International Journal of Differential Equations
Volume 2011 (2011), Article ID 619623, 22 pages
http://dx.doi.org/10.1155/2011/619623
Research Article

On Spectrum of the Laplacian in a Circle Perforated along the Boundary: Application to a Friedrichs-Type Inequality

G. A. Chechkin,1,2 Yu. O. Koroleva,1,3 L.-E. Persson,2,3 and P. Wall3

1Department of Differential Equations, Faculty of Mechanics and Mathematics, Moscow Lomonosov State University, Moscow 119991, Russia
2Narvik University College, Postboks 385, 8505 Narvik, Norway
3Department of Engineering Science and Mathematics, Luleå University of Technology, 971 87 Luleå, Sweden

Received 24 May 2011; Accepted 30 August 2011

Academic Editor: Mayer Humi

Copyright © 2011 G. A. Chechkin et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

In this paper, we construct and verify the asymptotic expansion for the spectrum of a boundary-value problem in a unit circle periodically perforated along the boundary. It is assumed that the size of perforation and the distance to the boundary of the circle are of the same smallness. As an application of the obtained results, the asymptotic behavior of the best constant in a Friedrichs-type inequality is investigated.