摘要
1878年,W. K. Clifford将高维空间中的几何与代数结合起来,引入了几何代数,后人以他的名字命名为Clifford代数.Clifford代数是一个可以结合但不可交换的代数,Clifford分析这个数学分支就是在Clifford代数An(R)上进行经典的函数理论分析,例如：研究正则函数,超正则函数以及k-超正则函数的基本性质；研究Cauchy型奇异积分算子的性质；研究各种边值问题等等.Clifford分析是实分析和复分析的自然推广.当n=0时,Clifford分析就是实分析；当n=1时,Clifford分析就是单复分析；当n=2时,Clifford分析就是四元数分析.因此Clifford分析是一个活跃的数学分支,它在许多数学领域内都具有重要的理论和应用价值.
在经典的函数理论分析中,研究Cauchy型积分的性质是非常重要的,它是解决各类边值问题的基本工具之一.Cauchy型积分是一类奇异积分,它在偏微分方程理论,奇异积分方程理论以及广义函数理论中有着广泛的应用.尤其是在偏微分方程和奇异积分方程的边值问题中,应用Cauchy型积分这个工具可以使得偏微分方程和奇异积分方程的处理显得特别地简练.Cauchy型积分算子的换序问题在奇异积分算子的正则化和奇异积分算子的合成中起着至关重要的作用.有了Cauchy型积分算子的换序公式,我们就可以解决闭光滑流形上具有B-M核的奇异积分方程的各种边值问题.因此Cauchy型积分算子的换序问题是解决许多问题的核心.在单复分析及多复分析中,Cauchy型积分算子的性质和换序问题解决得很彻底并且广泛地应用于弹性力学,流体力学以及高维奇异积分和积分方程中.但是在Clifford分析中,由于Clifford代数的不可交换性,有着同样重要性的Cauchy型积分算子的性质和换序问题却没有得到彻底解决.这给Cauchy型积分算子的合成和正则化带来了很大的挑战,从而影响了Clifford分析中积分方程和偏微分方程边值问题的发展.
1998年,黄沙证明了Clifford分析中Cauchy型积分的P-B (Poincare-Bertrand)置换公式,得到了很好的结论.在黄沙工作的基础上,本文另辟蹊径,给出了Clifford分析中累次奇异积分算子在Cauchy主值意义下更具体的一种新定义.然后利用Cauc-hy型奇异积分算子的性质证明了几个比较简单的情况下的两个奇异积分算子的换序公式.接下来又证明了一个关于被积表达式的不等式,即Clifford分析中的函数和微元乘积的不等式.这个不等式在本文中有着重要的意义.最后再利用此不等式和前面的结果证明了Clifford分析中关于一元函数及二元函数的Cauchy型奇异积分算子的P-B(Poincare-Bertrand)置换公式.
另外,本文还研究了一类Rn空间中的高阶奇异Teodorescu算子.通过这类高阶奇异算子,我们可以得到非齐次Dirac方程的解的积分表达式,从而可以解决许多边值问题.本文着重研究了这类高阶奇异Teodorescu算子的有界性,Holder连续性以及它的广义微商.同时还研究了它关于积分区域的边界曲面摄动的稳定性并给出了误差估计.最后用这个算子给出Rn空间中的一个广义Hn方程组的解的积分表达式.
全文共包括八个部分：
1.绪论.介绍了Clifford分析的历史背景,意义和研究现状,同时简单地介绍了一下我们的工作.
2.第一章.讨论了Clifford分析中一个Cauchy型奇异积分算子和普通积分算子的换序问题.首先证明了Clifford分析中两个普通积分算子在Liapunov曲面上的换序公式,然后在此基础上证明了Cauchy型奇异积分算子和普通积分算子的换序公式.证明过程中先证明两个累次积分在Cauchy主值意义下是收敛的,然后将两个累次积分分别分成两部分N1,N2和N1*,N2*,先证明N1=N1*,再证明
3.第二章.研究了Clifford分析中两个Cauchy型奇异积分算子的换序问题.先将两个累次积分分别分解为几个Cauchy型奇异积分算子与一个函数的和,从而证明了这两个累次积分是有意义的.然后再分别将两个累次积分分为四个部分,第一部分是挖掉奇点后的区域上的积分,另外几部分是带有奇点的区域上的积分.首先证明第一部分的值相等,再证明剩下的部分的差的极限为零.
