Fractional superharmonic functions and the Perron method for nonlinear integro-differential equations

By Janne Korvenpää, Tuomo Kuusi, Giampiero Palatucci. 2016

66 citations recorded by the source corpus.

We deal with a class of equations driven by nonlocal, possibly degenerate, integro-differential operators of differentiability order $$s\in (0,1)$$s∈(0,1) and summability growth $$p>1$$p>1, whose model is the fractional p-Laplacian with measurable coefficients. We state and prove several results for the corresponding weak supersolutions, as comparison principles, a priori bounds, lower semicontinuity, and many others. We then discuss the good definition of (s, p)-superharmonic functions, by also proving some related properties. We finally introduce the nonlocal counterpart of the celebrated Perron method in nonlinear Potential Theory.

Subfields: Applied Mathematics, Computational Theory and Mathematics

View Paper: https://www.semanticscholar.org/paper/7c3381fd5f2db48f892a027b7a0245cbcb0c2897

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In Applied Mathematics

In Computational Theory and Mathematics

Citations in the corpus

Equivalence of solutions to fractional p-Laplace type equations (2016, methodology): “Our main result, using the recent results in [15] and [16], states that solutions defined via comparison and viscosity solutions are exactly the same for the class of kernels Ker(Λ), see Section 2.”
Hölder regularity for weak solutions to nonlocal double phase problems (2021, background): “We also refer to [20, 22, 27, 28, 33, 34, 35] and references therein for various regularity results for nonlocal problems of fractional p-Laplacian type.”
Hölder estimates for viscosity solutions of equations of fractional p-Laplace type (2014, methodology): “In [27] and [26], Perron’s method is successfully applied to more general equations of this type.”
The obstacle problem for nonlinear integro-differential operators (2016, methodology): “sp (Rn) is what one expects in the nonlocal framework considered here (see [23]).”
Hölder regularity for nonlocal double phase equations (2019, methodology): “In this respect, let us observe that one cannot plainly applied the results for p-fractional minimizers as obtained in the recent paper [23] together with those in the forthcoming paper [21], whose proofs seem to be feasible only for a restrict class of kernels which cannot include a general modulat”