Inserted: 15 nov 2018
Last Updated: 15 nov 2018
Journal: Nonlinear Analysis
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 $p\in(1,\infty)$, whose model is the fractional $p$-Laplacian operator with measurable coefficients. We review several recent results for the corresponding weak solutionssupersolutions, as comparison principles, a priori bounds, lower semicontinuity, boundedness, H\"older continuity up to the boundary, and many others. We then discuss the good definition of $(s,p)$-superharmonic functions, and the nonlocal counterpart of the Perron method in nonlinear Potential Theory, together with various related results. We briefly mention some basic results for the obstacle problem for nonlinear integro-differential equations. Finally, we present the connection amongst the fractional viscosity solutions, the weak solutions and the aforementioned $(s,p)$-superharmonic functions, together with other important results for this class of equations when involving general measure data, and a surprising fractional version of the Gehring lemma.
We sketch the corresponding proofs of some of the results presented here, by especially underlining the development of new fractional localization techniques and other recent tools. Various open problems are listed throughout the paper.
Keywords: fractional Sobolev spaces, nonlocal tail, comparison estimates, Perron Method, integro-differential operators, Harnack inequalities, Caccioppoli inequalities, fractional viscosity solutions