TY - JOUR

T1 - Weak and viscosity solutions of the fractional laplace equation

AU - Servadei, Raffaella

AU - Valdinoci, Enrico

PY - 2014/1/1

Y1 - 2014/1/1

N2 - 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.

AB - 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.

KW - Fractional Laplacian

KW - Integrodifferential operators

KW - Regularity theory

KW - Viscosity solutions

KW - Weak solutions

UR - http://www.scopus.com/inward/record.url?scp=84890847878&partnerID=8YFLogxK

U2 - 10.5565/PUBLMAT_58114_06

DO - 10.5565/PUBLMAT_58114_06

M3 - Article

AN - SCOPUS:84890847878

VL - 58

SP - 133

EP - 154

JO - Publicacions Matematiques

JF - Publicacions Matematiques

SN - 0214-1493

IS - 1

ER -