Alexander Soldatov

Alexander Soldatov
Belgorod National Research University · Department of Mathematical Ananlysis

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82
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746
Citations

Publications

Publications (82)
Article
For the Bitsadze equation with low-order coefficients admitting a power singularity at a fixed point of the domain, we investigate a boundary-value problem with the Riemann–Hilbert data for the solution itself and for its partial derivatives. We prove the Fredholm solvability of that problem and present its index formula.
Article
Full-text available
We consider the Riemann–Hilbert boundary value problem for the Moisil–Theodoresco system in a multiply connected domain bounded by a smooth surface in the three–dimensional space. We obtain a criterion for the Fredholm property of the problem and a formula for its index æ = m − s, where s is the number of connected components of the boundary and m...
Article
We examine a piecewise analytic function that is defined in sectors of a disk whose real and imaginary parts obey contact conditions on adjacent boundary parts. Under the assumption of the power behavior, the sharp asymptotics of this function are established at the center of the disk.
Chapter
We considered an elliptic second order system on the plane consisting of two equations with constant (and only leading) coefficients. An explicit representation of the general solution of this system is given via the so-called J-analytic functions. A classification of systems with respect to the Dirichlet problem is given. Explicit expressions for...
Article
Full-text available
For the Lamé system from the flat anisotropic theory of elasticity, we introduce generalized double-layer potentials in connection with the function-theory approach. These potentials are built both for the translation vector (the solution of the Lamé system) and for the adjoint vector functions describing the stress tensor. The integral representat...
Conference Paper
For the elliptic equation 2l− th order with constant (and only) real coefficients considered boundary value problem of the job normal derivatives the (kj − 1)− order, j = 1, …, l, where 1 ≤ k1< … <k1 ≤ 2l − 1. When kj = j it moves to the Dirichlet problem, and when kj = j + 1 - in the Neumann problem. In this paper, the study is carried out in spac...
Article
Full-text available
We consider a first order strictly hyperbolic system of n equations with constant coefficients in a bounded domain. It is assumed that the domain is strictly convex relative to characteristics, so that the projection along each characteristic is an involution having two fixed singular points. The natural statement of boundary value problems for suc...
Article
Full-text available
The integrals with homogeneous-difference kernels are considered on a smooth contour. The boundary properties of the integrals are described in the Hölder space. An analogue of the known Sokhotski-Plemelj formula is obtained. Moreover, the differentiation formula of these integrals is also given.
Chapter
An analogue of the Schwarz problemfor the Moisil–Teodorescu system is considered in a domain D. It is shown that this problem has Fredholm property in the Hoelder class\(C^\mu (\overline{D})\). If the domain D is homeomorphic to a ball, then the problem is investigated in detail. In particular its index is equal to \(-1\) in this case.
Article
A new integral representation of the general solution of the Moisil–Théodorescu system in a bounded multiply connected domain with a smooth boundary is obtained.
Article
Full-text available
In this article we obtain a new integral representation of the general solution of the Moisil-Teodorescu system in a multiply connected domain. Also we give applications of this representation to Riemann-Hilbert problem.
Article
We consider a classical problem on linear conjugation problem for bi-analytic functions on smooth contour. We obtain explicit formula of a solution to a problem and describe necessary and sufficient conditions of its solvability.
Article
For an elliptic operator of order 2l with constant (and only leading) real coefficients, we consider a boundary value problem in which the normal derivatives of order (kj −1), j = 1,..., l, where 1 ≤ k1 < ··· < kl, are specified. It becomes the Dirichlet problem for kj = j and the Neumann problem for kj = j + 1. We obtain a sufficient condition for...
Article
We consider the classical linear conjugation problem for analytic functions on a piecewise smooth curve in the entire scale of weighted Hölder spaces. We derive a closed-form power-logarithmic asymptotics of the solution of this problem at the corner points of the curve under the assumption that the right-hand side of the problem admits a similar a...
Article
An estimate of the spectral radius of functional operators generated by operators of multiplication and shift operators in the space of continuous vector functions on the interval is given. It is assumed that shifts have fixed points only at both ends of the interval and belong to one type, i.e., they are either left or right shifts.
Article
We consider a mixed problem of plane isotropic elasticity in a half-plane in which the displacement vector and the normal component of the stress tensor are alternately specified on successive intervals of the real axis. We derive a closed-form expression for the solution of this problem, which is similar to the well-known Keldysh–Sedov formula for...
Article
For a generalized Cauchy–Riemann system whose coefficients admit higher-order singularities on a segment, we obtain an integral representation of the general solution and study a boundary value problem combining the properties of the linear conjugation problem and the Riemann–Hilbert problem in function theory.
