Cauchy-type Integrals Research Articles

Direct and Inverse Problems for the Simple Layer Logarithmic Potential

With the help of Cauchy-type integrals, a new derivation of integral equations for solving logarithmic potential problems is given. A variant of the inverse logarithmic potential problem with two starlike solutions is analyzed; it contains fewer parameters than those considered by I.M. Rapoport. A criterion for the interior simple layer potential to be identically constant is obtained.

On the Wiener-Hopf solution of water-wave interaction with a submerged elastic or poroelastic plate.

A solution to the problem of water-wave scattering by a semi-infinite submerged thin elastic plate, which is either porous or non-porous, is presented using the Wiener-Hopf technique. The derivation of the Wiener-Hopf equation is rather different from that which is used traditionally in water-waves problems, and it leads to the required equations directly. It is also shown how the solution can be computed straightforwardly using Cauchy-type integrals, which avoids the need to find the roots of the highly non-trivial dispersion equations. We illustrate the method with some numerical computations, focusing on the evolution of an incident wave pulse which illustrates the existence of two transmitted waves in the submerged plate system. The effect of the porosity is studied, and it is shown to influence the shorter-wavelength pulse much more strongly than the longer-wavelength pulse.

On the Derivatives of Cauchy-Type Integrals in the Polydisk

In the paper the formulas for derivatives of the Cauchy type integral K[f] of smooth functions f on the distinguished boundary of a polydisc are given. These formulas express the derivatives of orderm from K[f] through the derivatives of lower order (Theorem 1). We use them for the estimates of smoothness of derivatives of Cauchy type integral in terms of the Holder’s ordinary scale (Theorem 2) and in terms of anisotropic Holder’s classes (Theorem 5).

Meta-arrest of a fast propagating crack in elastic wave metamaterials with local resonators

In this study, the dynamic effective mass and effects of local resonators on the crack growth are investigated. Based on Wiener-Hopf method and factorization using Cauchy-type integral, the energy release ratio which denotes crack splitting resistance is derived and the corresponding radiated waves are discussed. The results show that at variance with traditional periodic mass-spring lattice chain, the radiated energy of the present elastic wave metamaterial with local resonators is more sensitive to low crack speeds. The energy barrier before the stable propagates is larger, which means that the ability to resist the crack speed is enhanced. The lower energy release ratio in the subsonic and supersonic regions indicates that the elastic wave metamaterials with local resonators has better crack resistance and shows the meta-arrest property. The present results might have potential application to improve the crack resistance of advanced materials and structures.

Differential Equations in Banach Algebras

In a complex Banach algebra that is not assumed to be commutative, nth-order linear differential equations with constant coefficients are considered. The corresponding algebraic characteristic equation of the nth degree is assumed to have n distinct roots for which the Vandermonde matrix is invertible. Analogues of Sylvester’s and Vieta’s theorems are proved, and a contour integral of Cauchy type is studied.

Parabolic Equations with Changing Direction of Time

A theorem about the behavior of Cauchy-type integrals at the endpoints of the integration contour and at discontinuity points of the density is stated, and its application to boundary value problems for 2n-order parabolic equations with a changing direction of time are described. The theory of singular equations, along with the smoothness of the initial data, makes it possible to specify necessary and sufficient conditions for the solution to belong to Holder spaces. Note that, in the case n = 3, the smoothness of the initial data and the solvability conditions imply that the solution belongs to smoother spaces near the ends with respect to the time variable.

Singular Integral Operators and Elliptic Boundary-Value Problems. Part I

The monograph consists of three parts. Part I is presented here. In this monograph, we develop a new approach (mainly based on papers of the author). Many results are published for the first time here. Chapter 1 is introductory. It provides the necessary background from functional analysis (for completeness). In this monograph, we mostly use weighted HOlder spaces; they are considered in Chap. 2. Chapter 3 plays the key role: in weighted HOlder spaces, we consider estimates of integral operators with homogeneous difference kernels, covering potential-type integrals and singular integrals as well as Cauchy-type integrals and double layer potentials. In Chap. 4, similar estimates in weighted Lebesgue spaces are proved. Integrals with homogeneous difference kernels will play an important role in Part III of the monograph, which will be devoted to elliptic boundary-value problems. They naturally arise in integral representations of solutions of first-order elliptic systems in terms of fundamental matrices or their parametrices. The investigation of boundary-value problems for second-order and higher-order elliptic equations or systems is reduced to first-order elliptic systems.

