О функционале гауссовой кривизны в классе поверхностей положительной гауссовой кривизны

On the Gaussian curvature functional in the class of surfaces of positive Gaussian curvature

The paper establishes the form of the Gaussian curvature functional defined on the class of infinitely differentiable horizontal surfaces of positive Gaussian curvature. With respect to admissible surfaces, it is assumed that they admit a global semi-geodetic parametrization. The paper proves that the first variation of the functional on the class of variations of admissible surfaces admitting connections between the coefficients of the first quadratic form and their geodesic lines similar to the axisymmetric case is determined by the Gaussian curvature of the varied surface. The considerations of the type are closely connected with the problems of the study of the equilibrium forms having one of the main curvatures sufficiently small. In this case, the classical Laplace formula fails. Thus, there appear a necessity to take into account more subtle processes leading to the adequate description of the equilibrium state of the two-phased system. In particular, it is quite natural to introduce into the study an intermediate layer consisting of the molecules of the two different phases, one of the Maxwell’s ideas. The calculations of the work spended by the pressure forces for the formation of intermediate layer leads us to the necessity to introduce Gauss functional into consideration. Linear combination of mean curvature and gaussian curvature functionals gives possibility to construct variational solution of generalized Laplace equation.