REAL AND COMPLEX CLIFFORD ANALYSIS Advances in Complex Analysis and Its Applications VOLUME 5 Series Editor: C.C. Yang The Hong Kong University of Science& Technology, Hong Kong Advisory Board: Walter Bergweiler Kiel University, Germany George Csordas University of Hawaii, U.S.A. Paul Gauthier University of Montreal, Canada Phillip Griffiths Princeton, U.S.A. Irwin Kra State University of New York, U.S.A. Armen G. Sergeev Steklov Institute of Mathematics, Russia Wolfgang Tutschke University of Graz, Austria REAL AND COMPLEX CLIFFORD ANALYSIS By SHA HUANG Hebei Normal University, Shijiazhuang, People’s Republic of China YU YING QIAO Hebei Normal University, Shijiazhuang, People’s Republic of China GUO CHUN WEN Peking University, Beijing, People’s Republic of China 1 3 Library of Congress Cataloging-in-Publication Data Huang, Sha, 1939– Real and complex Clifford analysis / by Sha Huang, Yu Ying Qiao, Guo Chun Wen. p. cm. — (Advances in complex analysis and its applications ; v. 5) Includes bibliographical references and index. ISBN-13: 978-0-387-24535-5 (alk. paper) ISBN-10: 0-387-24535-9 (alk. paper) ISBN-13: 978-0-387-24536-2 (e-book) ISBN-10: 0-387-24536-7 (e-book) 1. Clifford algebras. 2. Functions of complex variables. 3. Differential equations, Partial. 4. Integral equations. I. Qiao, Yu Ying. II. Wen, Guo Chun. III. Title. IV. Series. QA199.H83 2006 512´.57—dc22 2005051646 AMS Subject Classifications: 26-02, 32-02, 35-02, 45-02 (cid:164) 2006 Springer Science+Business Media, Inc. All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, Inc., 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now know or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks and similar terms, even if the are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed in the United States of America. 9 8 7 6 5 4 3 2 1 SPIN 11161424 springeronline.com Contents Preface ...............................................ix Chapter I General Regular and Harmonic Functions in Real and Complex Clifford Analysis....................1 1. Real and Complex Clifford Algebra............................1 2. Cauchy Integral Formula of Regular Functions and Plemelj Formula of Cauchy Type Integrals in Real Clifford Analysis....4 3. Quasi-Permutations and Generalized Regular Functions in Real Clifford Analysis..............................................15 4. The Chain Rule and the Differentiation Rules of new Hypercomplex Differential Functions in Clifford Analysis......24 5. Regular and Harmonic Functions in Complex Clifford Analysis30 Chapter II Boundary Value Problems of Generalized Regular Functions and Hyperbolic Harmonic Functions in Real Clifford Analysis ................................41 1. The Dirichlet Problem of Regular Functions for a ball in Real Clifford Analysis..............................................41 2. The Mixed Boundary Value Problem for Generalized Regular Functions in Real Clifford Analysis............................49 3. A Nonlinear Boundary Value Problem with Haseman Shift for Regular Functions in Real Clifford Analysis...................60 4. The Dirichlet Problem of Hyperbolic Harmonic Functions in Real Clifford Analysis..............................................68 Chapter III Nonlinear Boundary Value Problems for Generalized Biregular Functions in Real Clifford Analysis .............................................. 75 vi Contents 1. A Nonlinear Boundary Value Problem for Biregular Functions in Real Clifford Analysis.........................................75 2. Nonlinear Boundary Value Problems for Generalized Biregular Functions With Haseman Shift in Real Clifford Analysis......87 3. A Nonlinear Boundary Value Problem for Biregular Function Vectors in Real Clifford Analysis..............................95 Chapter IV Boundary Value Problems of Second Order Partial Differential Equations for Classical Domains in Real Clifford Analysis................................105 1. Harmonic Analysis in Classical Domains for Several Complex Variables ................................................... 