Asymptotics of a condenser capacity and invariants of Riemannian submanifolds
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 \(C_D (\varepsilon)\).
scientific article; zbMATH DE number 1002094