Riemann-Hilbert problem for Moisil-Teodorescu system
We consider the boundary conditions for the solvability of the Riemann-Hilbert problem in a bounded simply connected domain with smooth boundary for the three-dimensional analogue of the Cauchy-Riemann system - elliptical system Moisil-Teodorescu. Presented for the first time the system was first proposed and investigated by the Romanian mathematicians. It is a linear system of partial differential equations of the first order for the four-vector function in three-dimensional space. The increased interest in the study of the system caused them a special significance both in mathematics and in physics. In the modern theory of elliptic boundary value problems of the central place is occupied by the problem of finding a solution to the Fredholm elliptic systems tasks. This problem was posed in the famous monograph AV Bitsadze. It is in progress A. Bitsadze first began the study of boundary value problems for three-dimensional analogues of the Cauchy-Riemann system. In these studies, the study focused mainly half. The aim of this work is to provide conditions for the solvability of the Riemann-Hilbert boundary value problem in a bounded simply connected domain of three-dimensional space. With the known conditions of solvability of classical Dirichlet and Neumann proved a necessary and sufficient condition for the solvability of the Riemann-Hilbert problem in the class of Holder. As an example, consider the analogue of the problem Schwartz theory of analytic functions for elliptic system under consideration and investigated its solvability. A criterion for the solvability of this problem in the class of functions. Using the integral representation of the general solution of the boundary value problem of Riemann-Hilbert problem is reduced to the study of a system of singular integral equations on the boundary. Because of the weak features of the kernel matrix corresponding Fredholm integral equation is uniquely soluble and when the orthogonality conditions.