### Journal Menu >> International Journal of Modern Nonlinear Theory and Application, 2013, 2, 170-175 http://dx.doi.org/10.4236/ijmnta.2013.23024 Published Online September 2013 (http://www.scirp.org/journal/ijmnta) General Boundary Value Problems for Nonlinear Uniformly Elliptic Equations in Multiply Connected Infinite Domains Guochun Wen1, Yanhui Zhang2, Dechang Chen3 1School of Mathematical Sciences, Peking University, Beijing, China 2Mathematic Department, Beijing Technology and Business University, Beijing, China 3Uniformed Services University of the Health Sciences, Bethesda, USA Email: wengc@math.pku.edu.cn, zhangyhchengd@yahoo.com.cn, dechang.chen@usuhs.edu Received May 1, 2013; revised June 29, 2013; accepted July 3, 2013 Copyright © 2013 Guochun Wen et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. ABSTRACT This article discusses the g eneral boundary value problem for the nonlinear uniformly elliptic equation of second order ,, ,,,in,zzz zzzuFzuuuGzuu D (0.1) and the boundary condition  1222onuczu cz, (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  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 fo r Problem G can be obtained. By using reduction to absurdity, we first d iscuss a priori estimates of solutions and solvability for a modified problem. Then we present results on solvability of Problem G. Keywords: General Boundary Value Problems; Nonlinear Elliptic Equations; Multiply Co nnected Infinite Domains 1. Formulation of Elliptic Equations and Boundary Value Problems Let be an -connected domain which in- cludes the infinite point and has the boundary D1N0Njj  in , where 201C. Without loss of generality, we assume that D is a circular domain in 1z, where the boundary consists of circles 1N01 1Nz , ,1,,jjjzzr jN   and . Note zDthat this article uses the same notations as in references [1-8]. We consider the nonlinear uniformly elliptic equation of second order  123,, ,,,,Re ,,,,,, ,,,, ,1,2,3.zzz zzzzz zzzjj zuFzuuuGzuuFQuAuAuAGGzuuQQzuuuAAzuujzz (1.1) This is the complex form of the nonlinear real equation ,,,,,,,0x y xx xyyyxyuu u uuu (1.2) with certain conditions (see ). We suppose that the Equation (1.1) satisfies Condition C, as described below. Condition C 1)  ,,, ,,,1,2,3jQzuwUAzuwjzD are measurable in  for all continuous functions ,uz wz in D and all measurable functions 0,2 ,pUz LD and satisfy  ,2 10,220,231 2,, ,,,, ,,,,,,,,0in,pppLAzuwDkLAzuwD kLAzuwDkAzuw D (1.3) in which 0001,2,,,1ppppkk are non-ne- gative constants. 2) The above functions are continuous in ,uw , for almost every and ,zDU1,2,3j0, 0jQA for .zD3) The Equation (1.1) satisfies the uniform ellipticity condition 1201,,, ,,,,2FzuwUF zuwUq UU (1.4) for almost every point ,zD any functions ,uz wzCD and where 12,,UU01q Copyright © 2013 SciRes. IJMNTA G. C. WEN ET AL. 171is a non-negative constant. 4) The function possesses the form ,,Gzuw12,,in ,Gzuw BwBuD (1.5) where are continuous functions in  ,uz wzD, ,2,pjLBk0 00,, 1,2,2Dkj pp   for a positive constant . 