105 2. The Dirichlet Problem of Second Order Complex Partial Differential Equations for Classical Domains in Complex Clifford Analysis.....................................................110 3. APseudo-ModifiedBoundaryValueProblemofSecondOrderReal Partial Differential Equations for a Hyperball in Real Clifford Analysis.....................................................117 Chapter V Integrals Dependent on Parameters and Sin- gular Integral Equations in Real Clifford Analysis....125 1. Cauchy’s Estimates of Integrals with One Parameter in Real Clif- ford Analysis................................................125 2. Three Kinds of Poincar´e-Bertrand Transformation Formulas of Singular Integrals with a Cauchy’s Kernel in Real Clifford Ana- lysis.........................................................134 3. The Composition Formula and Inverse Formula of Singular Integrals with a Cauchy’s Kernel in Real Clifford Analysis...145 4. The Fredholm Theory of a Kind of Singular Integral Equations in Real Clifford Analysis ....................................... 148 5. Generalized Integrals and Integral Equations in Real Clifford Analysis.....................................................153 Contents vii Chapter VI Several Kinds of High Order Singular Integrals and Differential Integral Equations in Real Clifford Analysis.....................................159 1. The Hadamard Principal Value and Differential Formulas of High Order Singular Integrals with One Singular Point in Real Clifford Analysis.....................................................159 2. The H¨older Continuity of High Order Singular Integrals in Real Clifford Analysis.............................................179 3. NonlinearDifferentialIntegralEquationsincludingThreeKindsof High Order Singular Integrals of Quasi-Bochner-Martinelli Type in Real Clifford Analysis.....................................184 4. A Kind of High Order Singular Integrals of Quasi-Bochner- Martinelli Type With two Singular Points and Poincar´e-Bertrand Permutation Formulas in Real Clifford Analysis..............189 Chapter VII Relation Between Clifford Analysis and Elliptic Equations ...................................197 1. Oblique Derivative Problems for Uniformly Elliptic Equations of Second Order................................................197 2. BoundaryValueProblemsofDegenerateEllipticEquationsofSec- ond Order...................................................209 3. The Schwarz Formulas and Dirichlet Problem in Halfspace and in a Ball .......................................................217 4. Oblique Derivative Problems for Regular Functions and Elliptic Systems in Clifford Analysis.................................228 References...........................................239 Index................................................249 Preface Cliffordanalysisisacomparativelyactivebranchofmathematicsthat has grown significantly over the past 30 years. It possesses both theo- retical and applicable values of importance to many fields, for example in problems related to the Maxwell equation, Yang-Mills theory, quan- tum mechanics, and so on. Since 1965, a number of mathematicians have made great efforts in real and complex Clifford analysis, rapidly expanding our knowledge of one and multiple variable complex anal- ysis, vector-valued analysis, generalized analytic functions, boundary value problems, singular differential and integral equations of several di- mension and harmonic analysis in classical domains (see Luogeng Hua’s monograph [26]1)). In recent years, more mathematicians have recog- nized the important role of Clifford analysis in harmonic analysis and wavelet analysis. Most of content of this book is based on the authors’ research results over the past twenty years. We present the concept of quasi-permutation as a tool and establish some properties of five kinds of quasi-permutations. Moreover, we use this tool to overcome the diffi- culty caused by the noncommutative property of multiplication in Clif- ford algebra, give the sufficient and necessary condition for generalized regular functions, and discuss the solvability for some boundary value problems. In Chapter I, we introduce the fundamentals of Clifford algebra in- cluding definitions, some properties, the Stokes theorem, Cauchy inte- gral formulas and Pompieu formulas for generalized regular functions and harmonic functions. We also definite quasi-permutation and study the property of quasi-permutations. We state the sufficient and nec- essary conditions for generalized regular functions. In the last section of this chapter, we consider regular and harmonic functions in complex Clifford analysis. In Chapters II and III, the Cauchy principle value and Plemelj for- mula of Cauchy type integrals are firstly studied. Next, the relation between linear and non-linear boundary value problems with Haseman shift for generalized