Communications in Mathematical Sciences

Volume 16 (2018)

Number 4

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

Pages: 913 – 961



Goro Akagi (Mathematical Institute, Tohoku University, Sendai, Miyagi, Japan; Helmholtz Zentrum München, Neuherberg, Germany; and Zentrum Mathematik, Technische Universität München, Germany)

Giulio Schimperna (Dipartimento di Matematica, Università di Pavia, Italy)

Antonio Segatti (Dipartimento di Matematica, Università di Pavia, Italy)

Laura V. Spinolo (Istituto di Matematica Applicata e Tecnologie Informatiche (IMATI), Pavia, Italy)


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.

2010 Mathematics Subject Classification

35B30, 35J20, 35R11

Dedicated to Gianni Gilardi on the occasion of his 70th anniversary, with friendship and admiration.

All authors are supported by the JSPS-CNR bilateral joint research project “VarEvol: Innovative Variational Methods for Evolution Equations”. GA is supported by JSPS KAKENHI Grant Number JP16H03946, JP16K05199, JP17H01095, by the Alexander von Humboldt Foundation and by the Carl Friedrich von Siemens Foundation. AS and GS are supported by the MIUR-PRIN Grant 2010A2TFX2 “Calculus of Variations” and LVS is supported by the MIUR-PRIN Grant “Nonlinear Hyperbolic Partial Differential Equations, Dispersive and Transport Equations: theoretical and applicative aspects”. AS, GS and LVS are also members of the GNAMPA (Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni) group of INdAM (Istituto Nazionale di Alta Matematica).

Received 2 June 2017

Accepted 17 March 2018

Published 31 October 2018