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

TL;DR

This paper proves that free boundaries in certain non-local parabolic obstacle problems are smoother than previously known, achieving $C^{2,eta}$ regularity once they are $C^1$, and establishes optimal regularity of solutions in moving domains.

Contribution

It demonstrates that $C^1$ free boundaries in parabolic fractional obstacle problems are actually $C^{2,eta}$ and introduces a boundary Harnack inequality for moving domains.

Findings

Free boundary regularity upgraded from $C^1$ to $C^{2,eta}$.

Established boundary Harnack inequality in moving domains.

Proved optimal regularity of solutions to nonlocal parabolic equations.

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.