Approximate Method of Riemann-Hilbert Problem for Elliptic Complex Equations of First Order in Multiply Connected Unbounded Domains ()
(0.1)
with the boundary conditions
(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.
1. Formulation of Elliptic Equations and Boundary Value Problem
Let
be an
-connected domain including the infinite point with the boundary
in
where
. Without loss of generality, we assume that
is a circular domain in
, where the boundary consists of
circles
, ![](https://www.scirp.org/html/15-7401141\74d3d12b-032e-44a3-b110-7ec5c2dd9aef.jpg)
and
. In this article, the notations are as the same in References [1-6]. We discuss the nonlinear uniformly elliptic complex equation of first order
(1.1)
which is the complex form of the real nonlinear elliptic system of first order equations
(1.2)
under certain conditions (see [3]). Suppose that the complex Equation (1.1) satisfies the following conditions, namely Condition C: 1) ![](https://www.scirp.org/html/15-7401141\c5995a59-51d0-46a6-9356-4dcf710f6ded.jpg)
are measurable in
for all continuous functions
on
and all measurable functions
and satisfy
(1.3)
where
are non-negative constants.
2) The above functions are continuous in
for almost every point
and
for ![](https://www.scirp.org/html/15-7401141\9d46df2a-913c-4d16-b2c6-e24a903fa09d.jpg)
3) The complex Equation (1.1) satisfies the uniform ellipticity condition, i.e. for any
, the following inequality in almost every point
holds:
(1.4)
in which
is a non-negative constant.
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
satisfying the boundary condition
(1.5)
where
and
satisfy the conditions
(1.6)
in which
are non-negative constants.
This boundary value problem for (1.1) with
and
will be called Problem
The integer
![](https://www.scirp.org/html/15-7401141\a59542cd-49d1-4447-b9cb-3a04413da784.jpg)
is called the index of Problem
and Problem ![](https://www.scirp.org/html/15-7401141\2c61207e-364e-432c-8c2c-9fc8b53d9a94.jpg)
Due to when the index
Problem
may not be solvable, when
the solution of Problem
is not necessarily unique. Hence we put forward some well posednesses of Problem
with modified boundary conditions.
Problem B1: Find a continuous solution
of the complex Equation (1.1) in
satisfying the boundary condition
(1.7)
where
![](https://www.scirp.org/html/15-7401141\77fec097-983d-45b1-b8e7-397f80a5124e.jpg)
(1.8)
in which
are unknown real constants to be determined appropriately. In addition, we may assume that the solution
satisfies the following side conditions (point conditions)
(1.9)
where
![](https://www.scirp.org/html/15-7401141\13b552da-0641-4fb6-bc44-1d90ca8f2841.jpg)
are distinct points, and
are all real constants satisfying the conditions
(1.10)
herein
is a nonnegative constant.
Now, we give the second well posed-ness of Problem
.
Problem B2: If the point condition (1.9) in Problem
is replaced by the integral conditions
![](https://www.scirp.org/html/15-7401141\336e4813-8b0a-49c9-813b-d967f51e00b9.jpg)
(1.11)
respectively, where
are real constants satisfying the conditions
(1.12)
in which
is a nonnegative real constant.
For convenience, we sometimes will subsume the integral conditions or the point conditions under boundary conditions.
2. A Priori Estimates of Solutions of Boundary Value Problem
First of all, we give a representation theorem of solutions for Problem
and for Problem ![](https://www.scirp.org/html/15-7401141\695554b7-d173-482b-8dc8-c250fe0fd226.jpg)
Theorem 2.1. Suppose that the complex Equation (1.1) satisfies Condition C, and
is any solution of Problem
(or Problem
) for (1.1). Then
is representable by
(2.1)
where
is a homeomorphism on
, which quasiconformally maps D onto an
-connected circular domain G with boundary
where the
are located in
by
and
is an analytic function in G,
and its inverse function
satisfy the estimates
(2.2)
(2.3)
(2.4)
in which
are non-negative constants, ![](https://www.scirp.org/html/15-7401141\53c848b1-fd0b-41dc-a3e3-ae45551c7bf5.jpg)
Proof. Similarly to Theorem 2.4, Chapter 2 in [3], we substitute the solution
of Problem
(or Problem
) into the coefficients of the complex Equation (1.1) and consider the following system
(2.5)
(2.6)
(2.7)
By using the continuity method and the principle of contracting mappings, we can find the solutions
(2.8)
where ![](https://www.scirp.org/html/15-7401141\4301ef70-c76c-4243-ab8c-81b07480a75b.jpg)
is a homeomorphism on
is a univalent analytic function, which conformally maps
onto an
-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.
Now, we derive a priori estimates of solutions for Problem
and for Problem
for the complex Equation (1.1).
