Non-local gradient dependent operators

Local behavior of fractional p-minimizers

“…Also, it is worth mentioning that other fractional Caccioppoli-type inequalities have been recently used in different contexts (see, for instance, [21,22,9]), although none of them takes into account the tails. 1 Let us finally comment some recent results in the literature. In [7] we prove Harnack-type inequalities with tail for weak supersolutions and solutions to (1.2).…”

“…These can be applied to obtain Hölder continuity of the solutions. However, the 1 We recently discovered that Kassmann proved similar Caccioppoli estimates with tail terms in the linear case, when p = 2; see [10].…”

“…Nonetheless, some partial results are known. Firstly, we would like to cite the higher regularity contributions for viscosity solutions in the case when s is close to 1 proven in the recent interesting paper [1]; see, also, [16]. Secondly, the analysis in the papers [3] and [18] considers the special case when p is suitably large -thus falling in the Morrey embedding case when concerning regularity.…”

Nonlinear commutators for the fractional p-Laplacian and applications

“…The fractional p-Laplacian has recently received quite some interest, for example we refer to [2,9,10,21,18,16,13,17,23]. Higher regularity is one interesting and very challenging question where only very partial results are known, e.g.…”

“…Higher regularity is one interesting and very challenging question where only very partial results are known, e.g. in [2] they obtain for s ≈ 1 estimates in C 1,α .…”