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We establish sharp interior and boundary regularity estimates for solutions to ∂ t u − L u = f ( t , x ) in I × Ω , with I ⊂ R and Ω ⊂ R n . The operators L we consider are infinitesimal generators of stable Lévy processes. These are linear nonlocal operators with kernels that may be very singular. On the one hand, we establish interior estimates, obtaining that u is C 2 s + α in x and C 1 + α 2 s in t , whenever f is C α in x and C α 2 s in t . In the case f ∈ L ∞ , we prove that u is C 2 s − ϵ in x and C 1 − ϵ in t , for any ϵ > 0 . On the other hand, we study the boundary regularity of solutions in C 1 , 1 domains. We prove that for solutions u to the Dirichlet problem the quotient u / d s is Hölder continuous in space and time up to the boundary ∂Ω, where d is the distance to ∂Ω. This is new even when L is the fractional Laplacian. [ABSTRACT FROM AUTHOR]

We study the interior regularity of solutions to the Dirichlet problem L u = g in Ω, u = 0 in R n ∖ Ω , for anisotropic operators of fractional type L u ( x ) = ∫ 0 + ∞ d ρ ∫ S n − 1 d a ( ω ) 2 u ( x ) − u ( x + ρ ω ) − u ( x − ρ ω ) ρ 1 + 2 s . Here, a is any measure on S n − 1 (a prototype example for L is given by the sum of one-dimensional fractional Laplacians in fixed, given directions). When a ∈ C ∞ ( S n − 1 ) and g is C ∞ ( Ω ) , solutions are known to be C ∞ inside Ω (but not up to the boundary). However, when a is a general measure, or even when a is L ∞ ( S n − 1 ) , solutions are only known to be C 3 s inside Ω. We prove here that, for general measures a , solutions are C 1 + 3 s − ϵ inside Ω for all ϵ > 0 whenever Ω is convex. When a ∈ L ∞ ( S n − 1 ) , we show that the same holds in all C 1 , 1 domains. In particular, solutions always possess a classical first derivative. The assumptions on the domain are sharp, since if the domain is not convex and the measure a is singular, we construct an explicit counterexample for which u is not C 3 s + ϵ for any ϵ > 0 – even if g and Ω are C ∞ . [ABSTRACT FROM AUTHOR]

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 in Ω, in , for some and , then u is and is up to the boundary ∂Ω for some , where . 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 . Namely, the norms of u and in the sets are controlled by and , respectively. These regularity results are crucial tools in our proof of the Pohozaev identity for the fractional Laplacian (Ros-Oton and Serra, 2012 [19,20]). [Copyright &y& Elsevier]