Real and Complex Clifford Analysis (Advances in Complex Analysis and Its Applications) [1 ed.] 0387245359, 9780387245355, 9780387245362

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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

13

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 ¤ 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 springeronline.com

SPIN 11161424

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 Analysis 30

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. A Pseudo-Modified Boundary Value Problem of Second Order Real Partial Differential Equations for a Hyperball in Real Clifford Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

Chapter V Integrals Dependent on Parameters and Singular Integral Equations in Real Clifford Analysis . . . . 125 1. Cauchy’s Estimates of Integrals with One Parameter in Real Clifford Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 2. Three Kinds of Poincar´e-Bertrand Transformation Formulas of Singular Integrals with a Cauchy’s Kernel in Real Clifford Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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. Nonlinear Differential Integral Equations including Three Kinds of High Order Singular Integrals of Quasi-Bochner-Martinelli Type in Real Clifford Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184 4. A Kind of High Order Singular Integrals of Quasi-BochnerMartinelli 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. Boundary Value Problems of Degenerate Elliptic Equations of Second 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 Clifford analysis is a comparatively active branch of mathematics that has grown significantly over the past 30 years. It possesses both theoretical and applicable values of importance to many fields, for example in problems related to the Maxwell equation, Yang-Mills theory, quantum 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 analysis, vector-valued analysis, generalized analytic functions, boundary value problems, singular differential and integral equations of several dimension and harmonic analysis in classical domains (see Luogeng Hua’s monograph [26]1)). In recent years, more mathematicians have recognized 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 difficulty caused by the noncommutative property of multiplication in Clifford 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 including definitions, some properties, the Stokes theorem, Cauchy integral 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 necessary 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 formula 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 functions and singular integral equations in real Clifford analysis is discussed. In addition, we discuss the existence, uniqueness and integral expressions 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 quasipermutation, 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 representations. In Chapter V, we first introduce Cauchy’s estimates for three kinds of integrals with parameters, and then discuss the Poincar´e-Bertrand permutation formulas, inverse formulas of singular integrals with Cauchy kernel, Fredholm theory, and the regularization theorem for singular integral equations on characteristic manifolds. In Chapter VI, we introduce the definitions of the Hadamard principle value, the H¨ older continuity, recursive formulas, calculation formulas, differential formulas and Poincar´e-Bertrand permutation formulas for six 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 investigations. 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 Clifford algebra is an associative and noncommutative algebraic structure, 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 discuss 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 hypercomplex structure. In the last section, we discuss some properties of harmonic functions in complex Clifford analysis.

1

Real and Complex Clifford Algebra

Let An (R) (or An (C)) be an real (or complex) Clifford algebra 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 ; e2 e3 , ..., en−1 en ; ...; e2 , ..., en . Hence an arbitrary element of the basis may be written as eA = eα1 , ..., eαh ; here A = {α1 , ..., αh } ⊆ {2, ..., n} and 2 ≤ α1 < α2 < · · · < αh ≤ n and when A = Ø (empty set) eA = e1 . So real (complex) Clifford algebra is com
posed of elements having the type a = A xA eA , in which xA (∈ R) are
real numbers and a = A CA eA , where CA (∈ C) are complex numbers (see [6]). In general, one has e2i = +1, i = 1, ..., s, e2i = −1, i = s+1, ..., n and ei ej + ej ei = 0, i, j = 2, ..., n, i = j. With different s we can get different partial differential equations (elliptic, hyperbolic, parabolic equations) from the regular function in Clifford analysis. In this book we let s = 1.

2

Chapter I Noting that the real vector space Rn consists of the elements z := x1 e1 + · · · + xn en ,

(1.1)

we can consider that the elements z := x1 e1 + · · · + x2 en and z = (x1 , ..., xn ) are identical, and denote by Rez = x1 the real part x1 of z. For z ∈ Rn , we define the conjugate as z = x1 e1 − · · · − xn en , thus zz = e1 (x21 + · · · + x2n ). The absolute value (or a norm) for an
element a = aA eA ∈ An (R) is taken to be A

|a| =



|a|2

=

 A

|aA |2 ,

(1.2)

then |z|2 = |z|2 = zz = zz for z ∈ Rn . If z = 0, z ∈ Rn , then we have z(

z z ) = ( 2 )z = 1, 2 |z| |z|

(1.3)

hence, all non-zero elements of Rn possess the inverse operation of multiplication. However, it is not true for all Clifford elements, for instance 1 + e123 . Definition 1.1 as follows:

For a ∈ An (R) (or An (C)), we give some calculations a′ =



xA e′A (or a′ =

A



cA e′A ),

A

where e′A = (−1)|A| eA and |A| = nA (see [7]) being the cardinality of A, i.e. when A = Ø, |A| = 0 and when A = {α2 , α3 , ..., αh } = Ø, then |A| = h.   = = cA eA ), a xA eA (or a A

A

where eA = (−1)|A|(|A|−1)/2 eA , and ′ = a = a′ = a



(−1)

|A|(|A|+1) 2

xA eA (or a =

A



(−1)

|A|(|A|+1) 2

cA eA )).

A

For any a, b ∈ An (R) (An (C)), an inner product is defined by < a, b >= [ab]1 ,

(1.4)

3

General Regular and Harmonic Functions

where [ab]1 is the coefficient of e1 in ab, and we can prove that [ab]1 = [ba]1 , so < a, b >= [ab]1 = [ba]1 . From (1.2), (1.4) we have |a|2 =< a, a >= Proposition 1.1

 A

|aA |2 .

(1.5)

Let a, b ∈ An (R)(An (C)). Then

|, |a|2 = [aa]1 = [aa]1 . ab = ba, |a| = |a′ | = |a| = |a

Definition 1.2 A Clifford number a ∈ An (R)(An (C)) is said to be conjugate, if aa = aa = |a|2 , and a is called the conjugate Clifford number of a. Denote by ℵ (ℵ(C)) the set of all conjugate Clifford numbers in An (R)(An (C)). Proposition 1.2 equivalent:

For any a ∈ An (C), the following conditions are

1)

a is conjugate.

2)

|ab| = |ba| = |b||a| for any b ∈ An (C).

3)

There exists b ∈ An (C)\{0}, such that ab = |a||b| or ba = |b||a|.

Proof It is sufficient to show that the assertion is valid for a = 0. We first prove 1) → 2). By Proposition 1.1, it is clear that |ab|2 = [abab]1 = [baab]1 = |a|2 [bb]1 = |a|2 |b|2 . Similarly we can verify |ba| = |b||a|. Setting b = a and using Proposition 1.1, we can get 2) → 1), 1) → 3). As for 3) → 1), we need to show that ab = |a||b| (b = 0), i.e. aa = |a|2 . There is no harm in assuming that |a| = |b|, then by Proposition 1.1, we get < a, b >= [ab]1 = ab = |a||b| = |a||b|. (1.6) Hence there exists t ∈ C such that b = ta and |t| = 1. From (1.6), it follows that t|a|2 =< a, ta >= |a|2 .

