Duke Mathematical Journal

A minimization problem with free boundary related to a cooperative system

Luis A. Caffarelli, Henrik Shahgholian, and Karen Yeressian

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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}.

Article information

Duke Math. J., Volume 167, Number 10 (2018), 1825-1882.

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35R35: Free boundary problems
Secondary: 35J60: Nonlinear elliptic equations

minimization Bernoulli-type free boundary free boundary system regularity


Caffarelli, Luis A.; Shahgholian, Henrik; Yeressian, Karen. A minimization problem with free boundary related to a cooperative system. Duke Math. J. 167 (2018), no. 10, 1825--1882. doi:10.1215/00127094-2018-0007. https://projecteuclid.org/euclid.dmj/1528185619

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