Abstract. We study the regularity up to the boundary of solutions to the Dirichlet problem for the fractional Laplacian. We prove that if u is a solution of (−∆)to the boundary ∂Ω for some α ∈ (0, 1), where δ(x) = dist(x, ∂Ω). For this, we develop a fractional analog of the Krylov boundary Harnack method.Moreover, under further regularity assumptions on g we obtain higher order Hölder estimates for u and u/δ s . Namely, the C β norms of u and u/δ s in the sets {x ∈ Ω : δ(x) ≥ ρ} are controlled by Cρ s−β and Cρ α−β , respectively. These regularity results are crucial tools in our proof of the Pohozaev identity for the fractional Laplacian [19,20].

Abstract. In this paper we prove the Pohozaev identity for the semilinear Dirichlet problem (−∆)s is the fractional Laplacian in R n , and Ω is a bounded C 1,1 domain. To establish the identity we use, among other things, that if u is a bounded solution then u/δ s | Ω is C α up to the boundary ∂Ω, where δ(x) = dist(x, ∂Ω). In the fractional Pohozaev identity, the function u/δ s | ∂Ω plays the role that ∂u/∂ν plays in the classical one. Surprisingly, from a nonlocal problem we obtain an identity with a boundary term (an integral over ∂Ω) which is completely local.As an application of our identity, we deduce the nonexistence of nontrivial solutions in star-shaped domains for supercritical nonlinearities.

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Abstract. We establish sharp regularity estimates for solutions to Lu = f in Ω ⊂ R n , being L the generator of any stable and symmetric Lévy process. Such nonlocal operators L depend on a finite measure on S n−1 , called the spectral measure.First, we study the interior regularity of solutions to Lu = f in B 1 . We prove that if f is C α then u belong to C α+2s whenever α + 2s is not an integer. In case f ∈ L ∞ , we show that the solution u is C 2s when s = 1/2, and C 2s−ǫ for all ǫ > 0 when s = 1/2.Then, we study the boundary regularity of solutions to Lu = f in Ω, u = 0 in R n \ Ω, in C 1,1 domains Ω. We show that solutions u satisfy u/d s ∈ C s−ǫ (Ω) for all ǫ > 0, where d is the distance to ∂Ω.Finally, we show that our results are sharp by constructing two counterexamples.

We study fine boundary regularity properties of solutions to fully nonlinear elliptic integro-differential equations of order $2s$, with $s\in(0,1)$. We consider the class of nonlocal operators $\mathcal L_*\subset \mathcal L_0$, which consists of infinitesimal generators of stable L\'evy processes belonging to the class $\mathcal L_0$ of Caffarelli-Silvestre. For fully nonlinear operators $I$ elliptic with respect to $\mathcal L_*$, we prove that solutions to $I u=f$ in $\Omega$, $u=0$ in $\mathbb R^n\setminus\Omega$, satisfy $u/d^s\in C^{s+\gamma}(\overline\Omega)$, where $d$ is the distance to $\partial\Omega$ and $f\in C^\gamma$. We expect the class $\mathcal L_*$ to be the largest scale invariant subclass of $\mathcal L_0$ for which this result is true. In this direction, we show that the class $\mathcal L_0$ is too large for all solutions to behave like $d^s$. The constants in all the estimates in this paper remain bounded as the order of the equation approaches 2. Thus, in the limit $s\uparrow1$ we recover the celebrated boundary regularity result due to Krylov for fully nonlinear elliptic equations.Comment: To appear in Duke Math.