For the Lavrent’ev—Bitsadze equation in a partially perforated model domain with a characteristic size of microinhomogeneities \(\varepsilon,\) we consider the problem with the third-kind boundary condition on the boundary of the cavities (the Fourier condition), which has a small parameter \(\varepsilon^\alpha\) as a multiplier in the coefficients, and the Dirichlet condition on the outer part of the boundary. For this problem, we construct a homogenized problem and prove the convergence of the solutions of the original problem to the solution of the homogenized problem in three cases. The subcritical case with \(\alpha1\) is characterized by the fact that dissipation at the boundary of the cavities is negligibly small, in the critical case with \(\alpha=1\) a potential appears in the equation due to dissipation, and in the supercritical case with \(\alpha1\) the dissipation plays the major role, it leads to degeneracy of the solution of the entire problem.

Журнал "Современная математика. Фундаментальные направления" издается на двух языках (русском и английском). Журнал ориентирован на публикацию обзорных статей и статей, содержащих оригинальные научные результаты.

The problems of interpolation of periodic functions and construction of biorthogonal systems are considered. Uniform shifts of the third Jacobi theta function are used as a basis. Explicit formulas for the nodal function and the function generating the biorthogonal system are obtained. Exact values of the lower and upper Riesz constants are found.

We obtain the conditions under which a given multilayer differential operator \(P(D)\) (polynomial \(P(\xi)\)) is more powerful than operator \(Q(D)\) (polynomial \(Q(\xi)\)). This is used to obtain estimates of monomials, which, in turn, using the theory of Fourier multipliers, is used to obtain coercive estimates of derivatives of functions through the differential operator \(P(D)\) applied to these functions.

The main goal of the work is to construct analogues of Christoffel symbols for infinitedimensional systems and on this basis to obtain geodesic equations for such systems. These analogies are of particular interest in terms of identifying the relationship between the dynamics of systems with an infinite number of degrees of freedom and Riemannian geometry, as well as geometry defined by the pseudo-Riemannian metric.

The paper is devoted to the Dirichlet problem in a flat bounded domain for a linear secondorder functional differential equation in the divergent form with dilation, contraction and rotation of the argument of the higher-order derivatives of the unknown function. We study the existence, the uniqueness and the smoothness of the generalized solution for all possible values of the coefficients and parameters of transformations in the equation.

In this paper, we study a problem of normal oscillations of a homogeneous mixture of several viscous compressible fluids filling a bounded domain of three-dimensional space with an infinitely smooth boundary. Two boundary conditions are considered: the no-slip condition and the slip condition without shear stresses. It is proved that the essential spectrum of the problem in both cases is a finite set of segments located on the real axis. The discrete spectrum lies on the real axis, except perhaps for a finite number of complex conjugate eigenvalues. The spectrum of the problem contains a subsequence of eigenvalues with a limit point at infinity and a power-law asymptotic distribution.

We call a foliation (M,F) on a manifold M chaotic if it is topologically transitive and the union of closed leaves is dense in M. A foliated manifold M is not assumed to be compact. The chaotic foliations can be considered as multidimensional generalization of chaotic dynamical systems in the sense of Devaney. For foliations covered by fibrations we prove that a foliation is chaotic if and only if its global holonomy group is chaotic. We introduce the concept of the integrable Ehresmann connection for a foliation as a natural generalization of the integrable Ehresmann connection for smooth foliations. A description of the global structure of foliations with integrable Ehresmann connection and a criterion for the chaotic behavior of such foliations are obtained. Applying the method of suspension, a new countable family of pairwise nonisomorphic chaotic foliations of codimension two on 3-dimensional closed and nonclosed manifolds is constructed.

In this paper, we study a complete second-order integro-differential operator equation in a Hilbert space. The difference-type kernel of an integral perturbation is a holomorphic semigroup bordered by unbounded operators. The asymptotic behavior of solutions of this equation is studied. Asymptotic formulas for solutions are proved in the case when the right-hand side is close to an almost periodic function. The obtained formulas are applied to the study of the problem of forced longitudinal vibrations of a viscoelastic rod with Kelvin-Voigt friction.

In this article, we give an overview of the most significant and important results on holomorphic extensions of functions along a fixed direction. We discuss the following geometric questions of multidimensional complex analysis: holomorphic extension along a bundle of complex straight line, the Forelly theorem; holomorphic continuation of functions with thin singularities along a fixed direction; holomorphic continuation of functions along a family of analytic curves.