0According to , 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 uz D satisfying the boundary conditions    121222,..Re, .zuczu cziez uczuczz   (1.6) Here  is a given unit vector at the point ,z and cos,cos,,zxi y z and are real functions. We assume z1,c and satisfy the con- ditions 2c01 02,,,,,CkCckCc 2,k (1.7) and  1cos ,0,,czn z in which 02 are non-negative con- stant, and is the unit outer normal at 121, ,,kknz. If then we assu- me that  1cos,0 on,1,jz jN0,nc***2d0,1, ,1jjjjczzuabbkjN2, (1.8) in which *ja is a point on j,N and are real constants. There is no harm in assuming that on *1, ,jbj N,n 1cos ,0ncz* 0,010NN and cos1z.c do not both vanish identically on 0** 1NN  We can see that the above bound ar y cond itio ns in clu d e some irregular oblique derivative boundary conditions. If on , then Problem G is the regular oblique derivative problem (Problem III). If and 1 on cos ,0ncos ,0nc0, then Problem G is the first bound ary valu e problem, i.e., the Dirichlet boundary value problem (Problem D), in which the boundary condition is  ****21d,1,1,,jzjjjauz rzczsbrabjN1. (1.9) 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  ,uz wz  123,,,,, ,Re ,,, ,,,,,,, ,1,2,3,,zzzzjj zwFzuwwGzuwFQwAwAuAGGzuwQQzuwwAAzuwj wu (1.10) satisfying the modified boundary conditions   121222 ,..Re, ,zuczucz hziez uczuczhzz  (1.11) and the point conditions: 0001,0,1,,,, 1,,jj juabjmaa ajm .(1.12) An explanation of the above conditions is given as follows. The bound ary  can be divided into two parts: 1cos ,0,0ncz and cos,0,0 ,ncz 1 such that 11,,,,,,, ,mlEa aaa    every component of  and  includes its initial point, but does not in- clude the terminal point, and there is at least one point on each component of so that  ,cos ,0.n The points mja1, ,jaj and j possess the following property. 1,, laj and j, when the direction of a at ,jjaa is the same as the direction of . ja and ja, when the direction of  at ,jjaa is opposite to the direction of . And (cos )n, changes the sign once on the two components of , with the end point j or j. And aa0, 1,bj,j in (1.12) are real constants satisfying the condition: m3j herein 3 is a non-negative constant. More- over, the undetermined function in (1.11) can be written as ,kkbhz ,,0,1,jj jhzhzzjl,. (1.13) In (1.13) *0,1, ,jj j l are non-degen- erate, multiply disjointed arcs, each of which consists of inner point s of 0,1, ,jjl, such that 0, 0zcos ,n on 000, ,,.jla E1,j In addition, ,ljzj0, 1,hjj are unknown real constants to be determined appropriately, and is a positive function on  and 0jz on j and Cz0j ,,0,1,,,kjl  in which 12 1 and 0k are non-negative constants. It is not difficult to see that the index of Problem H is given by 1arg1 .22mlKzN  (1.14) Copyright © 2013 SciRes. IJMNTA G. C. WEN ET AL. 172 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  1cos ,0,0ncz,,.E cos,n 010cz on , the conditions (1.12) and (1.13) can be replaced by  01,0,1,,,,,1,jjjjua bjmhzhzzjl,. (1.15) with 3,0,1,,jbkj m, (1.16) 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 dom ains. k,, 0zGzuu 2. A Priori Estimates of Solu t io n s o f Boundary Value Problems We first give a priori estimates of solutions of Problem H . 