正曲率环境中的单调性公式及其在两相自由边界问题中的应用
Monotonicity formulas in positively curved settings with applications to two-phase free boundary problems
- Massachusetts Institute of Technology（麻省理工学院）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文针对二维正曲率环境构造ACF泛函的变体并证明其几乎单调性公式，借此得到两相自由边界问题极小元的新利普希茨界，常数仅依赖总积分曲率，还恢复了特定情形下的已有利普希茨界。
AI中文摘要：
受Alt、Caffarelli和Friedman的单调性公式启发，我们在二维正曲率环境中考虑ACF泛函的自然变体，该变体在格林函数的次水平集而非圆盘上积分。我们证明了新泛函在凸平面域（极点在边界和内部两种情况），以及具有欧氏体积增长和非负高斯曲率的二维完备流形环境下的尖锐几乎单调性公式。所用工具包括施瓦茨-克里斯托费尔公式和利用共形坐标的里斯分解。作为结果，我们利用拟共形估计，在流形环境中为Alt、Caffarelli和Friedman的两相自由边界问题的极小元给出新的利普希茨界，其常数仅依赖总积分曲率，这与Teixeira和Zhang的工作形成对比——后者给出的常数依赖于黎曼曲率张量及其导数的逐点界。我们的方法在凸平面域情形下还恢复了Gemmer、Moon和Raynor的至诺伊曼边界的利普希茨界。
英文摘要：
Inspired by the monotonicity formula of Alt, Caffarelli, and Friedman, we consider a natural variant of the ACF functional in positively curved 2-dimensional settings where we integrate over sublevel sets of Green's function rather than disks. We prove sharp almost-monotonicity formulas for our new functional in the case of convex planar domains (both with the pole on the boundary and in the interior) and in the setting of 2-dimensional complete manifolds with Euclidean volume growth and nonnegative Gaussian curvature. The tools include the Schwarz-Christoffel formula and Riesz decomposition using conformal coordinates. As a consequence, using quasiconformal estimates, we give a new Lipschitz bound in the manifold setting for minimizers of the two-phase free boundary problem of Alt, Caffarelli, and Friedman, with the constant depending on only the total integral curvature, in contrast to work of Teixeira and Zhang, which gives constants depending on pointwise bounds for the Riemann curvature tensor and its derivatives. Our methods also recover the Lipschitz bound up to a Neumann boundary of Gemmer, Moon, and Raynor in the planar convex domain case.
