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.
Discontinuous Riemann-Hilbert Problem Elliptic Systems of First Order Complex Equations Esti-mates and Existence of Solutions Multiply Connected Domains1. 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
satisfies certain conditions, then (1.1) can be transformed into the complex form
where (see Section 4, Chapter 2 in [5] ). Its vector form is as follows:
where is the transposed matrix of. We discuss the first order complex system (1.3) in the form
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
where is as stated in (1.8) below, and are non-negative constants.
3) The complex system (1.4) satisfies the following ellipticity condition
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
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
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
and we call the index of Problem A.
For problem A, we will assume satisfy the conditions
in which is an open arc from the point to on are non-negative constants,. Moreover, we require that the solution possess the property
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
Here
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 points,
which 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
where are distinct points, and are all real constants satisfying the conditions
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
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
in which
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 and any measurable vector
where satisfy the condition
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
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
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 iteration,
we can find a sequence of analytic vectors, which satisfies the boundary conditions
From (2.13) and (2.14), we have
In accordance with Theorem 2.1, we can conclude
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
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:
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
where 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 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
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
The difference of the above equations for and is as follows:
From Condition C, we can derive that
and
Moreover, satisfies the homogeneous boundary conditions
Similarly to Theorem 3.3, Chapter I, [1] , we have
where are positive constants. Provided is small enough, so that we can obtain
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
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
It is clear that satisfies the homogeneous boundary conditions
Noting that satisfy, and
and then is a solution of Problem for the complex equation
hence we have
in which
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
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
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.
ReferencesWEN, G.C. AND BEGEHR, H. (1990) BOUNDARY VALUE PROBLEMS FOR ELLIPTIC EQUATIONS AND SYSTEMS. LONGMAN SCIENTIFIC AND TECHNICAL COMPANY, HARLOW.WEN, G.C. (1992) CONFORMAL MAPPINGS AND BOUNDARY VALUE PROBLEMS, TRANSLATIONS OF MATHEMATICS MONOGRAPHS 106. AMERICAN MATHEMATICAL SOCIETY, PROVIDENCE.WEN, G.C., TAI, C.W. AND TIAN, M.Y. (1996) FUNCTION THEORETIC METHODS OF FREE BOUNDARY PROBLEMS AND THEIR APPLICATIONS TO MECHANICS. HIGHER EDUCATION PRESS, BEIJING (CHINESE).WEN, G.C. (1986) LINEAR AND NONLINEAR ELLIPTIC COMPLEX EQUATIONS. SHANGHAI SCIENTIFIC AND TECHNICAL PUBLISHERS, SHANGHAI (CHINESE).WEN, G.C. (1999) APPROXIMATE METHODS AND NUMERICAL ANALYSIS FOR ELLIPTIC COMPLEX EQUATIONS. GORDON AND BREACH, AMSTERDAM.WEN, G.C. (1999 LINEAR AND NONLINEAR PARABOLIC COMPLEX EQUATIONS. WORLD SCIENTIFIC PUBLISHING CO., SINGAPORE CITY.WEN, G.C. AND ZOU, B.T. (2002) INITIAL-BOUNDARY VALUE PROBLEMS FOR NONLINEAR PARABOLIC EQUATIONS IN HIGHER DIMENSIONAL DOMAINS. SCIENCE PRESS, BEIJING.WEN, G.C. (2002) LINEAR AND QUASILINEAR COMPLEX EQUATIONS OF HYPERBOLIC AND MIXED TYPE. TAYLOR & FRANCIS, LONDON. HTTP://DX.DOI.ORG/10.4324/9780203166581HUANG, S., QIAO, Y.Y. AND WEN, G.C. (2005) REAL AND COMPLEX CLIFFORD ANALYSIS. SPRINGER VERLAG, HEIDELBERG.WEN, G.C. (2008) ELLIPTIC, HYPERBOLIC AND MIXED COMPLEX EQUATIONS WITH PARABOLIC DEGENERACY. WORLD SCIENTIFIC, SINGAPORE CITY.WEN, G.C., CHEN, D.C. AND XU, Z.L. (2008) NONLINEAR COMPLEX ANALYSIS AND ITS APPLICATIONS, MATHEMATICS MONOGRAPH SERIES 12. SCIENCE PRESS, BEIJING.WEN, G.C. (2010) RECENT PROGRESS IN THEORY AND APPLICATIONS OF MODERN COMPLEX ANALYSIS. SCIENCE PRESS, BEIJING.