WIAS Preprint No. 2076, (2015)

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


  • Ros-Oton, Xavier
  • Valdinoci, Enrico
    ORCID: 0000-0001-6222-2272

2010 Mathematics Subject Classification

  • 35B65 35R11


  • Regularity theory, integro-differential equations, fractional Laplacian, anisotropic media, rough kernels




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.

Appeared in

  • Adv. Math., 288 (2016) pp. 732--790.

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