发表机构
Sichuan University; Fudan University; Southwest Jiaotong University（四川大学; 复旦大学; 西南交通大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
针对奇异合作系统，通过尺度不变内禀Hessian估计与两区域论证，建立了弱解的最优W^{2,∞}局部正则性，解决了开放问题。
AI 中文摘要
在向量值自由边界问题的开创性工作（参考文献[Adv. Math. 280 (2015)]{ASUW15}）中，Andersson、Shahgholian、Uraltseva 和 Weiss 研究了奇异合作系统 \begin{equation*} \Delta\mathbf{u} = \frac{\mathbf{u}}{|\mathbf{u}|} \chi_{\{|\mathbf{u}|>0\}} \quad\text{in }\Omega\subset\mathbb{R}^n, \qquad \mathbf{u}:\Omega\longrightarrow \mathbb{R}^m, \qquad n,m\ge 2 \end{equation*} 的解与自由边界的正则性理论。在解的层面，右端项的有界性通过标准椭圆估计直接给出对每个 $1<p<\infty$ 的 $W^{2,p}_{\mathrm{loc}}$ 正则性，而相应的 $W^{2,\infty}_{\mathrm{loc}}$ 正则性问题在该论文第753页被明确留作开放问题。在本文中，我们通过为每个弱解建立尺度不变的内禀 Hessian 估计，给出肯定回答，确立了最优的 $W^{2,\infty}_{\mathrm{loc}}$ 正则性。与标量情形不同，经典的 Alt--Caffarelli--Friedman 单调性公式不能直接应用于当前系统。我们的证明反而将局部化 Newton 势分解与单位球面上到二次调和多项式的显式投影相结合，该投影分离出 Hessian 的对称无迹二次部分。一个关键的新观察是，所得矩阵系数在对数尺度上满足具有常系数的精确线性常微分方程。本工作的主要新颖之处在于一种所谓的两区域论证，基于重缩放解的仿射部分与二次部分之间的竞争，加上连接两个区域并产生系数一致界的连续性论证。所得的先验估计为自由边界精细结构的进一步研究提供了分析基础。
英文摘要
In the pioneer work of vectorial free boundary problems \cite[Adv. Math. 280 (2015)]{ASUW15}, Andersson, Shahgholian, Uraltseva, and Weiss investigated the regularity theory of solutions and the free boundary of a singular cooperative system \begin{equation*} Δ\mathbf{u} = \frac{\mathbf{u}}{|\mathbf{u}|} χ_{\{|\mathbf{u}|>0\}} \quad\text{in }Ω\subset\mathbb{R}^n, \qquad \mathbf{u}:Ω\longrightarrow \mathbb{R}^m, \qquad n,m\ge 2. \end{equation*} At the level of solutions, the boundedness of the right-hand side directly yields $W^{2,p}_{\mathrm{loc}}$ regularity for every $1<p<\infty$ by standard elliptic estimates, while the corresponding $W^{2,\infty}_{\mathrm{loc}}$ regularity question was explicitly left as an open problem on page~753 of that paper. In this paper, we give a positive answer by establishing optimal $W^{2,\infty}_{\mathrm{loc}}$ regularity through a scale-invariant interior Hessian estimate for every weak solution. Unlike the scalar case, the classical Alt--Caffarelli--Friedman monotonicity formula cannot be applied directly to the present system. Our proof instead combines a localized Newtonian potential decomposition with an explicit projection onto quadratic harmonic polynomials on the unit sphere, which isolates the symmetric trace-free quadratic part of the Hessian. A key new observation is that the resulting matrix coefficients satisfy an exact linear ordinary differential equations with constant coefficients on the logarithmic scale. The main novelty of this work lies in a so-called two-regime argument based on the competition between the affine and quadratic parts of the rescaled solution, together with a continuity argument that connects the two regimes and yields a uniform bound for the coefficient. The resulting a priori estimate provides an analytic basis for further study of the fine structure of the free boundary.
Comments36 pages
