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

Abstract

We deal with a class of equations driven by nonlocal, possibly degenerate, integro-differential operators of differentiability order s ∈ (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.Keywords Quasilinear nonlocal operators · fractional Sobolev spaces · fractional Laplacian · nonlocal tail · Caccioppoli estimates · obstacle problem · comparison estimates · fractional superharmonic functions · the Perron Method

The obstacle problem for nonlinear integro-differential operators

“…It is worth noticing that the summability assumption of u belonging to the tail space L p−1 sp (R n ) is what one expects in the nonlocal framework considered here (see [23]).…”

Perron’s Method and Wiener’s Theorem for a Nonlocal Equation

“…It is plain to verify that the solution of the obstacle problem is a weak supersolution. Thus it is lower semicontinuous by [22], again by the modification in Lemma 29. It is a solution in the open set {v > ψ} ∩ D where the obstacle does not hinder (under our assumption that ψ is continuous).…”

“…Furthermore, (local) weak supersolutions can be made lower semicontinuous in by changing them in a set of measure zero. See [22] for this regularity result. In the same way, (local) subsolutions are upper semicontinuous 3 .…”

Equivalence of solutions to fractional p-Laplace type equations

“…Our main result, using the recent results in [15] and [16], states that solutions defined via comparison and viscosity solutions are exactly the same for the class of kernels Ker(Λ), see Section 2.…”

“…([15, Theorem 12]). Let u be the lower semicontinuous representative of a weak supersolution of (2.4) satisfying…”