Approximate Solutions to the Discontinuous Riemann-Hilbert Problem of Elliptic Systems of First Order Complex Equations

Several approximate methods have been used to find approximate solutions of elliptic systems of first order equations. One common method is the Newton imbedding approach, i.e. the parameter extension method. In this article, we discuss approximate solutions to discontinuous RiemannHilbert boundary value problems, which have various applications in mechanics and physics. We first formulate the discontinuous Riemann-Hilbert problem for elliptic systems of first order complex equations in multiply connected domains and its modified well-posedness, then use the parameter extensional method to find approximate solutions to the modified boundary value problem for elliptic complex systems of first order equations, and then provide the error estimate of approximate solutions for the discontinuous boundary value problem.

Problem A Find a system of continuous solutions ( ) which satisfies the boundary condition where 0 1 kmj γ ≤ < when 0 kmj J = , and 1 0 There is no harm in assuming that the partial indexes k K of ( ) are not integers, and the partial indexes k K of ( ) and we call ( ) For problem A, we will assume Γ is an open arc from the point , min ,1 2 p δ τ α < − are small positive constants.In general, Problem A may not be solvable.Hence we propose a modified problem as follows.
Problem B Find a system of continuous solutions ( ) w z of the complex equation (1.4) in * D , which satisfies the modified boundary condition constants to be determined appropriately, and ( ) ( ) k X z are given below.We begin with the following function where are fixed points, which are not the discontinuous points from Z .Note that the positive direction applies to the boundary circles ( ) 12), Chapter V, [2], we see that Clearly, with certain modification on the symbols on some arcs on Γ , ( ) ( ) ( ) to be continuous.In this case, its index are integers.And we have the following: are solutions of the modified Dirichlet problems with the above boundary conditions for analytic functions, ( ) ( ) In addition, we may assume that the solution ( ) w z satisfies the following point conditions where ( ) ) is the modified Dirichlet boundary value problem for (1.4).It is easy to see that the solutions of (1.4) include the generalized hyperanalytic functions as special cases.In fact, if (1.4) is linear, and = ≤ ≤ − then the solutions of (1.4) are called generalized hyperanalytic functions.

Parameter Extension Method of the Discontinuous Riemann-Hilbert Problem for Elliptic Systems of First Order Complex Equations
We begin with the following estimates of the solution for problem B. Theorem 2.1 Suppose that the complex system (1.4) satisfies Condition C and the constants where ( ) Proof There is no harm in assuming that * 2 5 6 0.

W z w z k =
It can be seen that

( )
W z is a solution of the following boundary value problem , Following the proof of the Theorem 2.1 of Chapter VI in [1], we can derive the estimate From the above estimate, it immediately follows that the estimate (2.1) is true.
In addition, we assume that (1.4) satisfies the following condition: For any continuous vectors , w z w z and any measurable vector where ( ) ( ) in which ( ) are non-negative constants.Now, we prove that there exists a unique solution of the modified Riemann-Hilbert problem (Problem B) for analytic vectors by the parameter extensional method.
Theorem 2.2 Let 4 k in (1.11) be a sufficiently small positive constant.Then Problem B for analytic vectors has a solution.
Proof We consider the modified Riemann-Hilbert problem (Problem B′ ) for analytic vectors with the boundary conditions , ( ) vector of constants.When 0 t = , it is clear that Problem B′ for analytic vectors has a unique solution (see [1]).If Problem B′ with ( ) for analytic vectors is solvable, we shall prove that there exists a positive number δ independent of 0 t , such that Problem B′ for every In fact, the boundary conditions (2.9), (2.10) can be rewritten in the form . (2.12) Substituting the zero vector into the position of ( ) w z on the right hand side of (2.11) and (2.12), by the hypothesis, the boundary value problem (2.11), (2.12) for analytic vectors has a unique solution .
From (2.13) and (2.14), we have ) In accordance with Theorem 2.1, we can conclude ) , , , where N is a positive integer.This shows that ( ) Hence, there exists an analytic vector Problem B for analytic vectors is solvable.Next we prove the solvability of Problem B for the system (1.4).Theorem 2.3 Let the nonlinear elliptic system (1.4) satisfy Condition C, and 1 3 4 , , k k k in (1.6), (1.7), (1.11)  be sufficiently small positive constants.Then Problem B for the complex system (1.4) is solvable.
Proof We consider the nonlinear elliptic complex system with the parameter where ( ) w z is an analytic vector satisfying the boundary conditions Suppose that when ( ) ≤ < , Problem B for the system (2.19) has a unique solution.Then we shall prove that there exists a neighborhood of 19) is solvable.In fact, the complex system (2.19) can be written in the form , , , , .
Suppose that Problem B for (2.13) with ( ) ≤ < is solvable, by using the similar method as in the proof of Theorem 2.2, we can find a positive constant δ , so that for every .
The difference of the above equations for 1 n + and n is as follows: From Condition C, we can derive that .
Similarly to Theorem 3.3, Chapter I, [1], we , , , , , , , , where N is a positive integer.This shows that ( ) n m → ∞ Thus there exists a system of continuous functions w z is a solution of Problem B for the system (2.23), i.e. (2.19) for t E ∈ .
It is easy to see that the positive constant δ is independent of ( )

Error Estimates of Approximate Solutions of the Discontinuous Riemann Hilbert Problem for Elliptic Systems of First Order Complex Equations
In this section, we shall introduce an error estimate of the above approximate solutions.