Nonlinear commutators for the fractional p-Laplacian and applications

By Armin Schikorra. 2015

67 citations recorded by the source corpus.

We prove a nonlinear commutator estimate concerning the transfer of derivatives onto testfunctions for the fractional p-Laplacian. This implies that solutions to certain degenerate nonlocal equations are higher differentiable. Also, weakly fractional p-harmonic functions which a priori are less regular than variational solutions are in fact classical. As an application we show that sequences of uniformly bounded $$\frac{n}{s}$$ns-harmonic maps converge strongly outside at most finitely many points.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/e94d8f0cc1414fe3d9c33d94ddd55a14ee75966a

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Citations in the corpus

Fractional div-curl quantities and applications to nonlocal geometric equations (2017, background): “|s p dxdy |x− y|n ≡ kdsuk p Lp(Rn). The operator divs(|dsu|p−2dsui) is often referred to as fractional p-Laplacian (−∆)s p, whose regularity theory has received a lot of attention lately, see, e.g., [14, 24, 5, 36]. The strategy for half-harmonic maps into spheres from Section 3, that is W12,2-harmo”
On a new class of fractional partial differential equations (2014, background): “quot; with contributions by a number of authors: Brasco and Lindgren [5], Di Castro, Kuusi, and Palatucci [14,15], Korvenpaa, Kuusi, and Palatucci [28], Kuusi, Mingione and Sire [30,31] and Schikorra [39]. 1 Technically the non-local gradients in [33] were dened with integration over a bounded domai”
Fractional superharmonic functions and the Perron method for nonlinear integro-differential equations (2016, background): “, the strong three-commutators estimates introduced in [6,7] to deduce the regularity of weak fractional harmonic maps (see also [40]), the strong barriers and density estimates in [37, 39], the pseudo-differential commutator and energy estimates in [35, 36], and many other successful techniques whi”
Sharp commutator estimates via harmonic extensions (2016, methodology): “more, it would be interesting to see how our methods performs for nonlinear commutators; for example the Caldero`n commutators, see e.g. [11, 55], or nonlinear commutators related to the p-Laplacian, [45, 47]. Also, as S. Chanillo pointed out to us after reading an earlier version of this article, i”
Hölder regularity for nonlocal double phase equations (2019, background): “For this, one should extend the recent nonlocal Gehring-type theorems proven in [24,34].”
Equivalence of solutions to fractional p-Laplace type equations (2016, background): “ properties for equations like (1.1) with a measure are studied. There has also been some recent progress in terms of higher integrability results; in [17], [19], and [6] the case p= 2 is treated. In [22] and [3] the general case when p≥ 2 is considered. Worth mentioning is also that there are other”
The obstacle problem for nonlinear integro-differential operators (2016, background): “Nevertheless, some related regularity results have been very recently achieved in this context, in [4,12,23,13,25,26,27,28,41] and many others, where often a fundamental role to understand the nonlocality of the nonlinear operators L has been played by a special quantity,”