Fractional superharmonic functions and the Perron method for nonlinear integro-differential equations

TL;DR

This paper studies nonlocal, possibly degenerate integro-differential equations driven by fractional operators, establishing fundamental properties of solutions and introducing a nonlocal Perron method for nonlinear potential theory.

Contribution

It develops a comprehensive theory for fractional superharmonic functions and introduces a nonlocal Perron method for nonlinear integro-differential equations.

Findings

Comparison principles for weak supersolutions

A priori bounds and lower semicontinuity results

Definition and properties of nonlocal superharmonic functions

Abstract

We deal with a class of equations driven by nonlocal, possibly degenerate, integro-differential operators of differentiability order $s\in (0,1)$ and summability growth $p>1$, whose model is the fractional $p$-Laplacian with measurable coefficients. We state and prove several results for the corresponding weak supersolutions, as comparison principles, a priori bounds, lower semicontinuity, and many others. We then discuss the good definition of $(s,p)$-superharmonic functions, by also proving some related properties. We finally introduce the nonlocal counterpart of the celebrated Perron method in nonlinear Potential Theory.