Toward Pólya and Szegő's conjecture for logarithmic vs Newtonian capacity

Carrie Clark; Richard S. Laugesen

Home/Archive/arXiv

arXiv · 2609.35438

Toward Pólya and Szegő's conjecture for logarithmic vs Newtonian capacity

Carrie Clark·Richard S. Laugesen

View original source ↗Read source document ↗Export metadata ↓

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 $L^p$-norms of the potentials, via Baernstein star-function symmetrization techniques.

Explore related subjects

math.CA math-ph math.MP

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

Carrie Clark, Richard S. Laugesen. 2026-09-28. Toward Pólya and Szegő's conjecture for logarithmic vs Newtonian capacity. https://arxiv.org/abs/2609.35438

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

A bilinear approach to the finite field restriction problem, II

Let $P_3$ denote the three-dimensional paraboloid over a finite field of prime order in which $-1$ is not a square. We prove that the Fourier extension operator associated with $P_3$ maps $L^2$ to $L^r$ for $r>\frac{176}{51}=3.45098\ldots$. The argument combines the author's bilinear approach to the problem with point-line incidence estimates. We also prove that the extension operator associated with the paraboloid $P_6$ in six dimensions maps $L^2$ to $L^{8/3}$. This was previously known up to but not including the endpoint, and is the sharp $L^2$ estimate in six dimensions. Finally we observe that the endpoint restriction conjecture for $P_3$ in finite fields implies that the integer lattice points on the $3$-d Euclidean paraboloid are a $Λ(3)$ set.

math.CA↗

arXiv2026

Positive Cubature Compression and Optimal Hyperinterpolation Stability on a Conic Surface

We study the computational cost and stability of positive cubature and hyperinterpolation on a truncated conic surface with a Jacobi-type weight. A classical quadratic disk-to-cone map is used in a basis-free form as a measure-preserving $\mathbb Z_2$ quotient. It identifies the full degree-$m$ conic trace space with the even disk polynomials of degree at most $2m$ and converts positive degree-$2n$ cone cubature into centrally symmetric positive degree-$(4n+1)$ disk cubature, and conversely. This yields an exact transfer of Möller's lower bound and of the node excess above it, so that near-minimal disk formulas produce compressed non-product cone rules. The same quotient transfers reproducing kernels and hyperinterpolation operators. Every positive degree-$2n$ cone rule gives an exact $L^2$ sampling isometry on $Π_n(V)$; in particular, the weighted sampling matrix has condition number one, independently of the number and geometry of the nodes. For the unweighted radial case $γ=0$, if $Λ_n^V$ denotes the $C(V)\to C(V)$ Lebesgue constant of degree-$n$ hyperinterpolation, then every positive degree-$2n$ cone cubature rule satisfies the rule-independent sharp law $$c n \le Λ_n^V \le C n.$$ More generally, the upper bound $Λ_{n,γ}^V \le C_γn^{γ+1}$ holds when $γ$ is a nonnegative half-integer. Thus node compression preserves exact $L^2$ conditioning while the associated $L^\infty$ stability has the optimal universal order. Low-degree near-minimal rules and higher-degree optimized disk schemes illustrate the reduction in sampling cost.

math.CA↗

arXiv2026

On the rational solutions of generalized Abel equations

We study the nonconstant rational solutions of generalized Abel equations \[ x'=\sum_{i=1}^s A_i(t)x^{n_i},\qquad s\geq3,\qquad 1\leq n_1<\cdots<n_s, \] with polynomial coefficients over $\mathbb{R}$ or $\mathbb{C}$. Using the Newton diagram at infinity and the associated edge polynomials, we obtain, under simplicity and nonresonance conditions on the edge polynomials, the global upper bound $n_s-1$ for the number of nonconstant rational solutions. Over $\mathbb{R}$, the same approach yields the additional bound $\min\{n_s-1,2s\}$. We construct a family of equations showing that both bounds are sharp for every $s\geq3$. When $n_1=1$, the real bound improves to $\min\{n_s-1,2(s-1)\}$. We then specialize to the three-term case with $1<n_1<n_2<n_3$, where we obtain a finer classification of the rational solutions. Under an additional antipodal separation condition, we obtain the real bound $\min\{n_3-1,5\}$, and show that the absolute constant $5$ is sharp.

math.CA↗