Inverse norm estimation of perturbed Laplace operators and corresponding eigenvalue problems

Abstract

The purpose of this paper is to reveal an eigenvalue problem corresponding to a perturbed Laplace operator \(-\Delta - Q\) for a linear bounded operator \(Q\) on \( L^2(\Omega) \). To verify the invertibility of the perturbed operator and explicitly evaluate its inverse norm, we evaluate the eigenvalues of the revealed problem based on Liu's approach with certain a priori error estimates. The accuracy is further improved using Lehman-Goerisch's method. The proposed method is applied to inverse norm estimation for a system of elliptic equations.