Some Properties Of A Τ-type Operator And Its Applications In The Clifford Analysis

Clifford analysis studies the function defined on the Euclidean space Rnwith value in Clifford algebra space, and its main research object is the holomorphic function. T-type operator related to the holomorphic function has an important application in solving the partial differential equation. In this thesis, T-type operator(TR4[g])(x) is studied in the situation of n = 4 in Clifford analysis, and the boundedness of it on a bounded domain and unbounded domain, the H¨older continuity and the partial derivative properties in the sense of generalized derivative are discussed. Finally, the expression of the solution to the elliptic partial differential equations are given using the properties of(TR4[g])(x).In the first chapter, the algebra space Cl0,3(R) and the operation rules of the spaceCl0,3(R) are introduced, and some definitions and some lemmas are given.In the second chapter, the boundedness and the H¨older continuity of(TR4[g])(x) are proved, and partial derivatives of(TR4[g])(x) in the generalized derivative are discussed.In the third chapter, a boundary problem of the elliptic partial differential equations is showed. By transformation the real partial differential equations in R4is transformed into the equation xw =∑7i=0ci(x)ei= g(x) in Clifford analysis, and the solution to the equation is obtained using the properties of(TR4[g])(x) which has already been proved in the second chapter.