Clifford Analysis And Its Applications In PDEs

The dissertation focuses on the application of Clifford analysis in PDE. This in-cludes the following four aspects.(1) We establish a general existence theorem for the Cauchy-Riemann type non-linear partial differential systems.(2) We establish the existence theorem of the solution of the non-homogeneous Cauchy-Riemann equations in the analysis of several Clifford variables.(3) We establish the existence theorem of the solution of the non-homogeneous K-Cauchy-Futer equations in the analysis of several quternionic variables.(4) We establish the Octonionic Hermitian Clifford analysis, in particular we deal with the Dirichlet boundary value problem.This paper is divided into five chapters.The first chapter is the introduction, given the background of Clifford analysis and its application in partial differential equations. We also outline the main approaches and main conclusions.The second chapter studies the theory of nonlinear partial differential equations of higher orders in higher dimensional spaces. This is an extension of the result of Nijenhuis-Woolf (Ann. Math.1963) on the theory of first-order nonlinear partial d-ifferential equations based on an approach of Clifford analysis. The theory depends heavily on the boundedness of the Teodorescu operator in Holder spaces. As a right inverse operator of Dirac operator, the Teodorescu operator is a singular integral oper-ator. For the study of this singular integral operator, we introduce an effective tool—approach of oblique spherical coordinates.The third chapter analyzes the theoretical study of multi-Clifford variables, which is the theory of functions of several complex variables in the promotion of non-commutative realm. For nonhomogenous Cauchy-Riemann equations in multi-Clifford analysis, we give a concrete expression of the solution with compact support. This also yields the Hartogs phenomenon in multi-Clifford analysis. We establish the corresponding Bochner-Martinelli integral formula, which unifies the corresponding result in the the-ory of a single complex variable, several complex variables, and several quaternions.The fourth chapter studies the theory of κ-Cauchy-Fueter operator, which belongs to the theory of several quaternionic variables. For the study of non-homogeneous κ-Cauchy-Fueter equations, the classical method is based on algebraic geometry, which can produce a specific expression of solution at low dimensions. Our approach comes from the theory of several complex variables, which takes the advantages of explic-it expression of solutions in any dimension. We introduce a new technique through increasing the dimensions of spaces in consideration to simplify the κ-Cauchy-Fueter operator in purpose. This technique allows us to establish the corresponding Bochner-Martinelli integral formula.In the final chapter, we study the Octonionic Hermitian Clifford analysis. We introduce the Witt basis of Octonions, construct the Octonionic Hermitian Dirac oper-ator, and then establish the corresponding integral theory as well as the boundary value problem.