Method of the optimal control in the solution of a variational problem

The paper provides a complete solution for the variational problem of finding a revolution surface of minimum area in the metric $|x|^{-n+1}$, corresponding extreme metric for $p$-module of family of surfaces that separate boundary components of a spherical ring.
The surface area in the $n$-dimensional Euclidean space $R^n$, defined by the rotation of the curve $\gamma$ around the polar axis, calculated in the metric $\frac {1}{|x|^{n-1}}$, $x \in R^n$, $n \geq 3$, expressed by the formula

\begin{equation*} S(\gamma) = (n-1)\omega_{n-1} \int_{t_0}^{t_1} \sin^{n-2} \varphi (t) \sqrt{(\varphi^{'}(t))^2+(\rho^{'} (t))^2} dt, \end{equation*}
where $\omega_n$ is a volume of $n$-dimensional sphere of radius 1, $\gamma$ is the curve of the family of planar piecewise-smooth curves, given by the parametric equation $z(t)=e^{\rho (t) + i \varphi (t)}$, $t \in [t_0,t_1]$, is lying in the closed set $\overline{B_r} = \{ z: r \leq |z| \leq r(1+\delta), \varphi \in [ \varphi_0, \varphi_1 ] \}$, $( 0< \varphi_0 < \varphi_1 \leq \pi)$ and is connecting the point $z(t_0)=r(1+\delta)e^{i\varphi_0}$ and the point $z(t_1)=r(1+\delta_1)e^{i\varphi_1}$, $0 \leq \delta_1 \leq \delta$.
The problem is to find the infimum of the functional $S(\gamma)$ in the described class of curves with natural condition that we consider only curves for which in the points of differentiability $\varphi^{'}(t) \geq 0$ and $\rho^{'}(t) \leq 0$. The method of optimal controls by L. Pontryagin [2] is applied for search for optimal trajectories. The properties of the hyperelliptic integral of a special type, arising in the solution of the variational problem, were investigated.