Since a = 0, we obtain t = 1, that is a = b, therefore aa = ab = |a||b| = |a|2 .

4

Chapter I

By using the above result it is easy to extend Proposition 1.1, hence we have the following corollaries. Corollary 1.3

a, b ∈ ℵ(C) ⇒ ab ∈ ℵ(C).

Corollary 1.4

If a1 , ..., am ∈ ℵ(C) and b ∈ An (C), then

|a1 , ..., aj−1 baj ...am | = |a1 , |..., |aj−1 ||b||aj |...|am |.

2

Cauchy Integral Formula of Regular Functions and Plemelj Formula of Cauchy Type Integrals in Real Clifford Analysis Firstly we give the definition of A-value function.

Definition 2.1 Let Ω ⊂ Rn be an open connected set. The function f which is defined in Ω with values in An (R) can be expressed as f (x) =



eA fA (x),

A

where the function fA is a real-valued function. The set of C r -functions in Ω with values in An (R) is denoted by (r)

FΩ = {f |f : Ω → An (R), f (x) =

 A

fA (x)eA }.

We introduce also the Dirac operator ∂=

n 

ei

i=1

∂ (r) (r−1) : FΩ → FΩ , ∂xi

that is ∂f =

 i,A

ei eA

(2.1)

∂fA . ∂xi

The Dirac operator ∂ is defined as ∂ = e1

∂ ∂ ∂ − e2 − · · · − en . ∂x1 ∂x2 ∂xn

(2.2)

Since An (R) is not a commutative algebra, in general the expression ∂f = e1

∂f ∂f + · · · + en ∂x1 ∂xn

5

General Regular and Harmonic Functions is not same as f∂ =

∂f ∂f e1 + · · · + en . ∂x1 ∂xn (r)

Definition 2.2 Let Ω ⊂ Rn be a domain and f ∈ FΩ . Then f is left regular in Ω, if ∂f = 0 in Ω; and f is right regular in Ω, if f ∂ = 0 in Ω. For n = 2, ∂f = 2∂f /∂z = ∂f /∂x1 + e2 ∂f /∂x2 , thus under this condition the Clifford algebra is just the complex space and left regular
is equivalent to holomorphic. For n = 3, f (x) = eA fA = e1 f1 + e2 f2 + A

e3 f3 +e23 f23 . If we transform f1 , f2 , f3 , f23 into Φ1 , −Φ2 , −Φ3 , Φ0 , then ∂f = 0 is equivalent to the elliptic system in higher dimensional domains, i.e. the M − T equation. Liede Huang discussed the boundary value problems of the M − T equation in [28] and found a lot of applications, for example the problem of airplane wing shapes, momentum pressure, electromagnetism, laser technology and so on.

Notice that the product of Dirac operators ∂ and ∂ is equivalent to the Laplace operator ∂∂ = ∂∂ =

Definition 2.3

∂2 ∂2 = △. + · · · + ∂x2n ∂x21

(2.3)

(r)

Let Ω ⊂ Rn be a domain and f ∈ FΩ . If △f = 0,

(2.4)

then f is called a harmonic function. It is clear that we have the following theorem. (2)

Theorem 2.1 Let Ω ⊂ Rn be a domain and f ∈ FΩ . If f is harmonic in Ω, then ∂f is left regular in Ω, and f ∂ is right regular in Ω; and if f is regular in Ω, then f is harmonic in Ω.


Let Ω ⊂ Rn be a domain. If U = np=1 ⊕ ∧p w is the exterior algebra with the basis {dx1 , ..., dxn }, we consider the pth-differential
ψA,H (x)eA dxH , where x ∈ Ω ⊂ Rn . Furthermore, the form ψ(x) = A,H

function ψA,H (x) is assumed to be in set C r (Ω) (r ≥ 1). Integration of ψ(x) over the p-chain Γ ⊆ Ω is defined as 

Γ

ψ(x) =



A,H Γ

ψA,H (x)eA dxH .

(2.5)

6

Chapter I

Now we establish the Stokes theorem for functions in F (1) as follows: Denote by Γ ⊆ Ω an n-dimensional differentiable and oriented manifold. By means of the (n − 1)-forms dxi = dx1 ∧ · · · ∧ dxi−1 ∧ dxi+1 ∧ · · · ∧ dxn , i = 1, ..., n,

a An (R)-valued n − 1−form is introduced by putting dσ =

n  i=1

(−1)i−1 ei dxi .

If dS stands for the “classical” surface element and − → m=

n 

ei ni ,

i=1

where ni is the i-th component of the unit outward normal vector, then the An (R)-valued surface element dσ can be written as → dσ = − mdS. Furthermore the volume-element dx = dx1 ∧ · · · ∧ dxn is used. Now we state and prove the Stokes-Green theorem about the function in FΩr . Theorem 2.2 (see [6]) Let Ω be as stated above, M ⊆ Ω be an n(r) dimensional differentiable, oriented manifold, f, g ∈ FΩ (r ≥ 1), and Γ be an arbitrary n-chain on M . Then we have 

f dσg =



[(f ∂)g + f (∂g)]dx.

(2.6)

Γ

∂Γ

Proof Using the Stokes theorem for real-valued functions, we successively get 

f dσg =





∂Γ A,i,B

∂Γ

=



(−1)i−1 eA ei eB

A,i,B

=





Γ A,i,B

=



Γ

(−1)i−1 fA gB dxi eA ei eB 

Γ

(−1)i−1 ∂xi (fA gB )dx

[(∂xi fA )eA ei gB eB + fA eA ei eB (∂xi gB )]dx

[(f ∂)g + f (∂g)]dx.

General Regular and Harmonic Functions

7

Corollary 2.3 If f is right regular in Ω and g is left regular in Ω, then for any n-chain Γ on M ⊂ Ω, the following integral holds: 

f dσg = 0.

(2.7)

∂Γ

Corollary 2.4

Putting f = 1, the formula (2.6) yields 

∂gdx =

Γ



dσg,

∂Γ

(r)

where g ∈ FΩ (r ≥ 1). For any g, which is left regular in M , we have 

dσg = 0.

