Weighted Chui's conjecture

TL;DR

This paper extends bounds related to Chui's conjecture to Coulomb potentials with boundary charges, proves sharpness in 2D, and explores a related problem with charges in the unit disc.

Contribution

It establishes a boundary Coulomb potential bound analogous to Newman’s bound, proves its sharpness in 2D, and discusses a related charge placement problem.

Findings

The boundary Coulomb potential bound is valid for arbitrary positive boundary charges.

The bound is proven to be sharp in two-dimensional cases.

A related problem with charges in the unit disc is analyzed.

Abstract

The goals of this paper are threefold. First, we show that a counterpart of the Newman bound related to the Chui conjecture is valid in the case where the gradient of Coulomb potential is generated by arbitrary positive charges placed at the boundary of a unit ball. Second, we prove that our bound is sharp in the two-dimensional case. Finally, we discuss a related problem, where the unit charges are placed in the unit disc.