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A function theory method in boundary value problems in the plane. I: The smooth case
Abstract
A general (not necessarily local) boundary value problem is considered for an elliptic [] system on the plane of [ n]th order containing only leading terms with constant coefficients. By a method of function theory developed for elliptic [] systems of first order
[]
with a constant triangular matrix [], []; this problem is reduced to an equivalent system of integrofunctional equations on the boundary. In particular, a criterion that the problem be Noetherian and a formula for its index are obtained in this way. All considerations are carried out in the smooth case when the boundary of the domain has no corner points, while the boundary operators act in spaces of continuous functions.
... In this paper, within the framework of the general functional-theoretic approach developed in [16], necessary and sufficient conditions for the unique solvability of the Dirichlet problem for a second-order elliptic system with constant leading coefficients 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. ...
... is the operator of a nondegenerate affine transformation of a plane of the complex variable w that preserves the origin. Now, let us replace the function u on v = T 1,στ u; (16) then, ∂∂v + ∂ 2 (Tv) = 0, ...
... Thus, for the boundary function f (z) = a m z m , m 2, we have The summation of the solutions leads to a Poisson-type integral representation, which we present (see also [25,28,29]) for the solution u(z) of Equation (15), which is related to the solution v(z) of Equation (17) by way of relation (16). ...
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.
... As was shown in [7], all basic facts of the theory of analytic functions associated with the integral Cauchy formula can be extended to solutions of the system (14). According to [6], the integral operator I 0 : C μ (Γ) → C μ (D) is bounded and the following Sokhotsk-Plemelj formula is valid: ...
... Obviously, in the case of the scalar matrix J = i, the operator S 0 becomes a classical singular Cauchy operator, which we denote by S. As was shown in [6], under the assumption Γ ∈ C 1,ν , the difference S 0 − S is a compact operator in the space C μ (Γ), and all principal results of the classical theory of singular operators (see [4]) can also be applied to the operator ...
... under the conditions (19) on the real density ϕ, which is established in [6] (see also [8]). Using the representation (18) and the Sokhotski-Plemelj formula (11), we can reduce the problem (8), (9) to the following equivalent system of operator equations: ...
In a finite domain D of the complex plane bounded by a smooth contour Γ, we consider the Riemann–Hilbert boundary-value problem ReCU⁺ = f for the first-order elliptic system∂U∂y−A∂U∂x+azUz+bzUz¯=Fz
with constant leading coefficients. Here + means the boundary value of the function U on Γ, the constant matrices A1,A2 ∈ ℂl×l and the (l × l)-matrix coefficients a and b belong to the Hölder class Cμ, 0 < μ < 1, and (l × l)-matrix function C belongs to the class Cμ(Γ). We prove that in the class U ∈ Cμ(D¯) ∩ C¹(D), this problem is a Fredholm problem and its index is given by the formulaϰ=−∑j=1m1πargdetGΓj+2−ml.
... (1. 16) In fact, let, for example, α / ∈ A 1 . Then α 2 p 3 − α 5 p 2 = 0 and we have the equality α 2 2 χ = (α 2 2 p 1 − α 2 5 p 2 )p 2 . ...
... The boundary properties of integrals of the Cauchy type Iϕ with density from ϕ ∈ L p (Γ), 1 < p < ∞, were studied in [16]. In this case the J-analytic function φ = Iϕ belongs to the Hardy space H p (D), which can be introduced [18] in the following way. ...
... In this notation the Cauchy type integral as a linear operator ϕ → φ = Iϕ is bounded L p (Γ) → H p (D), the angular limiting values φ ± exist almost all in Γ, and the Sohocki-Plemelj formula holds [16]. Conversely, any function φ ∈ H p is representable as a Cauchy type integral with density ϕ ∈ L p (Γ). ...
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 representation of these solutions is obtained using the potentials. As a corollary, the first and the second boundary-value problems in various spaces (Hölder, Hardy, and the class of functions just continuous in a closed domain) are reduced to the equivalent system of the Fredholm boundary equations in corresponding spaces. Note that such an approach was developed in [19, 20] for common second-order elliptic systems with constant (higher-order only) coefficients. However, due to important applications, it makes sense to consider this approach in detail directly for the Lamé system. To illustrate these results, in the last two sections we consider the Dirichlet problem with piecewise-constant Lamé coefficients when contact conditions are given on the boundary between two media. This problem is reduced to the equivalent system of the Fredholm boundary equations. The smoothness of kernels of the obtained integral operators is investigated in detail depending on the smoothness of the boundary contours.
... Therefore, the solutions φ of the system (1) is called the function analytic in the Douglis sense or shortly Douglis analytic functions. It plays an important role for elliptic systems of second and high order [2,3]. For general first order elliptic systems, the problem (2) was investigated by many authors [4,5]. ...
... This theorem is analogue of the corresponding result of N. I. Mushelishvili [7] for usual analytic functions. Using the representation (3) we can reduce the problem (1), (2) to the corresponding equivalent system of singular equations on Γ with respect to an unknown (ϕ, ξ ) ∈ L p (Γ) × R n 1 satisfying (4). This system is the following: ...
