2015/02/03 by Xavier Ros-Oton, Xavier Ros‐Oton, Enrico Valdinoci +2 · 1 citation
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations #Numerical methods in inverse problems #math.AP
paper · pdf · doi:10.48550/arxiv.1502.00782
openalex publication_date 2015/02/03 · arxiv created 2015/10/31 · arxiv updated 2015/11/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We study the interior regularity of solutions to the Dirichlet problem Lu=g in Ω, u=0 in \Rn∖Ω, for anisotropic operators of fractional type Lu(x)= ∫0+∞ dρ∫Sn-1 da(ω) \frac 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 spectral measure 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^∞.