کد مقاله | کد نشریه | سال انتشار | مقاله انگلیسی | نسخه تمام متن |
---|---|---|---|---|
6425563 | 1633800 | 2016 | 59 صفحه PDF | دانلود رایگان |
We study the interior regularity of solutions to the Dirichlet problem Lu=g in Ω, u=0 in RnâΩ, for anisotropic operators of fractional typeLu(x)=â«0+âdÏâ«Snâ1da(Ï)2u(x)âu(x+ÏÏ)âu(xâÏÏ)Ï1+2s. Here, a is any measure on Snâ1 (a prototype example for L is given by the sum of one-dimensional fractional Laplacians in fixed, given directions).When aâCâ(Snâ1) and g is Câ(Ω), solutions are known to be Câ inside Ω (but not up to the boundary). However, when a is a general measure, or even when a is Lâ(Snâ1), solutions are only known to be C3s inside Ω.We prove here that, for general measures a, solutions are C1+3sâϵ inside Ω for all ϵ>0 whenever Ω is convex. When aâLâ(Snâ1), 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 Câ.
Journal: Advances in Mathematics - Volume 288, 22 January 2016, Pages 732-790