具有外部场的贪婪能量点
Greedy energy points with external fields
摘要：
本文在局部紧度量空间上引入了几种极值点列，并研究了它们的渐近性质。这些序列是通过贪婪算法定义的，通过最小化某一能量泛函，其表达式涉及一个外部字段。在欧氏空间$\mathbb{R}^{p}$，$p\geq 2$中也得到了一些结果.作为一个特殊的例子，给定闭集$A\subset\mathbb{R}^{p}$，下连续函数$f:\mathbb{R}^p\to(-\infty,+\infty]$和整数$m\geq 2$，我们研究了（在$A$和$f$上的适当条件下）序列$\{a_{i}\}_{1}^{\infty}\subset A$，其通过选择第一$m$点$a_{1},...,a_{m}$而归纳地构造，使得对于$x_{i}=a_{i}$、$1\leq i\leq m$和对于每个整数$N\geq 1$，函数\[ \sum_{1\leq i<j\leq m}\frac{1}{|x_{i}-x_{j}|^{s}}+(m-1)\sum_{i=1}^{m}f(x_{i}) \]在$A^{m}$上达到其最小值，选择点$a_{mN+1},...,a_{m(N+1)}$以最小化$A^{m}$上的表达式\[ \sum_{i=1}^{m}\sum_{l=1}^{mN}\frac{1}{|x_{i}-a_{l}|^{s}} +\sum_{1\leq i<j\leq m}\frac{1}{|x_{i}-x_{j}|^{s}}+((N+1)m-1)\sum_{i=1}^{m}f(x_{i}) \]。我们假设PDMATHAND。推广了G. Choquet关于点配置的最小能量也得到了，并构成了我们的分析的一个关键成分。
英文摘要：
In this paper we introduce several extremal sequences of points on locally compact metric spaces and study their asymptotic properties. These sequences are defined through a greedy algorithm by minimizing a certain energy functional whose expression involves an external field. Some results are also obtained in the context of Euclidian spaces $\mathbb{R}^{p}$, $p\geq 2$. As a particular example, given a closed set $A\subset\mathbb{R}^{p}$, a lower semicontinuous function $f:\mathbb{R}^p\to(-\infty,+\infty]$ and an integer $m\geq 2$, we investigate (under suitable conditions on $A$ and $f$) sequences $\{a_{i}\}_{1}^{\infty}\subset A$ that are constructed inductively by selecting the first $m$ points $a_{1},...,a_{m}$ so that the functional \[ \sum_{1\leq i<j\leq m}\frac{1}{|x_{i}-x_{j}|^{s}}+(m-1)\sum_{i=1}^{m}f(x_{i}) \] attains its minimum on $A^{m}$ for $x_{i}=a_{i}$, $1\leq i\leq m$, and for every integer $N\geq 1$, the points $a_{mN+1},...,a_{m(N+1)}$ are chosen to minimize the expression \[ \sum_{i=1}^{m}\sum_{l=1}^{mN}\frac{1}{|x_{i}-a_{l}|^{s}} +\sum_{1\leq i<j\leq m}\frac{1}{|x_{i}-x_{j}|^{s}}+((N+1)m-1)\sum_{i=1}^{m}f(x_{i}) \] on $A^{m}$. We assume here that $s\in[p-2,p)$. An extension of a result due to G. Choquet concerning point configurations with minimal energy is also obtained and constitutes a key ingredient in our analysis.
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