Search arXiv⌕ Search
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.CAmath-phmath.MP

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

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

P-adic Curved Kakeya Sets, Projection Theorem and Covering Numbers

Let $n,d\geq1$, and suppose that $E\subset \mathbb{Q}_p^n$ contains a degree-$d$ polynomial image of $\mathbb{Z}_p$ in every leading direction of a nonempty set $V\subset\mathbb{P}^{n-1}(\mathbb{Q}_p)$. We prove the sharp bound $\dim_H E\geq\dim_H V+1$. For joint block-valued polynomial evaluations, the total incidence dimension is at least one plus the supremum of the image dimensions. For analytic coefficient sets, Haar-almost every evaluation attains this supremum, and we bound the Hausdorff dimension of the parameters where the image dimension drops. For bounded coefficient sets and arbitrary fixed anisotropic block scales, we prove a quantitative covering comparison with a common Haar-null exceptional set. Outside this set, each evaluation's covering number dominates the maximum over any compact reference ball, up to any positive power loss at all sufficiently small scales. The proofs use the polynomial-to-line lifting map due to Nadjimzadah, together with Dhar's prime-power Kakeya set estimate and a tube maximal inequality derived from it.

math.CA↗
arXiv2026

Monotone Sobolev functions: approximation, critical points, and level sets

We give an affirmative answer to the planar local smoothing problem in Question~1.7 of D.~Ntalampekos and positive and negative answers to the basic approximation and level-set parts of his higher-dimensional Question~1.8. In every dimension $n\ge2$, each continuous Lebesgue monotone function in $W^{1,p}$ on a bounded open set admits uniform and strong $W^{1,p}$ approximation by monotone $C^{1,α}_{\loc}$ functions, with unchanged Sobolev boundary values and no increase of the $p$-Dirichlet energy, for $1<p<\infty$. The energy can be made strictly smaller unless the original function is $p$-harmonic. In the plane we obtain smooth local replacement at every isolated $p$-harmonic critical point, with arbitrarily small $C^1$ and Sobolev error. A point of gradient index $-m$ can be resolved into exactly $m$ nondegenerate saddles. Together with Ntalampekos's planar theorem, this gives smooth monotone density with fixed boundary values; at a nonsmooth $p$-harmonic minimizer the energy increase is unavoidable but can tend to zero. In dimensions $n\ge3$, an explicit Lipschitz monotone function has a nonmanifold point on every level in an interval, although it admits smooth monotone approximation. A product extension of a planar homogeneous $7$-harmonic function also rules out a general discrete exceptional set under the exact boundary and energy constraints. The answer to Question~1.8 thus distinguishes the five basic approximation properties from the stronger topological and exceptional-set conclusions. Applications include constrained integral functionals, strict $BV$ convergence of superlevel sets, nonlinear flux convergence, and stability of persistence diagrams under tameness assumptions.

math.CA↗
arXiv2026

Weak Endpoint Estimate for Commutators of Rough Maximal Singular Integral operators

Let $d\geq 2$ and $T^*_{Ω,b}$ be the commutator of the rough maximal singular integral on $\mathbb{R}^d$ defined by $$T^*_{Ω,b}f(x)=\sup_{\varepsilon>0}\left|\int_{|x-y|>\varepsilon}\bigl(b(x)-b(y)\bigr)\frac{Ω(x-y)}{|x-y|^d}f(y)\,dy\right|.$$ Assume that $Ω\in L(\log L)^2(\mathbb{S}^{d-1})$ has mean zero and that $b\in\operatorname{BMO}(\mathbb{R}^d)$. We prove that, for every $λ>0$, \[\bigl|\{x\in\mathbb{R}^d:T^*_{Ω,b}f(x)>λ\}\bigr|\lesssim_{Ω,b}\int_{\mathbb{R}^d}\frac{|f(x)|}λ\log\!\left(e+\frac{|f(x)|}λ\right)\,dx.\] The argument starts from a Calderón-Zygmund decomposition of $f$ and a dyadic linearization of the maximal truncation. A decomposition of $Ω$ by size, followed by regularization of the spatial cutoffs and a microlocal decomposition, reduces the problem to a family of estimates with quantitative decay. The required decay follows by interpolating localized $L^1$ and $L^3$ bounds and using an analytic-family argument for the commutators.

math.CA↗