Weak and viscosity solutions of the fractional laplace equation

Raffaella Servadei, Enrico Valdinoci

Research output: Contribution to journalArticle

65 Citations (Scopus)

Abstract

Aim of this paper is to show that weak solutions of the following fractional Laplacian equation (Equation) are also continuous solutions (up to the boundary) of this problem in the viscosity sense. Here s G (0, 1) is a fixed parameter, Ω is a bounded, open subset of ℝn (n ≥ 1) with C2-boundary, and (-Δ)s is the fractional Laplacian operator, that may be defined as (Equation) for a suitable positive normalizing constant c(n,s), depending only on n and s. In order to get our regularity result we first prove a maximum principle and then, using it, an interior and boundary regularity result for weak solutions of the problem. As a consequence of our regularity result, along the paper we also deduce that the first eigenfunction of (-Δ)s is strictly positive in Ω.

Original languageEnglish
Pages (from-to)133-154
Number of pages22
JournalPublicacions Matematiques
Volume58
Issue number1
DOIs
Publication statusPublished - 1 Jan 2014
Externally publishedYes

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