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 -