High Order Singular Integrals And Boundary Value Problems In Clifford Analysis

K.T.Vahlen~[1] set up Clifford algebra at the begin of last century. It is a kind of algebraic structure which is associative and incommutable. Clifford analysis is a branch of mathematics arising in 1970s. It studies theory of function from real vector space to real incommutable Clifford algebra. It plays an important role in many fields such as other mathematical branches, theory of fields and quantum mechanics and so on. Many mathematicians pay much attention to it. Robert P.Gilbert~[2], F.Brack, R.Relanghe, F.Sommen~[3], WenGuochun~[4], LeHuang Song~[5], Huangsha~[6-12] and so on did many works through comparing Clifford analysis with single variable, multiple variables complex analysis function theory. The studies to Clifford analysis were developed rapidly by the end of last century.In this paper, we consider high order singular integrals, function properties and boundary value problems in Clifford analysis. There are eight chapters in this paper.We discuss high order singular integrals in Clifford analysis from the first chapter to the third chapter. The integral theory is very important in real analysis, it is not only the theory base of modern mathematics, but also one of the main tools in math application. But some practical problems can't be solved by the normal integral. It is necessary to consider singular integrals and even high order singular integrals. Hadamard~[13] defined high order singular integrals by the think of the integral for finite part in real analysis. For example, we consider the high order singular integral Here ,Its finite part is defined like this:The main idea is to discard the unintegrable part and define the high order singular integral by use of the other parts. In paper[14], Lujianke defined directly high order singular integrals on the closed curve L like this: P'That i / (F7r%dt = / Prdt'That is> / (Ft^W* /, ifrL (t — t0)2 JLt — to JL t — t0 JL t — tocomparing —(-------) with (-------)',the singularity of the former is lower. Professort — t o t — toLu's theory about high order singular integrals was approved widely in the field ofmathematics.Moreover, it is used in practical work. By the think of induction, we use normal integral or weakly-singular integrals to define singular integrals and use the lower order singular integral to define high order singular integrals. Some results about high order singular integrals were proved in the document[14][15][16].Huangsha has discussed three kinds of high order singular integrals in Clifford analysis. Based on paper[17], by the think of Hadamard principal value of high order singular integrals and the think of induction,we discuss the inductive definition, computation formula, properties and Poincare-Bertrand exchange formula for high order singular integrals of quasi Bochner-Martinelli- type(in the following, it is abbreviated quasi B-M type) from the first chapter to the third chapter. We know integral operator and integral equation play an important role in solving boundary value problems and differential equations, so the content in this part is significant in mathematical theory and application.In the fourth chapter, we discuss the solvability of the second kind integral equation and the solution expression formula. The integral equation problem is another important problem in practice. Studying integral equation in Clifford analysis is blank space yet. Resorting to unit resolution ,M. Spivark gave the definition of improper integral defined on the manifold in n dimensional real space in his works <>, it can be found in paper[18]. This definition is the generalization of real improper integrals in one dimensional space, it permits that there are many singular points on the manifold. We define improper integrals using the similar method in Clifford analysis. Robert P. Gilbert introduced definition of the commutative factor in paper [2], then he solved partly the incommutable problem of multiplication in Clifford analysis. Resorting to this idea, we design integral operator kernel with commutative factor and study the solvability of the second kind integral equation and series expression formula of solution .We discuss boundary value problems for the two kinds of functions in the fifth chapter and the sixth chapter. The regular function in Clifford analysis is similar with the holomorphic function in single variable complex analysis,its definition was given firstly in paper[22]. The Cauchy type singular integral and Plemelj formula about regular function have been studied.In paper [20],the author discussed a spe-...