A free boundary problem for the parabolic Poisson kernel

Max Engelstein

doi:10.1016/j.aim.2017.04.032

A free boundary problem for the parabolic Poisson kernel

Max Engelstein

Research output: Contribution to journal › Article › peer-review

6 Scopus citations

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	https://doi.org/10.1016/j.aim.2017.04.032
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

Publisher link

10.1016/j.aim.2017.04.032

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title = "A free boundary problem for the parabolic Poisson kernel",

abstract = "We study parabolic chord arc domains, introduced by Hofmann, Lewis and Nystr{\"o}m [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.",

keywords = "Caloric measure, Free boundary problem, Parabolic PDE, Poisson kernel",

author = "Max Engelstein",

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: {\textcopyright} 2017 Elsevier Inc.",

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N2 - 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.

AB - 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.

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KW - Free boundary problem

KW - Parabolic PDE

KW - Poisson kernel

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UR - https://www.scopus.com/pages/publications/85019986830#tab=citedBy

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DO - 10.1016/j.aim.2017.04.032

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SP - 835

EP - 947

JO - Advances in Mathematics

JF - Advances in Mathematics

ER -

A free boundary problem for the parabolic Poisson kernel

Abstract

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