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

Abstract: In this paper, we study local minimizers of the energy functional$$J(\mathbf{u}) = \int_D (|\nabla \mathbf{u}|² + Q<sup>2(x)χ<em>{Ω</em>{\mathbf{u}}})\,dx$$ which give rise to a singular cooperative system. Here (\mathbf{u}=(u_1,\dots,u_m): D\to\R_+^m) is a vector-valued unknown function, (Ω{\mathbf{u}}:={|\mathbf{u}|>0}) is the positive set, (χ{Ω{\mathbf{u}}}) is the characteristic function of the positive set (Ω{\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.} \textbf{167}(10), 2018), where the regularity theory for minimizers and for the free boundary (\partialΩ{\mathbf{u}}) was established for nondegenerate Bernoulli weights, i.e., (Q(x)\ge Q{\rm min}>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 (\partialΩ_{\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 $\mathcal{H}<sup>{n-2}$-measure and is countably $(n-2)$-rectifiable.