Geometric approach in multidimensional potential theory
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)].
scientific article; zbMATH DE number 1746604