  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.

Analysis of PDEs