Higher regularity of the free boundary in the obstacle problem for the fractional heat operator

Abstract

We prove a higher regularity result for the free boundary in the obstacle problem for the fractional heat operator via a higher order boundary Harnack estimate. As a consequence, we show that if the obstacle is Hm+β, then the free boundary is Hm−1+α near non-degenerate free boundary points for some 0<α≤β. In particular, smooth obstacles yield smooth free boundaries near non-degenerate free boundary points.