Local behavior of fractional $p$-minimizers

By Agnese Di Castro, Tuomo Kuusi, Giampiero Palatucci. 2015

437 citations recorded by the source corpus.

We extend the De Giorgi-Nash-Moser theory to nonlocal, possibly degenerate integro-differential operators.

Subfields: Applied Mathematics, Modeling and Simulation

View Paper: https://www.semanticscholar.org/paper/d01c6b95f2993cd58cbc09a267573382f4e077bd

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In Applied Mathematics

In Modeling and Simulation

Citations in the corpus

Gradient regularity in mixed local and nonlocal problems (2022, methodology): “These techniques ultimately rely on those used in the nonlocal case [9–11, 30, 31, 44–47] for purely nonlocal operators.”
The second eigenvalue of the fractional p-Laplacian (2014, neutral): “As in [8], we define the nonlocal tail of a function φ by”
Hölder regularity for nonlocal double phase equations (2019, background): “It could be interesting to treat unbounded solutions, namely by truncation and dealing with the resulting error term as a right hand-side, in the same flavour of the papers [15,16], where the local boundedness for p-fractional minimizers is proven, given that a precise quantity, the so-called “nonlo”
Global H\"older regularity for the fractional $p$-Laplacian (2014, methodology): “2 from [7], and it could be deduced in the case p > 2−s/N using the results of [14], and in the case p > 1/(1 − s) using [15].”
A Hopf's lemma and a strong minimum principle for the fractional p-Laplacian (2016, background): “3 in [12], it is showed that right side of the above inequality is bounded by”
Stability of parabolic Harnack inequalities for symmetric non-local Dirichlet forms (2016, methodology): “ociated heat kernel and jumping kernel. See the equivalence condition (3) in main result Theorem 1.17. (ii) We adopt some PDE’s techniques from the recent study of fractional p-Laplacian operators in [CKP1] to derive some useful properties of the process X. We emphasis that, to get the stability of ”
The obstacle problem for nonlinear integro-differential operators (2016, methodology): “Then Corollary 1 says that u is a weak solution in Br0(x0), and consequently we can use the results in [13], by also noticing that in the proofs there it makes no difference to assume u ∈ W loc ( )∩L p−1 sp (Rn) instead of u ∈ Ws,p(Rn).”
On a new class of fractional partial differential equations (2014, methodology): “We can cite, for instance, the regularity theory of nonlocal equations developed by Fukushima [13] and Kassmann [16, 17] in the symmetric case, and Barles, Chasseigne and Imbert [3], Ca arelli and Silvestre [6], and Di Castro, Kuusi and Palatucci [10] where no such symmetry is assumed.”
Hölder regularity for weak solutions to nonlocal double phase problems (2021, methodology): “[41] G. Palatucci, A. Pisante, and Y. Sire, Subcritical approximation of a Yamabe-type nonlocal equation: a gamma- convergence approach, Ann.”
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.”
On the fractional p-Laplacian equations with weight and general datum (2016, neutral): “Notice that from [14], we know that |v(x) − v(y)|(v(x) − v(y))( ψ(x) v(x)p−1 − ψ(y) v(y)p−1 ) ≤ −C1| log(v(x)) − log(v(y))| ψ(y) + C2(ψ(x) − ψ(y)) .”
Two-sided estimates of heat kernels of jump type Dirichlet forms (2018, neutral): “This is a quite elaborate argument that we have borrowed from the paper of Di Castro, Kuusi, Palatucci [12] and that is implemented in Lemma 4.”