Geometric approach in multidimensional potential theory
type: Item
instance of: scholarly article
title: Geometric approach in multidimensional potential theory
zbMATH Open document ID: 0996.53028
published in: Siberian Advances in Mathematics
publication date: 2002-05-28T00:00:00Z
review text: The author establishes a correspondence between the classical potentials \[ \sigma (x)=\frac{1}{2\pi}\int\log |x-y|d\mu_y, \qquad \text{for} n=2, \] and \[ \sigma (x)=\int\frac{d\mu_y}{|x-y|^{n-2}}, \qquad \text{for} n\geq 3, \] and a class of non-regular multidimensional conformally flat metrics. This approach gives him a possibility to study weakly regular, conformally flat multidimensional metrics of nonnegative curvature in terms of the multidimensional potential theory and to use geometrical methods to solve some problems arising in potential theory. The two-dimensional version of this approach was used previously in [\textit{V. V. Slavskij}, Sib. Math. J. 30, No. 5, 811-823 (1989); translation from Sib. Mat. Zh. 30, No. 5, 187-201 (1989; Zbl 0689.53025)] and [\textit{Yu. G. Reshetnyak}, in Geometry IV. Non-regular Riemannian geometry. Encycl. Math. Sci. 70, 3-163 (1993; Zbl 0781.53050)].
Mathematics Subject Classification ID: 53C21
Mathematics Subject Classification ID: 31B15
zbMATH DE Number: 1746604
zbMATH Keywords: conformally flat metrics
zbMATH Keywords: nonnegative curvature
zbMATH Keywords: potential theory
author: V. V. Slavskij
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1746604