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 𝑅𝑛, defined by the rotation of the curve around the polar axis, calculated in the metric 1 |𝑥|𝑛-1, ∈ 𝑅𝑛, ≥ 3, expressed by the formula 𝑆( ) = (𝑛 - 1)𝜔𝑛-1 w 𝑡1 𝑡0 sin𝑛-2 '(𝑡)√︁('′(𝑡))2 + ( ′(𝑡))2𝑑𝑡, where is a volume of 𝑛-dimensional sphere of radius 1, is the curve of the family of planar piecewise-smooth curves, given by the parametric equation 𝑧(𝑡) = (𝑡)+𝑖'(𝑡), ∈ [𝑡0, 𝑡1], is lying in the closed set = {𝑧 : ≤ |𝑧| ≤ ≤ 𝑟(1 + ),' ∈ ['0,'1]}, (0