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$C^{2,α}$ regularity of free boundaries in parabolic non-local obstacle problems (2203.04055v2)
Published 8 Mar 2022 in math.AP
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 $C1$ it is actually $C{2,\alpha}$. To do so, we establish a boundary Harnack inequality in $C1$ 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.