Riemann-Hilbert Boundary Value Problems In C～n

Boundary value problems for analytic functions is an important branch of theory for complex variable functions and extensively applying in mechanics,physics, and engineering etc. In our domestic,the research work of such type problems has achieved further progress from the fifties,and at the same time using it into the fields of mathematic elastic mechanics,generalized analytic functions and partial differential equations.Riemann—Hilbert boundary value problem(abbreviated called R—H problem) is the basic one in the problems for analytic functions. Professor Lu jianke has made an explict introduction to R—H boundary value problems and its related problems in his book "Boundary value problems for analytic functions"where Plemelj formula play an important role.For the research of the R—H problems,the domestic scholars mainly adopt this method,from analytic functions to bi-analytic,tri-analytic,even polyanalytic functions, and the extension of the generalized analytic functions and systems of equations as well. On the other hand, considering assumption domain, we may discuss R—H boundary value problems with higher order singularity(Hadamard principal part value sense).Riemann—Hilbert boundary value problems for analytic functions of several complex variable is naturally issued from the one of one complex variable. yet,the research work about it has not enough. In this paper, the author mainly has done the following two aspects:first,using poission kernel to solve the Riemann—Hilbert boundary value problem in the unit polydisc and on the basis of it, extending the solution space from analytic function space into the meromorphic function space and the multiform function space. Secondly,discussing the nonlinear Cauchy—Riemann systems for generalized analytic functions.The thesis involes the theory of functions of several complex variable,generaliz-ed analytic functions,Clifford analysis,measure theory and Fourier analysis,for example,boundary value problems for functions of several complex variable in higher dimensional space,the discussion of inhomogeneous C—R system of generalized analyitic functions,nonlinear Dirichlet problems,multiple integral and integral transformation,the solvability and solvable conditions,etc.In the thesis,some importanttransition functions are used,e.g. Plemelj formula,functions as δ(x), T-operator. For methodology, we apply symmetric extension(e.g. coformal map,symmetric function) in one complex variable and adopt transition functions(e.g.functions as δ(x),etc.)inseveral complex variable.For the solution spaces to be asked,we do it from one complex variable to several complex variable,from analytic function space to the meromorphic function,multiform function and monogenic function spaces. Thedifficulties we occurred in analysis of the problem is the condition domains,such as polydisc into the common domains,and so forth.The thesis consists of five chapters. Chapter 1 is preface,we introduce the history and background of the problem and the work of internel and extemel researchers up till now. In chapter 2,we shall give an exact introduction to Fourier series which occurred in analysis, multiplicative measure and Fubini theorem in measure theory and some details for the theory of the generalized analytic function. Chapter 3 is the main part of this paper, we make a concrete and minute discussion of the method, the solution expression and solvability condition for the R—H boundary value problem. Chapter 4 is continuation of the preceding chapter,we make an inquiry into the R—H boundary value problem for generalized analytic functions.In chapter 5, we discuss the nonlinear half-Dirichlet problems for first order elliptic equations.