A minimization problem for a cooperative system with a degenerate Bernoulli weight

In this paper, we study local minimizers of the energy functional$J(\mathbf{u})={\int }_D(\mathrm{\mid }\mathrm{\nabla }\mathbf{u}{\mathrm{\mid }}^2+Q^2(x){\chi }_{{\mathrm{\Omega }}_{\mathbf{u}}})dx$ which give rise to a singular cooperative system. Here $\mathbf{u}=(u_1,\dots ,u_m):D\to {\mathbb{R}}_{+}^m$ is a vector-valued unknown function, ${\mathrm{\Omega }}_{\mathbf{u}}:=\{\mathrm{\mid }\mathbf{u}\mathrm{\mid }>0\}$ is the positive set, ${\chi }_{{\mathrm{\Omega }}_{\mathbf{u}}}$ is the characteristic function of the positive set ${\mathrm{\Omega }}_{\mathbf{u}}$, and $Q(x)$ is the Bernoulli weight function. This problem was first introduced by Caffarelli, Shahgholian, and Yeressian in the poineer work ({\it Duke Math. J.} $\mathbf{\text{167}}$(10), 2018), where the regularity theory for minimizers and for the free boundary $\mathrm{\partial }{\mathrm{\Omega }}_{\mathbf{u}}$ was established for nondegenerate Bernoulli weights, i.e., $Q(x)\ge Q_{\mathrm{m}\mathrm{i}\mathrm{n}}>0$. The present paper studies the same free boundary problem but for a degenerate Bernoulli weight. The main difficulty lies in the coupling between the nondegeneracy of $Q(x)$ and the vector-valued nature of the problem. We show that the the free boundary set $\mathrm{\partial }{\mathrm{\Omega }}_{\mathbf{u}}\cap \{Q(x)=0\}$ decomposes into a nondegenerate part and a degenerate part. Nondegenerate points admit nontrivial blow-up limits; degenerate points have zero weighted density. We classify nondegenerate points into single-phase, multi-phase non-branching, and multi-phase branching points, analyzing their geometric structure. The branching-point singular structure for multi-phase points is purely vectorial and new in this setting. Finally, following Naber–Valtorta (Ann. Math. 185, 2017) and Edelen–Engelstein (Trans. Amer. Math. Soc. 371, 2019), we prove the nondegenerate set has locally finite ${\mathcal{H}}^{n-2}$-measure and is countably $(n-2)$-rectifiable.