Higher-order boundary regularity estimates for nonlocal parabolic equations

Abstract

We establish sharp higher-order Hölder regularity estimates up to the boundary for solutions to equations of the form ∂tu- Lu= f(t, x) in I× Ω where I⊂ R, Ω ⊂ Rn and f is Hölder continuous. The nonlocal operators L that we consider are those arising in stochastic processes with jumps, such as the fractional Laplacian (- Δ) s, s∈ (0 , 1). Our main result establishes that, if f is Cγ is space and Cγ / 2 s in time, and Ω is a C2 , γ domain, then u/ ds is Cs + γ up to the boundary in space and u is C1 + γ / 2 s up the boundary in time, where d is the distance to ∂Ω. This is the first higher order boundary regularity estimate for nonlocal parabolic equations, and is new even for the fractional Laplacian in C∞ domains.