4.第三章.研究了Clifford分析中一个普通积分算子和以普通积分算子的积分变量为奇点的Cauchy型奇异积分算子的换序问题.首先证明了几个相关的奇异积分算子的性质,并利用这些性质证明了两个累次积分是有意义的.然后巧妙地将积分区域分为几部分,从而将积分算子分成带有奇性的部分和不带奇性的部分.我们证明了带有奇性的部分的极限是零,并且不带奇性的部分相等.这样我们就证明了普通积分算子和以普通积分算子的积分变量为奇点的Cauchy型奇异积分算了的换序公式.
5.第四章.研究了Clifford分析中关于一元函数的两个Cauchy型奇异积分算子的换序问题,其中第二个Cauchy型奇异积分算子的奇点是第一个Cauchy型奇异积分算子的积分变量.这个问题的结论与前几章大不相同,这是因为当两个算子换序后,会多出一个函数项,这与复分析中的结果是一致的.在证明过程中,我们首先证明了一个带有微元的不等式.然后利用这个不等式证明了我们所讨论的两个累次奇异积分算子是有意义的.同时利用这个不等式和挖掉奇点的方法证明了换序公式,即Clifford分析中关于一元函数的Cauchy型奇异积分算于的P-B(Poincare-Bertrand)置换公式.
6.第五章.利用前面的结果讨论了Clifford分析中关于含有两个高维变量的函数的Cauchy型奇异积分算子的P-B(Poincare-Bertrand)置换公式.先给出了关于二元函数的Cauchy型奇异积分算子的定义,讨论了累次奇异积分算子的收敛性.然后将累次奇异积分算子分解为几个部分,对不同的部分利用前面的结论证明了关于二元函数的Cauchy型奇异积分算子的P-B(Poincare-Bertrand)置换公式.
7.第六章.研究了Rn空间中的一类高阶奇异Teodorescu算子的性质,分为三块内容：
(1).利用几个不等式证明了这类算子有界性.又通过证明几种特殊情况下这类算子的Holder连续性证明了算子在整个Rn空间中的Holder连续性,同时根据定义得到了它的广义微商.
(2).利用几个重要的不等式研究了这类算子关于积分区域的边界曲面摄动的稳定性并给出了误差估计.
(3).利用变量替换将广义H。方程组转换为一个Clifford分析中向量值的广义Di-rac方程.然后利用高阶奇异Teodorescu算子给出广义Dirac方程的解的积分表达式,从而得到了广义Ⅱn方程组的解的积分表达式.
8.结论.总结了论文的结论和有待解决的问题.
Clifford algebra is named after W. K. Clifford who introduced geometric algebra by the combination of the high-dimensional geometry with algebra in 1878. It is an associative and incommutable algebra. Clifford Analysis is a branch of mathematic study which is to execute typical functional theory analysis on Clifford algebra An(R), such as the study of the properties of regular functions, hypermongenic functions and k-hypermongenic functions; the study of the properties of the Cauchy-type singular integral operators and the study of various boundary value problems. Clifford analysis is the natural extension of complex analysis. When n = 0, Clifford analysis is the real analysis; when n= 1, Clifford analysis is the complex analysis; when n= 2, Clifford analysis is the quaternionic analysis. Therefore as an active branch of mathematical study, it has significant theoretical and applied value in various fields of mathematical study.
It is very important to study the properties of Cauchy-type integral in the typical functional theory analysis and it is one of the basic tools to solve various boundary value problems. Cauchy-type integral is a type of singular integral. It is widely used in the partial differential equational theories, the singular integral equational theories and the general functional theories, especially in the boundary value problems of the partial differential equations and the singular integral equations. It simplifies and stresses the process by using Cauchy-type integral. The transformation problem of Cauchy-type operators is crucial in the regularization and composition of the singular integral oper-ators. With the transformation formula, we can solve various boundary problems of the singular integral equation which has B-M core and is on the closed smooth manifold. As a result, the transformation problem is the core problem in the salvation of many problems. In complex analysis and multi-complex analysis, the nature and transfor-mation problem of Cauchy type integral operators are defined and solved thoroughly. It is widely used in elastic mechanics, fluid mechanics, hyper-dimensional singular integrals and integral equations. However, the nature and transformation problem of Cauchy-type integral operators have not been defined and solved although it is also very important and that is because Clifford algebra is incommutable. Consequently, it put us in great trouble in the composition and regularization of Cauchy type integral op-erators so that it has impeded the development of the boundary problems of the integral equations and the partial differential equations in Clifford analysis.
In 1998 Huang Sha proved the P-B(Poincare-Bertrand) transformation formula of the Cauchy-type integrals in Clifford analysis. On the base of Huang Sha's works, this dissertation finds a new approach. It first gives a new definition of the the Cauchy-type integrals in Clifford analysis. Then it proves the transformation formula of the iter-ated integral in several fairly simple cases by using Cauchy singular integral operators' nature; then it proves a very important inequation about integrated element i.e. a in-equation with a differential element; and further it proves the P-B(Poincare-Bertrand) transformation formula about Clifford values functions with one or two variables by using that inequation and the aforementioned results.
