A minimum principle for potentials with application to Chebyshev constants

Abstract

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 $U_K^{\mu }=\int K(x,y)d\mu (x)$, where $\mu$ is a Borel measure supported on $A$. Setting $P_K(\mu )={\inf }_{y\in A}{U}^{\mu }(y)$, the $K$-polarization of $\mu$, the principle is used to show that if $\{{\nu }_N\}$ is a sequence of measures on $A$ that converges in the weak-star sense to the measure $\nu$, then $P_K({\nu }_N)\to P_K(\nu )$ as $N\to \infty$. The continuous Chebyshev (polarization) problem concerns maximizing $P_K(\mu )$ over all probability measures $\mu$ supported on $A$, while the $N$-point discrete Chebyshev problem maximizes $P_K(\mu )$ only over normalized counting measures for $N$-point multisets on $A$. We prove for such kernels and sets $A$, that if $\{{\nu }_N\}$ is a sequence of $N$-point measures solving the discrete problem, then every weak-star limit measure of ${\nu }_N$ as $N\to \infty$ is a solution to the continuous problem.

References in corpus (1)

• Extremal Polarization Configurations for Integrable Kernels