(2.8)

∂Γ

This is called the Cauchy theorem. Next let Ω ⊂ Rn be a domain and the boundary ∂Ω be a differentiable, oriented, compact Liapunov surface. The so-called Liapunov surface is a kind of surface satisfying the following three conditions (see [19]). 1. Through each point in ∂Ω, there is a tangent plane. 2. There exists a real constant number d > 0, such that for any point N0 ∈ ∂Ω, we can construct a sphere E with the center at N0 and radius d, and E is divided into two parts by ∂Ω, one included in ′ the interior of E denoted by ∂Ω , the other in the exterior of E; and each straight line parallel to the normal direction of ∂Ω at N0 , ′ with ∂Ω intersects at most a point. 3. If the angle θ(N1 , N2 ) between outward normal vectors through N1 , N2 is an acute angle, and r12 is the distance between N1 and N2 , then there exist two numbers b, α (0 ≤ α ≤ 1, b > 0) independent of N1 , N2 , such that θ(N1 , N2 ) ≤ br12 for all points N1 , N2 ∈ ∂Ω. From 3 we get that θ is continuous at the point N ∈ ∂Ω (see [29]2)). Let N0 ∈ ∂Ω be a fixed point, and establish a polar coordinate system with the origin at N0 and the outward normal direction of ∂Ω at N0

8

Chapter I ′

as the direction of the positive xn axis. Then the surface ∂Ω may be written in the form ξn = ξn (ξ1 , ..., ξn−1 ), (2.9) and ξn has first order partial derivatives on ξi (i = 1, ..., n − 1). ′

Let d > 0 be small enough such that bdα ≤ 1, and for any point N ∈ ∂Ω , θ0 = θ(N0 , N ) as stated before, and r0 = |N0 N | be the distance from N0 to N (r0 < d). We can obtain 1 1 cos θ0 ≥ 1 − θ02 ≥ 1 − b2 r02α ≥ 0, 2 2 and then

1 1 ≤ ≤ 1 + b2 r02α ≤ 2. cos θ0 1 − 21 b2 r02α

Thus we have the formula

1 cos θ0 ≥ . 2

(2.10)

Moreover we introduce a local generalized spherical coordinate at N0 as follows: ξn−1 = ρ0 cos ϕ1 cos ϕ2 ... cos ϕn−3 cos ϕn−2 , ξn−2 = ρ0 cos ϕ1 cos ϕ2 ... cos ϕn−3 sin ϕn−2 , ..., ξ2 = ρ0 cos ϕ1 sin ϕ2 , ξ1 = ρ0 sin ϕ1 , where ρ0 is the length of the projection of r0 on the tangent plane of ∂Ω at N0 , and ϕi satisfy the conditions |ϕj | ≤

π , j = 1, 2, ..., n − 3, 0 ≤ ϕn−2 < 2π, 2

hence we get 



→0 , xn ) ≥ 1 ,  D(ξ1 , ..., ξn−1 )  ≤ ρn−2 , cos(− m 0 2  D(ρ0 , ϕ1 , ..., ϕn−2 ) 

(2.11)

→0 is a normal vector through N . where − m

A function f (y) : ∂Ω → An (R) is said to be H¨older continuous on ∂Ω, if f (y) satisfies |f (y1 ) − f (y1 )| ≤ M1 |y1 − y2 |α , y1 , y2 ∈ ∂Ω (0 < α < 1).

(2.12)

9

General Regular and Harmonic Functions

α the set of all H¨ older continuous function on ∂Ω with the Denote by H∂Ω index α.

Now we consider the Cauchy type integral, Cauchy singular integral and the Plemelj formula. Definition 2.4

The integral Φ(x) =

1 ωn

or



∂Ω

E(x, y)n(y)f (y)dsy ,

(2.13)



1 E(x, y)dσy f (y) ωn ∂Ω is called the Cauchy type integral, where Ω and ∂Ω are as before, n(y) is the normal vector through y, dSy is the area difference E(x, y) = y−x 2π n/2 , and ωn = Γ(n/2) is the area of the unit sphere in Rn . When n |y − x| α , it is clear that the integral is well defined. x ∈ Rn \∂Ω, f (x) ∈ H∂Ω Φ(x) =

Definition 2.5 If x0 ∈ ∂Ω, construct a sphere E with the center at x0 and radius δ > 0, where ∂Ω is divided into two parts by E, and the part of ∂Ω lying in the interior of E is denoted by λδ . If lim Φδ = I, in δ→0

which



1 E(x0 , y)dσy f (y), (2.14) ωn ∂Ω−λδ then I is called the Cauchy principal value of singular integral and denoted by I = Φ(x0 ). Φδ (x0 ) =

Now we prove the Cauchy-Pompjeu integral formula. Theorem 2.5 Let Ω, ∂Ω be as stated above, and Ω = Ω ∪ ∂Ω. Then for each x ∈ Ω, f ∈ FΩr (r ≥ 1), we have 1 f (x) = ωn



1 ωn



and f (x) =

1 E(x, y)dσy f (y) − ωn ∂Ω

∂Ω

f (y)dσy E(x, y) −

1 ωn



E(x, y)(∂f )(y)dy,

(2.15)

(f ∂)(y)E(x, y)dy.

(2.16)

Ω



Ω

Proof We also use the notations as before and denote by x a point in Ω and the hypercomplex number x = x1 e1 + · · · + xn en . For n ≥ 3, set E(x, y) =

1 1 ∂(|y − x|2−n ) = (|y − x|2−n )∂, 2−n 2−n

10

Chapter I

and for n = 2, E(x, y) = ∂(ln|y − x|) = (ln|y − x|)∂. In both instances, ∂(E(x, y)) = (E(x, y))∂ = 0, y = x. So both |y − x|2−n (n ≥ 3) and ln |y − x| (n = 2) are harmonic. If Ωε = {y ∈ Ω : |y − x| > ǫ}, where x ∈ Ω is fixed and the ε-ball: |y − x| < ε about x lies completely within Ω, by the Stokes theorem one has 

E(x, y)(∂f )(y)dy =

Ωε



∂Ω

E(x, y)dσy f (y)−



|y−x|=ε

E(x, y)dσy f (y).

About the fixed point x, f (y) may be approximated as f (y) = f (x) + O(ε), in which lim O(ε) = 0. Using the Stokes theorem we have ε→0



E(x, y)dσy = ωn .

|y−x|=ε

Hence when ε → 0, we can obtain the Cauchy-Pompieu representation (2.15). A calculation analogous to the above verifies the representation (2.16). Corollary 2.6 Let Ω, ∂Ω be as stated above, and f be a left regular function defined on Ω. Then Φ(x0 ) =

⎧ ⎨ f (x0 ), ⎩ 0,

x0 ∈ Ω, x0 ∈ Rn \Ω,

(2.17)

here Φ(x0 ) is defined by Definition 2.5. Theorem 2.7 Suppose that Ω, ∂Ω are as stated before, and f ∈ α , x ∈ ∂Ω. Then H∂Ω 0

1 = ωn



1 ωn



∂Ω

E(x0 , y)dσy f (y)

1 E(x0 , y)dσy [f (y)−f (x0 )]+ f (x0 ), 2 ∂Ω

(2.18)

where the first integral is defined by Cauchy’s principal value, and the second integral is defined by a generalized integral.

