AI 中文总结
本文解决了$C$-闭集在$p\in(0,1)$时的$L_p$ Minkowski问题，并建立了$p\in[0,1]$时解的连续性，主要贡献在于存在性证明与连续性结果。
AI 中文摘要
设$C$为$\n\mathbb{R}^n$中具有非空内部的尖闭凸锥，$S^{n-1}$表示$\n\mathbb{R}^n$中的单位球面。$C$-闭集的$L_p$ Minkowski问题是：对于实数$p$和定义在$\Omega_{C^\circ}=S^{n-1}\cap \mathrm{int} C^{\circ}$上的非零有限Borel测度$\mu$，确定是否存在$C$-闭集$\mathds{A}$使得$\mu$是$\mathds{A}$的$L_p$表面积测度。本文中，我们将对$p\in (0,1)$且$\mu$为$\Omega_{C^\circ}$上的非零有限Borel测度的情况解决该问题。此外，我们在若干设置下建立了$p\in [0, 1]$时$L_p$ Minkowski问题解的连续性。
英文摘要
Let $C$ be a pointed closed convex cone in $\mathbb{R}^n$ with nonempty interior, and let $S^{n-1}$ denote the unit sphere in $\mathbb{R}^n$. The $L_p$ Minkowski problem for $C$-close sets is to determine, for a real number $p$ and a nonzero finite Borel measure $μ$ defined on $Ω_{C^\circ}=S^{n-1}\cap \mathrm{int} C^{\circ}$, whether there exists a $C$-close set $\mathds{A}$ such that $μ$ is the $L_p$ surface area measure of $\mathds{A}$. In this paper, we will solve the problem for $p\in (0,1)$ and for $μ$ being a nonzero finite Borel measure on $Ω_{C^\circ}$. Moreover, we establish the continuity of solutions to the $L_p$ Minkowski problem for $p\in [0, 1]$ in several settings.