Article
We consider the classical Riemann–Hilbert problem in a simply connected domain bounded by a piecewise smooth contour in the entire scale of weighted H¨older spaces. By using an appropriate refinement of the Kellogg theorem on a conformal mapping of this domain onto a disk, we provide a complete description of the solvability situation for this prob...
Conference Paper
We give an analogue of the well known Keldish-Sedov formula for solution of the orthotropic plane elasticity satisfying mixed boundary conditions in the upper half-plane. The case of isotropic medium is also received.
Conference Paper
Boundary value problems, more precisely Dirichlet’s problem for a string equation, or for an equivalent system of first order equations have been first studied in the first half of last century ([1] – [9]). The interest to these problems has been big ever since, see e.g. [10, 11]. All these papers have looked into the boundary value problems in a f...
Article
We consider the classical linear conjugation problem for analytic functions on piecewise-smooth curve in the whole scale of weighted Hölder spaces and describe its solvability in dependence on a weight order.
Article
We obtain sufficient conditions for the fundamental Faddeev–Marchenko theorem to be true. In addition, we derive a representation of the solution of the inverse Sturm–Liouville problem on the entire line on the basis of the solution of a boundary value problem for the Jost functions and the corresponding singular integral equation.
Article
Matrix kernels of generalized double-layer potentials representing homogeneous functions of zero degree are described in explicit form in terms of elastic moduli and the roots of the characteristic equation in the upper half-plane (more precisely, in terms of the sum and product of these roots). The cases of orthotropic and isotropic media are cons...
Article
We consider the Dirichlet problem for harmonic functions on two-dimensional stratified sets, which are assumed for simplicity to be complexes. We show that under certain conditions this problem is Fredholm in the Holder space and in weighted Holder spaces of functions satisfying the Hölder condition outside any neighbourhood of the vertex set of th...
Article
Second order elliptic systems with constant leading coefficients are considered. It is shown that the Bitsadze definition of weakly connected elliptic systems is equivalent to the known Shapiro–Lopatinskiy condition with respect to the Dirichlet problem for weakly connected elliptic systems. An analogue of potentials of double layer for these syste...
Article
Connected with the function-theoretic approach, generalized potentials of double layer are introduced for the Lamé system of plane anisotropic elasticity theory. These potentials are constructed for the displacement vector - a solution of the Lamé system, and as well for the conjugate vector-functions describing the stress tensor. There are obtaine...
Conference Paper
It is considered inverse Sturm - Liouville problem −y"+q(x)y = k 2 y on the whole axis. Sufficient conditions are given under which the fundamental Faddeev - Marchenko theorem is valid. Using Jost functions this problem is reduced to the Marcushevich problem on real axis which is explicitly solved.
Conference Paper
The representation of general solutions of Lame system of plane elasticity is given with the help of so-called Douglis analytic functions. Using this representation generalized potentials of double layer are introduced and the basic boundary value problems for Lame system are reduced to equivalent Fredholm integral equations on the boundary.
Article
For Lavrentiev -Bitsadze equation we consider boundary value problems with Dirichlet dates on part or whole boundary of mixed domain. We formulate results results on existence and uniqueness of solution for these problems in scale of Holder weighted classes.
Article
The existence and uniqueness issues are discussed for several boundary value problems with Dirichlet data for the Lavrent’ev-Bitsadze equation in a mixed domain. A general mixed problem (according to Bitsadze’s terminology) is considered in which the Dirichlet data are relaxed on a hyperbolic region of the boundary inside a characteristic sector wi...
Article
We examine a piecewise analytic function that is defined in sectors of a disk whose real and imaginary parts obey contact conditions on adjacent boundary parts. Under the assumption of the power behavior, the sharp asymptotics of this function is established at the center of the disk.
Article
On a smooth closed surface, we consider integrals of the Cauchy type with kernel depending on the difference of arguments. They cover both double-layer potentials for second-order elliptic equations and generalized integrals of the Cauchy type for first-order elliptic systems. For the functions described by such integrals, we find sufficient condit...
Article
This paper considers the Schwarz problem that consists in finding a J-analytic function by its real part on the boundary. The Fredholm solvability of this problem is proved. The integral representation of J-analytic functions by Cauchy-type integrals with real density is obtained.
Article
We consider nonlocal boundary value problems for three harmonic functions each of which is defined in its own domain. A contact condition is posed on the common part of the boundaries of these domains, and the Dirichlet or Neumann data (or mixed boundary conditions) are given on the remaining parts of the boundary. We prove the unique solvability o...
Article
The Lame system of general anisotropic plane elasticity is considered. A representation of a general solution or the system through a so-called Douglis analytic function is given. The cases of orthotropic and isotropic media are also considered.