On Stresses of a Strip under Concentrated Loads at Longitudinal Borders

The problem of plane elasticity on the stress state of strip S of constant width 2c whose opposite borders are loaded by concentrated forces is analyzed. An analytical solution to this problem is found via methods in the theory of functions of a complex variable. The stress components at an arbitrary point of the strip are defined in terms of two regular functions, Φ(z) and Ψ1(z). To find these functions, the conformal mapping of domain S onto lower half-plane ζ is used. The half-plane problem is solved via classical technique based on Cauchy-type integrals. Exact analytical expressions for functions Φ(ζ) and Ψ1(ζ) are obtained, which are then converted by the inverse conformal transformation into the required formulas for Φ(z) and Ψ1(z). Since functions Ψ1(z) and Φ'(z) were found to be related, the stresses in strip S were specified by function Φ(z) and its derivative Φ'(z). Graphs of normal and shear stresses on the lines parallel to the borders of the strip are presented. The stresses along the axis of the strip are compared to Filon’s data. The solution satisfies the differential equilibrium equations, boundary conditions, and the continuity equation.

Biorthogonal Systems of Analytic Functions Generated by a Regular Triangle

We consider the properties of biorthogonal systems induced by a convolution operator with Carleman kernel for a regular triangle. This is a perturbed singular operator with fixed singularities. We describe its set of anti-invariant points. To this end, we regularize the operator using a Carleman linear convolution shift that maps each triangle side to itself and changes its orientation, with the middle points of the sides being the fixed points of the shift. We search for a solution in the form of a Cauchy-type integral with unknown density. For this, both the theory of the Carleman boundary-value problem and the method of locally conformal gluing are used in an essential manner. We also apply the theory of elliptic functions that are generated by the corresponding doubly periodic group determined by the triangle as ‘half’ of the fundamental set. Using the method of contracting mappings in a Banach space, we study the corresponding homogeneous Fredholm integral equation of the second kind with regard to its solvability. Its fundamental system of solutions contains a single function; the fundamental system of solutions of the conjugated equation contains only the constant function. This makes it possible to use this equation for the construction of a system of biorthogonally conjugated analytic functions. More precisely, we consider a system of successive derivatives of a certain rational function determined by the Carleman kernel for the triangle and investigate the approximating properties of this system, as well as those of the corresponding biorthogonally conjugated system. This is a system of Cauchy-type integrals over the triangle boundary with a density which is invariant under the considered Carleman shift. Nontrivial decompositions of zero are obtained using the system of successive derivatives of the given rational function. The results are applied to the representation of some classes of analytical functions by means of the corresponding biorthogonal series.

On Existence of Solutions of Nonlinear Equilibrium Problems on Shallow Inhomogeneous Anisotropic Shells of the Timoshenko Type

We prove the existence of solutions to the geometrically nonlinear boundary value problem for elastic inhomogeneous anisotropic shallow shells with rigidly clamped edges in the framework of the S.P. Timoshenko shear model. The boundary value problem is reduced to one nonlinear operator equation, whose solvability is established using the principle of compressed maps. The investigation method is based on integral representations for generalized displacements containing holomorphic functions determined by boundary conditions involving the theory of Cauchy type integrals with real densities.

Cauchy Type Integral in Bicomplex Setting and Its Properties

In this paper a bicomplex analogue of a Cauchy type integral in case of a hyperbolic curve of integration is studied. We derive the corresponding Plemelj–Sokhotski formulas for their limit values for all densities satisfying a $${\mathbb {BC}}$$ -Holder-type condition.

Some properties of a Cauchy type integral in a three-dimensional commutative algebra with one-dimensional radical

In the paper we consider a certain analog of the Cauchy type integral taking values in a three-dimensional commutative algebra over the field of complex numbers with one-dimensional radical. We have established sufficient conditions for the existence of limiting values for such an integral. It is also shown that analogues of Sokhotskii–Plemelj formulas hold.

Support design for permanent mine roadways in tectonically active areas

The authors propose an analytical method of support design for arbitrary cross-section mine roadways subjected to the action of tectonic forces and seismic loads of earthquakes. As distinguished from the existing methods, the new technique allows taking into account influence of a close interface of different rocks. The seismic analysis includes the action of the waves reflected from the interface and the waves refracted through it. The method is based on an analytical solution of 2D elastic problem on stress state of an elastic ring supporting an arbitrary hole in one of two half-infinite media with the common boundary. The solution of the problem has been obtained using the theory of analytical functions of complex variables, Cauchy type integrals, conformal mapping and complex series. The examples of the support design are given.

