Vol 24, No 1 (2018)

Multidimensional hyperbolic-elliptic equations describe important physical, astronomical and geometric processes. It is known that vibrations of elastic membranes in space according to the Hamiltonian principle can be modeled by a multidimensional wave equation. Assuming that the membrane is in equilibrium in the bending position, the Hamiltonian principle also yields the multidimensionalLaplaceequation. Consequently, the vibrations of elastic membranes in space can be modeled as the multidimensional Lavrentiev — Bitsadze equation. When studying these applications, it becomes necessary to obtain an explicit representation of the boundary value problems being studied. The author has previously studied the Dirichlet problem for multidimensional hyperbolic-elliptic equations, where a unique solvability of this problem is shown, which essentially depends on the height of the entire cylindrical region under consideration. In this paper we investigate a Dirichlet type problem in the cylindrical domain for the multidimensional Lavrentiev — Bitsadze equation and obtain an explicit form of its classical solution. In this case, the unique solvability depends only on the height of the hyperbolic part of the cylindrical domain, and a criterion for the uniqueness of the solution is given.

In the paper the well-posed characteristic problem is considered for the hyperbolic differential equation of the third order with nonmultiple characteristics. The regular solution of the characteristic problem for the hyperbolic differential equation of the third order with the nonmultiple characteristics is constructed in an explicit form. The well-posed characteristic problem is considered for one system of hyperbolic differential equations of the third order. The regular solution of the characteristic problem for the one system of hyperbolic differential equations of the third order is constructed. The theorem for the Hadamard’s well-posedness characteristic problem for the one system of hyperbolic differential equations is considered as the result of the research.

The structure of variability and intraspecific differentiation of Satureja hortensis L. (Lamiaceae) are studied on the basis of a complex of morphological features along a high-altitude gradient. Biomorphological peculiarities in four geographically remote locations are studied. The volume of each sample is 30 plants. The dimensional and quantitative features are taken into account. The dry mass of plants, foliage and reproductive effort are determined. A decrease in the dimensional and quantitative features of Satureja hortensis with a set of altitudes above sea level is noted. In the studied populations a high variability of the number of lateral branches per plant (CV = 21,6–26,0 %) is observed. With a set of heights, the percentage of foliage (28,5–35,2 %) and reproductive effort (6,6–11,0 %) is increase. As a result of the conducted studies, it was revealed that a significant contribution to variability of the Satureja hortensis features is made by a complex of abiotic and biotic factors caused by a high-altitude gradient. The correlation coefficient reflects the negative relationship of all the counted features with a high-altitude gradient, except on of the inflorescences mass. In the studied populations a high degree of variability on the weight characteristics of separate fractions and shoot as a whole is observed.

In this paper we consider the size dependent of surface tension in nanocapillary. Based on the analogueo f the Gibbs-Tolman-Koenig-Buff differential equation it is shown that for sufficiently small values of the capillary radius the Tolman’s equation for the surface tension holds. Taking into account the size dependent of the surface tension and the contact angle the problem of the capillary rise is discussed.