Contemporary Mathematics. Fundamental Directions

In a number of previous works it was found that for binomial functional equations of the form \[\hspace{-1.5cm} a(x)u(\alpha(x)) - \lambda u(x) = v(x),\quad x \in X,\] where \(\alpha:X \to X\) is an invertible mapping of the set \(X\) into itself, a situation typical for differential equations is possible: the equation is solvable for any right-hand side and there is no uniqueness of the solution. As in the case of differential equations, the question arises of formulating well-posed boundary value problems, i.e., of specifying additional conditions under which the solution exists and is unique. In this paper, we discuss the question of what kind of additional conditions lead to well-posed boundary-value problems for the equations under consideration.

The aim of the work is to construct multidimensional vector fields that are represented by autonomous systems of ordinary differential equations and have specified topological structures in specified limited simply connected domains of the phase space, provided that these structures can be specified by topological structures of projections of the sought vector fields onto coordinate planes. This problem is an inverse problem of the qualitative theory of ordinary differential equations. The results of this work can be used to construct mathematical models of dynamic systems in various fields of science and technology. In particular, for mechanical systems with an arbitrary finite number of degrees of freedom, such vector fields can represent kinematic equations of program motions and be used to obtain control forces and moments implementing these motions.

For a class of boundary-value problems with a parameter that are elliptic in the sense of Agranovich-Vishik, we establish the equality of the η-invariant defined in terms of the Melrose regularization and the spectral η-invariant of the Atiyah-Patodi-Singer type defined using the analytic continuation of the spectral η-function of the operator.

We consider the problem of constructing a system of differential equations from a given set of constraint equations and reducing them to the form of Lagrange equations with dissipative forces that ensure stabilization of the constraints. We determine the dissipative function from the equations of constraint disturbances. We use modified Helmholtz conditions to represent differential equations in the form of Lagrange equations. We give the solution of the Bertrand problem of determining the central force under the action of which a material point performs stable motion along a conic section.

In this paper, we study the initial-boundary value problem for a second-order nonhomogeneous hyperbolic equation in a half-strip of the plane with constant coefficients, containing a mixed derivative, with zero and nonzero potentials. This equation is the equation of transverse oscillations of a moving finite string. We consider the case of fixed ends (Dirichlet conditions). We assume that the roots of the characteristic equation are simple and lie on the real axis on different sides of the origin. We formulate our previously proven theorems on finite formulas for a generalized solution in the case of homogeneous and nonhomogeneous problems. Then, based on these formulas, we prove theorems on finite formulas for a classical solution or, in other words, a solution almost everywhere. In the second part of the paper, we formulate theorems on generalized solution of the initial-boundary value problem with ordinary potential and potential of general type, which we had proved earlier. These results are based on the idea of treating an equation with a potential as an inhomogeneity in an equation without a potential. This idea was previously used by A. P. Khromov and V. V. Kornev in the case of equation without mixed derivative. Further, on the basis of formulas for generalized solution to the problem with potentials, we prove theorems on the corresponding formulas for classical solutions for these two types of potentials.

We study the problem of small motions of a system of immiscible ideal fluids with a free surface consisting of two domains: a domain of elastic ice and a domain of crushed ice. Elastic ice is modeled by an elastic plate. By crushed ice we mean weighty particles of some substance floating on the free surface. We also assume that the interface between the fluid layers is a weighty surface. Using the method of orthogonal projection of boundary conditions and the introduction of auxiliary problems, we reduce the original initial-boundary value problem to an equivalent Cauchy problem for a second-order differential equation in a Hilbert space. We obtain the conditions under which there is a strong-in-time solution of the initial-boundary value problem describing the evolution of this hydraulic system. We prove statements about the structure of the spectrum of the problem and about the basis property of the system of eigenfunctions.