11

General Regular and Harmonic Functions Proof Let λδ be as stated above. Then

=



∂Ω−λδ



∂Ω−λδ

E(x0 , y)dσy f (y)

E(x0 , y)dσy [f (y) − f (x0 )] +



∂Ω−λδ

E(x0 , y)dσy f (x0 ). (2.19)

α and (2.11), (2.12), we can get Because f ∈ H∂Ω

|E(x0 , y)dσy [f (y) − f (x0 )]| ≤  

≤  ≤|



M |dSy| |y − x0 |n−1−α

D(ξ1 , ..., ξn−1 )  M |dρ0 dϕ1 ...dϕn−2 |  D(ρ0 , ϕ1 , ..., ϕn−2 ) |y − x0 |n−1−α

1 M ′ 2ρn−2 dρ0 dϕ1 ...dϕn−2 | ≤ M 1−α dρ0 , |y − x0 |n−1−α 0 ρ0

in which the integral of the last function is convergent, so lim



δ→0 ∂Ω−λδ

E(x0 , y)dσy [f (y)−f (x0 )] =



∂Ω

E(x0 , y)dσy [f (y) − f (x0 )].

Calculating the second item, and setting Dout = {∂[(D(x0 , δ)) ∪ Ω]} ∩ (Rn − Ω), by the Cauchy formula (see Corollary 2.6), we have 

E(x0 , y)dσy f (x0 ) = ωn f (x0 ),

∂Ω−λδ +Dout

and



E(x0 , y)dσy f (x0 ) =

Dout

=



Dout



E(x0 , y)

Dout

y−x0 dSy f (x0 ) |y−x0 |

1 dSy f (x0 ), |y − x0 |n−1

and from the condition about ∂Ω, we obtain lim



δ→0 Dout

1 E(x0 , y)dσy f (x0 ) = ωn f (x0 ), 2

12

Chapter I

so lim



δ→0 ∂Ω−λδ

E(x0 , y)dσy f (x0 )



= lim

δ→0 ∂Ω−λδ +Dout

− lim



δ→0 Dout

E(x0 , y)dσy f (x0 )

1 E(x0 , y)dσy f (x0 ) = ωn f (x0 ). 2

Summarizing the above discussion and letting δ → 0 in (2.19), we can get the required result. Next we consider the limit of Cauchy’s integral when x → x0 , x0 ∈ ∂Ω from Ω+ = Ω and Ω− = Rn \Ω; we first give a Hile’s lemma. Lemma 2.8 Then

where

Suppose that t, x ∈ Rn , n(≥ 2) and m(≥ 0) are integers.    x t  Pm (x, t)   |x|m+2 − |t|m+2  ≤ |x|m+1 |t|m+1 |x − t|,

Pm (x, t) =

⎧ m
⎪ ⎪ |x|m−k |t|k , ⎨ ⎪ ⎪ ⎩

k=0

1,

(2.20)

m = 0, m = 0.

Proof The proof can be seen in [19]. We rewrite (2.13) in the form 

1 E(x, y)dσy f (y) ωn ∂Ω   1 1 E(x, y)dσy [f (y)−f (x0 )]+ E(x, y)dσy f (x0 ) = ωn ∂Ω ωn ∂Ω  1 E(x, y)dσy f (x0 ). = F (x) + ωn ∂Ω (2.21) From Corollary 2.6, we get Φ(x) =

1 ωn



∂Ω

E(x, y)dσy f (x0 ) =

⎧ ⎨ f (x0 ), ⎩ 0,

x ∈ Ω+ , x ∈ Ω− .

(2.22)

Now we need to consider the limit of (2.14), it suffices to study the first item in (2.21).

13

General Regular and Harmonic Functions

α , 0 0, there exists a ρ > 0 such that when δ < ρ, |I1 | < ε/2. Given a fixed δ such that 0 < δ < ρ, thus we can estimate I2 . By |E(x, y) − E(x0 , y)| ≤

n−1  |y − x0 |k k=1

1 |y − x|k |y − x0 |n



|x − x0 |,

we get |I2 | =

1 ωn



≤

1 ωn

L

∂Ω−λδ

δ

|E(x, y)−E(x0 , y)||dσy ||f (y)−f (x0 )|

(n − 1)M2k n−2 2ρ0 dρ0 M1 |x − x0 |. ρn0

Here L is a positive constant because of ∂Ω being bounded. Thus when |x − x0 | is small enough we can get |I2 | < ε/2. Hence F (x) converges to F (x0 ) on ∂Ω as x → x0 . Note that δ is independent of x0 , and ′ x , x0 ∈ ∂Ω, thus it is easy to see that F (x) is uniformly continuous on ∂Ω. Next consider x → x0 along the direction of the tangent plane of ∂Ω at x0 . When |x − x0 | is small enough, we can choose a point y(∈ ∂Ω) such that |x − y|, |y − x0 | are also small enough, and x lies in the direction of the tangent plane of ∂Ω at y. Taking account that F (x) on ∂Ω is uniformly continuous, we can derive that for any ε > 0, when |x − x0 |, |y − x0 | are small enough, the following inequalities hold: ε ε |F (y) − F (x0 )| ≤ , |F (x) − F (y)| ≤ , 2 2

General Regular and Harmonic Functions

15

thus |F (x) − F (x0 )| ≤ |F (x) − F (y)| + |F (y) − F (x0 )| ≤ ε. This completes the proof. α , Theorem 2.10 Let Ω, ∂Ω, Ω+ , Ω− be as stated above, f (x) ∈ H∂Ω + − 0 < α < 1, x0 ∈ ∂Ω, and denote by Φ (x0 ), Φ (x0 ) the limits of Φ(x), when x → x0 in Ω+ , Ω− respectively. Then

 ⎧ 1 ⎪ + ⎪ ⎪ ⎨ Φ (x0 ) =

or

ωn

1 E(x0 , y)dσy f (y) + f (x0 ), 2 ∂Ω

ωn

1 E(x0 , y)dσy f (y) − f (x0 ), 2 ∂Ω

 ⎪ 1 ⎪ − ⎪ ⎩ Φ (x0 ) =

⎧ + − ⎪ ⎨ Φ (x0 ) − Φ (x0 ) = f (x0 ),  2 ⎪ ⎩ Φ− (x0 ) + Φ− (x0 ) = E(x0 , y)dσy f (y).

ωn

(2.24)

(2.25)

∂Ω

Proof By using Cauchy’s integrals (2.13), (2.21), (2.22) and Theorem 2.5, when x → x0 in Ω+ , Ω− , Φ(x) possesses the limits Φ+ (x0 ) =

1 ωn

Φ− (x0 ) =

1 ωn



∂Ω

E(x0 , y)dσy [f (y) − f (x0 )] + f (x0 ),

∂Ω

E(x0 , y)dσy [f (y) − f (x0 )],



respectively. Form (2.21) it follows that (2.24) is valid. Similarly we can prove (2.25). The formula (2.25) is called the Plemelj formula for left regular functions, which is generally used to discuss boundary value problems.

