## Duke Mathematical Journal

### A minimization problem with free boundary related to a cooperative system

#### Abstract

We study the minimum problem for the functional $\begin{equation*}\int_{\Omega}(\vert\nabla\mathbf{u}\vert^{2}+Q^{2}\chi_{\{\vert\mathbf{u}\vert\gt 0\}})\,dx\end{equation*}$ with the constraint $u_{i}\geq0$ for $i=1,\ldots,m$, where $\Omega\subset\mathbb{R}^{n}$ is a bounded domain and $\mathbf{u}=(u_{1},\ldots,u_{m})\in H^{1}(\Omega;\mathbb{R}^{m})$. First we derive the Euler equation satisfied by each component. Then we show that the noncoincidence set $\{\vert u\vert\gt 0\}$ 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 $\Omega\cap\partial\{\vert u\vert\gt 0\}$.

#### Article information

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

Dates
Revised: 13 January 2018
First available in Project Euclid: 5 June 2018

https://projecteuclid.org/euclid.dmj/1528185619

Digital Object Identifier
doi:10.1215/00127094-2018-0007

Mathematical Reviews number (MathSciNet)
MR3827812

Zentralblatt MATH identifier
06928112

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

#### Citation

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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