Nonlocal equations with measure data

TL;DR

This paper develops a comprehensive potential theory for nonlinear nonlocal equations with measure data, including existence, regularity, and estimates, extending classical local results to the nonlocal setting.

Contribution

It introduces a natural function class for solving the Dirichlet problem and establishes optimal Wolff potential estimates for solutions of nonlocal equations with measure data.

Findings

Established existence and regularity results for nonlocal equations with measure data.

Proved optimal nonlinear Wolff potential estimates for solutions.

Extended classical local potential theory results to nonlocal operators, including fractional p-Laplacian.

Abstract

We develop an existence, regularity and potential theory for nonlinear integrodifferential equations involving measure data. The nonlocal elliptic operators considered are possibly degenerate and cover the case of the fractional $p$-Laplacean operator with measurable coefficients. We introduce a natural function class where we solve the Dirichlet problem, and prove basic and optimal nonlinear Wolff potential estimates for solutions. These are the exact analogs of the results valid in the case of local quasilinear degenerate equations established by Boccardo & Gallou\"et \cite{BG1, BG2} and Kilpel\"ainen & Mal\'y \cite{KM1, KM2}. As a consequence, we establish a number of results which can be considered as basic building blocks for a nonlocal, nonlinear potential theory: fine properties of solutions, Calder\'on-Zygmund estimates, continuity and boundedness criteria are established via Wolff potentials. %In particular, optimal Lorentz spaces continuity criteria follow. A main tool is the introduction of a global excess functional that allows to prove a nonlocal analog of the classical theory due to Campanato \cite{camp}. Our results cover the case of linear nonlocal equations with measurable coefficients, and the one of the fractional Laplacean, and are new already in such cases.