Inverse norm estimation of perturbed Laplace operators and corresponding eigenvalue problems

TL;DR

This paper introduces a new eigenvalue-based method to estimate the inverse norm of perturbed Laplace operators, aiding in the numerical analysis of semi-linear elliptic systems and their eigenvalue problems.

Contribution

It develops a novel approach combining Liu's method and the Temple-Lehman-Goerisch method for verifying invertibility and estimating inverse norms of perturbed Laplace operators.

Findings

Effective estimation of inverse norms for perturbed Laplace operators.

Application to Dirichlet boundary value problems in Lotka-Volterra systems.

Validation of the method's efficacy in numerical proofs.

Abstract

In numerical existence proofs for solutions of the semi-linear elliptic system, evaluating the norm of the inverse of a perturbed Laplace operator plays an important role. We reveal an eigenvalue problem to design a method for verifying the invertibility of the operator and evaluating the norm of its inverse based on Liu's method and the Temple-Lehman-Goerisch method. We apply the inverse-norm's estimation to the Dirichlet boundary value problem of the Lotka-Volterra system with diffusion terms and confirm the efficacy of our method.