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Minkowski空间中离边界的距离函数

The distance function from the boundary in a Minkowski space

G. Crasta, A. Malusa

备注：
34 pages
发表信息：
Trans. Amer. Math. Soc. 359 (2007), pp. 5725-5759
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摘要：

设空间$\mathbb{R}^n$具有Minkowski结构$M$（即$M\colon \mathbb{R}^n \to [0,+\infty)$是以原点为内点的紧致凸集的规范函数，其边界为类$C^2$），并设$d^M(x,y)$是与$M$相关联的（非对称）距离。给定类$C^2$的开域$Ω\subset\mathbb{R}^n$，令$d_Ω(x) := \inf\{d^M(x,y); y\in\partialΩ\}$是点$x\inΩ$到$Ω$的边界的Minkowski距离。我们证明了$d_Ω$到$\mathbb{R}^n$的一个适当的扩张（它起到了$\partial Ω$的带符号Minkowski距离的作用）是在$\partial Ω$的管状邻域中的$C^2$类，并且$d_Ω$是在$\partialΩ$的切割轨迹之外的$C^2$类（即$d_Ω$在$Ω$中的不可微点集的闭包）。此外，我们证明了切割轨迹的$\partial Ω$有Lebesgue测度为零，$Ω$可以分解，这套消失的措施，到测地线从$\partialΩ$开始，并进入$Ω$沿法线方向（相对于闵可夫斯基距离）。我们明确计算雅可比行列式的变量的变化，关联到每一个点$x\in Ω$外的切割轨迹对$(p(x), d_Ω(x))$，其中$p(x)$表示（唯一）的投影$x$ $\partialΩ$，我们应用这些技术分析的Monge-Kantorovich型偏微分方程所产生的问题，在最优运输理论和形状优化。

英文摘要：

Let the space $\mathbb{R}^n$ be endowed with a Minkowski structure $M$ (that is $M\colon \mathbb{R}^n \to [0,+\infty)$ is the gauge function of a compact convex set having the origin as an interior point, and with boundary of class $C^2$), and let $d^M(x,y)$ be the (asymmetric) distance associated to $M$. Given an open domain $Ω\subset\mathbb{R}^n$ of class $C^2$, let $d_Ω(x) := \inf\{d^M(x,y); y\in\partialΩ\}$ be the Minkowski distance of a point $x\inΩ$ from the boundary of $Ω$. We prove that a suitable extension of $d_Ω$ to $\mathbb{R}^n$ (which plays the röle of a signed Minkowski distance to $\partial Ω$) is of class $C^2$ in a tubular neighborhood of $\partial Ω$, and that $d_Ω$ is of class $C^2$ outside the cut locus of $\partialΩ$ (that is the closure of the set of points of non--differentiability of $d_Ω$ in $Ω$). In addition, we prove that the cut locus of $\partial Ω$ has Lebesgue measure zero, and that $Ω$ can be decomposed, up to this set of vanishing measure, into geodesics starting from $\partialΩ$ and going into $Ω$ along the normal direction (with respect to the Minkowski distance). We compute explicitly the Jacobian determinant of the change of variables that associates to every point $x\in Ω$ outside the cut locus the pair $(p(x), d_Ω(x))$, where $p(x)$ denotes the (unique) projection of $x$ on $\partialΩ$, and we apply these techniques to the analysis of PDEs of Monge-Kantorovich type arising from problems in optimal transportation theory and shape optimization.

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