Greedy energy points with external fields

Abstract

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 ${\mathbb{R}}^p$, $p\ge 2$. 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 $m\ge 2$, 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 $x_i=a_i$, $1\le i\le m$, and for every integer $N\ge 1$, 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.

arXiv commentSection with numerical experiments added, typos corrected, minor modifications in the text, new reference added. The space X is assumed to be metric (instead of Hausdorff) to ensure that the space of positive measures supported on a compact subset of X, endowed with the weak-star topology, satisfies the first axiom of countability, 22 pages

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