regular and biregular functions, vector-valued func- tionsandsingularintegralequationsinrealCliffordanalysisisdiscussed. In addition, we discuss the existence, uniqueness and integral expres- sions of their solutions. By using quasi-permutation as a tool, we study the Dirichlet and mixed boundary value problems for generalized reg- x Preface ular functions, and give the Schwarz integral formulation of hyperbolic harmonic functions in real Clifford analysis. In Chapter IV, the theory of harmonic analysis in classic domains studied by Luogeng Hua is firstly introduced. Moreover, using quasi- permutation, we investigate two boundary value problems for four kinds of partial differential equations of second order in four kinds of classical domains of real and complex Clifford analysis, prove the existence and uniqueness of the regular solutions, and give their integral representa- tions. In Chapter V, we first introduce Cauchy’s estimates for three kinds of integrals with parameters, and then discuss the Poincar´e-Bertrand permutationformulas,inverseformulasofsingularintegralswithCauchy kernel, Fredholm theory, and the regularization theorem for singular integral equations on characteristic manifolds. InChapterVI,weintroducethedefinitionsoftheHadamardprinciple value, the Ho¨lder continuity, recursive formulas, calculation formulas, differentialformulasandPoincar´e-Bertrandpermutationformulasforsix kinds of high order singular integrals of quasi-Bochner-Martinelli type with one and two singular points, and then prove the unique solvability of the corresponding non-linear differential integral equations in real Clifford analysis. In Chapter VII, we use the method of Clifford analysis to solve some boundary value problems for some uniformly and degenerate elliptic systems of equations. It is clear that when n = 2, the functions in real Clifford analysis are the functions in the theory of one complex variable, hence the results in this book are generalizations of the corresponding results in complex analysis of one complex variable. In this book, we introduce the history of the problems as elaborately as possible, and list many references for readers’ guidance. After reading the book, it will be seen that many questions about real and complex analysis remain for further investiga- tions. Finally the authors would like to acknowledge the support and help of NSFC, Mr. Pi-wen Yang, Lili Wang, Nanbin Cao and Yanhui Zhang. Shijiazhuang and Beijing December, 2005 Sha Huang, Yu Ying Qiao and Guo Chun Wen Hebei Normal University and Peking University CHAPTER I GENERAL REGULAR AND HARMONIC FUNCTIONS IN REAL AND COMPLEX CLIFFORD ANALYSIS Cliffordalgebraisanassociativeandnoncommutativealgebraicstruc- ture, that was devised in the middle of the 1800s. Clifford analysis is an important branch of modern analysis that studies functions defined on Rn with values in a Clifford algebra space. In the first section of this chapter, we define a Clifford algebra. In the second section, we dis- cuss the Cauchy type integral formula of regular functions and Plemelj formula of Cauchy type integral in real Clifford analysis. In the third section, we introduce the conception of quasi-permutation posed by Sha Huang, and from this we get an equivalent condition for regular and general regular functions. In the fourth section, we establish a new hy- percomplex structure. In the last section, we discuss some properties of harmonic functions in complex Clifford analysis. 1 Real and Complex Clifford Algebra Let A (R) (or A (C)) be an real (or complex) Clifford algebra n n over an n-dimensional real vector space Rn with orthogonal basis e := {e1,...,en}, where e1 = 1 is a unit element in Rn. Then An(R) (An(C)) has its basis e1,...,en;e2e3,...,en−1en;...;e2,...,en. Hence an arbitrary element of the basis may be written as e = e ,...,e ; here A α1 αh A = {α1,...,αh} ⊆ {2,...,n} and 2 ≤ α1 < α2 < ··· < αh ≤ n and when A = Ø (empty set) eA = e1. So real (co(cid:1)mplex) Clifford algebra is com- posed of elements having the type a = x e , in which x (∈ R) are (cid:1) A A A A real numbers and a = C e , where C (∈ C) are complex numbers A A A A (see[6]). Ingeneral,onehase2 = +1, i = 1,...,s,e2 = −1, i = s+1,...,n i i and e e +e e = 0, i,j = 2,...,n,i (cid:4)= j. With different s we can get dif- i j j i ferent partial differential equations (elliptic, hyperbolic, parabolic equa- tions) from the regular function in Clifford analysis. In this book we let s = 1.