Regional fractional Laplacians: Boundary regularity

Abstract

We study boundary regularity for solutions to a class of equations involving the so called regional fractional Lapacians (−∆) s Ω , with Ω ⊂ R N . Recall that the regional fractional Laplacians are generated by symmetric stable processes which are not allowed to jump outside Ω. We consider weak solutions to the equation|x−y| N +2s dy = f (x), for s ∈ (0, 1) and Ω ⊂ R N , subject to zero Neumann or Dirichlet boundary conditions. The boundary conditions are defined by considering w as well as the test functions in the fractional Sobolev spaces H s (Ω) or H s 0 (Ω) respectively. While the interior regularity is well understood for these problems, little is known in the boundary regularity, mainly for the Neumann problem. Under optimal regularity assumptions on Ω and provided f ∈ L p (Ω), we show that w ∈ C 2s−N/p (Ω) in the case of zero Neumann boundary conditions. As a conse-As what concerned the Dirichlet problem, we obtain w/δ 2s−1 ∈ C 1−N/p (Ω), provided p > N and s ∈ (1/2, 1), where δ(x) = dist(x, ∂Ω).To prove these results, we first classify all solutions, with some growth control, when Ω is a half-space and the right hand side is zero. We then carry over a fine blow up and some compactness arguments to get the results.2010 Mathematics Subject Classification. 35R11, 42B37.