A minimization problem with free boundary related to a cooperative system

TL;DR

This paper investigates a free boundary minimization problem for a vector-valued function, reducing it to a scalar problem to analyze boundary regularity, with potential implications for systems in geometric analysis.

Contribution

It introduces a method to reduce a vector-valued free boundary problem to a scalar case, facilitating the study of boundary regularity for systems.

Findings

Reduction of vector problem to scalar problem

Similar boundary regularity results as scalar case

Potential for advancing free boundary theory for systems

Abstract

We study the minimum problem for the functional $\int_{\Omega}\bigl( \vert \nabla \mathbf{u} \vert^{2} + Q^{2}\chi_{\{\vert \mathbf{u}\vert>0\}} \bigr)dx$ with the constraint $u_i\geq 0$ for $i=1,\cdots,m$ where $\Omega\subset\mathbb{R}^{n}$ is a bounded domain and $\mathbf{u}=(u_1,\cdots,u_m)\in H^{1}(\Omega;\mathbb{R}^{m})$. Using an array of technical tools, from geometric analysis for the free boundaries, we reduce the problem to its scalar counterpart and hence conclude similar results as that of scalar problem. This can also be seen as the most novel part of the paper, that possibly can lead to further developments of free boundary regularity for systems.