3

Quasi-Permutations and Generalized Regular Functions in Real Clifford Analysis

Clifford numbers that we use in Clifford analysis do not have the commutative property, creates a lot of difficulties in handling some problems. In this section we first introduce the conception of quasi-permutation proposed by Sha Huang (see [29]4),5),6)), and then we give some equivalent conditions of regular functions and generalized regular functions by the above quasi-permutation.

16

Chapter I

3.1

The quasi-permutation and its properties

Definition 3.1 Let A = {h1 , h2 , ..., hk }, 1 ≤ h1 < h2 < ... < hk , here hi (i = 1, ..., k), and m be natural numbers. For the arrangement mA =

⎧ ⎨ A \ {m},

⎩ {g , g , ..., g 1 2 k+1 },

m ∈ A, m ∈ A,

where gi ∈ A ∪ m, 1 ≤ g1 < g2 < ... < gk+1 , we call mA the first class quasi-permutation for arrangement mA, and define mm = 1, m1 = m. If there are p natural numbers hi ∈ [2, m], then the sign δmA = (−1)p is called the sign of the first class quasi-permutation mA. Property 3.1 1. If mA = B, then mB = A. 2. mA = Am. 3. Let m = 1, then δmA = 1. 4. If mA = B and m = 1, then δmA = −δmB . 5. em eA = δmA emA . The proof is easily obtained by Definition 3.1. Definition 3.2 Let A be as stated in Definition 3.1 and B = {l1 , l2 , ..., lm }, 1 ≤ l1 < l2 < ... < lm ≤ n, lj (1 ≤ j < m) be natural numbers. We call the arrangement BA = (B ∪ A)\(B ∩ A) with the natural order as the second class quasi-permutation for arrangement BA, and m δBA =



j=1

δ lj A

as the sign of the second class quasi-permutation. And we let BB = 1, B1 = B Property 3.2.

1. If BA = C, then CA = B.

2. BA = AB. 3. Let A = {h1 , h2 , ..., hr } = {1}. Then δAA = (−1) 4. eB eA = δBA eBA .

r(r+1) 2

.

17

General Regular and Harmonic Functions 5. If a =


C

aω =

 

(

A

Proof.




aC eC , A = CB, ω =

B

ωB eB , then

aC δCB ωB )eA .

C

Noting that


aω =

C

aC eC





=

C B




=

B,C


B

ωB eB

aC ωB eC eB

aC ωB δCB eCB ,

we can write A = CB, and notice for every C, A and B monogamy, so aω =

 A

aC δCB ωB eA .

C

Definition 3.3 Let j be a natural number and D = {h1 , h2 , ..., hs }, 1 < h1 < h2 < ... < hs ≤ n, hp (1 ≤ p ≤ s) be natural numbers. The arrangement Dj is called the third class quasi-permutation of arrangement jD, and the sign εjD =

⎧ ⎨ (−1)s

for any p ∈ [1, s], hp = j,

⎩ (−1)s−1 for some p ∈ [1, s], h = j, p

ε1D = 1,

is called the sign of the third class quasi-permutation. Property 3.3

If j, D are as stated above, then ej eD = εjD eD ej .

Definition 3.4 Let the arrangement D = {h1 , ..., hr } and C be natural number arrangement with natural order as above. Then the arrangement CD is called the fourth class quasi-permutation of arrangement DC and the sign εDC =

r 

p=1

ε hp C

18

Chapter I

is called the sign of the fourth class quasi-permutation. Property 3.4

If C, D are as stated above, then eD eC = εDC eC eD .

Property 3.5 Let C = {h1 , ..., hr }, D = {k1 , ..., kp } as stated in Definition 3.1, where hi , kj are all not equal to 1 and q elements in C and D are the same. Then eD eC = (−1)rp−q eC eD , εDC = (−1)rp−q , and when r or p is an even number, then eD eC = (−1)q eC eD , εDC = (−1)q . Definition 3.5 Let C, D, E, M be the arrangement of natural numbers as in Definition 3.1. The arrangement CDEM is called the fifth class quasi-permutation of arrangement EDCM , and if CM = {h1 , ..., hr }, EM = {b1 , ..., bp }, νEDC = εED εEC εDC , then we call µEDC = (−1)

r(r+1)+p(p+1) 2

νEDC

the sign of the fifth class quasi-permutation. Property 3.6 Let E, D, M be the arrangement of natural numbers as in Definition 3.1, ED = {h1 , ..., hr } = A (hj = 1), M A = C, CD = {b1 , ..., bp } = B (bj = 1). Then 1. δED δCM = µEDC δCD δEM . 2. δCD δEM = µEDC δED δCM . 3. eE eD eC eM = νEDC (−1)

p(p+1) 2

δCD δEM .

Proof We calculate eE eD eC eM in two ways, and obtain eE eD eC eM = δED eED δCM eCM = δED δCM eED eCM = (−1)

r(r+1) 2

δED δCM ,

where ED = M EDM = CM , and then eE eD eC eM = νEDC eC eD eE eM = νEDC δCD δEM eCD eEM = νEDC (−1)

p(p+1) 2

δCD δEM ,

19

General Regular and Harmonic Functions

where CD = M EDD = EM . From the above equality, we can derive δED δCM = (−1)

r(r+1)+p(p+1) 2

νEDC δCD δEM

and the proof of 1 is finished. Similarly we can prove 2, 3. Property 3.7 Let D, B be the natural arrangement of natural numbers, j(= m) be natural numbers, and jD = A = {h1 , ..., hr } = mB, jB = mD = {b1 , ..., bp }. Then δmD δjB = −δjD δmB . Proof We prove it according to the following three cases: 1. m, j ∈ D. 2. m, j ∈ D. 3. D includes one of m and j. If m, j ∈ D, then m, j ∈ N \ B, and there exists a set E, such that D = E ∪ {j, m}, B = E \ {j, m}, hence em eD ej eB = δmD emD δjB ejB = δmD δjB (−1)

r(r+1) 2

,

moreover we have em eD ej eB = νmDj ej eD em eB = νmDj δjD δmB ×ejD emB = −δjD δmB (−1)

r(r+1) 2

.

From this, we get 1. Similarly we can prove 2, 3.

3.2

Regular functions and generalized regular functions

By the method using the sign of quasi-permutation, in the following we give equivalent conditions for regular functions and generalized regular functions.

