对数凹测度的Christoffel问题中凸光滑解的存在性与唯一性

Existence and Uniqueness of Convex Smooth Solutions to a Christoffel Problem for Log-Concave Measures

摘要: 本文研究$n$ 维球面${\mathbb{S}}^n\subseteq {ℝ}^{n+1}$ 上的如下非线性椭圆方程，证明其存在唯一的光滑凸解：$\Delta S+nS={\text{e}}^{\varphi \left(S^2+{\left|\nabla S\right|}^2\right)}{S}^{p}f\left(\xi \right)$ 该方程对应对数凹测度${\text{e}}^{-\varphi \left({\left|x\right|}^2\right)}\text{d}x$ 的Christoffel型问题。解的存在性采用经典连续性方法证明，该方法的核心是解的先验估计；唯一性借助极大值原理得到，关键在于对线性化方程的核空间开展分析；凸性由秩不变定理推得，核心是矩阵${\left(S_{ij}+{\delta }_{ij}S\right)}_{n\times n}$ 小特征值所满足的微分不等式。本文主要结论进一步丰富了对数凹测度${\text{e}}^{-\varphi \left({\left|x\right|}^2\right)}\text{d}x$ 的几何分析相关研究成果。

Abstract: In the present paper, we prove that there exists a unique convex smooth solution to the following nonlinear elliptic equation on n-sphere ${\mathbb{S}}^n\subseteq {ℝ}^{n+1}$ ,$\Delta S+nS={\text{e}}^{\varphi \left(S^2+{\left|\nabla S\right|}^2\right)}{S}^{p}f\left(\xi \right)$ which is a Christoffel type problem for log-concave measures ${\text{e}}^{-\varphi \left({\left|x\right|}^2\right)}\text{d}x$ . The existence is based on the well-known continuous method whose core factor is a priori bounds of solutions. The uniqueness is based on a Maximum Principle whose core factor is the analysis of kernel of the linearized equation. The convexity follows from constant rank theorem whose core factor is a differential inequality for the small eigenvalue of ${\left(S_{ij}+{\delta }_{ij}S\right)}_{n\times n}$ . The main result of the present paper is to get more results on the geometric analysis of log-concave measures ${\text{e}}^{-\varphi \left({\left|x\right|}^2\right)}\text{d}x$ .