Harnack inequality for nonlocal problems with non-standard growth

Abstract

We prove a full Harnack inequality for local minimizers, as well as weak solutions to nonlocal problems with non-standard growth. The main auxiliary results are local boundedness and a weak Harnack inequality for functions in a corresponding De Giorgi class. This paper builds upon a recent work on regularity estimates for such nonlocal problems by the same authors.

Nonlocal Harnack inequality for fractional elliptic equations with Orlicz growth

“…We mention that, very recently, Chaker, Kim, and Weidner [13] proved similar Harnack inequalities for (1.1) that are robust as $$s \nearrow 1$$. Their approach is based on [12, 36] and quite different from the one used in this paper.…”

Nonlocal Harnack inequalities in the Heisenberg group

“…In the same spirit, it could be interesting to understand if our methods do apply in non-Euclidean settings for even more general nonlocal nonstandard growth equations, as the one recently considered in [10,40].…”

Regularity results for mixed local and nonlocal double phase functionals

“…Such mixed local and nonlocal problems have an anisotropic feature, and they naturally link to other kinds of problems, namely nonlocal problems with nonstandard growth. Recently, the methods and results in [29,30] have been extended to nonlocal problems with various nonstandard growth conditions [7,8,9,27,32,33,42,46]; see also [14,15,16] for the extensions of the methods in [23]. Specifically, in [9] local boundedness and Hölder continuity results were proved for the nonlocal double phase problem…”