Abstract
In this paper, a class of elliptic systems of four 1st order differential equations of the orthogonal type in R3 is considered. For such systems we study the issue of regularizability of the Riemann – Hilbert boundary value problem in an arbitrary limited simply-connected region with a smooth boundary in R3. Using the coefficients of the elliptic system and the matrix of the boundary operator, a special vector field is constructed, and its not entering the tangent plane in any point of the boundary provides the Lopatinski condition of the regularizability of the boundary value problem. The obtained condition permits to prove that the set of regularizable Riemann – Hilbert boundary value problems for the considered class of systems has two components of homotopic connectedness, and the index of an arbitrary regularizable problem equals to minus one.
Highlights
In this paper, a class of elliptic systems of four 1st order differential equations of the orthogonal type in R3 is considered
A class of elliptic systems of four 1st order differential equations of the orthogonal type in R3 is considered. For such systems we study the issue of regularizability of the Riemann – Hilbert boundary value problem in an arbitrary limited -connected region with a smooth boundary in R3
Using the coefficients of the elliptic system and the matrix of the boundary operator, a special vector field is constructed, and its not entering the tangent plane in any point of the boundary provides the Lopatinski condition of the regularizability of the boundary value problem
Summary
Следовательно, задача (1), (2) в классе регуляризуемых краевых задач гомотопна задаче для системы (1) с граничным условием (y ∈ ∂Ω). Аналогично доказывается, что каждая задача из множества I– гомотопна задаче для системы (20) с граничным условием u( 1u 2 1. Тогда неравенство ν( y); P( y, t) = (1 − t) ν( y); P( y) − t < 0 выполняется при всех y ∈ ∂Ω и любом t ∈ [0; 1] и полученная задача гомотопна задаче для системы (1) с граничным условием: a3 1 |. В статье в терминах матрицы граничного оператора и коэффициентов системы описано условие Лопатинского регуляризуемости краевой задачи Римана – Гильберта для класса эллиптических систем четырех дифференциальных уравнений первого порядка ортогонального типа в R3. – доказать, что I имеет две компоненты гомотопической связности; – для каждой компоненты указать представителя; – вычислить индекс произвольной задачи из множества I. Что индекс задачи, являясь гомотопическим инвариантом, в рассматриваемом случае не различает компоненты гомотопической связности множества I
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