A fundamental solution matrix for elliptic systems of the second order with constant leading coefficients is constructed. It is used to obtain an integral representation of functions belonging to the Hölder class in a closed domain with a Lyapunov boundary. In the case of an infinite domain, these functions have power-law asymptotics at infinity. The representation is used to study a mixed-contact boundary value problem for a second-order elliptic system with piecewise constant leading coefficients. The problem is reduced to a system of integral equations that are Fredholm in the domain and singular at its boundary.

In the unit disc, we consider the Dirichlet problem for second order elliptic system with only higher order constant coefficients. We establish the unique solvability of this problem under the assumption that the problem is Fredholm, and obtain an explicit formula for the solution.

The classes of C^1 -smooth domains bounded by a contour that is Lyapunov outside any neighborhood of its certain point such that the derivative of a conformal mapping onto the unit disk is continuous at this point are described. The description is given in terms of some spaces for a unit tangent vector on the boundary contour. Corresponding results for piecewise smooth domains are obtained as a consequence.

We consider the generalized Neumann problem for a 2lth-order elliptic equation with constant real higher-order coefficients in an infinite domain containing the exterior of some circle and bounded by a sufficiently smooth contour. It consists in specifying of the (kj-1){(k_j - 1)}th-order normal derivatives where 1≤k1...kl≤2l{1\le k_1 < ... < k_l \le 2l}; for kj=j{k_j = j} it turns into the Dirichlet problem, and for kj=j+1{k_j = j +1} into the Neumann problem. Under certain assumptions about the coefficients of the equation at infinity, a necessary and sufficient condition for the Fredholm property of this problem is obtained and a formula for its index in Holder spaces is given.

An explicit expression (in polar coordinates) for the fundamental solution matrix of the Lamé system of the plane anisotropic theory of elasticity is given. It is shown that the operator of convolution with this matrix in a finite domain with Lyapunov boundary is bounded in the Hölder spaces. A similar result is also established for an infinite domain in the corresponding weighted Hölder spaces (with a power-law behavior at infinity).

Графдины (ГД) являются двухмерной углеродной наноструктурой, содержащей атомы углерода с sp - и sp 2 -гибридизацией, причем sp -гибридизированные атомы образуют сопряженные связи, входящие в состав линейных цепей, соединяющих 6-членные углеродные циклы. Результаты сканирующей и просвечивающей электронной микроскопии (СЭМ и ПЭМ), рентгеновской фотоэлектронной) спектроскопии (РФЭС) и рамановской спектроскопии показали, что ГД имеют однородную поверхность и содержат сопряженные –С≡С–С≡С-связи. Исследована водород-адсорбционная способность ГД и проведен сравнительный анализ адсорбции водорода в ГД, графенах, графеновых нанотрубках и графеновых структурах, сформированных на цеолитах. Показано существенное влияние подложки, на которой формируется углеродная наноструктура, на ее адсорбционную способность. Рассмотрена возможность и перспективность синтеза графенов на катализаторах для повышения их эффективности в процессах гидрирования.

Sufficient conditions are given that ensure the validity of the main Faddeev–Marchenko theorem on the reconstruction of the potential of the Sturm–Liouville equation on the entire line based on given relations between the Jost functions. These conditions are formulated in terms of the so-called reflection coefficient. In the framework of the functional-theoretic approach, these relations are written in the form of a boundary value problem for the Jost functions. The unique solvability of this problem and the singular integral equation corresponding to it in the appropriate weighted Hölder classes is established. As a consequence, by solving this problem, an explicit formula for the potential is obtained in terms of the Sturm–Liouville equations, alternative to the well-known Levitan–Faddeev–Marchenko formula.

Singular integral operators with piecewise continuous matrix coefficients defined on segments of the real axis in weighted Lebesgue and Holder spaces are considered. In contrast to the classical case, the singular Cauchy operator together with the noncompact integral operators of a special form whose kernels are approximately homogeneous of degree –1 are among these operators. Similar operators arise in many applications. The Fredholm property criterion is established for these operators as well as a formula for the index. Examples of singular integral equations arising in the study of boundary value problems for forward-backward parabolic equations are displayed.

In weighted Hölder spaces, classes of smooth arcs and piecewise smooth contours are introduced that are invariant under power mappings. The boundary properties of conformal mappings are described in terms of these classes by analogy with Kellogg’s classical theorem.

