Nonlocal Equations with Measure Data

By Tuomo Kuusi, Giuseppe Mingione, Yannick Sire. 2014

253 citations recorded by the source corpus.

We develop an existence, regularity and potential theory for nonlinear integrodifferential equations involving measure data. The nonlocal elliptic operators considered are possibly degenerate and cover the case of the fractional p-Laplacean operator with measurable coefficients. We introduce a natural function class where we solve the Dirichlet problem, and prove basic and optimal nonlinear Wolff potential estimates for solutions. These are the exact analogs of the results valid in the case of local quasilinear degenerate equations established by Boccardo and Gallouët (J Funct Anal 87:149–169, 1989, Partial Differ Equ 17:641–655, 1992) and Kilpeläinen and Malý (Ann Scuola Norm Sup Pisa Cl Sci (IV) 19:591–613, 1992, Acta Math 172:137–161, 1994). As a consequence, we establish a number of results that can be considered as basic building blocks for a nonlocal, nonlinear potential theory: fine properties of solutions, Calderón–Zygmund estimates, continuity and boundedness criteria are established via Wolff potentials. A main tool is the introduction of a global excess functional that allows us to prove a nonlocal analog of the classical theory due to Campanato (Ann Mat Pura Appl (IV) 69:321–381, 1965). Our results cover the case of linear nonlocal equations with measurable coefficients, and the one of the fractional Laplacean, and are new already in such cases.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/317f639db7b65a2efaeeba889b7302a01dc30c28

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Applied Mathematics

In Mathematical Physics

Citations in the corpus

Nonlinear commutators for the fractional p-Laplacian and applications (2015, unknown): “The fractional p-Laplacian has recently received quite some interest, for example we refer to [2, 9, 10, 21, 18, 16, 13, 17, 23].”
Fractional superharmonic functions and the Perron method for nonlinear integro-differential equations (2016, background): “This is one of the novelty with respect to the analog of the definition of supersolutions in the local case, say when s = 1, and it is necessary since here one has to use in a precise way the definition in (10) to deal with the long-range interactions; see Remark 8 below, and also, the regularity es”
Nonlocal Diffusion and Applications (2015, background): “Many fundamental topics slipped completely out of these notes: just to name a few, the topological methods and the fine regularity theory in the fractional cases are not presented here, the fully nonlinear or singular/degenerate equations are not taken into account, only very few applications are di”
Hölder regularity for weak solutions to nonlocal double phase problems (2021, background): “Such approaches are further applied to several research areas including obstacle problems [29] and measure data problems [30].”
The effect of the Hardy potential in some Calder\'on-Zygmund properties for the fractional Laplacian (2015, background): “lt; N N−2s and (−∆)s 2 (u) ∈ Lr(Ω) for all r∈ 1, N N−s . Moreover, according to Theorem 5 (C) of Charter 5 in [33], we conclude that u∈ Ws1,q1 0 (Ω) for all s1 <sand for all q1 < N N−s. See too [27] and [2] for a simple proof. Remark 4.11. As a consequence of this result, together with the wea”
Higher Hölder regularity for the fractional p-Laplacian in the superquadratic case (2017, background): “Here we seize the opportunity to mention the paper [26], in which regularity for equations of the type (1.”
The Dirichlet Problem for the fractional p-Laplacian evolution equation (2015, background): “aviour for t near the extinction time is an open problem. • The problem with right-hand side, ∂tu + Lp,su = f is interesting. Existence and uniqueness of solutions is granted by the usual theory. See [38] for recent results on the elliptic problem. • Some authors have considered operators like Lp,s ”
On the fractional p-Laplacian equations with weight and general datum (2016, background): “The argument considered in [23] seems to be complicated to be adapted to our case.”
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”
The Brezis–Nirenberg problem for the fractional p-Laplacian (2015, unknown): “[32], and Lindgren [34].”
Regularity results and Harnack inequalities for minimizers and solutions of nonlocal problems: A unified approach via fractional De Giorgi classes (2016, background): “The above scale-invariant quantity has already been considered in several papers, such as [26, 27, 9, 44].”