AI 中文总结
本文研究带退化 Bernoulli 权的合作系统极小化问题，将自由边界分解为非退化与退化部分，分类非退化点并证明其具有局部有限测度与可数可求长性。
AI 中文摘要
本文研究能量泛函 $J(\mathbf{u}) = \int_D (|\nabla \mathbf{u}|^2 + Q^2(x)\chi_{\Omega_{\mathbf{u}}})\\,dx$ 的局部极小元，该泛函产生一个奇异合作系统。这里 $\mathbf{u}=(u_1,\dots,u_m): D\to\R_+^m$ 是向量值未知函数，$\Omega_{\mathbf{u}}:=\{|\mathbf{u}|>0\}$ 是正集，$\chi_{\Omega_{\mathbf{u}}}$ 是正集 $\Omega_{\mathbf{u}}$ 的特征函数，$Q(x)$ 是 Bernoulli 权函数。此问题最早由 Caffarelli、Shahgholian 和 Yeressian 在开创性工作（{\it Duke Math. J.} \textbf{167}(10), 2018）中引入，其中对非退化 Bernoulli 权（即 $Q(x)\ge Q_{\rm min}>0$）建立了极小元和自由边界 $\partial\Omega_{\mathbf{u}}$ 的正则性理论。本文研究相同的自由边界问题，但针对退化 Bernoulli 权。主要困难在于 $Q(x)$ 的非退化性与问题的向量值性质之间的耦合。我们证明自由边界集 $\partial\Omega_{\mathbf{u}}\cap\{Q(x)=0\}$ 分解为非退化部分和退化部分。非退化点允许非平凡的 blow-up 极限；退化点具有零加权密度。我们将非退化点分类为单相、多相非分支和多相分支点，并分析其几何结构。多相点的分支点奇异结构是纯向量性的，并且在此设定中是新的。最后，遵循 Naber--Valtorta (Ann. Math. 185, 2017) 和 Edelen--Engelstein (Trans. Amer. Math. Soc. 371, 2019) 的方法，我们证明非退化集具有局部有限的 $\mathcal{H}^{n-2}$-测度，并且是可数 $(n-2)$-可求长的。
英文摘要
In this paper, we study local minimizers of the energy functional$J(\mathbf{u}) = \int_D (|\nabla \mathbf{u}|^2 + Q^2(x)χ_{Ω_{\mathbf{u}}})\,dx$ 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 $\mathcal{H}^{n-2}$-measure and is countably $(n-2)$-rectifiable.
Comments61 pages, four figures, comments are welcome
