$$C^{2,\alpha }$$ regularity of free boundaries in parabolic non-local obstacle problems

Abstract

We study the regularity of the free boundary in the parabolic obstacle problem for the fractional Laplacian \((-\Delta )^s\) (and more general integro-differential operators) in the regime \(s>\frac{1}{2}\). We prove that once the free boundary is \(C^1\) it is actually \(C^{2,\alpha }\). To do so, we establish a boundary Harnack inequality in \(C^1\) and \(C^{1,\alpha }\) (moving) domains, providing that the quotient of two solutions of the linear equation, that vanish on the boundary, is as smooth as the boundary. As a consequence of our results we also establish for the first time optimal regularity of such solutions to nonlocal parabolic equations in moving domains.