State University, Chelyabinsk, Russia. E-mail: iam@csu.ru

Note: [] Corresponding author: Arlen M. Il'in, Chelyabinsk State University, Br. Kashirinyh St., 129, 454021, Chelyabinsk, Russia. Tel.: +7 351 7997250; Fax: +7 351 7420925; E-mail: iam@csu.ru.

Abstract: The Neumann problem in two-dimensional domain with a narrow slit is studied. The width of the slit is a small parameter 0<ε�1. The complete asymptotic expansion with respect to ε for the eigenvalue of the perturbed problem converging to a simple eigenvalue of the limiting problem is constructed by means of the method of the matching asymptotic expansions. It is shown that the regular perturbation theory can formally be applied in a natural way up to terms of order ε2. However, the result obtained in that way is false. The correct result can be obtained only by means of inner asymptotic expansion. We also show that the eigenvalue of the perturbed problem can be both more and less than the eigenvalue of the limiting problem subject to the position and geometry of the slit.

Keywords: singular perturbation, asymptotics, eigenvalues, Neumann problem

DOI: 10.3233/ASY-2009-0976

Journal: Asymptotic Analysis, vol. 67, no. 3-4, pp. 167-189, 2010