20

Chapter I

Theorem 3.8

If f : Ω → An (R), then f (x) =

regular function if and only if fAx1 =

n 

m=2


A

fA (x)eA is a left

δmA fmAxm ,

where ( )xm = ∂( )/∂xm . Proof By the definition of quasi-permutations, we have ∂f (x) = =

n 

n   ∂f ∂fA ei ei eA = ∂xi ∂xi i=1 i=1 A





n ∂fA  ∂fA  ∂fA δiA eiA δiA eiA = + δ1A e1A , ∂x ∂x ∂xi i 1 i=2 i,A A



when 2 ≤ i ≤ n; if iA = B, then A = iB, δiA = −δiB . So ∂f (x) =



δ1A e1A

A







n  ∂f ∂fA − δiB eB iB . ∂x1 ∂xi B i=2

Again because iA = B, iB = A (2 ≤ i ≤ n), we can verify that the elements of the sets {A} and {B} possess the same quantity and same form. From this formula it follows that ∂f (x) =

 A





n ∂f ∂fA  δiA eA iA . − eA ∂x1 i=2 ∂xi

Finally by the definition of ∂f (x), the proof is completed. Especially, when n = 3, the equivalent condition is the system of equations (3.0), Chapter VII. Theorem 3.9

Let f : Ω → A. Then f (x) =

regular function if and only if n 

m=1

δmB


A

fA (x)eA is a left

∂fB = 0, ∂xm

where mA = B. Proof By Theorem 3.8 and δiA = −δiB (i = 1, iA = B, iB = A), the theorem is easily proved.

21

General Regular and Harmonic Functions Let w =


A

(−1)

r(r+1) 2

(−1)

r(r+1) 2

wA eA . Then w =

eA (see Definition 1.1).




wA eA , if A = {h1 , ..., hr }, eA =

A

When r = 0, 3 (mod 4), i.e.

= 1, we denote A by A and the A is called the A-type index. r(r+1)

When r = 1, 2 (mod 4), i.e. (−1) 2 = −1, we denote A by A and the A is called the A-type index. Hence we can write 

ω=

A

ωA eA −



ωA eA

A

Definition 3.6 Let Ω ⊂ Rn be a bounded domain, and w(x) = (r) wA (x)eA ∈ FΩ (r ≥ 1). If w(x) satisfies the system of first order


A

equations

∂w = aw + bw + l, x ∈ Ω,

(3.1)

then w(x) is called a generalized regular function in Ω, where a(x) =



aC (x)eC , b(x) =

C



bC (x)eC , l(x) =

C



lA (x)eA .

A

By Theorems 3.8 and 3.9, we know 

∂w =

n 

δmB wBxm eA ,

mB=A m=1

and by Property 3.2, at the same time letting w =


M

we have


aw+bw+l = (

C




+(

C

=




C,A

−

+




aC eC )(

M

wM eM +




bC eC )(

M

C,A




C,A

M

wM eM −

aC wM δCM eA +










C,A

bC wM δCM eA +

A


M

wM eM ,

wM eM )


M

wM eM ) +

aC wM δCM eA +




wM eM +

lA eA =

(aC − bC )wM δCM eA +

C,A


A





A

lA eA




C,A

bC wM δCM eA

(aC +bC )wM δCM eA

lA eA ,

22

Chapter I

where CM = A, CM = A. Thus (3.1) becomes n 

δmB wBxm eA =

A m=1

+



(aC + bC )wM δCM eA

C,A




C,A

(aC − bC )wM δCM eA +




lA eA ,

A

in which mB = A, CM = A, CM = A. From the above discussion, we have the following theorem. That is to say for any A, the following formula (3.2) is true. Theorem 3.10 Let Ω ⊂ Rn be a bounded domain, and w(x) =
(r) wM (x)eM ∈ FΩ (r ≥ 1). Then w(x) is a generalized regular funcM

tion in Ω if and only if w(x) satisfies the elliptic system of first order equations n 

δmB wBxm =

m=1 mB=A

+



(aC + bC )wM δCM

C CM =A



C CM =A

(3.2)

(aC − bC )wM δCM + lA .

In addition, we can also derive the relation between a generalized regular function and an elliptic system of second order equations. Theorem 3.11 Let Ω ⊂ Rn be a bounded domain, and w(x) = (r) wM (x)eM ∈ FΩ (r ≥ 2). Then w(x) is a generalized regular function


M

in Ω if and only if the components of w(x) satisfy the elliptic system of second order equations △wD = +




j,C



δjD δCM (aC + bC )xj wM

j,C

δjD δCM (aC + bC )wM xj +

×δCM (aC − bC )xj wM + ×(aC − bC )wM xj +

n


j=1




j,C




j,C

δjD

δjD δCM

δjD lAxj ,

where j = 1, ..., n, jD = A, CM = A, CM = A.

(3.3)

23

General Regular and Harmonic Functions Proof

Let w(x) =


M

wM (x)eM be a generalized regular function.

From Theorem 3.10, it is clear that wM (x) satisfies (3.2), and for a fixed index D, we consider the term wD in the left-hand side of equation (3.2), i.e. the equation corresponding to eA satisfying that jD = A. Suppose that the term corresponding to wD is δjD wDkj , and this equation is said to be the equation for wD . Then we find partial derivatives with respect to xj for the equation including WD , and then multiply this equation by δjD . Finally sum these derivatives according to the index j, we get ⎛

n 

δjD ⎝

j=1

=

n 

n 

j=1

(δjD )2 ωDx2 + j

j=1

⎞

δmB ωBxxm ⎠xj 

(3.4)

δjD δmB ωBxxm xj .

j,m,j=m

Noting that mB = A, A = jD (j = 1, ..., n) in (3.4) only include a part of all indexes(real subset) of sets A, and that from Corollary 3.7 we have δjD δmB wBxm xj = −δmD δjB wBxj xm , then 

j,m,j=m

δjD δmB wBxm xj = 0 (jD = A, mB = A, j = m);

moreover we can derive n 

δjD

n 

δmB wBxm

m=1

j=1



xj

= △wD .

(3.5)

In fact we consider the right side of equation for w(x) in (3.2). By using the same method (i.e. we find the partial derivatives with respect to xj and multiply by δjD , and then sum them by j) we can get the expression of wD in the right side of equation (3.4): n 

j=1

+





(aC + bC )wM δCM

C

C

=



δj D [

(aC − bC )wM δCM + lA ]xj

δj D δCM [(aC + bC )wM ]xj

j, C

+

 j,C

δjD δCM [(aC − bC )wM ]xj

24

Chapter I

+

n 

δjD lAxj =

j=1

+





δjD δCM (aC + bC )xj wM

j,C

δjD δCM (aC + bC )wM xj

j,C

+

δjD δCM (aC − bC )xj wM



δjD δCM (aC − bC )wM xj +

j, C

+

(3.6)



j, C

n 

δjD lAxj .

j=1

Here CM = A, CM = A, jD = A. From equations (3.4), (3.5), (3.6), it follows that the formula (3.3) is true.

