Equivalence of solutions to fractional p-Laplace type equations

By Janne Korvenpää, Tuomo Kuusi, Erik Lindgren. Journal des Mathématiques Pures et Appliquées, 2016

107 citations recorded by the source corpus.

In this paper, we study different notions of solutions of nonlocal and nonlinear equations of fractional $p$-Laplace type $${\rm P.V.} \int_{\mathbb R^n}\frac{|u(x)-u(y)|^{p-2}(u(x)-u(y))}{|x-y|^{n+sp}}\,dy = 0.$$ Solutions are defined via integration by parts with test functions, as viscosity solutions or via comparison. Our main result states that for bounded solutions, the three different notions coincide.

Subfields: Applied Mathematics

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In Applied Mathematics

Citations in the corpus

On the regularity theory for mixed local and nonlocal quasilinear elliptic equations (2021, unknown): “For lower semicontinuity results of supersolutions, we refer to Korvenpää-Kuusi-Lindgren [29].”
Hölder regularity for nonlocal double phase equations (2019, unknown): “d vice versa. 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 g”
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.”
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.”