Regional fractional Laplacians: Boundary regularity

• Published in: Journal of Differential Equations Vol. 320; pp. 598 - 658
• Main Author: Fall, Mouhamed Moustapha
• Format: Journal Article
• Language: English
• Published: Elsevier Inc 25.05.2022
• Subjects: Boundary regularity; Censored fractional Laplacians; Censored processes; Gradient estimates; Hölder regularity; Regional fractional Laplacian; 35R11; Censored fractional Laplacians; Regional fractional Laplacian; Boundary regularity; Hölder regularity; Censored processes; 42B37; Gradient estimates
• ISSN: 0022-0396; 1090-2732
• DOI: 10.1016/j.jde.2022.02.040

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 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)Ωsw(x)=p.v.∫Ωw(x)−w(y)|x−y|N+2sdy=f(x), for s∈(0,1) and Ω⊂RN, 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 Hs(Ω) or H0s(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∈Lp(Ω), we show that w∈C2s−N/p(Ω‾) in the case of zero Neumann boundary conditions. As a consequence for 2s−N/p>1, w∈C1,2s−Np−1(Ω‾). As what concerned the Dirichlet problem, we obtain w/δ2s−1∈C1−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.