Abstract
We study parabolic chord arc domains, introduced by Hofmann, Lewis and Nyström [14], 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 [26] 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.
| Original language | English (US) |
|---|---|
| Pages (from-to) | 835-947 |
| Number of pages | 113 |
| Journal | Advances in Mathematics |
| Volume | 314 |
| DOIs | |
| State | Published - Jul 9 2017 |
| Externally published | Yes |
Bibliographical note
Funding 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.
Publisher Copyright:
© 2017 Elsevier Inc.
Keywords
- Caloric measure
- Free boundary problem
- Parabolic PDE
- Poisson kernel