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15 July 2018 A minimization problem with free boundary related to a cooperative system
Luis A. Caffarelli, Henrik Shahgholian, Karen Yeressian
Duke Math. J. 167(10): 1825-1882 (15 July 2018). DOI: 10.1215/00127094-2018-0007

Abstract

We study the minimum problem for the functional Ω(|u|2+Q2χ{|u|>0})dx with the constraint ui0 for i=1,,m, where ΩRn is a bounded domain and u=(u1,,um)H1(Ω;Rm). First we derive the Euler equation satisfied by each component. Then we show that the noncoincidence set {|u|>0} is (locally) nontangentially accessible. Having this, we are able to establish sufficient regularity of the force term appearing in the Euler equations and derive the regularity of the free boundary Ω{|u|>0}.

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Luis A. Caffarelli. Henrik Shahgholian. Karen Yeressian. "A minimization problem with free boundary related to a cooperative system." Duke Math. J. 167 (10) 1825 - 1882, 15 July 2018. https://doi.org/10.1215/00127094-2018-0007

Information

Received: 1 November 2016; Revised: 13 January 2018; Published: 15 July 2018
First available in Project Euclid: 5 June 2018

zbMATH: 06928112
MathSciNet: MR3827812
Digital Object Identifier: 10.1215/00127094-2018-0007

Subjects:
Primary: 35R35
Secondary: 35J60

Rights: Copyright © 2018 Duke University Press

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Vol.167 • No. 10 • 15 July 2018
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