The Dirichlet problem for thep-fractional Laplace equation

By Giampiero Palatucci. Nonlinear Analysis, 2018

69 citations recorded by the source corpus.

Abstract We deal with a class of equations driven by nonlocal, possibly degenerate, integro-differential operators of differentiability order s ∈ ( 0 , 1 ) and summability growth p ∈ ( 1 , ∞ ) , whose model is the fractional p -Laplacian operator with measurable coefficients. We review several recent results for the corresponding weak solutions/supersolutions, as comparison principles, a priori bounds, lower semicontinuity, boundedness, Holder continuity up to the boundary, and many others. We then discuss the good definition of ( s , p ) -superharmonic functions, and the nonlocal counterpart of the Perron method in nonlinear Potential Theory, together with various related results. We briefly mention some basic results for the obstacle problem for nonlinear integro-differential equations. Finally, we present the connection amongst the fractional viscosity solutions, the weak solutions and the aforementioned ( s , p ) -superharmonic functions, together with other important results for this class of equations when involving general measure data, and a surprising fractional version of the Gehring lemma. We sketch the corresponding proofs of some of the results presented here, by especially underlining the development of new fractional localization techniques and other recent tools. Various open problems are listed throughout the paper.

Subfields: Applied Mathematics, Computational Theory and Mathematics

View Paper: https://www.semanticscholar.org/paper/8e0e52209cdf7c74a9900dbdb8fc98d5339b243a

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In Applied Mathematics

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Citations in the corpus

Fractional double-phase patterns: concentration and multiplicity of solutions (2020, background): “for which several existence and multiplicity results have been obtained in this last decade; see for instance [8,13,34,61,65,71] and the references therein, and [36,48] for some interesting regularity results.”
Hölder regularity for nonlocal double phase equations (2019, background): “The measurable kernels Ksp and Ktq essentially behave like (s, p) and (t, q)-kernels, respectively; see [17, 31] for the basics on fractional Sobolev spaces.”
Hölder regularity for weak solutions to nonlocal double phase problems (2021, background): “For a general overview of the history and related topics, we refer to the survey paper [38] and the monographs [3, 32].”