Asymptotics for eigenelements of Laplacian in domain with oscillating boundary: multiple eigenvalues

Abstract

We study the asymptotic behavior of the solutions of a spectral problem for the Laplacian in a domain with rapidly oscillating boundary. We consider the case where the eigenvalue of the limit problem is multiple. We construct the leading terms of the asymptotic expansions for the eigenelements and verify the asymptotics.

Keywords:

• Spectral problems
• Oscillating boundary
• Asymptotic expansions
• 2000 Mathematics Subject Classifications:
• 35J25
• 35B40

Acknowledgments

This article has been partially written during the stay of Gregory A. Chechkin in the Blaise Pascal University (Clermont-Ferrand, France) in June–July 2004. He wants to express deep thanks for the hospitality, for the support and for perfect conditions to work. The final version was completed when Gregory A. Chechkin was visiting the Bashkir State Pedagogical University (Ufa, Russia) in November 2006. We would like to thank the referee for his helpful comments and suggestions. The work of G. A. Chechkin was partially supported by RFBR (06-01-00441) and by the program “Leading Scientific Schools” (HIII-2538.2006.1). The work of R. R. Gadyl'shin was partially supported by RFBR (06-01-00138).