TY - JOUR

T1 - The Dirichlet problem for nonlocal operators with singular kernels: Convex and nonconvex domains

AU - Ros-Oton, X.

AU - Valdinoci, E.

PY - 2016/1/22

Y1 - 2016/1/22

N2 - We study the interior regularity of solutions to the Dirichlet problem Lu=g in Ω, u=0 in RnØΩ, for anisotropic operators of fractional typeLu(x)=∫0+∞dρ∫Sn-1da(ω)2u(x)-u(x+ρω)-u(x-ρω)ρ1+2s. Here, a is any measure on Sn-1 (a prototype example for L is given by the sum of one-dimensional fractional Laplacians in fixed, given directions).When a∈C∞(Sn-1) and g is C∞(Ω), solutions are known to be C∞ inside Ω (but not up to the boundary). However, when a is a general measure, or even when a is L∞(Sn-1), solutions are only known to be C3s inside Ω.We prove here that, for general measures a, solutions are C1+3s-ε inside Ω for all ε>0 whenever Ω is convex. When a∈L∞(Sn-1), we show that the same holds in all C1,1 domains. In particular, solutions always possess a classical first derivative.The assumptions on the domain are sharp, since if the domain is not convex and the measure a is singular, we construct an explicit counterexample for which u is not C3s+ε for any ε>0 - even if g and Ω are C∞.

AB - We study the interior regularity of solutions to the Dirichlet problem Lu=g in Ω, u=0 in RnØΩ, for anisotropic operators of fractional typeLu(x)=∫0+∞dρ∫Sn-1da(ω)2u(x)-u(x+ρω)-u(x-ρω)ρ1+2s. Here, a is any measure on Sn-1 (a prototype example for L is given by the sum of one-dimensional fractional Laplacians in fixed, given directions).When a∈C∞(Sn-1) and g is C∞(Ω), solutions are known to be C∞ inside Ω (but not up to the boundary). However, when a is a general measure, or even when a is L∞(Sn-1), solutions are only known to be C3s inside Ω.We prove here that, for general measures a, solutions are C1+3s-ε inside Ω for all ε>0 whenever Ω is convex. When a∈L∞(Sn-1), we show that the same holds in all C1,1 domains. In particular, solutions always possess a classical first derivative.The assumptions on the domain are sharp, since if the domain is not convex and the measure a is singular, we construct an explicit counterexample for which u is not C3s+ε for any ε>0 - even if g and Ω are C∞.

UR - https://www.scopus.com/inward/record.uri?eid=2-s2.0-84947460565&doi=10.1016%2fj.aim.2015.11.001&partnerID=40&md5=0899c66377715329f97e977154ef7072

U2 - 10.1016/j.aim.2015.11.001

DO - 10.1016/j.aim.2015.11.001

M3 - Article

VL - 288

SP - 732

EP - 790

JO - Advances in Mathematics

JF - Advances in Mathematics

SN - 0001-8708

ER -