Regional fractional Laplacians: Boundary regularity
2022; Elsevier BV; Volume: 320; Linguagem: Inglês
10.1016/j.jde.2022.02.040
ISSN1090-2732
Autores Tópico(s)Nonlinear Differential Equations Analysis
ResumoWe 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.
Referência(s)