Approximate Method of Riemann-Hilbert Problem for Elliptic Complex Equations of First Order in Multiply Connected Unbounded Domains

In this article, we discuss the approximate method of solving the Riemann-Hilbert boundary value problem for nonlinear uniformly elliptic complex equation of first order   , , in , z z w F z w w D  (0.1) with the boundary conditions       Re on , t w t r t        (0.2) in a multiply connected unbounded domain D, the above boundary value problem will be called Problem A. If the complex Equation (0.1) satisfies the conditions similar to Condition C of (1.1), and the boundary condition (0.2) satisfies the conditions similar to (1.5), then we can obtain approximate solutions of the boundary value problems (0.1) and (0.2). Moreover the error estimates of approximate solutions for the boundary value problem is also given. The boundary value problem possesses many applications in mechanics and physics etc., for instance from (5.114) and (5.115), Chapter VI, [1], we see that Problem A of (0.1) possesses the important application to the shell and elasticity.


Formulation of Elliptic Equations and Boundary Value Problem
Let be an -connected domain including the D  1 N   infinite point with the boundary 0 where .Without loss of generality, we assume that is a circular domain in and .In this article, the notations are as the same in References [1][2][3][4][5][6].We discuss the nonlinear uniformly elliptic complex equation of first order w F z w w which is the complex form of the real nonlinear elliptic system of first order equations   , , , , , , , 0, 1,2 under certain conditions (see [3]).Suppose that the complex Equation (1.1) satisfies the following conditions, namely Condition C: 1) , , 1 ,2 , , where   0 0 0 , 2 , , p p p p k k   1 are non-negative constants.
2) The above functions are continuous in w   for almost every point , z D U    and 3) The complex Equation (1.1) satisfies the uniform ellipticity condition, i.e. for any 1 2 , the following inequality in almost every point holds: F z w U F z w U q U U    (1.4) in which is a non-negative constant.
  0 1 q  Problem A: The Riemann-Hilbert boundary value problem for the complex Equation (1.1) may be formulated as follows: Find a continuous solution of (1.1) on

 
w z D satisfying the boundary condition where   1, , z z    and satisfy the conditions in which are non-negative constants.
  This boundary value problem for (1.1) with and will be called Problem where , , in which are unknown real constants to be determined appropriately.In addition, we may assume that the solution w z satisfies the following side conditions (point conditions) , , , , , , , , , , , 4,5 Proof.Similarly to Theorem 2.4, Chapter 2 in [3], we substitute the solution of Problem 1 (or Problem 2 ) into the coefficients of the complex Equation (1.1) and consider the following system By using the continuity method and the principle of contracting mappings, we can find the solutions where -connected circular domain , and is an analytic function in .We can verify that satisfy the estimates (2.2) and (2.3).Moreover noting that is a homeomorphic solution of the Beltrami complex Equation (2.7), which maps the circular domain onto the circular domain with the condition and in accordance with the result in Lemma 2.1, Chapter 2, [3], we see that the estimate (2.4) is true.
are non-negative constants only dependent on 0 0 , , , , , q p k K D  and respectively.0 0 Proof.On the basis of Theorem 2.1, the solution , , , , , , where where of Problem satisfies the estimate then from the representation (2.1) of the solution and the estimates (2.2)-(2.4)and (2.15), the estimates (2.9) and (2.10) can be derived.

  w z
It remains to prove that (2.15) holds.For this, we first verify the boundedness of Suppose that (2.16) is not true.Then there exist sequences of functions There is no harm   in assuming that Obviously 1, 1, 2, .
Applying the Schwarz formula, the Cauchy formula and the method of symmetric extension (see Theorem 1.4, Chapter 1, [3]), the estimates On the basis of the uniqueness theorem (see Theorem 2.4), we conclude that This contradiction proves that (2.16) holds.using the method which leads from where are as stated in Theorem 2.2, 0 , , from Theorem 2.4, it follows that .If it is easy to see that satisfies the complex equation and boundary conditions or , , and according to the proof of Theorem 2.2, we have , .
From the above estimates, it immediately follows that (2.22) holds.
Next, we prove the uniqueness of solutions of Problem 1 and Problem 2 B for the complex Equation (1.1).For this, we need to add the following condition: For any continuous functions , w z w z on D and , , , According to the representation (2.1), we have are as stated in (2.11)-(2.13).In accordance with Theorem 2.2, it can be derived that , .w z w z z D  

The Continuity Method of Solving Boundary Value Problem
Next, we discuss the modified Riemann-Hilbert boundary value problems (Problem 1 and Problem 2 ) for the nonlinear elliptic complex Equation (1.1) in the (N+1)-connected unbounded domain as stated in Section 1, here we use the Newton imbedding method of another form and give an error estimate, which is better than that as stated before.In the following, we only deal with Problem 1 , because by using the same method, Problem can be discussed.
where   A z is any measurable function in and is an analytic function in and satisfies the boundary conditions From Theorem Theorem 2.2, We see that  in the right hand side of (1.22).By Condition , it is obvious that Noting the (3.5) has a solution Applying the successive iteration, we can find out a sequence of functions: The difference of the above equations for 1 n  and n is as follows: .
, it is not difficult to see that there exists a unique solution .
Moreover, satisfies the homogeneous boundary conditions On the basis of Theorem 2.3, we have where , , for where is a positive integer.This shows that as Following the completeness of the Banach space where .
It is clear that satisfies the homogeneous boundary conditions , , , , , , , and according to Theorem 2.2, it can be concluded where  0 0 , and we choose that is the solution of Problem 1 for (2.

2 Theorem 3 . 1 .Proof
Suppose that the nonlinear elliptic Equation (1.1) satisfies Condition C and (1.6), (1.10), on B We introduce the nonlinear elliptic complex equation with the parameter of Problem 1 for the complex Equation (3.1), which possesses the form B From Condition , on C D , it can be seen that Copyright © 2013 SciRes.AM non-negative constants as stated in (1.3), (1.6) and (1.10).From (4.5) and (4.6), it follows , ,

. A Priori Estimates of Solutions of Boundary Value Problem
k 2 we derive a priori estimates of solutions for Problem 1 and for Problem for the complex Equation (1.1).
2 B Theorem 2.2.Under the same conditions as in Theorem 2.1, any solution of Problem (or Problem ) for (1.1) satisfies the estimates Problem 1 (or Problem 2 ) can be expressed the formula as in (2.1), hence the boundary value problem 1 can be transformed into the boundary value problem (Problem ) for analytic functions Problem 1 for the complex Equation (1.18) has a unique solution, we shall prove that there exists a neighborhood of

Error Estimates of Approximate Solutions for Boundary Value Problem
