Higher Sobolev regularity for the fractional p-Laplace equation in the superquadratic case

By Lorenzo Brasco, Erik Lindgren. 2015

133 citations recorded by the source corpus.

Abstract We prove that for p ≥ 2 , solutions of equations modeled by the fractional p-Laplacian improve their regularity on the scale of fractional Sobolev spaces. Moreover, under certain precise conditions, they are in W loc 1 , p and their gradients are in a fractional Sobolev space as well. The relevant estimates are stable as the fractional order of differentiation s reaches 1.

Subfields: Applied Mathematics, Computational Theory and Mathematics

View Paper: https://www.semanticscholar.org/paper/f9ceb7195efc3ac3c383fa3402a0e4790e74ecf3

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In Applied Mathematics

In Computational Theory and Mathematics

Citations in the corpus

Gradient regularity in mixed local and nonlocal problems (2022, background): “In [9], a related nonlocal quantity, called snail, was considered, namely (3.”
Fractional superharmonic functions and the Perron method for nonlinear integro-differential equations (2016, background): “It is worth citing the higher regularity contributions in [2,9], together with the viscosity approach in the recent paper [27], and [4] for related existence and uniqueness results in the case when p goes to infinity.”
On a new class of fractional partial differential equations (2014, background): “c complex interpolation theory" of Caffarelli and Stinga [11], and a "divergence form elliptic real interpolation theory" 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 ”
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 the regularity theory for mixed local and nonlocal quasilinear elliptic equations (2021, background): “See also Brasco-Lindgren-Scikorra [7] for higher regularity results.”
Regularity results and Harnack inequalities for minimizers and solutions of nonlocal problems: A unified approach via fractional De Giorgi classes (2016, background): “irichlet problem has been then proved by A. Iannizzotto, S. Mosconi and M. Squassina in [35]. The H¨older continuity of ﬁrst derivatives is, to the best of our knowledge, still unknown. See the paper [8] by L. Brasco and E. Lindgren for regularity results in fractional Sobolev spaces of higher order”
Recent developments in problems with nonstandard growth and nonuniform ellipticity (2021, background): “After the analog of the nonlocal p-Laplacian has been considered (see, for example, [34,35,76,77,113]), several papers have been devoted to the study of nonlocal operators having non polynomial growth with respect to the underlying Gagliardo norm.”
The Brezis–Nirenberg problem for the fractional p-Laplacian (2015, methodology): “Regularity of solutions was obtained in Brasco and Lindgren [5], Di Castro et al.”
Hölder regularity for nonlocal double phase equations (2019, methodology): “[6] L. Brasco, E. Lindgren, A. Schikorra: Higher Hölder regularity for the fractional pLaplacian in the superquadratic case.”
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, Tail(u;x 0 , r) := ✓”
Higher Hölder regularity for the fractional p-Laplacian in the superquadratic case (2017, methodology): “er, we need to employ a blow-up argument similar to that used by Caﬀarelli and Silvestre in [12], see Proposition 6.2 below. Noteworthy is that the methods in this paper are quite similar to those in [7] by the ﬁrst two authors, where we only test with δhu(x) |h|ϑ , and iterate to obtain higher diﬀe”