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  ,,zuzwzuz u,, 0Gzuw of Problem H for (1.10) with satisfies the estimates  01,2 1,,,pzzzzSuCuz DLuuDM   (2.1) 2*2123,SuMkM kkk  in which 0min,1 2,p 02,pp1100,,,,,,MMqpk KD22000,,,,, 123,,kkkk,.MMqpk KD Proof First of all , we prove that the solution uz of Problem H satisfies the estimate 113300,,,,,,SCuzD MMqpkKD . Suppose that the estimate (2.3) is not true. Then there exist sequences of coefficients *123 12,,,,,,,,nnnnnn njnjnQAAAcc bb*123 12,,,,,,,, of (1.10), (1.11), (1.12) and (1.15) satisfying the same conditions of jjQA AAc cbb,nQ1,nA2,nA3n, such that A in weakly converge D0000,,,QAAAto respectively, and 123 *12,,,njnccb,nn jnb*,, on uniformly converge to 01020 00,,jjbbcc respectively, and the corresponding bou nda ry valu e problems 1232Re,0 in,nn nnnzzzz zuQuAuAuAA D (2.4) 121222,cos,0on,d0,jnnnnnn nucu chcz ncs** 01,1,,,1,0,1,,,1, 2,jjnjjnua bjNua bjmn  (2.6) have the continuously differentiable solutions 1, 2,nuzn with the property that 1,nnHCuD as There is no harm in .nassuming that Denote 1,1,2,nHn,1,2,nnnnUuH It is clear that the function nwz Unz is a solution of the following Rie- mann-Hilbert boundary value problem 11Re,in ,nn nnnnnznz nnwQwAwAAAuA  3D (2.7) ** 01,1,,,1,0,1,,,nj jnnj jnuabjNua bjmn 1,2, (2.8) where the index of zn is 12KN ml, and ,1nCw z D showing that n on wz D is bounded. According to the method in the proof of Theorem 4.7, Chapter I , we can obtain that nwz satisfies the estimate 0,2 4,,nnpnznzLwCw DLwwDM  , (2.9) in which 4400,,,,,,MMqpk KD and then   2 (2.5) *0212Re djznnnawzUz zuzHz  satisfies 01,2 5,,nnpnzznzzSUCU DLUUDM  , (2.10) where 5500,,,,,.MMqpk KD Hence from nUz and nzU, we can choose the subsequences knUz and knzU, which uniformly converge to 0 and 0UzzU in D respectively, such that 0 is a solution of the following boundary value problem Uz00 00122Re0,0in,zzzz zUQUAUAUA D (2.11) 100 100022,cos,0 on,Ucuhczn (2.12) *010,1,,,10,0,1,, .jjUa jNUajm  (2.13) By the uniqueness of solutions of Problem H (see Theorem 2.3 below), we see that 0Uz on D. However from 1,1,CU zDn it can be derived that 10.,1CUzD This contradiction proves that (2.3) is true. Afterwards, using the method of deriving (2.9) from 1,1,CUDn we can obtain the estimate (2.1). The estimate (2.2) can be concluded from (2.1). Theorem 2.2 Let the Equation (1.1) satisfy Condition Copyright © 2013 SciRes. IJMNTA G. C. WEN ET AL. Copyright © 2013 SciRes. IJMNTA 173dary conditions: C and  in (1.3), (1.7) be a sufficiently small positive constant. Then any solution  ,wz uz of Problem H for (1.10) satisfies the estimates 123Re, ,zzwQwAwAuAGzD (2.16)  12Re, ,zwzcuczh zz  (2.17)  6*,,C wzDCuzDMk, (2.14) ** 01,1,,,1,0,1,,,1, 2,jjnjjnua bjNua bjmn  (2.18) 00,2,27 *,,pzzpzLwwDLuDMk , (2.15) where 0,p are as stated in Theorem 2.1, 000,,,,, ,6,7,MMpk KDjjjq *1230 ,,kkkkkCwD CuD . 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 ,, 0Gzuw. For this, we need to consider the follo- wing condition Proof It is easy to see that  ,wz uz of Problem H for (1.10) satisfies the following equation and boun-  0112211 221 212,20 0,, ,,,,Re,,,,,1,2,,,2,zz zjjp jFzu uUFzuuUAuuAuuAAzuuUjL ADkpp  (2.19) for any continuously differentiable functions 1,1,juz CDj2 and any measurable function 0,2 ,pUz LD where 0min,1 2,p H for (1.10). By the above conditions, we see that 12uzuzuz is a solution of the following boundary value problem Problem 12Re0,,zzzzzuQuAuAuz 2ppp,k00 0 are constants as stated in Section 1. We can prove the uniqueness of solutions of Problem H for (1.1). D (2.20)  122,uczuzHz z, (2.21) Theorem 2.3 Let the second order nonlinear Equation (1.1) satisfy Condition C and (2.19) with 2 in . Then the solution of Problem H for (1.10) with 0AD,, 0zGzuu  is unique. *010,1, ,, 10,0,1, ,,jjuajNua hm  (2.22) Proof Let be two solutions of Problem  12,uzuz with     012111 11211211 212212 222 212122120,2 2Re, ,,, ,,,Re,,,,,,,,, ,,,,for ,0for1,,,1,2,0inzzzz zzzzzzz zzzzzzz zzzpjQuuFzuu uFzuu uAuuFzuuuFzuuuFzuu uFzuuuuz uzuuAuzuzz DQqL ADjA     ,D,, ,,GGzuzwz is coninuous and bounded with where 001 are non-negative constants. Accor- ding to the proof of Theorem 2.6, Chapter I, , and using the extremum principle of solutions for (2.20) (see Chapter 3, ), we can prove that in , and then in . ,,qpk 1uz u0uzD2zD  ,2,2 1,2 2,, ,,,,pppLGzuzwzDLBDCwDLBDCuD,, (3.1) 3. Solvability of Boundary Value Problems We first prove a lemm a. Lemma 3.1. If satisfies the condition stated in Condition then the nonlinear mapping : ,,Gzuw,C T  ,2pCD CDLD defined by where 02.pp Proof In order to prove that the mapping T: ,2pCD CDLD defined by ,,GGzuzwz  is continuous, we choose any sequence of functions ,0,1,2,nnuz CDn,,nnwzuz wz G. C. WEN ET AL. 174 such that 00,,nnCwwDCuu D0 as Similarly to Lemma 2.2.1 , we can prove that possesses the property that .nC G00,, ,,nnnzuwG zuw,2 ,0aspnLCD n . (3.2) 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1,2,pwz uzWD0 with the constant 002ppp as stated before. 2) When min ,1, Problem H for (1.10) has a solution ,wz uz, where 01,2,pwz WD pro- vided that 08,232 0,,mpjjMLADCcb  (3.3) is sufficiently small. Proof 1) In this case, the algebraic equation for t becomes , Problem H for the Equation (1.10) has a solution ,uz,wz where 9,23,21,2220,,, ,mpp pjj,MLADLBDtLBDtLcb t   (3.4) with 967MMM, where 67,MM0 are constants as stated in (2.14) and (2.15). Because ,1,0. Equa- tion (3.4) has a unique solution 10 Now we introduce a bounded, closed and convex subset tM*Bof the Banach space  ,CDCD whose elements are of the form satisfying the condition  ,wz uz  10,,,,.wzuzCDCwz DCuz DM (3.5) We choose a pair of fu nctions  *,wz uzB and substitute it into the appropriate positions of , ,,,z,,FzuwwGzuw in (1.10) and the boundary condition (1.11) to obtain ,,,,,,,,zzwFzuwuwwGzuw  (3.6) 12Re, ,zwzc zuczz  (3.7) where 123,,,,,Re,,,, ,,,,,.zzzFzuwuwwQzuwwwAzuwwAzuwuAzuw   In accordance with the method in the proof of The- orem 1.2.5 , we can prove that the boundary value problem (3.6), (3.7) and (1.15) has a unique solution ,wz uz. Denote by ,,wuTw zuz the mapping from ,wz uz to Noting that  ,wz uz.,2210 0110 0,,,pLAuD MkCcuMk , provided that the positive number  is sufficiently small, and noting that the coefficients of complex Equa- tion (3.6) satisfy the same conditions as in Condition C, from Theorem 2.2, we can obtain 00,2,29,2 32,2098 ,21,2298 ,2110 ,221010,,,,,,,,, ,,,,mpzzpzpjpjpp pp.wDLwwDCuDLuD MLADCcb L GDCMML BDCwDL BDCuDMMLBDML BDMM          (3.8) This shows that T maps onto a compact subset in Next, we verify that *B.