Riesz energy deformation through insulated strips
arXiv:2603.04649
The paper shows that Riesz energies with exponents differing by one appear as limiting cases of a family of infinite‑strip energies with Neumann boundary conditions, and uses this connection to obtain refined error estimates and to suggest an approach to the Pólya‑Szegő capacity conjecture.
Abstract
For compact sets in Euclidean space, Riesz energies whose exponents differ by are shown to arise as the endpoint cases of a one-parameter family of infinite-strip energies as the strip thickness increases from to , under Neumann boundary conditions. An approach is suggested to a capacity conjecture of Pólya and SzegÅ.
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)