Theorem 2.2. Under the same conditions as in Theorem 2.1, any solution
of Problem
(or Problem
) for (1.1) satisfies the estimates
(2.9)
(2.10)
where
![](https://www.scirp.org/html/15-7401141\3de8e42c-34e5-45bf-8eb5-4ac4e2ea3a79.jpg)
are non-negative constants only dependent on
and
respectively.
Proof. On the basis of Theorem 2.1, the solution
of Problem
(or Problem
) can be expressed the formula as in (2.1), hence the boundary value problem
can be transformed into the boundary value problem (Problem
) for analytic functions
(2.11)
![](https://www.scirp.org/html/15-7401141\c63cd313-50fc-4d76-b35c-1ae7e836f4bb.jpg)
(2.12)
(2.13)
where
![](https://www.scirp.org/html/15-7401141\3c36d8a0-9a10-4291-ac10-e789852d2914.jpg)
By (2.2)-(2.4), it can be seen that
satisfy the conditions
(2.14)
where
If we can prove that the solution
of Problem
satisfies the estimate
(2.15)
in which
![](https://www.scirp.org/html/15-7401141\24fc7ffd-74ab-41bd-8bf9-e59a40db7528.jpg)
![](https://www.scirp.org/html/15-7401141\0c19ce25-3a54-43bf-9f66-e828440fb29b.jpg)
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.
It remains to prove that (2.15) holds. For this, we first verify the boundedness of
, i.e.
(2.16)
Suppose that (2.16) is not true. Then there exist sequences of functions
satisfying the same conditions as
which uniformly converge to
on L respectively. For the solution
of the boundary value problem (Problem
) corresponding to
we have
as
There is no harm in assuming that
Obviously
satisfies the boundary conditions
(2.17)
(2.18)
Applying the Schwarz formula, the Cauchy formula and the method of symmetric extension
(see Theorem 1.4, Chapter 1, [3]), the estimates
(2.19)
can be obtained, where
. Thus we can select a subsequence of
which uniformly converge to an analytic function
in
, and
satisfies the homogeneous boundary conditions
(2.20)
(2.21)
On the basis of the uniqueness theorem
(see Theorem 2.4), we conclude that
Howeverfrom
it follows that there exists a point
such that
This contradiction proves that (2.16) holds. Afterwards using the method which leads from
to (2.19), the estimate (2.15) can be derived.
Similarly, we can verify that any solution
of Problem
satisfies the estimates (2.9) and (2.10).
Theorem 2.3. Under the same conditions as in Theorem 2.1, any solution
of Problem
(or Problem
) for (1.1) satisfies
(2.22)
where
are as stated in Theorem 2.2,
![](https://www.scirp.org/html/15-7401141\e53667a7-5a3e-4e6b-b3d8-1829ef71ba5a.jpg)
Proof. If
i.e.
from Theorem 2.4, it follows that
. If
it is easy to see that
satisfies the complex equation and boundary conditions
(2.23)
(2.24)
(2.25)
Noting that
![](https://www.scirp.org/html/15-7401141\6ff4d4d8-4c18-4fec-80eb-aa5fcfc734b5.jpg)
and according to the proof of Theorem 2.2, we have
(2.26)
From the above estimates, it immediately follows that (2.22) holds.
Next, we prove the uniqueness of solutions of Problem
and Problem
for the complex Equation (1.1). For this, we need to add the following condition: For any continuous functions
on
and
there is
(2.27)
where
. When (1.1) is linear, (3.27) obviously holds.
Theorem 2.4. If Condition C and (2.27) hold, then the solution of Problem
(or Problem
) for (1.1) is unique.
Proof. Let
be two solutions of Problem
for (1.1). By Condition
and (2.27), we see that
is a solution of the following boundary value problem
(2.28)
(2.29)
(2.30)
where
![](https://www.scirp.org/html/15-7401141\38758d9f-faee-4ed4-b795-b9347807c428.jpg)
and
According to the representation (2.1), we have
(2.31)
where
are as stated in Theorem 2.1. It can be seen that the analytic function
satisfies the boundary conditions of Problem ![](https://www.scirp.org/html/15-7401141\ecd60c20-bb37-4ae9-950a-b5187f1be1f0.jpg)
(2.32)
(2.33)
where
are as stated in (2.11)-(2.13). In accordance with Theorem 2.2, it can be derived that
Hence,
i.e. ![](https://www.scirp.org/html/15-7401141\5af25999-d775-41e7-9863-5a6c6e1cb3c6.jpg)
3. The Continuity Method of Solving Boundary Value Problem
Next, we discuss the modified Riemann-Hilbert boundary value problems (Problem
and Problem
) 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
, because by using the same method, Problem
can be discussed.
