Applied Mathematics
Vol.5 No.10(2014), Article ID:46591,11 pages DOI:10.4236/am.2014.510148

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

Guochun Wen1*, Yanhui Zhang2, Dechang Chen3#

1School of Mathematical Sciences, Peking University, Beijing, China

2Mathematics Department, Beijing Technology and Business University, Beijing, China

3Uniformed Services University of the Health Sciences, Bethesda, USA

Email: wengc@math.pku.edu.cn, zhangyanhui@th.btbu.edu.cn, dechang.chen@usuhs.edu

Received 23 March 2014; revised 23 April 2014; accepted 30 April 2014

ABSTRACT

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.

Keywords:Discontinuous Riemann-Hilbert Problem, Elliptic Systems of First Order Complex Equations, Estimates and Existence of Solutions, Multiply Connected Domains

1. Introduction

Let be an -connected bounded domain in with the boundary

. Without loss of generality, we assume that is a circular domain in, bounded by the -circles and. In this article, the notations are the same as in references [1] -[12] . If the first order elliptic system with unknown real functions

(1.1)

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

(1.2)

where (see Section 4, Chapter 2 in [5] ). Its vector form is as follows:

(1.3)

where is the transposed matrix of. We discuss the first order complex system (1.3) in the form

(1.4)

in which with, with with

We assume (1.4) satisfies the following conditions:

Condition C 1) are continuous in for almost every point

2) The above functions are measurable in for all systems of continuous functions in and any systems of measurable functions in and satisfy

(1.5)

(1.6)

where is as stated in (1.8) below, and are non-negative constants.

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

(1.7)

where are non-negative constants.

For convenience, and are used to indicate   and respectively, and we define the following:

in which and are stated as in (1.12), (2.1) below, and

and are non-negative constants.

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 of (1.4), which satisfies the boundary condition

(1.8)

in which with for and

are the first kind of discontinuous points of on.

Denote by and the left limit and right limit of as on, and

(1.9)

where when, and when. There is no harm in assuming that the partial indexes of on are not integers, and the partial indexes of on are integers. Set

(1.10)

and we call the index of Problem A.

For problem A, we will assume satisfy the conditions

(1.11)

in which is an open arc from the point to on are non-negative constants,. Moreover, we require that the solution possess the property

(1.12)

in, where in and 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 of the complex equation (1.4) in, which satisfies the modified boundary condition

(1.13)

Here

(1.14)

in which are unknown real constants to be determined appropriately, and, if is an odd integer. More description on and are given below. We begin with the following function

where denotes the partial index on, are fixed pointswhich are not the discontinuous points from. Note that the positive direction applies to the boundary circles. Similarly to (1.7)-(1.12), Chapter V, [2] , we see that

Clearly, with certain modification on the symbols on some arcs on, on is seen to be continuous. In this case, its index

are integers. And we have the following:

in which are solutions of the modified Dirichlet problems with the above boundary conditions for analytic functions, are real constants, and.

In addition, we may assume that the solution satisfies the following point conditions

(1.15)

where are distinct points, and are all real constants satisfying the conditions

(1.16)

for a positive constant. Problem B with in, on and is called Problem.

If then Problem B for (1.4) 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 and then the solutions of (1.4) are called generalized hyperanalytic functions.

2. 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 in

(1.6), (1.7), (1.11) are small enough. Then any solution of Problem B for (1.4) satisfies the estimate

(2.1)

where with in,

with are non-negative constants.

Proof There is no harm in assuming that Let It can be seen that is a solution of the following boundary value problem

(2.2)

(2.3)

(2.4)

in which

(2.5)

Following the proof of the Theorem 2.1 of Chapter VI in [1] , we can derive the estimate

(2.6)

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 and any measurable vector

(2.7)

where satisfy the condition

(2.8)

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 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) for analytic vectors with the boundary conditions

(2.9)

(2.10)

where

in which is a real parameter, and is any vector of real functions, and is any vector of constants. When, it is clear that Problem for analytic vectors has a unique solution (see [1] ). If Problem with for analytic vectors is solvable, we shall prove that there exists a positive number independent of, such that Problem for every has a unique solution. In fact, the boundary conditions (2.9), (2.10) can be rewritten in the form

(2.11)

(2.12)

Substituting the zero vector into the position of 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 and Using the successive iterationwe can find a sequence of analytic vectors, which satisfies the boundary conditions

(2.13)

(2.14)

From (2.13) and (2.14), we have

(2.15)

(2.16)

In accordance with Theorem 2.1, we can conclude

(2.17)

where with, and Choosing a positive constant, such that it is not difficult to see that

and

for where is a positive integer. This shows that

Hence, there exists an analytic vector such that

(2.18)

Thus is a solution of Problem with. From this we can derive that Problem with i.e. 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 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:

(2.19)

where is any measurable vector in and Applying Theorem 2.2, we see that Problem B for (2.19) with is solvable, and the solution can be expressed as

(2.20)

where is an analytic vector satisfying the boundary conditions

(2.21)

(2.22)

Suppose that when, Problem B for the system (2.19) has a unique solution. Then we shall prove that there exists a neighborhood of so that for every and any function Problem B for (2.19) is solvable. In fact, the complex system (2.19) can be written in the form

(2.23)

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, so that for every, there exists a sequence of solutions satisfying

(2.24)

The difference of the above equations for and is as follows:

(2.25)

From Condition C, we can derive that

and

Moreover, satisfies the homogeneous boundary conditions

(2.26)

(2.27)

Similarly to Theorem 3.3, Chapter I, [1] , we have

(2.28)

where are positive constants. Provided is small enough, so that we can obtain

(2.29)

for every Thus

for where is a positive integer. This shows that as Thus there exists a system of continuous functions in, such that

By Condition C, it follows that is a solution of Problem B for the system (2.23), i.e. (2.19) for. It is easy to see that the positive constant is independent of. Hence Problem B for the system (2.19) with is solvable. Correspondingly we can derive that when, Problem B for (2.19) is solvable. Especially Problem B for (2.19) with and, namely Problem B for the system (1.4) has a solution.

3. 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.

Theorem 3.1 Under the same conditions as in Theorem 2.3, let be a solution of Problem B for the complex system (1.4) satisfying Condition C in, and be its approximation as stated in the proof of Theorem 2.3 with Then we have the following error estimate

(3.1)

where with as in (2.28), and as in (1.6),(1.7), (1.11) and (1.16).

Proof From (1.4) and (2.24) with, we have

(3.2)

It is clear that satisfies the homogeneous boundary conditions

(3.3)

Noting that satisfy, and

and then is a solution of Problem for the complex equation

(3.4)

hence we have

(3.5)

in which

(3.6)

where the non-negative constants are as stated in (2.28), (1.5), (1.11) and (1.12). Moreover according to the proof of Theorem 2.3, we can derive

(3.7)

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

where and is the solution of Problem B for (2.24) with and Finally, we obtain

(3.8)

This shows that (3.1) holds. If the positive constant is small enough, so that when is sufficiently large and is close to 1, then the right hand side becomes very small.

Note: The opinions expressed herein are those of the authors and do not necessarily represent those of the Uniformed Services University of the Health Sciences and the Department of Defense.

References

NOTES

*Deceased.

#I am very grateful for the guidance and help of Professor Guochun Wen, who served as my adviser for many years. 1 will always remember him because he inf1uenced me greatly.