Vol. 13, No. 2, 2020

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Regularity estimates for elliptic nonlocal operators

Bartłomiej Dyda and Moritz Kassmann

Vol. 13 (2020), No. 2, 317–370
Abstract

We study weak solutions to nonlocal equations governed by integrodifferential operators. Solutions are defined with the help of symmetric nonlocal bilinear forms. Throughout this work, our main emphasis is on operators with general, possibly singular, measurable kernels. We obtain regularity results which are robust with respect to the differentiability order of the equation. Furthermore, we provide a general tool for the derivation of Hölder a priori estimates from the weak Harnack inequality. This tool is applicable for several local and nonlocal, linear and nonlinear problems on metric spaces. Another aim of this work is to provide comparability results for nonlocal quadratic forms.

Keywords
nonlocal Dirichlet forms, Hölder regularity estimates, weak Harnack inequality
Mathematical Subject Classification 2010
Primary: 31B05, 35B05, 35B45, 35R11, 47G20
Secondary: 60J75
Milestones
Received: 25 November 2015
Revised: 8 December 2017
Accepted: 9 April 2018
Published: 19 March 2020
Authors
Bartłomiej Dyda
Department of Pure and Applied Mathematics
Wrocław University of Science and Technology
Wrocław
Poland
Moritz Kassmann
Fakultät für Mathematik
Universität Bielefeld
Bielefeld
Germany