The Riemann-Hilbert problem for the first order elliptic system is considered in the Hardy-Smirnov space. This system is reduced to an equivalent Fredholm integral system on the boundary in the space Lp.
... is called the generalized Cauchy kernel for Eq. (4.2), and the following relations are valid (see [13,76]): ...
We consider the Riemann–Hilbert-type boundary-value problems in the case of several unknown functions and obtain solvability conditions and index formulas for the linear conjugation problem, the Riemann–Hilbert problem, and the Riemann–Hilbert–Poincaré problem on a plane cut along several regular arcs in various weighted functional classes. We also examine the general differential boundary-value problem for analytic functions and boundary-value problems with shifts and complex conjugation on a cut plane.
... We present first some necessary fundamental results of the theory of generalized analytic functions [53], [6], [7], [8] in the form convenient for our purposes. A modern consistent exposition of this theory was given by A.Soldatov [48], [49], [50]. ...
Two-dimensional Yang-Mills equations on Riemann surfaces and Bogomol’ny equation are studied using methods of the theory of Riemann-Hilbert problem. In particular, representations of solutions in terms of connections are given and solvability conditions of arising Riemann-Hilbert problems are established.
... We present first some necessary fundamental results of the theory of generalized analytic functions and vectors [124], [31], [90] in the form convenient for our purposes. A modern consistent exposition of this theory was gives by A.Soldatov [115], [116]. ...
The paper considers the Riemann boundary value problem in the half-plane in the space Lp(ρ), where weight function ρ(x) has infinite number of zeros. A necessary and sufficient condition is obtained for the normal solvability and Noetherianness of the considered problem. If the problem is solvable, solutions are represented in an explicit form.
The Dirichlet problem for sixth order improperly elliptic equation is considered. The functional class of boundary functions, where this problem is normally solvable is determined. If the roots of the characteristic equation satisfy some conditions, the number of linearly independent solutions of homogeneous problem and the number of linearly independent solvability conditions of in-homogeneous problem are obtained. Solutions of homogeneous problem and solvability conditions of in-homogeneous problem are obtained in explicit form.
The Dirichlet problem for sixth-order improperly elliptic equation is considered. The functional class of boundary functions, where this problem is normally solvable is determined. If the roots of the characteristic equation satisfy some conditions, the number of linearly independent solutions of the homogeneous problem and the number of linearly independent solvability conditions of the inhomogeneous problem are determined. Solutions of the homogeneous problem and solvability conditions of the inhomogeneous problem are obtained in explicit form.
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 the generalized potentials of a double layer are derived and their applications to solution of the Dirichlet problem are described. The results are illustrated by the example of the Lamé system of plane elasticity theory.
A solution of the Dirichlet problem for an elliptic systemof equations with constant coefficients and simple complex characteristics in the plane is expressed as a double-layer potential. The boundary-value problem is solved in a bounded simply connected domain with Lyapunov boundary under the assumption that the Lopatinskii condition holds. It is shown how this representation is modified in the case of multiple roots of the characteristic equation. The boundary-value problem is reduced to a system of Fredholm equations of the second kind. For a Hölder boundary, the differential properties of the solution are studied.
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 systems is introduced in the frame of functional theoretic approach. With the help of these potentials all solutions are described in the Holder and Hardy classes as well as in the class of all continuous functions.
We consider second-order elliptic systems on the plane with constant (and only leading) matrix coefficients. We show that for these systems the notion of being weakly coupled (in the sense of A.V. Bitsadze) is equivalent to the well-known complementarity condition for the Dirichlet problem. In the framework of the function theoretic approach, we introduce analogs of double-layer potentials for solutions of weakly coupled systems. By using these potentials, we obtain a complete description of solutions of weakly elliptic systems in Hölder classes as well as in the Hardy classes h
p
(D) and
.
For elliptic systems of the second and third orders with an interior supersingular point, we find the integral representation of their solutions and the corresponding inversion formulas.The obtained integral representations can be applied in studying the asymptotic behavior of solutions as r = |z| → 0 and also in studying boundary-value problems.
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 carried out in domains with piecewise smooth boundaries and in weighted spaces. A Noetherian criterion and an index formula for this problem are established, and the asymptotics of its solution in a neighborhood of corner points is described.
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 study the Fredholm solubility of these problems. (In particular, we obtain an analytic expression for the index.) We work in weighted Hölder and Hardy spaces.
We consider boundary-value problems for general elliptic systems in a bounded multiply connected domain with a smooth boundary in the plane. This includes boundary-value problems for homogeneous elliptic systems and for systems elliptic in the sense of Petrovskii or Douglis-Nirenberg. The systems under consideration need not be properly elliptic. We obtain necessary and sufficient conditions for the system to be Noetherian and give index formulae with applications to geometry and hydrodynamics.
The space indicated in the title is introduced and studied.
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.
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 hydromechanics.
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.
