The Properties And Boundary Value Problem Of Holomorphic Cliffordian Functions In Clifford Analysis

W. K. Clifford introduced a kind of geometric algebra-Clifford algebra, which combinedthe high-dimensional geometry and algebra. It is an incommutable and associative algebra.Clifford analysis is the classical functional theory analysis on the Clifford algebra. It is thehigh-dimensional generalization of the real analysis, complex analysis, quaternionic analysisetc. As an active branch of the mathematical study, it has significant theoretical and applicablevalue in mathematics and other subjects. For example, the study of the Cauchy type integral isan important foundation of the B. V. P study in the P. D. E, singular integral equation, and gener-alized functional theory. Moreover, it is also widely used in various modern science technologysuch as elastic mechanics, ?uid mechanics, robotics, calculated biology, chemistry and com-puters. The introduction of the Dirac operator D =∑xi in the n dimensional Euclideanspace and the development of its null solutions– regular functions are the milestones of Cliffordanalysis. Compared with the classical functional theory in complex analysis, many researchersin home and abroad give a lot of study. For example, the Cauchy integral formula, the bound-ary value properties, and the series extension for regular functions, Morera theorem, RiemannB. V. P etc. Later, H. Leutwiler advanced the generalization of different functional classes inClifford space by way of some modifications to the Dirac operator. Therefore, the appearanceof many new functional classes such as hypermonogenic functions, hyperharmonic functions,k monogenic functions, k hypermonogenic functions enriches the research in Clifford analysis.In Clifford analysis, the solutions of the P. D. E D?mf(x) = 0 (Or : f(x)D?m = 0) arecalled holomorphic Cliffordian functions, which belong to a new kind of natural generalizationof the regular function. Regular functions must be holomorphic Cliffordian functions, but thereverse saying is not necessarily true. For instance, xn is the holomorphic Cliffordian functionand the hypermonogenic function, but it is a pity that it doesn't belong to regular functions.However, from the definition of the holomorphic Cliffordian function, we can easily find thatthe function can be transformed into the regular function after being performed by the Laplaceoperator at most m times. As one kind of the spaces being widely applied in the high dimen-sional functional theory, it further enlarges the research view and developing direction of theresearchers in the real and complex analysis on Clifford algebra.This paper is made up of four parts:In chapter 1, the preliminaries, some main lemmas and a local generalized sphere coordi-nates transformation are given. Here we also estimate two inequalities, which are the base of some integral estimates as follows. Firstly, we introduce the P. D. E D?mf(x) = 0 on the realClifford algebra R0,2m+1, whose solutions are holomorphic Cliffordian functions, and the kernelfunction and its properties of the integral representation of holomorphic Cliffordian functions.Then, the integral representation and Plemelj formula for the holomorphic Cliffordian functionsin the bounded domain are given, which lay a foundation for the unbounded case.In chapter 2, some simple properties of holomorphic Cliffordian functions are proved fromthe regular function's degree. Firstly, we point out that the space of holomorphic Cliffordianfunctions can form a right R0,2m+1 module, but it cannot make a left one. Secondly, we givetwo equal conditions with the help of the first class of Quasi-Permutation defined by Sha Huangwhich not only make us judge this kind of complicated functions more simply, but also build forus the relations between regular functions and holomorphic Cliffordian functions. Lastly, wediscuss and calculate the extension theorem of 2m+1 times continuously differentiable func-tions defined in the bounded domain ? in R2m+2 and with values in Clifford algebra R0,2m+1 byway of the Cauchy integral formula and the Plemelj formula of the Cauchy type integral usingsome small techniques.In chapter 3, we introduce the Cauchy type integral, Cauchy principle value and the Plemeljformula of 2m+1 times continuously differentiable functions defined in unbounded domains Uin R2m+2 and with values in Clifford algebra R0,2m+1. As we all know, these two main partscan be reduced to the corner stones in holomorphic functional theory and which can also beclassified as the significant foundations in the discussion of extension theorem and B. V. P withsimilarity to the classical holomorphic functional theory. Firstly, we define the Cauchy typeintegral of holomorphic Cliffordian functions and its Cauchy principal value, and we also provethat the Cauchy type integral is convergent under the meaning of Cauchy principal value over theboundary. Here we divide the boundary ?U of the unbounded integral domain U ? R2m+2 intobounded and unbounded parts. In the unbounded part, we give some elegant integral estimationand some additional conditions to the kernel and density functions, then we prove that thisintegral is convergent under the meaning of Cauchy principal value using the similar method ofregular functions and k monogenic functions in the unbounded domain. As to the bounded part,we divide the integral into the normal and the weak singularity items, and we just have to provethe convergence of the weak singular item, and this result could be easily obtained using thesimilar method of regular functions and k monogenic functions in the bounded domain. Whenwe discuss the Plemelj formula, we mainly prove that the sum of the weak singularity items of the Cauchy type integral of holomorphic Cliffordian functions is continuous, and which issolved by way of some significant integral estimation in chapter 1 and some methods in thischapter above.In chapter 4, one kind of boundary value problem of 2m+1 times continuously differen-tiable functions defined in the bounded domain ? in R2m+2 and with values in Clifford algebraR0,2m+1isdiscussed.Therearethenonlinearandlinearcaseshere.Astothenonlinearcase,firstly, some related operators are defined whose bounded properties about theβmodule areproved; then we change the B. V. P into integral equation problem; lastly, the problem is solvedby way of the integral equation theory and the Schauder fixed point theorem under a certainsolvable condition. In the linear case, the existence and uniqueness of the problem have beenproved by way of the Banach fixed point theorem-compressed mapping principle in high di-mension.Lastly, we have consumed the related functional progress in holomorphic Cliffordian func-tions, whose more research will be expected later.