Regularity for solutions of non local parabolic equations

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Abstract

We study the regularity of solutions of parabolic fully nonlinear nonlocal equations. We prove C α regularity in space and time and, under different assumptions on the kernels, C 1,α in space for translation invariant equations. The proofs rely on a weak parabolic ABP and the classic ideas of Tso (Commun. Partial Diff. Equ. 10(5):543–553, 1985) and Wang (Commun. Pure Appl. Math. 45(1), 27–76, 1992). Our results remain uniform as σ → 2 allowing us to recover most of the regularity results found in Tso (Commun. Partial Diff. Equ. 10(5):543–553, 1985).

Communicated by O.Savin.