1. The Dirichlet problem for nonlocal operators with kernels: Convex and nonconvex domains
- Author
-
Ros-Oton, Xavier and Valdinoci, Enrico
- Subjects
Regularity theory ,35R11 ,35B65 ,rough kernels ,fractional Laplacian ,integro-differential equations ,anisotropic media - Abstract
We study the interior regularity of solutions to a Dirichlet problem for anisotropic operators of fractional type. A prototype example is given by the sum of one-dimensional fractional Laplacians in fixed, given directions. We prove here that an interior differentiable regularity theory holds in convex domains. When the spectral measure is a bounded function and the domain is smooth, the same regularity theory applies. 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.
- Published
- 2015
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