Higher-order boundary regularity estimates for nonlocal parabolic equations

Abstract

We establish sharp higher-order Holder regularity estimates up to the boundary for solutions to equations of the form $$\partial _tu-Lu=f(t,x)$$ in $$I\times \Omega $$ where $$I\subset \mathbb {R}$$ , $$\Omega \subset \mathbb {R}^n$$ and f is Holder continuous. The nonlocal operators L that we consider are those arising in stochastic processes with jumps, such as the fractional Laplacian $$(-\Delta )^s$$ , $$s\in (0,1)$$ . Our main result establishes that, if f is $$C^\gamma $$ is space and $$C^{\gamma /2s}$$ in time, and $$\Omega $$ is a $$C^{2,\gamma }$$ domain, then $$u/d^s$$ is $$C^{s+\gamma }$$ up to the boundary in space and u is $$C^{1+\gamma /2s}$$ up the boundary in time, where d is the distance to $$\partial \Omega $$ . This is the first higher order boundary regularity estimate for nonlocal parabolic equations, and is new even for the fractional Laplacian in $$C^\infty $$ domains.