Spectral boundary value problems for Laplace--Beltrami operator: moduli of continuity of eigenvalues under domain deformation

TL;DR

This paper investigates how eigenvalues of Laplace--Beltrami boundary value problems on manifolds change continuously with domain deformation, providing estimates of their moduli of continuity under specific geometric conditions.

Contribution

It establishes resolvent continuity of the operators and derives estimates for the continuity moduli of eigenvalues and eigenfunctions under domain deformation measured by the Hausdorff--Pompeiu metric.

Findings

Eigenvalues depend continuously on domain shape under specified conditions

Continuity moduli of eigenvalues and eigenfunctions are explicitly estimated

Results apply to domains with boundaries locally represented as graphs

Abstract

The paper is pertaining to the spectral theory of operators and boundary value problems for differential equations on manifolds. Eigenvalues of such problems are studied as functionals on the space of domains. Resolvent continuity of the corresponding operators is established under domain deformation and estimates of continuity moduli of their eigenvalues $\slash$ eigenfunctions are obtained provided the boundary of nonperturbed domain is locally represented as a graph of some continuous function and domain deformation is measured with respect to the Hausdorff--Pompeiu metric.