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Bulletin of the American Mathematical Society

ISSN 1088-9485(online) ISSN 0273-0979(print)



Minimal surfaces and free boundaries: Recent developments

Authors: Luis A. Caffarelli and Yannick Sire
Journal: Bull. Amer. Math. Soc. 57 (2020), 91-106
MSC (2010): Primary 35A01, 35R35
Published electronically: June 28, 2019
Original Version: Original version posted June 28, 2019
Corrected Version: Current version corrects a misplaced character
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Abstract: Free boundaries occur in a lot of physical phenomena and are of major interest both mathematically and physically. The aim of this contribution is to describe new ideas and results developed in the last 20 years or so that deal with some nonlocal (sometimes called anomalous) free boundary problems. Actually, such free boundary problems have been known for several decades, one of the main instances being the thin obstacle problem, the so-called (scalar) Signorini free boundary problem. We will describe in this survey some new techniques that allow to deal with long-range interactions. We will not try to be exhaustive since the literature on this type of problem has been flourishing substantially, but rather we give an overview of the main current directions of research. In particular, we want to emphasize the link, very much well-known in the community, between minimal surfaces, their ``approximation'' by the Allen-Cahn equation and free boundary problems.

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Additional Information

Luis A. Caffarelli
Affiliation: Department of Mathematics, University of Texas at Austin, 2515 Speedway Stop C1200, Austin, Texas

Yannick Sire
Affiliation: Department of Mathematics, Johns Hopkins University, 3400 N. Charles Street, Baltimore, Maryland 21218

Received by editor(s): May 8, 2019
Published electronically: June 28, 2019
Additional Notes: The first author is supported by NSF DMS-1540162
The second author is partially supported by the Simons Foundation
Article copyright: © Copyright 2019 American Mathematical Society