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A function theory method in elliptic problems in the plane. II. the piecewise smooth case
Abstract
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.
... where Φ ± (z) are analytic functions in Π ± , has been investigated in weighted spaces L p (ρ), p ∈ (1, ∞), ρ(x) = N k=1 |x k − x| α k by many authors; let's note some of them: Khvedelidze B.V. [1]- [3], Simonenko I.B. [4], [5], Tovmasyan N.E. [8], [9], Soldatov A.P. [10], [11], Kazarian K., Soria F., Spitkovsky I. [12], Kazarian K.S. [13], [14]. Also see [15] - [18]. ...
The paper considers the Riemann boundary value problem in the half-plane in the class of functions that are -continuous with respect to the weight , when the weight function has infinite number of zeros. Necessary and sufficient conditions for solvability of the problem are established. If the problem is solvable, solutions are represented in an explicit form.
... 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.
Рассматриваются сингулярные интегральные операторы на кусочно гладкой кривой в весовых лебеговых и гёльдеровых пространствах с кусочно непрерывными матричными коэффициентами. В отличие от классического случая эти операторы помимо сингулярного оператора Коши содержат также некомпактные интегральные операторы специального вида, которые определяются ядром, приближенно однородным степени −1 относительно расстояний до узлов кривой. Подобные операторы возникают во многих приложениях. Получен критерий фредгольмовости этих операторов и приводится формула их индекса.
Рассматривается интегро-дифференциальное уравнение гиперболического типа в ограниченной по переменной x области D = {(x, t) : 0 < x < l, t > 0}. Сначала исследуется прямая задача. Для прямой задачи исследуется обратная задача определения ядра интегрального члена интегро-дифференциального уравнения на основе имеющейся дополнительной информации о решении прямой задачи при x = 0. Интегральное уравнение, полученное относительно u(x, t), трижды дифференцируется по t, и решение данной задачи с использованием дополнительного условия сводится к решению системы интегральных уравнений относительно неизвестных функций. К этой системе применяется принцип сжатых отображений в пространстве непрерывных функций с весовыми нормами. Доказана теорема о глобальной однозначной разрешимости обратной задачи, и получена оценка условной устойчивости решения обратной задачи.
The paper considers the Riemann boundary value problem in the half-plane in the space L¹(ρ). The weight function ρ(x) has infinite number of zeros on the real axis. The boundary condition is understood in the sense of L¹(ρ). A necessary and sufficient condition is obtained for the normal solvability of the considered problem. The solutions are represented in explicit form.
The goal of this note is to consider the application of the complex interpolation space of Morrey spaces. Actually, the boundedness of Riesz potentials acting on Morrey spaces, which is obtained by Adams, is refined by means of the complex interpolation.
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.
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.
In simply connected complex-shaped domains ℬ a Riemann-Hilbert problem with discontinuous data and growth condidions of a solution at some points of the boundary is considered. The desired analytic function ℱ(z) is represented as the composition of a conformal mapping of ℬ onto the half-plane H+ and the solution ℘ of the corresponding Riemann-Hilbert problem in H+ Methods for finding this mapping are described, and a technique for constructing an analytic function ℘+ in H+ in the terms of a modified Cauchy-type integral. In the case of piecewise constant data of the problem, a fundamentally new representation of ℘+ in the form of a Christoffel-Schwarz-type integral is obtained, which solves the Riemann problem of a geometric interpretation of the solution and is more convenient for numerical implementation than the conventional representation in terms of Cauchytype integrals.
We consider a Dirichlet-type problem in the upper half-plane for the improperly elliptic equation , with boundary functions from the class L (ρ) (ρ = (1 + |x|), α ≥ 0). The solutions of the problem are obtained in explicit form.
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 of these problems.
CONTENTS Introduction Chapter I. General elliptic boundary-value problems § 1. The solubility of general elliptic boundary-value problems in domains with conic points § 2. The asymptotic behaviour of the solutions of a general boundary-value problem in the neighbourhood of a conic boundary point § 3. General boundary-value problems in non-smooth domains Chapter II. Boundary-value problems for the equations of mathematical physics in non-smooth domains § 1. Boundary-value problems for the system of elasticity theory § 2. Problems of hydrodynamics in domains with a non-smooth boundary § 3. The biharmonic equation Chapter III. Second-order elliptic equations in domains with a non-smooth boundary § 1. Boundary-value problems for second-order elliptic equations in an arbitrary domain § 2. Boundary-value problems in domains with isolated non-regular points on the boundary § 3. Second-order elliptic equations in domains with edges § 4. Boundary-value problems in domains that are diffeomorphic to a polyhedron Chapter IV. Parabolic and hyperbolic equations and systems in non-smooth domains § 1. Parabolic equations and systems in non-smooth domains § 2. Hyperbolic equations and systems in domains with singular points on the boundary References
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.