Riesz energy deformation through insulated strips

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 Pólya and Szegő.

Energy and kernel error estimates improved in the case q=1 as strip degenerates (t -> 0): Theorem 3.3(a): error term O(t) eliminated when q=1, using improved Proposition 4.1(d). Proposition 4.1(d): error term O(t) eliminated when q=1, by simplified proof. Proposition 5.1: O(1) part of error term eliminated from kernel estimate (13) when q=1. Corollary 5.2: error term O(1) improved to O(1/r)