The role played by explicit formulas for solving boundary value problems for elliptic equations and systems is well known. In this paper, explicit formulas for a general solution of the Dirichlet problem for second-order elliptic systems in the unit disk are given. In addition, an iterative method for solving this problem for systems with respect to two unknown functions is described, and an integral representation of the Poisson type is obtained by applying this method.

We establish the solvability of a Volterra integro-differential equation with logarithmic kernel in a class of weighted spaces on a finite interval with power singularities at the endpoints of the interval.

Graphdiynes (GDYs) are two-dimensional carbon nanostructures containing sp - and sp 2 -hybridized carbon atoms that form conjugated bonds in the linear chains connecting six-membered carbon rings. The results of scanning and transmission electron microscopy (SEM and TEM), X-ray photoelectron spectroscopy (XPS), and Raman spectroscopy showed that GDYs have a uniform surface and contain conjugated –С≡С–С≡С bonds. The hydrogen-adsorption capacity of GDYs was studied, and a comparative analysis of hydrogen adsorption in GDYs, graphenes, graphene nanotubes, and graphene structures formed on zeolites was performed. The substrate on which the carbon nanostructure is formed was shown to have a significant effect on the adsorption capacity of the latter. The possibility and prospects for the synthesis of graphenes on catalysts to increase their efficiency in hydrogenation processes are considered.

An explicit expression (in polar coordinates) for the fundamental solution matrix of the Lamé system of the plane anisotropic theory of elasticity is given. It is shown that the operator of convolution with this matrix in a finite domain with Lyapunov boundary is bounded in the Hölder spaces C^μ→ C^2,μ . A similar result is also established for an infinite domain in the corresponding weighted Hölder spaces (with a power-law behavior at infinity).

We construct the fundamental matrix of solutions to the Lamé system with constant main coefficients in the general anisotropic case. Using the fundamental matrix, we obtain an integral representation of functions of a Hölder class in a closed domain with Lyapunov boundary. In the case of an infinite domain, the representation is described within the framework of weighted Hölder spaces (of functions with power behavior at infinity). Based on this representation, we can reduce the mixed contact boundary value problem for the Lamé system with piecewise constant main coefficients to a system of integral equations that are Fredholm in the domain and are singular on the boundary. As a result, we establish a Fredholm criterion and indicate the index for this problem.

An explicit expression (in polar coordinates) for the fundamental solution matrix of the Lamé system of the plane anisotropic theory of elasticity is given. It is shown that the operator of convolution with this matrix in a finite domain with Lyapunov boundary is bounded in the Hölder spaces. A similar result is also established for an infinite domain in the corresponding weighted Hölder spaces (with a power-law behavior at infinity).

The paper consideres a boundary value problem for a fourth-order elliptic equation with constant real coefficients in a multiply connected domain, in which the function and its normal third-order derivative on the boundary of this domain are specified. A convenient Fredholmity criterion is given, and a formula for the index of this problem is presented. Classes of equations for which the Fredholmity criterion is especially simple are indicated, and the exact values of the index are calculated.

We consider the Dirichlet problem for the inhomogeneous Lamé system in the plane with constant leading coefficients in a (finite or infinite) domain, bounded by a Lyapunov contour. For domains of finite diameter we use the weighted Hölder class of functions with power behavior at infinity. We propose an equivalent reduction of the problem to a system of Fredholm integral equations in the domain and on the boundary contour.

The Riemann-Hilbert problem for the first order elliptic system is considered in the Hardy-Smirnov space. This system is reduced to an equivalent Fredhoim integral system on the boundary in the space L-p.

Для эллиптического уравнения 2l -го порядка с постоянными старшими вещественными коэффициентами в бесконечной области, содержащей внешность некоторого круга и ограниченной достаточно гладким контуром, рассмотрена обобщенная задача Неймана. Она заключается в задании нормальных производных ( k j - 1 ) {(k_j - 1)} -го порядков, где 1 ≤ k 1 < . . . < k l ≤ 2 l {1\le k_1 ; при k j = j {k_j = j} она переходит в задачу Дирихле, а при k j = j + 1 {k_j = j +1} - в задачу Неймана. При некоторых предположениях относительно коэффициентов уравнения на бесконечности получено необходимое и достаточное условие фредгольмовости этой задачи и приведена формула ее индекса в гельдеровских пространствах.

In this paper, we consider the Dirichlet problem for harmonic functions on a two-dimensional complex of a special type. We prove that this problem is a Fredholm problem in the Hölder class and its index is zero.