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