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


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

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