A free boundary problem for the parabolic Poisson kernel

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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 languageEnglish (US)
Pages (from-to)835-947
Number of pages113
JournalAdvances in Mathematics
StatePublished - Jul 9 2017
Externally publishedYes

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.


  • Caloric measure
  • Free boundary problem
  • Parabolic PDE
  • Poisson kernel


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