Sectio aurea conditions for Mityuk's radius of two-connected domains

A. V. Kazantsev

Učënye zapiski Kazanskogo universiteta. Seriâ Fiziko-matematičeskie nauki, Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, Tome 159 (2017) no. 1, pp. 33-46

Résumé

Connection of an exterior inverse boundary value problem with the critical points of some surface is one of the central themes in the theory of exterior inverse boundary value problems for analytic functions. In the simply connected case, such a surface is defined by the inner mapping radius; in the multiply connected one, by the function $\Omega(w)$ such that $\mathrm M(w)=(2\pi)^{-1}\ln\Omega(w)$ is Mityuk's version of a generalized reduced module. In the present paper, the relation between the curvature of the surface $\Omega=\Omega(w)$ with the Schwarzian derivatives of the mapping functions and with the Bergman kernel functions $k_0(w,\overline\omega)$ and $l_0(w,\omega)$ is established for an arbitrary multiply connected domain. When passing to two-connected domains, due to the choice of the ring as a canonical domain, we construct the conditions for the critical points of Mityuk's radius to concentrate on the golden section circle of the ring. Finally, we show that the minimal collection of the critical points of the Mityuk radius in the two-connected case, consisting of one maximum and one saddle, is attained for the linear-fractional solution of the exterior inverse boundary value problem.

Détail

Keywords: exterior inverse boundary value problem, multiply connected domain, Gakhov equation, Mityuk's radius, inner mapping (conformal) radius, hyperbolic derivative.