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arXiv · 1510.05704

A Free Boundary Problem for the Parabolic Poisson Kernel

Max Engelstein
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Abstract

We study parabolic chord arc domains, introduced by Hofmann, Lewis and Nyström, and prove a free boundary regularity result below the continuous threshold. More precisely, we show that a Reifenberg flat, parabolic chord arc domain whose Poisson kernel has logarithm in VMO must in fact be a vanishing chord arc domain (i.e. satisfies a vanishing Carleson measure condition). This generalizes, to the parabolic setting, a result of Kenig and Toro and answers in the affirmative a question left open in the aforementioned paper of Hofmann et al. A key step in this proof is a classification of "flat" blowups for the parabolic problem.

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Max Engelstein. 2015-10-19. A Free Boundary Problem for the Parabolic Poisson Kernel. https://arxiv.org/abs/1510.05704

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