We study the regularity up to the boundary of solutions to the Neumann problem for the fractional Laplacian. We prove that if u is a weak solution of (−∆) s u = f in Ω, Nsu = 0 in Ω c , then u is C α up tp the boundary for some α > 0. Moreover, in case s > 1 2 , we then show that u ∈ C 2s−1+α (Ω). To prove these results we need, among other things, a delicate Moser iteration on the boundary with some logarithmic corrections.Our methods allow us to treat as well the Neumann problem for the regional fractional Laplacian, and we establish the same boundary regularity result.Prior to our results, the interior regularity for these Neumann problems was well understood, but near the boundary even the continuity of solutions was open.

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In this work we prove the uniqueness of solutions to the nonlocal linear equation $$L \varphi - c(x)\varphi = 0$$ L φ - c ( x ) φ = 0 in $$\mathbb {R}$$ R , where L is an elliptic integro-differential operator, in the presence of a positive solution or of an odd solution vanishing only at zero. As an application, we deduce the nondegeneracy of layer solutions (bounded and monotone solutions) to the semilinear problem $$L u = f(u)$$ L u = f ( u ) in $$\mathbb {R}$$ R when the nonlinearity is of Allen–Cahn type. To our knowledge, this is the first work where such uniqueness and nondegeneracy results are proven in the nonlocal framework when the Caffarelli–Silvestre extension technique is not available. Our proofs are based on a nonlocal Liouville-type method developed by Hamel, Ros-Oton, Sire, and Valdinoci for nonlinear problems in dimension two.