Riesz energy deformation through insulated strips

TL;DR

This paper explores how Riesz energies for compact sets can be viewed as limits of strip energies with increasing thickness, offering insights into a classical capacity conjecture and boundary conditions.

Contribution

It introduces a new perspective linking Riesz energies to strip energies and proposes an approach to a longstanding capacity conjecture.

Findings

Riesz energies with exponents differing by 1 are limits of strip energies.

The method applies Neumann boundary conditions to relate energies.

Provides a new approach to Pólya and Szegő's capacity conjecture.

Abstract

For compact sets in Euclidean space, Riesz energies whose exponents differ by $1$ are shown to arise as the endpoint cases of a one-parameter family of infinite-strip energies as the strip thickness increases from $0$ to $\infty$, under Neumann boundary conditions. An approach is suggested to a capacity conjecture of P\'{o}lya and Szeg\H{o}.