Quantitative estimates on localized finite differences for the fractional Poisson problem, and applications to regularity and spectral stability

TL;DR

This paper develops new quantitative estimates for localized finite differences in solutions to fractional Poisson problems, leading to improved understanding of regularity and spectral stability under domain perturbations.

Contribution

It introduces novel estimates for localized finite differences in fractional Poisson solutions and applies them to regularity and spectral stability analyses.

Findings

Enhanced regularity results in Besov spaces

Quantitative stability estimates for solutions under domain changes

Spectral stability estimates for eigenvalues and eigenfunctions

Abstract

We establish new quantitative estimates for localized finite differences of solutions to the Poisson problem for the fractional Laplace operator with homogeneous Dirichlet conditions of solid type settled in bounded domains satisfying the Lipschitz cone regularity condition. We then apply these estimates to obtain (i)~regularity results for solutions of fractional Poisson problems in Besov spaces; (ii)~quantitative stability estimates for solutions of fractional Poisson problems with respect to domain perturbations; (iii)~quantitative stability estimates for eigenvalues and eigenfunctions of fractional Laplace operators with respect to domain perturbations.