O. V. Vlasiuk | Searcharxiv

arXiv2016

Generating Point Configurations via Hypersingular Riesz Energy With an External Field

For a compact $ d $-dimensional rectifiable subset of $ \mathbb{R}^{p} $ we study asymptotic properties as $ N\to\infty $ of $N$-point configurations minimizing the energy arising from a Riesz $ s $-potential $ 1/r^s $ and an external field in the hypersingular case $ s\geq d$. Formulas for the weak$ ^* $ limit of normalized counting measures of such optimal point sets and the first-order asymptotic values of minimal energy are obtained. As an application, we derive a method for generating configurations whose normalized counting measures converge to a given absolutely continuous measure supported on a rectifiable subset of $ \mathbb{R}^{p} $. Results on separation and covering properties of discrete minimizers are given. Our theorems are illustrated with several numerical examples.

math.CA↗

arXiv2016

A minimum principle for potentials with application to Chebyshev constants

For "Riesz-like" kernels $K(x,y)=f(|x-y|)$ on $A\times A$, where $A$ is a compact $d$-regular set $A\subset \mathbb{R}^p$, we prove a minimum principle for potentials $U_K^μ=\int K(x,y)dμ(x)$, where $μ$ is a Borel measure supported on $A$. Setting $P_K(μ)=\inf_{y\in A}U^μ(y)$, the $K$-polarization of $μ$, the principle is used to show that if $\{ν_N\}$ is a sequence of measures on $A$ that converges in the weak-star sense to the measure $ν$, then $P_K(ν_N)\to P_K(ν)$ as $N\to \infty$. The continuous Chebyshev (polarization) problem concerns maximizing $P_K(μ)$ over all probability measures $μ$ supported on $A$, while the $N$-point discrete Chebyshev problem maximizes $P_K(μ)$ only over normalized counting measures for $N$-point multisets on $A$. We prove for such kernels and sets $A$, that if $\{ν_N\}$ is a sequence of $N$-point measures solving the discrete problem, then every weak-star limit measure of $ν_N$ as $N \to \infty$ is a solution to the continuous problem.

math.CA↗