Asymptotic property and convergence estimation for the eigenelements of the Laplace operator

Abstract

In this article, we provide a rigorous derivation of asymptotic expansions for eigenfunctions and we establish convergence estimation for both eigenvalues and eigenfunctions of the Laplacian. We address the integral equation method to investigate the interplay between the geometry, boundary conditions and spectral properties of the eigenelements of the Laplace operator under deformation of the domain. The asymptotic formula and convergence estimation are tested by numerical examples.

Keywords:

• Eigenvalue problem
• Shape deformation
• Boundary integral equation
• asymptotic expansions
• Convergence estimation
• 2000 Mathematics Subject Classifications:
• 35C15
• 45C05
• 35P05
• 34L15

Acknowledgements

The author is grateful to the editor and the anonymous referees for their valuable comments and helpful suggestions which have much improved the presentation of the article.