О НЕУСТОЙЧИВОСТИ РЕШЕНИЙ ОДНОРОДНОЙ КРАЕВОЙ ЗАДАЧИ РИМАНА ДЛЯ КВАЗИГАРМОНИЧЕСКИХ ФУНКЦИЙ В КРУГОВЫХ ОБЛАСТЯХ

K.M Rasulov,Sh.S Khankishieva

doi:10.14529/mmph180306

Abstract

We consider the boundary value problem of Riemann type (the conjugation problem) in the classes of piecewise quasiharmonic functions. A homogeneous problem of a Riemann type problem in the classes of piecewise quasiharmonic functions of the second kind in circular domains is studied in details. In particular, in this case a clear solution method is developed for a homogeneous problem of Riemann type, the logical essence of which consists in reducing the solution of the homogeneous problem under consideration to a sequential solution of the common homogeneous Riemann problem for analytic functions and two second-order linear differential Euler equations. Moreover, instability of the solutions of the homogeneous problem is determined with respect to the change in the radius value of the considered circular domain, and a complete picture of its solvability for different values of the index of the problem and the radius of the circular domain is constructed. It is proved that the main reason for the instability of the solutions of a homogeneous problem of Riemann type in classes of piecewise quasiharmonic functions of the second kind in circular domains with respect to the change in the radius value of the considered circular domain is the fact that the number of linear independent analytic solutions of homogeneous differential Euler equations, to which the desired Riemann type problem is reduced, depends essentially on the radius of the considered circular region. In this paper we use the methods of the theory of functions of a complex variable, the theory of integral equations, and the analytic theory of differential equations.

Highlights

Рассмотрим на плоскости комплексного переменного z = x + iy область T +, ограниченную замкнутым гладким контуром L, причем для определенности будем считать, что точка z = 0 принадлежит T +
Причем общее решение задачи ρ20 , задаваемое формулой (10), будет линейно зависеть не более чем от l произвольных комплексных постоянных, причем l

Summary

Introduction

О неустойчивости решений однородной краевой задачи Римана для квазигармонических функций в круговых областях где G(t) – заданная на Lr функция, G(t) ∈ H (Lr ) и G(t) ≠ 0 на Lr . A(T + ) ∩ H (2) (L) и A(T − ) ∩ H (2) (L) соответственно, общее решение исходной задачи ρ20 также можно найти по формуле (10), где φ + (z) и φ − (z) – общие решения неоднородных дифференциальных уравнений (11) и (12) соответственно.

Results

Conclusion