Regional fractional Laplacians: Boundary regularity

We study boundary regularity for solutions to a class of equations involving the so called regional fractional Laplacians (−Δ)Ωs, with Ω⊂RN. Recall that are generated by Lévy-type processes which not allowed jump outside Ω. consider weak equation (−Δ)Ωsw(x)=p.v.∫Ωw(x)−w(y)|x−y|N+2sdy=f(x), s∈(0,1) and Ω⊂RN, subject zero Neumann or Dirichlet conditions. The conditions defined considering w as well test functions in Sobolev spaces Hs(Ω) H0s(Ω) respectively. While interior is understood these problems, little known regularity, mainly problem. Under optimal assumptions on Ω provided f∈Lp(Ω), we show w∈C2s−N/p(Ω‾) case As consequence 2s−N/p>1, w∈C1,2s−Np−1(Ω‾). what concerned problem, obtain w/δ2s−1∈C1−N/p(Ω‾), p>N s∈(1/2,1), where δ(x)=dist(x,∂Ω). To prove results, first classify all having certain growth at infinity when half-space right hand side zero. then carry over fine blow up some compactness arguments get results.