Calculus of Variations and Geometric Measure Theory
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T. Kuusi - G. Mingione - Y. Sire

Nonlocal equations with measure data

created by mingione on 29 Jun 2014
modified on 29 Apr 2015


Published Paper

Inserted: 29 jun 2014
Last Updated: 29 apr 2015

Journal: Comm. Math. Phys.
Volume: 337
Pages: 1317-1368
Year: 2015


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. 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.


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