Vol 65, No 4 (2019): Proceedings of the S.M. Nikolskii Mathematical Institute of RUDN University

New Results

On One Damping Problem for a Nonstationary Control System with Aftereffect

Adkhamova A.S., Skubachevskii A.L.


We consider a control system described by the system of differential-difference equations of neutral type with variable matrix coefficients and several delays. We establish the relation between the variational problem for the nonlocal functional describing the multidimensional control system with delays and the corresponding boundary-value problem for the system of differential-difference equations. We prove the existence and uniqueness of the generalized solution of this boundary-value problem.

Contemporary Mathematics. Fundamental Directions. 2019;65(4):547-556
pages 547-556 views

Translational Rectilinear Motion of a Solid Body Carrying a Movable Inner Mass

Bardin B.S., Panev A.S.


We consider the motion of the mechanical system consisting of the case (a solid body) and the inner mass (a material point). The inner mass circulates inside the case on a circle centered at the center of mass of the case. We suppose that absolute value of the velocity of circular motion of the inner mass is constant. The case moves translationally and rectilinearly on a flat horizontal surface with forces of viscous friction and dry Coulomb friction on it. The inner mass moves in vertical plane. We perform the full qualitative investigation of the dynamics of this system. We prove that there always exist a unique motion of the case with periodic velocity. We study all possible types of such a periodic motion. We establish that for any initial velocity, the case either reaches the periodic mode of motion in a finite time or asymptotically approaches to it depending on the parameters of the problem.

Contemporary Mathematics. Fundamental Directions. 2019;65(4):557-592
pages 557-592 views

On Asymptotics of the Density of States for Hypoelliptic Almost Periodic Systems

Bezyaev V.I.


In this paper, we find the asymptotics of integrated density of states with remainder estimate for hypoelliptic systems with almost periodic (a.p.) coefficients. We use the approximate spectral projector method for matrix a.p. operators with continuous spectrum.

Contemporary Mathematics. Fundamental Directions. 2019;65(4):593-604
pages 593-604 views

Absence of Solutions for Some Nonhomogeneous Elliptic Inequalities

Galakhov E.I., Salieva O.A.


By means of the modified method of test functions, we obtain sufficient conditions of absence of nontrivial solutions for some classes of semilinear elliptic inequalities of higher order and quasilinear elliptic inequalities containing nonhomogeneous terms (independent of the unknown function).

Contemporary Mathematics. Fundamental Directions. 2019;65(4):605-612
pages 605-612 views

Boundary-Value Problems for Differential-Difference Equations with Incommeasurable Shifts of Arguments Reducible to Nonlocal Problems

Ivanova E.P.


We consider boundary-value problems for differential-difference equations containing incommeasurable shifts of arguments in higher-order terms. We prove that in the case of finite orbits of boundary points generated by the set of shifts of the difference operator, the original problem is reduced to the boundary-value problem for differential equation with nonlocal boundary conditions.

Contemporary Mathematics. Fundamental Directions. 2019;65(4):613-622
pages 613-622 views

Application of Contemporary Proof of the Sforza Formula to Computation of Volumes of Hyperbolic Tetrahedra of Special Kind

Krasnov V.A.


In this paper, we use the contemporary proof (by Abrosimov and Mednykh) of the Sforza formula for volume of an arbitrary non-Euclidean tetrahedron to derive new formulas that express volumes of hyperbolic tetrahedra of special kind (orthoschemes and tetrahedra with the symmetry group S 4) via dihedral angles.

Contemporary Mathematics. Fundamental Directions. 2019;65(4):623-634
pages 623-634 views

Strongly Elliptic Differential-Difference Equations with Mixed Boundary Conditions in a Cylindric Domain

Liiko V.V., Skubachevskii A.L.


We consider strongly elliptic differential-difference equations with mixed boundary conditions in a cylindrical domain. We establish the connection between such problems and nonlocal mixed problems for strongly elliptic differential equations, and prove the uniqueness of solutions.

Contemporary Mathematics. Fundamental Directions. 2019;65(4):635-654
pages 635-654 views

Smoothness of Generalized Solutions of the Second and Third Boundary-Value Problems for Strongly Elliptic Differential-Difference Equations

Neverova D.A.


In this paper, we investigate qualitative properties of solutions of boundary-value problems for strongly elliptic differential-difference equations. Earlier results establish the existence of generalized solutions of these problems. It was proved that smoothness of such solutions is preserved in some subdomains but can be violated on their boundaries even for infinitely smooth function on the right-hand side. For differential-difference equations on a segment with continuous right-hand sides and boundary conditions of the first, second, or the third kind, earlier we had obtained conditions on the coefficients of difference operators under which there is a classical solution of the problem that coincides with its generalized solution. Also, for the Dirichlet problem for strongly elliptic differential-difference equations, the necessary and sufficient conditions for smoothness of the generalized solution in Holder spaces on the boundaries between subdomains were obtained. The smoothness of solutions inside some subdomains except for ε-neighborhoods of angular points was established earlier as well. However, the problem of smoothness of generalized solutions of the second and the third boundary-value problems for strongly elliptic differential-difference equations remained uninvestigated. In this paper, we use approximation of the differential operator by finite-difference operators in order to increase the smoothness of generalized solutions of the second and the third boundary-value problems for strongly elliptic differential-difference equations in the scale of Sobolev spaces inside subdomains. We prove the corresponding theorem.

Contemporary Mathematics. Fundamental Directions. 2019;65(4):655-671
pages 655-671 views

On the Algebra of Operators Corresponding to the Union of Smooth Submanifolds

Poluektova D.A., Savin A.Y., Sternin B.Y.


For a pair of smooth transversally intersecting submanifolds in some enveloping smooth manifold, we study the algebra generated by pseudodifferential operators and (co)boundary operators corresponding to submanifolds. We establish that such an algebra has 18 types of generating elements. For operators from this algebra, we define the concept of symbol and obtain the composition formula.

Contemporary Mathematics. Fundamental Directions. 2019;65(4):672-682
pages 672-682 views

On Initial-Boundary Value Problem on Semiaxis for Generalized Kawahara Equation

Faminskii A.V., Martynov E.V.


In this paper, we consider initial-boundary value problem on semiaxis for generalized Kawahara equation with higher-order nonlinearity. We obtain the result on existence and uniqueness of the global solution. Also, if the equation contains the absorbing term vanishing at infinity, we prove that the solution decays at large time values.

Contemporary Mathematics. Fundamental Directions. 2019;65(4):683-699
pages 683-699 views

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