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arXiv · 2203.04055

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

Teo Kukuljan
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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.

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Teo Kukuljan. 2022-03-08. $C^{2,\alpha}$ regularity of free boundaries in parabolic non-local obstacle problems. https://arxiv.org/abs/2203.04055

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