Article
The space indicated in the title is introduced and studied.
Article
The analogue of the Privalov theorem is established for solutions of elliptic systems of second order in Lipshitz domains. The corresponding result is also obtained with respect to the weighted Hölder spaces. The notion of conjugate functions to solutions of elliptic systems is introduced and representation formulas for them are received. For so ca...
Article
The authors develop a functional-theoretic approach to solving boundary-value problems for the Lamé system of elasticity theory. Special attention is paid to the case of a plane orthotropic medium.
Article
For an elliptic 2lth-order equation with constant (and only leading) real coefficients, we consider the boundary value problem in which the (k j − 1)st normal derivatives, j = 1,..., l, are specified, where 1 ≤ k 1 < ... < k l . If k j = j, then it becomes the Dirichlet problem; and if k j = j + 1, then it becomes the Neumann problem. We obtain a s...
Article
A second order elliptic system is considered for the determination of Hardy spaces. Hardy spaces are defined by the weakly connected systems and provides a smooth parameterization with sequence of contours that converge, stating that there exists a homeomorphic mapping that converges to the identity mapping. The definition of Hardy spaces in domain...
Article
The representation of general solutions of Lame system of plane elasticity is given with the help of so-called Douglis analytic functions. Using integral representation of these functions the basic boundary value problems for Lame system are reduced to equivalent singular integral equations on the boundary. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA,...
Article
A general boundary value problem, encompassing from a unified viewpoint a broad circle of local and nonlocal boundary value problems, is studied for elliptic systems with real, constant (and only leading) matrix coefficients. A method is given for the equivalent reduction of this problem to a system of boundary equations. The considerations are car...
Article
We consider boundary-value problems in the upper half-plane for second-order elliptic systems with constant higher coefficients. Using the Bitsadze transformation, we reduce these problems to equivalent problems for analytic functions. This approach enables us to obtain explicit formulae for the solutions of basic boundary-value problems and to stu...
Article
The Hardy space of solutions to first-order elliptic systems has been defined. For a particular solution to an elliptic system, it was proved that there exist almost everywhere angular limit values, which determine an analytical function. It was proved that if the domain is bounded by a piecewise Lyapunov contour without cusp point and the matrix c...
Article
The solutions to the problem of Bitsadze-Samarskii type for second-order elliptic systems in the plane, are described. The system of equations is a bounded domain with a piecewise smooth boundary Γ=δD without cuspidal points. The solution considers the case when the coefficients b s are defined only on a part of Γ. To solve the problem, the compati...
Article
The Dirichlet problem for the elliptic system of second order with constant and only leading coefficients is considered in a piecewise smooth domain on the plane. The Fredholm criterion and index formula for this problem in Holder spaces are given. In particular, the examples of the elliptic systems are found such that the index of Dirichlet proble...
Article
A survey on the theory of hyperanalytic functions in the sense of Douglis is presented. Some applications of hyperanalytic functions are also given, in particular, to the description of solutions of elliptic systems in the plane. Special attention is devoted to some systems arising in the plane elasticity theory and linearized Stokes system of hydr...
Article
The family of functions analytic in the Douglis sense is considered in angular domains. These functions are connected by some functional relations on the lateral sides of the angles. It is assumed that the right-hand sides of these relations permit given power-logarithmic asymptotic at the vertexes of the angles. The problem under what conditions i...
Article
We study classes of elliptic systems on the plane for which the first and second boundary value problems are Fredholm. For systems of two equations with two unknown functions, we give a homotopy classification. We pay special attention to the Lamé system of anisotropic plane elasticity in the orthotropic case.
Article
A general (not necessarily local) boundary value problem is considered for an elliptic [$ l \times l$] system on the plane of [$ n$]th order containing only leading terms with constant coefficients. By a method of function theory developed for elliptic [$ s \times s$] systems of first order [$\displaystyle \frac{\partial \Phi}{\partial y} - J\frac{...
Article
The matrix case of a boundary value problem of linear conjugation in the theory of analytic functions, for arbitrary piecewise smooth curves, is treated, as is the corresponding adjoint problem. For these problems Noetherian theorems are proved, a family of canonical functions is constructed, and the behavior of these solutions at corner points of...
Article
The classical posed mixed-contact problem for an elliptic system with piecewise constant coefficients in a domain with piecewise smooth boundary has the Fredholm property, that is, the homogeneous problem has finitely many linearly independent solution and the inhomogeneous problem is solvable if a finite number of linearly independent orthogonalit...

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