4

The Chain Rule and Differentiation Rules of new Hypercomplex Differential functions in Clifford Analysis

There are three ways to discuss general holomorphic functions in complex analysis: one is Cauchy’s method that is based on the differentiability; another one is the Weierstrass method that is based on power series; the third one is the Riemann method that is based on the Cauchy-Riemann equations. In the case of a quaternion, F. Sommen [72]1) uses a special differential form that avoide the differentiability in Cauchy’s method. Conversely in 1990, H. Malonek [48] gave an imitation of Cauchy’s classical method by using the new hypercomplex structure, first giving a relation between new hypercomplex differentiable functions and monogenic functions, a hypercomplex structure that possesses an obvious advantage. In this section, we establish a new chain rule and new differential rule in Clifford analysis from the view of the new hypercomplex structure of Malonek. 4.1.

Hypercomplex Differential

Let λ=

m 

k=1

λk ek , (m ≤ n, λk are real numbers)

25

General Regular and Harmonic Functions

be a hypercomplex number and the set of all λ make up a space Am (R). Then Rm ∼ = Am (R), x = (x1 , ..., xm ) ∈ Rm , and zk = xk − x1 ek , k = − → → 2, ..., m, z = (z2 , ..., zm ). Denote by H m−1 the set of all − z , and dem−1 note the topological product of m − 1 spaces An (R) by An (R), then → → H m−1 ⊂ Am−1 (R). For any − z ∈ H m−1 define the norm of − z by n 1 → → → − z  = (− z ,− z )2 = (

m 

1

z k zk ) 2 = (mx21 +

k=2

m 

1

x2k ) 2 ,

k=2


− → − → → → where the inner product (− z , ξ ) = ( ξ ,− z)= m k=2 z k ξk .

(R), Anp−1 (R)) the set of linear Definition 4.1 Denote by L(Am−1 n − → m−1 p−1 mappings from An (R) to An (R). A bounded mappings l ∈ (R), Anp−1 (R)) is said to be A linear (or linear for short), if L(Am−1 n − → − → u , v ∈ Am−1 (R), λ, µ ∈ An (R); then we have n − → − − →→ − →→ → l (λ→ u + µ− v ) = λ l (− u ) + µ l (− v ). Lemma 4.1 (see [48]) Every A linear mapping l ∈ L(H m−1 , An (R)) → has a unique expression l(− z ) = z2 A2 + · · · + zm Am , where Ak ∈ An (R), k = 2, ..., m. Proof have and then

Let l ∈ L(H m−1 , An (R)) be an A linear mapping. Then we l(0, 0, ..., 1, ..., 0) = Ak ∈ An (R), k = 2, ..., m,

′

→ l(− z ) = z2 A2 + · · · + zm Am .

If there are other Ak , k = 2, ..., m, such that

→ l(− z ) = z2 A2 + · · · + zm Am , ′

then

′

→ → → → 0 = l(− z −− z ) = l(− z ) − l(− z) ′

′

= z2 (A2 − A2 ) + · · · + zm (Am − Am ), ′

and we get Ak = Ak , k = 2, ..., m.

→ Definition 4.2 (see [48]) Let − a ∈ H m−1 and f be a continuous → → mapping from some domain of − a to An (R). The function f = f (− z) is called the left hypercomplex differential, for short hypercomplex differential, if there exists an A linear mapping l ∈ L(H m−1 , An (R)) such that → → → → |f (− a + △− z ) − f (− a ) − l(△− z )| = 0, lim − → − → △ z  △ z →0

26

Chapter I

→ and l = l(− z ) is called the left hypercomplex differential for f (for short → → → → a ) or l(− z ) = f ′ (− a )(− z ). hypercomplex differential), written by l = f ′ (− → Lemma 4.2 (see [72]1)) If f (− z ) is a hypercomplex differential funcm−1 → to An (R), then the differential f ′ (− a ) is unique. tion from H 4.2

A-chain rule of hypercomplex differentiation

→ → a ), and f ′ (− a ) is given, Theorem 4.3 If f : H m−1 → An (R), b = f (− → then for any ε > 0, there exists a number δ > 0 such that when − z − − → − → − → a  < δ, the inequality |f ( z ) − f ( a )| < ε is valid, and we say that f is → continuous at − a. Proof

From the conditions, we can get lim → z →0 △−

→ → → → → |f (− a + △− z ) − f (− a ) − f ′ (− a )(△− z )| = 0, − → △ z 

and then → → → → → a + △− z ) − f (− a ) − f ′ (− a )(△− z )| = 0. lim |f (−

→ z →0 △−

→ → a )(∆− z ) = △z2 A2 + On the basis of Lemma 4.1, it is easy to see that f ′ (− · · · + △zm Am , where Ak (k = 2, ..., m) are fixed Clifford numbers that → → → are independent with ∆− z , hence lim f ′ (− a )(∆− z ) = 0 and → z →0 ∆− → → → a + △− z ) − f (− a )| lim |f (−

→ z →0 △− =

→ → → → → → → a + △− z ) − f (− a ) − f ′ (− a )(△− z ) + f ′ (− a )(△− z )| = 0. lim |f (−

→ z →0 △−

− → (R), Anp−1 (R)). Then there exists an Theorem 4.4 Let l ∈ L(Am−1 n M > 0 such that

→ where − u=



− →→ → →  l (− u ) ≤ M − u , − u ∈ Am−1 (R), n m


i=2

→ |ui |2 , − u = (u2 , ..., um ), ui ∈ An (R), i = 2, ..., m.

− → → Definition 4.3 Let − a ∈ Anp−1 (R) and f be a continuous mapping − → − →→ → from some domain of − a to Am−1 (R). Then f = f (− z ) is said to n

General Regular and Harmonic Functions

27

→ be A differential at the point − a ; if there exists an A linear mapping p−1 m−1 l ∈ L(An (R), An (R)) such that − →→ − →→ − → → →  f (− a + △− z ) − f (− a ) − l (△− z ) = 0, lim − → → △ z  z →0 △− − → − →→ − →→ then l = l (− z ) is called the left differential, written by l (− z) = −−→ − − → − → ′ − ′ − → → (f )( a ) or l = (f )( a ). − →→ − → Theorem 4.5 Let f : Anp−1 (R) → Am−1 (R) and f ′ (− a ) be given. n − →− → − → Then f ( z ) is continuous at a . − → (R) is A differenLemma 4.6 (see [72]1)) If f : Anp−1 (R) → Am−1 n − →′ − → tiable, then f ( a ) is unique. − → → (R) be A differentiable at − a Theorem 4.7 Let f : Anp−1 (R) → Am−1 n − →− m−1 → and g : An (R) → An (R) be An (R) differentiable at f ( a ). Then the − → → composite function g ◦ f : Anp−1 (R) → An (R) is differentiable at − a , and −−→ → − →→ − → → a ) = g ′ ( f (− a )) ◦ ((f ′ )(− a )). (g ◦ f )′ (− − →→ − → − →→ − − →→ − → Proof Suppose that b = f (− a ), → w = f (− z ), λ = f ′ (− a ), µ = − →− ′ → g ( f ( a )), and let − → → → → → → △− z =− z −− a , △− w =− w − b, − →→ − →→ − →→ − − → → ϕ (− z ) = f (− z ) − f (− a ) − λ (− z −→ a ),

− → − → → → → ψ(− w ) = g(− w ) − g( b ) − µ(− w − b ),

− →→ − →→ − →→ − → ρ(− z ) = g ◦ f (− z ) − g ◦ f (− a ) − µ ◦ λ (− z −→ a ), where

− →→ − →→ − − →→ µ ◦ λ (− z −→ a ) = g ′ ( f (− a )) ◦ ( f ′ (− a )).