*BT in is a continuous operator. In fact, we arbitrarily select a sequence in such that *B,nwznuz*B,00,,0asnn .w DCuu Dn Cw (3.9) By Lemma 3.1, we can see that ,20 0,,,, ,01,2,3as .pjnnjLAzuwA zuwDjn  (3.10) Moreover, from 00 00,,,,,nn nnwuTwu wu Twu , it is clear that 0nn0,wwuu is a solution of Problem H for the following equation: 00000000,,,,,,,,,,,,,, in,n nnnnnzzznnww FzuwuwwFzuwuwwGzu wGzuwD  (3.11) 010Reon ,nnzw wczuuhz  (3.12) Copyright © 2013 SciRes. IJMNTA G. C. WEN ET AL. 175**00110,1,,110,0,1,, .nj jnj juaua jNuaua jm0, (3.13) In accordance with the method in proof of Theorem 2.2, we can obtain the estimate 000,20 00,2011,22200 0,2330 0,20010,,,,,,,,,,,,,,,,,,,, ,npnnzznpnzpnnnpnnpnn nCw wDLw ww wDCuu DLuuDMLAzuwuAzuwuDLAzuwAzuwDLGzuwGzuwDCczuu      (3.14) in which 111100 0,,,,, .MMqpk KD From (3.9), (3.10) and the above estimate, we obtain 00,,0.nnn as On the basis of the Schauder fixed-point theorem, there exists a function CwwDCuu D  ,,wz uzwzuzCD ,,.zT wzuz such that wz u And from Theorem 2.2, it is easy to see that  1,2,pwz uzWD0, 1.0 and  is a solution of Problem H for the Equation (1.10) with the condition , ,wz uz In addition, using a method similar to the above, we see that if 12,, ReGzuwBw Bu in , where D,2 001,,,1,pjLBDk j  2, then the above solvability result still holds. 2) Secondly, we discuss the case, where min ,1.tM In this case, (3.4) has the solution 10 provided that 8M in (3.3) is small enough. We consider a closed and convex subset in the Banach space *B ,CDCD i.e.,  *10,,,,BwzuzCDCwDCuDM  . Applying a similar method as before, we can verify that there exists a solution 00 of Problem H for (1.10) with the condition 11,2 ,2,ppwz uzWDWDmin ,1. Moreover, if 12,, ReGzuwBw Bu in , where D1, ,2 0,,1,2,pj then under the same condition, we can derive the above solvability result by a similar method. LBDk jFrom the above theorem, the next result can be derived. Theorem 3.3 Under the same conditions as in The- orem , Problem G has 3.2 1luz solvability conditions, and the general solution includes arbitrary real constants. 1mProof Let the solution ,wz uz0, zl of Problem for (1.10) be substituted into the boundary condition (1.11). If t he function , i.e. then we have Hhz, ,0,, 0,1,jhz jzwz u in and the function 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 . Duz1lGAlso note that the real constants 0, 1,,jbj m in (1.12) 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. REFERENCES  I. N. Vekua, “Generalized Analytic Functions,” Pergamon, Oxford, 1962.  G. C. Wen, “Linear and Nonlinear Elliptic Complex Equations,” Shanghai Scientific and Technical Publishers, Shanghai, 1986. (in Chinese)  G. C. Wen, “Conformal Mappings and Boundary Value Problems,” American Mathematical Society, Providence, 1992.  G. C. Wen, “Approximate Methods and Numerical Analysis for Elliptic Complex Equations,” Gordon and Breach, Amsterdam, 1999.  G. C. Wen, D. C. Chen and Z. L. Xu, “Nonlinear Com- plex Analysis and its Applications, Mathematics Mono- graph Series 12,” Science Press, Beijing, 2008.  G. C. Wen, “Recent Progress in Theory and Applications of Modern Complex Analysis,” Science Press, Beijing, 2010.  G. C. Wen and C. C. Yang, “On General Boundary Value Problems for Nonlinear Elliptic Equations of Second Or- der in a Multiply Connected Domain,” Acta Applicandae Mathematicae, Vol. 43 No. 2, 1996, pp. 169-189. doi:10.1007/BF00047923  G. C. Wen, “Irregular Oblique Derivative Problems for Second Order Nonlinear Elliptic Equations on Infinite Domains,” Electronic Journal of Differential Equations, Vol. 2012, No. 142, 2012, pp. 1-8. Copyright © 2013 SciRes. IJMNTA