General Boundary Value Problems for Nonlinear Uniformly Elliptic Equations in Multiply Connected Infinite Domains

This article discusses the general boundary value problem for the nonlinear uniformly elliptic equation of second order     , , , , , in , zz z zz z u F z u u u G z u u D   (0.1) and the boundary condition     1 2 2 2 on u c z u c z   ,     (0.2) in a multiply connected infinite domain with the boundary D  . The above boundary value problem is called Problem G. Problem G extends the work [8] in which the equation (0.1) includes a nonlinear lower term and the boundary condition (0.2) is more general. If the complex equation (0.1) and the boundary condition (0.2) meet certain assumptions, some solvability results for Problem G can be obtained. By using reduction to absurdity, we first discuss a priori estimates of solutions and solvability for a modified problem. Then we present results on solvability of Problem G.


Boundary Value Problems
Let be an -connected domain which includes the infinite point and has the boundary Without loss of generality, we assume that D is a circular domain in 1 z  , where the boundary consists . Note z   D that this article uses the same notations as in references [1][2][3][4][5][6][7][8].We consider the nonlinear uniformly elliptic equation of second order . with certain conditions (see [3]).We suppose that the Equation (1.1) satisfies Condition C, as described below.3) The Equation (1.1) satisfies the uniform ellipticity condition is a non-negative constant.
4) The function possesses the form where are continuous functions in According to [7], we introduce the general boundary value problem for the Equation (1.1) in D as follows.
Problem G Find a continuously differentiable solution of the second order Equation (1.1) in Here  is a given unit vector at the point and are real functions.We assume and in which   0 2 are non-negative con- stant, and is the unit outer normal at and are real constants.There is no harm in assuming that on We can see that the above boundary conditions include some irregular oblique derivative boundary conditions.If on , then Problem G is the regular oblique derivative problem (Problem III).If and 1 on One problem regarding the well posed-ness of Problem G for (1.1) can be formulated as follows: Problem H Find a system of continuous functions of the equation , (1.10) satisfying the modified boundary conditions and the point conditions: , (1.12) An explanation of the above conditions is given as follows.The boundary  can be divided into two parts: E a a a a and   includes its initial point, but does not include the terminal point, and there is at least one point on each component of so that and j possess the following property.
  l are non-degenerate, multiply disjointed arcs, each of which consists of inner points of are unknown real constants to be determined appropriately, and is a positive function on  and If on , then In this case, Problem H for (1.1) is called Problem O or Problem IV, which includes the Dirichlet problem, the Neumann problem and the regular oblique derivative problem as its special cases.We note that except the case where and on , the conditions (1.12) and (1.13) can be replaced by in which 3 is a non-negative constant.Also note that [4,7] discuss the corresponding problem for the equation (1.1) with in the bounded domains.

A Priori Estimates of Solutions of Boundary Value Problems
We first give a priori estimates of solutions of Problem H.
, , , Proof First of all, we prove that the solution   u z of Problem H satisfies the estimate Suppose that the estimate (2.3) is not true.Then there e x i s t s e q u e n c e s o f c o e f f i c i e n t s , , , , , , , , o f (1.10), (1.11), (1.12) and (1.15) satisfying the same conditions of respectively, and , , , Re , 0 in , have the continuously differentiable solutions Re , in , , , where the index of , and  , D is bounded.According to the method in the proof of Theorem 4.7, Chapter I [4], we can obtain that , , in which , , Re 0, 0 in , By the uniqueness of solutions of Problem H (see Theorem 2.3 below), we see that This contradiction proves that (2.3) is true.Afterwards, using the method of deriving (2.9) from 1  , 1 , C and  in (1.3), (1.7) be a sufficiently small positive constant.Then any solution , , By using the same method as in the proof of Theorem 2.1, we can obtain the estimates (2.14) and (2.15).Now we discuss the uniqueness of solutions of Problem H for the nonlinear elliptic Equation (1.1) with

 
, , 0 G z u w  .For this, we need to consider the following condition Proof It is easy to see that  of Problem H for (1.10) satisfies the following equation and boun- for any continuously differentiable functions H for (1.10).By the above conditions, we see that is a solution of the following boundary value problem Problem Re 0, , 0 are constants as stated in Section 1.We can prove the uniqueness of solutions of Problem H for (1.1).
 is coninuous and bounded with where 0 0 1 are non-negative constants.According to the proof of Theorem 2.6, Chapter I, [4], and using the extremum principle of solutions for (2.20) (see Chapter 3,[3]), we can prove that in , and then in .

Solvability of Boundary Value Problems
We first prove a lemma.Lemma 3.1.If satisfies the condition stated in Condition then the nonlinear mapping : defined by where 0

p p  
Proof In order to prove that the mapping T : is continuous, we choose any sequence of functions  as Similarly to Lemma 2.2.1 [5], we can prove that possesses the property that , 0as And the inequality (3.1) is obviously true.Theorem 3.2.Let the complex Equation (1.1) satisfy Condition C, and the positive constant  in (1.3) and (1.7) be small enough. 1) When 0 , 1 , , , M M 0 are constants as stated in (2.14) and (2.15).Because , 1, where F z u w u w w In accordance with the method in the proof of Theorem 1.2.5 [5], we can prove that the boundary value problem (3.6), (3.7) and (1.15) has a unique solution provided that the positive number  is sufficiently small, and noting that the coefficients of complex Equation (3.6) satisfy the same conditions as in Condition C, from Theorem 2.2, we can obtain .
This shows that T maps onto a compact subset in Next, we verify that T in is a continuous operator.In fact, we arbitrarily select a sequence in such that By Lemma 3.1, we can see that .
In accordance with the method in proof of Theorem 2.2, we can obtain the estimate In addition, using a method similar to the above, we see that if

( 1 . 1 )
This is the complex form of the nonlinear real equation   , , , , , , , 0 x y xx xy yy x y u u u u u u   (1.2) Q z u w U A z u w j  z D are measurable in for all continuous functions     , u z w z in D and all measurable functions

Theorem 2 . 1
Suppose the second order nonlinear Equation (1.10) satisfies Condition C, and  in (1.3), (1.7) is small enough.Then any solution a solution of the following Riemann-Hilbert boundary value problem 1 1 4) has a unique solution 10 Now we introduce a bounded, closed and convex subset t M  * B of the Banach space     , w w G z u w in (1.10) and the boundary condition(1.11)to obtain

0 and
is a solution of Problem H for the Equation (1.10) with the condition , 

Theorem 3 . 3
condition, we can derive the above solvability result by a similar method.From the above theorem, the next result can be derived.Under the same conditions as in Theorem Problem for (1.10) be substituted into the boundary condition(1.11).If the function , i.e. then we have H is just a solution of Problem G for (1.1).Hence the total number of above equalities is just the number of solvability conditions of Problem .12) and (1.15) are arbitrarily chosen.This shows that the general solution of Problem G for (1.1) includes the 1  m arbitrary real constants as stated in the theorem.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.
then Problem G is the first boundary value problem, i.e., the Dirichlet boundary value problem (Problem D), in which the boundary condition is