A note on quasilinear equations with fractional diffusion

2020/03/29 by Abdellaoui, Boumediene, Ochoa, Pablo, Peral, Ireneo · 1 citation
#35B65 #35D40 #35J62 #Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2003.13069

Abstract

In this paper, we study the existence of distributional solutions of the following non-local elliptic problem \lbrace (-Δ)^su + |∇ u|^p =f in Ω u=0 in ℝ^N∖ Ω, s ∈ (1/2, 1). . We are interested in the relation between the regularity of the source term f, and the regularity of the corresponding solution. If p<2s, that is the natural growth, we are able to show the existence for all f∈ L¹(Ø). In the subcritical case, that is, for p < p_*:=N/(N-2s+1), we show that solutions are C^{1, α} for f ∈ L^m, with m large enough. In the general case, we achieve the same result under a condition on the size of the source. As an application, we may show that for regular sources, distributional solutions are viscosity solutions, and conversely.

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• Global regularity results for the fractional heat equation and application to a class of non-linear KPZ problems

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