Theorem 3.1. Suppose that the nonlinear elliptic Equation (1.1) satisfies Condition C and (1.6), (1.10), on
. Then Problem
for (1.1) has a solution ![](https://www.scirp.org/html/15-7401141\9e384276-701e-4741-96d0-f175f191aef1.jpg)
Proof We introduce the nonlinear elliptic complex equation with the parameter
:
(3.1)
where
is any measurable function in
and
When
, it is not difficult to see that there exists a unique solution
of Problem
for the complex Equation (3.1), which possesses the form
(3.2)
where
is an analytic function in
and satisfies the boundary conditions
(3.3)
(3.4)
From Theorem Theorem 2.2, We see that
![](https://www.scirp.org/html/15-7401141\1963dd1b-06c4-43ff-9527-4363639771ca.jpg)
Suppose that when
, Problem
for the complex Equation (1.18) has a unique solution, we shall prove that there exists a neighborhood of
so that for every
and any function
Problem
for (1.18) is solvable. In fact, the complex Equation (3.2) can be written in the form
(3.5)
We arbitrarily select a function
in particular
on
. Let
be replaced into the position of
in the right hand side of (1.22). By Condition
, it is obvious that
![](https://www.scirp.org/html/15-7401141\e28f48d7-83a5-41d3-baec-acc3ba58c1cc.jpg)
Noting the (3.5) has a solution
Applying the successive iteration, we can find out a sequence of functions:
which satisfy the complex equations
(3.6)
The difference of the above equations for
and n is as follows:
(3.7)
From Condition
, on
, it can be seen that
![](https://www.scirp.org/html/15-7401141\b2082ca0-380d-4574-8511-5bac51b1875a.jpg)
![](https://www.scirp.org/html/15-7401141\eb617587-a752-46d6-a995-fc64ee2bd2a5.jpg)
and
![](https://www.scirp.org/html/15-7401141\19e20e8c-77bb-43bb-8627-efec882bf5a9.jpg)
Moreover,
satisfies the homogeneous boundary conditions
(3.8)
(3.9)
On the basis of Theorem 2.3, we have
(3.10)
where ![](https://www.scirp.org/html/15-7401141\541a027c-3aff-4ca2-8668-38b2222233d1.jpg)
is as stated in (2.22). Provided
is small enough, so that
it can be obtained that
(3.11)
for every
Thus
(3.12)
for
where
is a positive integer. This shows that
as
Following the completeness of the Banach space
there is a function
such that when ![](https://www.scirp.org/html/15-7401141\4020123c-2c65-489e-b7b7-49869641b26a.jpg)
![](https://www.scirp.org/html/15-7401141\e2122bc4-078f-4c53-b89f-eb7108cf4045.jpg)
By Condition
and (1.6), (1.10), from the above formula it follows that
is a solution of Problem
for (3.5), i.e. (3.1) for
. It is easy to see that the positive constant
is independent of
. Hence from Problem
for the complex Equation (3.1) with
is solvable, we can derive that when
, Problem
for (3.1) are solvable, especially Problem
for (3.2) with
and
, namely Problem
for (1.1) has a unique solution.
4. Error Estimates of Approximate Solutions for Boundary Value Problem
In this section, we shall introduce an error estimate of the above approximate solutions of the boundary value problem and can give the following error estimate of the approximate solutions.
Theorem 4.1 Let
be a solution of Problem
for the complex Equation (1.1) satisfying Condition
and (1.6), (1.10) on
, and
be its approximation as stated in the proof of Theorem 2.2 with
Then we have the following error estimate
(4.1)
where
![](https://www.scirp.org/html/15-7401141\5fab64f8-3fba-4bc6-9334-6c8c7131ffc3.jpg)
and
are constants in (1.3), (1.6) and (1.10).
Proof From (1.1) and (2.23) with
, we have
(4.2)
It is clear that
satisfies the homogeneous boundary conditions
(4.3)
(4.4)
Noting that
satisfy
, and
![](https://www.scirp.org/html/15-7401141\93754cf2-bc39-4a13-b25e-e0e55c37a541.jpg)
![](https://www.scirp.org/html/15-7401141\53eb3740-dc52-4c76-bde1-c346d6d1faa1.jpg)
and according to Theorem 2.2, it can be concluded
![](https://www.scirp.org/html/15-7401141\55a19a05-4c84-4f29-b967-b51ff7da530a.jpg)
(4.5)
where
and
(4.6)
where
are non-negative constants as stated in (1.3), (1.6) and (1.10). From (4.5) and (4.6), it follows
![](https://www.scirp.org/html/15-7401141\959185e7-77e1-4276-80f1-5b4de6a92c2d.jpg)
where
and we choose that
is the solution of Problem
for (2.22) with
and
Due to
is a solution of Problem
for the complex equation
(4.7)
hence
(4.8)
Finally, we obtain
![](https://www.scirp.org/html/15-7401141\728b3648-1de0-462e-b874-eb7e73afdd77.jpg)
(4.9)
this shows that (4.1) 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 small.