Structure of One-Phase Free Boundaries in the Plane
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1412.4106.pdf
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Author(s)
Jerison, David S
Kamburov, Nikola Angelov
Date Issued
November 2015
Journal
International Mathematics Research Notices
Publisher
Oxford University Press (OUP)
Citation
Jerison, David and Nikola Kamburov. “Structure of One-Phase Free Boundaries in the Plane.” International Mathematics Research Notices 2016, 19 (November 2015): 5922–5987 © 2015 The Author(s)
Version
Original manuscript
Abstract
We study classical solutions to the one-phase free boundary problem in which the free boundary consists of smooth curves and the components of the positive phase are simply connected. We characterize the way in which the curvature of the free boundary can tend to infinity. Indeed, if curvature tends to infinity, then two components of the free boundary are close, and the solution locally resembles an entire solution discovered by Hauswirth, Hélein, and Pacard, whose free boundary has the shape of a double hairpin. Our results are analogous to theorems of Colding and Minicozzi characterizing embedded minimal annuli, and a direct connection between our theorems and theirs can be made using a correspondence due to Traizet.
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
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Creative Commons Attribution-Noncommercial-Share Alike
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DOI of Published Version
https://doi.org/10.1093/IMRN/RNV339
