Volume 4, Issue 3

Consideration is given to a family of minimal surfaces bounded by the broken lines in which are locally injectively projected onto the coordinate plane. The considered family is bijectively mapped by means of the Enepper–Weierstrass representation onto a set of circular polygons of a certain type. The parametrization of this set is constructed, and the dimension of the parameter domain is established.

Solutions to the initial-boundary value problem for a nonlinear Timoshenko equation are considered. Conditions on the initial data and nonlinear term are given so that solutions to the problem under consideration do not exist for all t > 0. An upper estimate of the t -interval of the existence of solutions is obtained. An estimate of the growth rate of the solutions is given.

Analogues of the well-known Kolosov–Muskhelishvili formulas of general representations are obtained for nonhomogeneous equations of statics in the case of the theory of elastic mixtures. It is shown that in this theory the displacement and stress vector components, as well as the stress tensor components, are represented through four arbitrary analytic functions. The usual Cauchy–Riemann conditions are generalized for homogeneous equations of statics in the theory of elastic mixtures.

We investigate the basic boundary value problems of the connected theory of elastothermodiffusion for three-dimensional domains bounded by several closed surfaces when the same boundary conditions are fulfilled on every separate boundary surface, but these conditions differ on different groups of surfaces. Using the results of papers [Kupradze, Gegelia, Basheleishvili, and Burchuladze, Three-dimensional problems of the mathematical theory of elasticity and thermoelasticity, North-Holland Publishing Company, 1979, Russian original, 1976–Mikhlin, Multi-dimensional singular integrals and integral equations, 1962], we prove theorems on the existence and uniqueness of the classical solutions of these problems.

Classical theorems on differential inequalities [Coddington and Levinson, Theory of ordinary differential equations, Mc Graw-Hill Book Company, Inc., 1995, Hartman, Ordinary differential equations, John Wiley & Sons, 1964, Walter, Differential and integral inequalities, Springer-Verlag, 1970] are generalized for initial value problems of the kind and where ƒ : C ([ a, b ]; R n ) → L loc (] a, b ]; R n ) is a singular Volterra operator, c 0 ∈ R n , h : [ a, b ] → [0, +∞[ is continuous and positive on ] a, b ], ‖ · ‖ is a norm in R n , and [ u ] + and [ u ] – are respectively the positive and the negative part of the vector u ∈ R n .

The following Riemann–Hilbert problem is solved: find an analytical function Φ from the Smirnov class E p ( D ), whose angular boundary values satisfy the condition Re[( a ( t ) + ib ( t ))Φ + ( t )] = ƒ( t ). The boundary Γ of the domain D is assumed to be a piecewise smooth curve whose nonintersecting Lyapunov arcs form, with respect to D , the inner angles with values ν k π , 0 < ν k ≤ 2.