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## The Singular Case of Riemann-Hilbert Boundary Value Problem

Wen Guo-chunDepartment of Mathematics
In this paper, we consider the nonlinear uniformly elliptic complex equation of first orderand suppose that Eq. (l)satisfies the condition C in an (N+ l)-connected domain G (See Ref. [8])The singular case of Riemann-Hilbert problem (Problem A). Find a solution w(z) of Eq. (1) which is continuous on the (N+l)-connected closed domain G, and satisfies the boundary condition:(2) Re [λ-(t)w(t)]=r(t), t∈Г where |λ(t)|=l, λ(t) ∈Cv(Г), r(t)∈Cv(Г),Theorem I. Let Eq. (1) satisfy the condition C. Then the total number of solvable conditions for Problem A is not greater than N-χ, and this estimate is accurate.To prove the above theorem, we introduce the modified boundary value problem with the boundary condition are the unknown constants to be determined appropriately and this problem is called Problem B. In particular, if χ =0,λ(z)=1, then Problem B is a general modified Dirichlet problem.Theorem Ⅱ. Suppose that Eq. (1) satisfies the condition C, then Problem B is solvable, and under more restrictions on Eq. (1). the solution of Problem B is unique.We construct an integral operator of problem B for multiply connected domain, qive a priori estimates of problem B for Eq. (1), and use the Leray-Schauder theorem, and so Theorem Ⅱ is proved. Theorem Ⅰ is easily derived from Theorem Ⅱ.
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 【References】
 Chinese Journal Full-text Database 3 Hits
 1 Wen Guochun Department of Mathematics;Some Nonlinear Boundary Value Problems for Nonlinear Elliptic Systems of Several First Order Equations in a Multiply Connected Domain[J];Acta Scicentiarum Naturalum Universitis Pekinesis;1983-04 2 Huang Sha Gao Shan Zhi (Hebei Teacher's Unirersity);A CLASS OF NONLINEAR BOUNDARY VALUE PROBLEM FOR NONLINEAR ELLIPTIC SYSTEMS OF SECOND ORDER WITH SEVERAL UNKNOW FUNCTIONS IN A MULTIPLY CONNECTED DOMAIN[J];Pure and Applied Mathematics;1985-00 3 Wang　Minghua(Chongqing　Normal　College) Wang　Dacheng(Chongqing　Jaotong　Institute);On　RiemannHilbert　Boundary　Value　Problems　for　EllipticEquations　of　First　Order　of　2n　Unknown　Functions[J];JOURNAL OF SOUTHWEST NATIONALITIES COLLEGE(NATURAL SCIENCE EDITION);1998-02
 【Co-citations】
 Chinese Journal Full-text Database 10 Hits
 1 Wen Guo-chun Tai Chung-wei Department of Mathematics;The Oblique Derivative Boundary Value Problem for Elliptic Complex Equation of First Order in a Multiply-Connected Domain[J];Acta Scicentiarum Naturalum Universitis Pekinesis;1981-03 2 Wen Guochun Department of Mathematics;Green's Functions of Multiply Connected Domain and the Oblique Derivative Problems for Elliptic Systems of Second Order[J];Acta Scicentiarum Naturalum Universitis Pekinesis;1982-06 3 Tai Chungwei, Wen Guochun (Department of Mathematics);The Poincare Boundary Value Problem for the Linear Elliptic Complex Equation of Second Order in a Multiply Connected Domain[J];Acta Scicentiarum Naturalum Universitis Pekinesis;1983-02 4 Han Houde, Department of Mathematics;The Plane Elastic Problem with an Arbitrary Shape Boundany Crack[J];Acta Scicentiarum Naturalum Universitis Pekinesis;1983-02 5 Wen Guochun Department of Mathematics;Some Nonlinear Boundary Value Problems for Nonlinear Elliptic Systems of Several First Order Equations in a Multiply Connected