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Toward Pólya and Szegő's conjecture for logarithmic vs Newtonian capacity
Abstract
Pólya and Szegő conjectured in 1945 that the disk achieves the largest electrostatic capacity among all planar sets with given logarithmic capacity. This question generalizes naturally to balls and Newtonian capacities in higher dimensions. The conjecture remains open. This paper establishes a conditional result: the conjecture would follow if among all lower dimensional sets in a horizontal strip with Neumann boundary conditions, the transverse component of the Dirichlet integral of the equilibrium potential could be shown minimal for the lower dimensional ball. As supporting evidence toward this latter property, the paper shows extremality of the ball for Lp-norms of the potentials, via Baernstein star-function symmetrization techniques.
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BibTeX
@misc{clark2026toward,
title = {{Toward Pólya and Szegő's conjecture for logarithmic vs Newtonian capacity}},
author = {Carrie Clark and Richard S. Laugesen},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.35438},
eprint = {2609.35438},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.35438},
}
RIS
TY - UNPB
TI - Toward Pólya and Szegő's conjecture for logarithmic vs Newtonian capacity
AU - Carrie Clark
AU - Richard S. Laugesen
PY - 2026
DA - 2026/09/28
PB - arXiv
DO - 10.48550/arxiv.2609.35438
UR - https://arxiv.org/abs/2609.35438
AB - Pólya and Szegő conjectured in 1945 that the disk achieves the largest electrostatic capacity among all planar sets with given logarithmic capacity. This question generalizes naturally to balls and Newtonian capacities in higher dimensions. The conjecture remains open. This paper establishes a conditional result: the conjecture would follow if among all lower dimensional sets in a horizontal strip with Neumann boundary conditions, the transverse component of the Dirichlet integral of the equilibrium potential could be shown minimal for the lower dimensional ball. As supporting evidence toward this latter property, the paper shows extremality of the ball for L^p-norms of the potentials, via Baernstein star-function symmetrization techniques.
ER -