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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 Riemann-Hilbert 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.

Let

in

satisfies certain conditions, then (1.1) can be transformed into the complex form

where

where

in which

We assume (1.4) satisfies the following conditions:

Condition C 1)

2) The above functions are measurable in

where

3) The complex system (1.4) satisfies the following ellipticity condition

where

For convenience,

in which

The so-called Riemann-Hilbert boundary value problem for the complex system (1.4) may be formulated as follows.

Problem A Find a system of continuous solutions

in which

Denote by

where

and we call

For problem A, we will assume

in which

in

In general, Problem A may not be solvable. Hence we propose a modified problem as follows.

Problem B Find a system of continuous solutions

Here

in which

constants to be determined appropriately, and

where

which are not the discontinuous points from

Clearly, with certain modification on the symbols on some arcs on

to be continuous. In this case, its index

are integers. And we have the following:

in which

In addition, we may assume that the solution

where

for a positive constant

If

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

(1.6), (1.7), (1.11) are small enough. Then any solution

where

Proof There is no harm in assuming that

in which

Following the proof of the Theorem 2.1 of Chapter VI in [

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

where

in which

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

Proof We consider the modified Riemann-Hilbert problem (Problem

where

in which

vector of constants. When

Substituting the zero vector

solution

we can find a sequence

From (2.13) and (2.14), we have

In accordance with Theorem 2.1, we can conclude

where

and

for

Hence, there exists an analytic vector

Thus

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

Proof We consider the nonlinear elliptic complex system with the parameter

where

solvable, and the solution

where

Suppose that when

Suppose that Problem B for (2.13) with

The difference of the above equations for

From Condition C, we can derive that

and

Moreover,

Similarly to Theorem 3.3, Chapter I, [

where

for every

for

By Condition C, it follows that

In this section, we shall introduce an error estimate of the above approximate solutions.

Theorem 3.1 Under the same conditions as in Theorem 2.3, let

where

Proof From (1.4) and (2.24) with

It is clear that

Noting that

and then

hence we have

in which

where the non-negative constants

From (3.6) and (3.7), it follows that

where

This shows that (3.1) holds. If the positive constant

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