Asymptotics of a condenser capacity and invariants of Riemannian submanifolds
type: Item
instance of: scholarly article
title: Asymptotics of a condenser capacity and invariants of Riemannian submanifolds
zbMATH Open document ID: 0874.31004
DOI: 10.1007/BF02433453
published in: Selecta Mathematica. New Series
publication date: 1997-11-13T00:00:00Z
review text: This paper deals with a problem of classical mathematical physics which is more than 100 years old. In 1877, G. Kirchhoff considered the following problem. Let \(D\) be a unit disc in \(\mathbb{R}^2\) and \(\varepsilon>0\) be a fixed number. Consider the corresponding condenser \(\Omega_{D, \varepsilon} =\{(x_1, x_2, x_3) \in\mathbb{R}^3, |x_3 |= \varepsilon\), \((x_1,x_2) \in D\}\). Find the behaviour of the capacity \(C_D (\varepsilon)\) of \(\Omega_{D, \varepsilon}\) when \(\varepsilon \to 0\). This problem was considered in [\textit{G. Polya} and \textit{G. Szegö}, Isoperimetric Inequalities in Mathematical Physics, Princeton Univ. Press (1951; Zbl 0044.38301)]. In this paper the problem has been discussed in a submanifold of a Riemannian manifold. Further, one should solve this boundary value problem for a class of functions having singularities along the boundary \(\partial D\) and then compute the contribution of this solution to \(C_D (\varepsilon)\).
Mathematics Subject Classification ID: 31C12
Mathematics Subject Classification ID: 35J25
Mathematics Subject Classification ID: 53C20
zbMATH DE Number: 1002094
zbMATH Keywords: condenser
zbMATH Keywords: behaviour of the capacity
zbMATH Keywords: Riemannian manifold
Wikidata QID: Q115392192
author: Yan Soibelman
MaRDI profile type: Publication
cites work: Q5736892
cites work: Isoperimetric Inequalities in Mathematical Physics. (AM-27)
Recommended article: Q4717182
Recommended article: Conformal capacities and conformally invariant functions on Riemannian manifolds
Recommended article: Q4291253
Recommended article: Q4783276
Recommended article: Conformal capacities and extremal metrics
Recommended article: Riemannian submersions, \(\delta\)-invariants, and optimal inequality
Recommended article: Riemannian submersions and isoperimetric inequalities
Recommended article: A Riemannian invariant and its applications to submanifold theory
Recommended article: Q3810093
Recommended article: Q4934602
description: scientific article; zbMATH DE number 1002094