Besides, this dissertation also studies a high-order singular Teodorescu operator in the space Rn. By this kind of operator, we get the integral expression of the solution to the non-homogeneous Dirac equation so that many boundary value problems can be solved. It stresses on the boundedness, Holder continuity and the general differential of the high-order singular Teodorescu operator. Meanwhile it studies the perturbation stability concerning boundary surface of the integral domain and gives its error estima-tion. Ultimately, it gives an integral expression of a general Hn system in the space of Rn by using that operator.
The dissertation consists of eight parts:
1. Introduction. In this part, it recalls the history, academic significance and status que of Clifford analysis.
2. Chapter 1. It discusses the transformation problems of the Cauchy-type singular integral operator and the general integral operator. Firstly, it proves the transformation formula of the two general integral on the Liapunov surface in Clifford analysis; then based on that it proves the transformation formula of the Cauchy-type singular integral operators and the general operators. In the aforementioned process, at the outset, it proves that the two iterated integrals are convergent according to the Cauchy principal values. The two iterated integral are separated into N1, N2and N1*, Ar2*, and first it proves N1= N1*, and then proves (?) N2 - (?) N2*= 0.
3. Chapter 2. It studies the transformation problems of two Cauchy-type singular integral operators. Firstly, it separates the two iterated integrals into several Cauchy type singular integral operators and the sum of a function respectively to prove that the two iterated integrals are well defined. Then each of the aforementioned integrals are, respectively, separated into four parts. Part 1 are the integrals in the domain where the singular points are eliminated and the rest parts are integrals in the domain where the singular points exist. At the beginning it proves the values in the first part are equal and then the limit of the differences of the rest parts is zero.
4. Chapter 3. It studies the transformation problem of a general integral operator with the Cauchy type singular integral operators whose singular point is the integral variable of a general integral. Firstly, it proves the nature of several relevant singular integral operators and then it proves that the two iterated integrals are well defined by using that nature. Next, it divides cleverly the integral domain into several parts. Naturally, the integral operators are grouped into two parts. One is with the singular integrals and the other is with non-singular operators. And it proves that the limit of the part with the singular operators is zero and the part without singularity are equal. Thus, it proves the transformation formula of a general integral with the Cauchy type integral whose singular point is the integral variable of a general integral.
5. Chapter 4. It studies the transformation problem of two Cauchy type singular integral operators, among which the singular point of the second Cauchy type inte-gral is the integral variable of the first Cauchy singular integral. The conclusion of this problem is extremely different from those of the aforementioned chapters. Af-ter the transformation of the two operators, it has an extra function which agrees with the results in the multi-complex analysis. In the proving process, first it proves the existence of an inequation with differential elements. Then it proves that the iterated singular integrals under discussion are well defined. Meanwhile, by using that inequa-tion and eliminating the singular points, it proves the transformation formula, i.e., the P-B(Poincare-Bertrand) transformation formula of the Cauchy type singular integral operators in Clifford analysis.
6. Chapter 5. It discusses the P-B(Poincare-Bertrand) transformation formula of the Cauchy singular integral operators in the two dimensional Clifford function by us-ing the results above. First, it defines the Cauchy singular integral operators in the two dimensional Clifford function and then discusses the convergence of the iterated singu-lar integrals. Next it separates the iterated singular integral operators into several parts. Then for each part it proves the correctness of the P-B(Poincare-Bertrand) transfor-mation formula of Cauchy singular integral operators in the two dimensional Clifford function by using the conclusion it has already got.
7. Chapter 6. It studies the nature of a high-order singular Teodorescu operator in the space of Rn and this chapter consists of three sections:
(1). First by using several inequations it proves the boundedness of this operator. Then it proves the Holder continuity of the operator in several special cases to sup-port that the Holder continuity of the operator in the whole space of Rn also exists. Meanwhile, it gets the general differential elements according to the definition.
(2). By using several important inequations, it studies the perturbation stability concerning the boundary surface in the integral domain and it gives the error estimation too.
(3). First by using variable replacement, it transforms a general Hn equation sys-tem into a general vector value Dirac equation in Clifford analysis. Then it gives the integral expression of the general Dirac equation by using high-order singular Teodor-escu operators so that it gets the integral expression of the generalⅡn equation system.
8. Conclusions. The conclusions will be drawn and the problems get to be solved will be summarized in this part.
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