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.
Bibliographical noteFunding Information:
This research was partially supported by the National Science Foundation's Graduate Research Fellowship, Grant No. (DGE-1144082). We thank Abdalla Nimer for helpful comments regarding Section 5 and Professor Tatiana Toro for helping us overcome a technical difficulty in Section 6. Finally, we owe a debt of gratitude to Professor Carlos Kenig who introduced us to free boundary problems and whose patience and guidance made this project possible.
© 2017 Elsevier Inc.
- Caloric measure
- Free boundary problem
- Parabolic PDE
- Poisson kernel