Domain[J];Acta Scicentiarum Naturalum Universitis Pekinesis;1983-04 6 Wen Guochun Department of Mathematics;Some Boundary Value Problems for A Class of Nonlinear Elliptic Systems of Several Second Order Equations[J];Acta Scicentiarum Naturalum Universitis Pekinesis;1984-02 7 Wen Guochun Department of Mathematics;Some Nonlinear Poincare Problems for Nonlinear Elliptic Systems of Several Second Order Equations[J];Acta Scicentiarum Naturalum Universitis Pekinesis;1984-06 8 Wen Guochun (Department of Mathematics);The Generalized Modified Compound Boundary ValueProblem with Complex Conjugate Values anda Priori Estimate of Its Solutions[J];Acta Scicentiarum Naturalum Universitis Pekinesis;1985-05 9 Wen Guochun Department of Mathematics;The Irregular Oblique Derivative Problem for Nonlinear Elliptic Equation of Second Order[J];Acta Scicentiarum Naturalum Universitis Pekinesis;1986-05 10 Wang Jian-ding;Semi-analytical Function[J];Journal of Beijing Polytechnic University;1983-03
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 1 Tao Fangming (Shanghai Jiaotong Univ.200030);The Interaction Between Antiplane Elastic Line Inclusion and Crack[A];[C];2002
 【Co-references】
 Chinese Journal Full-text Database 4 Hits
 1 Wen Guochun Department of Mathematics;On the Linear and Nonlinear Compound Boundary Value Problems with Displacement[J];Acta Scicentiarum Naturalum Universitis Pekinesis;1982-02 2 Wen Guochun Department of Mathematics;Green's Functions of Multiply Connected Domain and the Oblique Derivative Problems for Elliptic Systems of Second Order[J];Acta Scicentiarum Naturalum Universitis Pekinesis;1982-06 3 WEN GUOCHUN(Beijing University) FANG AINONG(Hunan University);THE COMPLEX FORM AND SOME BOUNDARY VALUE PROBLEMS FOR NONLINEAR ELLIPTIC SYSTEMS OF SECOND ORDER[J];Chinese Annals of Mathematics,series A;1981-02 4 Wen Guochun;The Poincare Boundary Value Problem with Nonpositive Indices for Linear Elliptic Systems of Second Order[J];Journal of Mathematical Research and Exposition;1982-01
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 Chinese Journal Full-text Database 6 Hits
 1 Wen Guochun Department of Mathematics;Some Boundary Value Problems for A Class of Nonlinear Elliptic Systems of Several Second Order Equations[J];Acta Scicentiarum Naturalum Universitis Pekinesis;1984-02 2 Wen Guochun Department of Mathematics;Some Nonlinear Poincare Problems for Nonlinear Elliptic Systems of Several Second Order Equations[J];Acta Scicentiarum Naturalum Universitis Pekinesis;1984-06 3 Wen Guochun (Peking University);SOME DISCONTINUOUS BOUNDARY VALUE PROBLEMS FOR NONLINEAL ELLIPTIC EQUATIONS OF SECOND ORDER[J];Pure and Applied Mathematics;1985-00 4 Li Shengxun (Hebei Chemi cal Engineering Institute)Huang Sha (Hebei Teacher's University);THE FUNDAMENTAL THEOREM OF THE QUASICONFORMAL MAPPING AND A BOUN DARY VALUE PROBLEM FOR SYSTEM OF COMPLEX EQUATIONS[J];Journal of Mathematics;1991-01 5 Hu Minde (Logistics Engineering College);THE MIXED BOUNDARY VALUE PROBLEM FOR NONLINEAR ELLIPTIC SYSTEMS OF SECOND ORDER[J];Journal of Mathematics;1991-01 6 Wang　Minghua(Chongqing　Normal　College) Wang　Dacheng(Chongqing　Jaotong　Institute);On　RiemannHilbert　Boundary　Value　Problems　for　EllipticEquations　of　First　Order　of　2n　Unknown　Functions[J];JOURNAL OF SOUTHWEST NATIONALITIES COLLEGE(NATURAL SCIENCE EDITION);1998-02
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