Asymptotic behavior of repeated eigenvalues of perturbed self-adjoint elliptic operators

Article type: Research Article

Authors: Dabrowski, Alexander^{; *}

Affiliations: Johann Radon Institute, Altenberger Straße 69, 4040 Linz, Austria. E-mail: alexander.dabrowski@ricam.oeaw.ac.at

Correspondence: [*] Corresponding author. E-mail: alexander.dabrowski@ricam.oeaw.ac.at.

Abstract: A variational characterization for the shift of eigenvalues caused by a general type of perturbation is derived for second order self-adjoint elliptic differential operators. This result allows the direct extension of asymptotic formulae from simple eigenvalues to repeated ones. Some examples of particular interest are presented theoretically and numerically for the Laplacian operator for the following domain perturbations: excision of a small hole, local change of conductivity, small boundary deformation.

Keywords: Eigenvalue perturbation, domain perturbation, repeated eigenvalue, Laplacian eigenvalues, asymptotic expansion

DOI: 10.3233/ASY-201669

Journal: Asymptotic Analysis, vol. 126, no. 1-2, pp. 187-202, 2022