Oblique Derivative Problem Solution for the Lavrentyev-Bitsadze Equation in a Half-Plane

The paper solves the boundary value problem of an oblique derivative for the Lavrent'ev – Bitsadze equation in a half-plane. The Lavrent'ev – Bitsadze equation is an equation of mixed (elliptic-hyperbolic) type. Mixed-type equations arise when solving many applied problems (for example, when simulating transonic flows of a compressible medium).In the paper, the domain of ellipticity is a half-plane, and that of hyperbolicity is its adjacent strip. On one of the straight lines bounding the strip, an oblique derivative is specified (in the direction that forms an acute angle with this straight line), and on the other straight line, which is the interface between the strip and the half-plane, the solutions are matched by boundary conditions of the fourth kind. In the hyperbolicity strip, the solution is represented by the d'Alembert formula, and in the half-plane, where the equation is elliptic, the bounded solution is represented by the Poisson integral with unknown density. For this unknown density of the Poisson integral, a singular integral equation is obtained, which is reduced to the Riemann boundary value problem with a shift for holomorphic functions. The solution of the Riemann problem is reduced to the solution of two functional equations. Solutions of these functional equations and the Sokhotsky formula for an integral of Cauchy type allowed us to find the unknown density of the Poisson integral. This allowed us to find a solution to the oblique derivative problem as the sum of a functional series (up to an arbitrary constant term).

РАСЧЕТ МНОГОСЛОЙНЫХ ОБДЕЛОК КОМПЛЕКСОВ ПАРАЛЛЕЛЬНЫХ ПОДВОДНЫХ ТОННЕЛЕЙ С УЧЕТОМ ПОСЛЕДОВАТЕЛЬНОСТИ СООРУЖЕНИЯ

The analytical design methodfor multilayer non-circular linings of complexes mutually influencing underwater tunnels with the sequence of their driving is proposed. The method is based on the solution of the corresponding elasticity theory plane problem based on the application of the analytical functions of complex variable apparatus, conform mapping and properties of Cauchy-type integrals. The method is implemented in the form of a complete calculation algorithm and the original computer program. Specific results of design illustrating the effect of the sequence of construction of a complex consisting of three underwater tunnels on the stress state and load - bearing capacity of multilayer underground structures are presented.

Method of mechanical quadratures for solving singular integral equations of various types

The method of mechanical quadratures is proposed as a common approach intended for solving the integral equations defined on finite intervals and containing Cauchy-type singular integrals. This method can be used to solve singular integral equations of the first and second kind, equations with generalized kernel, weakly singular equations, and integro-differential equations. The quadrature rules for several different integrals represented through the same coefficients are presented. This allows one to reduce the integral equations containing integrals of different types to a system of linear algebraic equations.

Nonlinear harmonic analysis of integral operators in weighted grand Lebesgue spaces and applications

In this article, we give a boundedness criterion for Cauchy singular integral operators in generalized weighted grand Lebesgue spaces. We establish a necessary and sufficient condition for the couple of weights and curves ensuring boundedness of integral operators generated by the Cauchy singular integral defined on a rectifiable curve. We characterize both weak and strong type weighted inequalities. Similar problems for Calderon–Zygmund singular integrals defined on measured quasimetric space and for maximal functions defined on curves are treated. Finally, as an application, we establish existence and uniqueness, and we exhibit the explicit solution to a boundary value problem for analytic functions in the class of Cauchy-type integrals with densities in weighted grand Lebesgue spaces.

Singular Integral Operators and Elliptic Boundary-Value Problems. I

The book consists of three Parts I-III and Part I is presented here. In this book, we develop a new approach mainly based on the author’s papers. Many results are published here for the first time. Chapter 1 is introductory. The necessary background from functional analysis is given there for completeness. In this book, we mostly use weighted Ho¨lder spaces, and they are considered in Ch. 2. Chapter 3 plays the main role: in weighted Ho¨lder spaces we consider there estimates of integral operators with homogeneous difference kernels, which cover potential-type integrals and singular integrals as well as Cauchy-type integrals and double layer potentials. In Ch. 4, analogous estimates are established in weighted Lebesgue spaces. Integrals with homogeneous difference kernels will play an important role in Part III of the monograph, which will be devoted to elliptic boundary-value problems. They naturally arise in integral representations of solutions of first-order elliptic systems in terms of fundamental matrices or their parametrixes. Investigation of boundary-value problems for second-order and higher-order elliptic equations or systems is reduced to first-order elliptic systems.

Riemann–Hilbert problem in the class of Cauchy type integrals with densities of grand Lebesgue Spaces

The present paper deals with a solution of the Riemann–Hilbert problem with piecewise continuous coefficients in the class of Cauchy type integrals with densities in grand Lebesgue spaces. Necessary and sufficient solvability condition is established. In solvability case the solutions in explicit form are constructed.

On the asymptotic expansion of certain plane singular integral operators

We discuss the problem of the asymptotic expansion for some operators in a general theory of pseudo-differential equations on manifolds with borders. Using the distribution theory one obtains certain explicit representations for these operators. These limit distributions are constructed with the help of the Fourier transform, the Dirac mass-function and its derivatives, and the well-known distribution related to the Cauchy type integral.