Regional fractional Laplacians: Boundary regularity.

We study boundary regularity for solutions to a class of equations involving the so called regional fractional Laplacians (− Δ) Ω s , with Ω ⊂ R N. Recall that the regional fractional Laplacians are generated by Lévy-type processes which are not allowed to jump outside Ω. We consider weak solutions to the equation (− Δ) Ω s w (x) = p. v. ∫ Ω w (x) − w (y) | x − y | N + 2 s d y = 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 0 s (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 2 s − N / p (Ω ‾) in the case of zero Neumann boundary conditions. As a consequence for 2 s − N / p > 1 , w ∈ C 1 , 2 s − N p − 1 (Ω ‾). As what concerned the Dirichlet problem, we obtain w / δ 2 s − 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 having a certain growth at infinity 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. [ABSTRACT FROM AUTHOR]