The Neumann problem for the fractional Laplacian: regularity up to the boundary

Regional fractional Laplacians: Boundary regularity

“…We prove in this paper that, under optimal regularity assumptions on Ω and f , the solutions to (1.5) and (1.6) are Hölder continuous up to the boundary. Our result for solutions to (1.5), improves those obtained in [2], since under the same assumptions, we obtain that u ∈ C min(2s−N/p,1−ε) (Ω), for all s, ε ∈ (0, 1). Moreover for 2s > 1, we also obtain Hölder estimates, up to the boundary, of the gradient of both solutions to (1.5) and (1.6), provided 2s − N/p > 1.…”

“…We notice that prior to the recent paper [2] and the present paper, even the continuity of solutions to (1.5) up to the boundary was an open question.…”

“…However, the boundary regularity for problem (1.5) is less understood and the only paper, beside the present one, dealing with this appeared few days ago. Indeed, Audrito, Felipe-Navarro and Ros-Oton showed recently in [2] that if Ω is of class C 1 and f ∈ L p (Ω), p > N 2s , then u ∈ C α (Ω), for some α > 0. Moreover for s ∈ (1/2, 1) and 2s − N/p > 2s − 1 they obtain u ∈ C 2s−1+α (Ω), for some α > 0.…”

“…Moreover for s ∈ (1/2, 1) and 2s − N/p > 2s − 1 they obtain u ∈ C 2s−1+α (Ω), for some α > 0. We would like to mention that the authors in [2] considered an other interesting fractional order equation with exterior Neumann condition.…”

“…We then carry out a blow up and some compactness arguments to obtain Hölder continuity up to the boundary. We note that the classification of the solutions on the half-space was left as a challenging open problem in [2]. To achieve it, we use several ingredients of independent interest, including some reflection principles, Hardytype inequalities and the Caffarelli-Silvestre extension [7].…”