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

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.