Regularity theory for general stable operators: Parabolic equations

Abstract

We establish sharp interior and boundary regularity estimates for solutions to ∂tu−Lu=f(t,x) in I×Ω, with I⊂R and Ω⊂Rn. 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 C2s+α in x and C1+α2s in t, whenever f is Cα in x and Cα2s in t. In the case f∈L∞, we prove that u is C2s−ϵ in x and C1−ϵ in t, for any ϵ>0.On the other hand, we study the boundary regularity of solutions in C1,1 domains. We prove that for solutions u to the Dirichlet problem the quotient u/ds 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.