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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Quantitative stratification for some free-boundary problems


Authors: Nick Edelen and Max Engelstein
Journal: Trans. Amer. Math. Soc. 371 (2019), 2043-2072
MSC (2010): Primary 35R35
DOI: https://doi.org/10.1090/tran/7401
Published electronically: October 26, 2018
MathSciNet review: 3894044
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Abstract: In this paper we prove the rectifiability of and measure bounds on the singular set of the free-boundary for minimizers of a functional first considered by Alt-Caffarelli [J. Reine Angew. Math. 325 (1981), pp. 105-144]. Our main tools are the Quantitative Stratification and Rectifiable-Reifenberg framework of Naber-Valtorta [Ann. of Math. (2) 185 (2017), pp. 131-227], which allow us to do a type of ``effective dimension-reduction''. The arguments are sufficiently robust that they apply to a broad class of related free-boundary problems as well.


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Additional Information

Nick Edelen
Affiliation: Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts, 02139-4307
Email: nedelen@mit.edu

Max Engelstein
Affiliation: Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts, 02139-4307
Email: maxe@mit.edu

DOI: https://doi.org/10.1090/tran/7401
Received by editor(s): February 24, 2017
Received by editor(s) in revised form: September 12, 2017
Published electronically: October 26, 2018
Additional Notes: The first author was supported by NSF grant DMS-1606492. The second author was partially supported by NSF Grant No. DMS-1440140 while the author was in residence at MSRI in Berkeley, California, during Spring 2017.
Article copyright: © Copyright 2018 American Mathematical Society