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- ItemThe Dirichlet problem for nonlocal operators with kernels: Convex and nonconvex domains(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2015) Ros-Oton, Xavier; Valdinoci, EnricoWe 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.