A Free Boundary Problem for the Parabolic Poisson Kernel

TL;DR

This paper investigates the regularity of free boundaries in parabolic chord arc domains, showing that certain flatness and kernel conditions imply a vanishing Carleson measure, extending prior elliptic results to the parabolic setting.

Contribution

It extends the regularity theory of free boundaries to parabolic domains by proving that Reifenberg flat domains with VMO Poisson kernel are vanishing chord arc domains.

Findings

Reifenberg flat, parabolic chord arc domains with VMO Poisson kernel are vanishing chord arc domains.

Classification of flat blowups for the parabolic problem.

Generalization of elliptic free boundary regularity results to the parabolic case.

Abstract

We study parabolic chord arc domains, introduced by Hofmann, Lewis and Nystr\"om, 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.