A minimization problem with free boundary related to a cooperative system. (English) Zbl 1395.35226

Caffarelli, Luis A.; Shahgholian, Henrik; Yeressian, Karen

Duke Math. J. 167, No. 10, 1825-1882 (2018).

The paper is focused on the Bernoulli-type free boundary problem: Minimize the functional \[ J(u)=\displaystyle\int_{\Omega}(|\nabla {\mathbf u}|^2+Q^2\chi_{\{|\mathbf{u}|>0\}})\,dx \] over \(H^1(\Omega;\mathbb{R}^m)\), where \(\mathbf{u}=(u_1,\ldots,u_m)\), \(\Omega\) is a bounded domain in \(\mathbb{R}^n\) and \(Q\) is a smooth enough function, under the constraint that \(\mathbf{u}=\mathbf{g}\) on \(\partial\Omega\) and the sign constraints \(u_i\geq 0\) a.e. in \(\Omega\) for \(i=1,\ldots,m\), with \(\mathbf{g}\in H^1(\Omega;\mathbb{R}^m)\) be such that \(g_i\geq 0\) a. e. in \(\Omega\) for \(i=1,\ldots,m\). The authors investigate the existence of an absolute minimizer, the general structure and initial regularity of minimers, and the optimal linear growth of minimizers near the free boundary. A preliminary local analysis of the minimizers and the free boundary is also presented. The authors prove that the noncoincidence set \(\{|\mathbf{u}|>0\}\) is locally a nontangentially accesible domain, and by using the reduction to a scalar problem, they obtain that the flatness of the free boundary implies its regularity. A Pohozaev-type identity is then proved and a Weiss-type monotonicity formula is also given, which establishes the homogeneity of blow-up limits. Finally, the authors classify all homogeneous global minimizers of the above problem, and they obtain the structure of the free boundary \(\Omega\cap\partial\{|\mathbf{u}|>0\}\) and its higher regularity close to regular points when the data of the problem is accordingly regular.

Reviewer: Rodica Luca (Iaşi)

MSC:

35R35	Free boundary problems for PDEs
35J60	Nonlinear elliptic equations

Keywords:

minimization; Bernoulli-type free boundary; free boundary; system; regularity

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