Then we have

In order to prove

→ → − ϕ (− z ) lim − → − → = 0, → → z −− a )→0  z − a  (− → |ψ(− w )| lim − → = 0. − → − → → w − b )→0  w − b  (− → |ρ(− z )| lim − → − → = 0, − → − → ( z − a )→0  z − a 

(4.1) (4.2)

(4.3)

28

Chapter I

we consider − →→ − →→ − →→ − →→ → → → ρ(− z ) = g ◦ f (− z ) − g ◦ f (− a ) − µ ◦ [ f (− z ) − f (− a)−− ϕ (− z )]

− →→ − →→ − →→ − →→ → → = g ◦ f (− z )−g ◦ f (− a )−µ ◦ [ f (− z ) − f (− a )]+µ ◦ [− ϕ (− z )]

− →→ → → = ψ ◦ f (− z )+µ(− ϕ (− z )).

(4.4)

It is sufficient to prove − →→ |ψ ◦ f (− z )| = 0, lim − → − → → → z −− a )→0  z − a  (−

(4.5)

→ → |µ(− ϕ (− z ))| lim − → − → = 0. − → − → ( z − a )→0  z − a 

(4.6)

For any fixed number ε > 0, from (4.1) there exists a δ > 0 such that when − → − →→ − →→ → − w − b  =  f (− z ) − f (− a ) < δ, the following inequality holds: − →→ − →→ − →→ |ψ ◦ f (− z )| ≤ ε f (− z ) − f (− a ). Moreover from Theorem 4.3, we see that for the δ as stated above there → → exists a δ1 > 0, such that when − z −− a  < δ1 , we have − →→ − →→  f (− z ) − f (− a ) < δ. Again by Theorem 4.2, there exists M > 0, such that − →→ − → →  λ (− z −→ a ) ≤ M − z −− a . Hence − →→ − →→ − →→ − →→ − → → |ψ ◦ f (− z )| ≤ ε f − z ) − f (− a ) ≤ ε− ϕ (− z ) + ε λ (− z −→ a ) → → → → ≤ ε− ϕ (− z ) + M ε− z −− a .

(4.7) In addition from (4.1), (4.7), it follows that (4.5) holds. Again from Theorem 4.2, we get → → → → µ(− ϕ (− z ))| M − ϕ (− z ) ≤ . − → − → − → − → z − a z − a Noting (4.1) and (4.4), we see that (4.6) and (4.3) are true, i.e. − →→ − →→ − − →→ − → → a ) = µ ◦ λ (− z −→ a ) = g ′ ( f (− a )) ◦ ( f ′ (− a )). (g ◦ f )′ (−

General Regular and Harmonic Functions

29

4.3. Differentiation rules First of all, we give a notation for a project function. A function − → − →→ → → z ), ..., fm (− z )) : H p−1 → Am−1 (R) can uniquely f (= f (− z ) = (f2 (− n → determine m − 1 component functions fi (− z ) : H p−1 → An (R), i = → → → 2, ..., m. If − π : Am−1 (R) → Am−1 (R) is an identical mapping − π (− u) = n n m−1 − → − → u = (u2 , ..., um ), then πi : An (R) → An (R), πi ( u ) = ui , i = 2, ..., m − → is called the i-th project function. From this we can get fi = πi ◦ f , i = 2, ..., m. (R) → An (R) is an A linear mapping, then Theorem 4.8 If f : Am−1 n → → → → → f ′ (− a )(− z ) = f (− z ), for short f ′ (− a ) = f (− z ). − → − → (R). Then f is Theorem 4.9 Let f = (f2 , ..., fm ) : H p−1 → Am−1 n → A differentiable at − a ∈ H p−1 , if and only if every fi (i = 2, ..., m) is → hypercomplex differentiable at − a , and − →′ − ′ − → f (→ a ) = (f2′ (− a ), ..., fm (→ a )).

Proof Suppose that every fi (i = 2, ..., m) is hypercomplex differen− →→ − →→ ′ (− → → → → → tiable at − a . If λ (− z ) = (f2′ (− a ), ..., fm a ))(− z ), then f (− a + △− z)− − →− − → ′ → − → − → − → − → − → − → → f ( a ) − λ (△ z ) = (f2 ( a + △ z ) − f2 ( a ) − f2 ( a )(△ z ), ..., fm (− a + ′ → → → → △− z ) − fm (− a ) − fm (− a )(△− z )), and then − →− →− − → → → − → − a )− λ (△− z ) (→ lim  f ( a +△ z )− f− → △ z  → z →0 △− → − → → − − → ′− → m f (−
i a +△ z )−fi ( a )−fi ( a )(△ z ) , ≤ lim − → △ z  → z →0 i=2 △− − → → namely f is A differentiable at − a , and the equality in this theorem is − → → true. Conversely if f is differentiable at − a ∈ H m−1 , then from Theo− → rems 4.4 and 4.5, we see that fi = πi ◦ f is hypercomplex differentiable → at − a i. Theorem 4.10 = S(u, v).

If S : A2n (R) → An (R), S(u, v) = u + v, then S ′ (a, b)

Theorem 4.11 = ub + av.

Let q : A2n (R)→An (R), q(u, v) = uv. Then q ′ (a, b)(u, v)

30

Chapter I

Proof

Suppose λ(u, v) = ub + av. Then lim

(△u,△v)→0

=

q(a + △u, b + △v) − q(a, b) − λ(△u, △v) (△u, △v)

B△u△v △u△v  ≤ lim (△u,△v)→0 (△u,△v)→0 (△u, △v) (△u)2 + (△v)2 lim

≤

B △u2 + △v2  = 0, (△u,△v)→0 2 (△u)2 + (△v)2 lim

where B > 0 is a constant, hence q ′ (a, b)(u, v) = ub + av. Theorem 4.12 Let f, g : H m−1 → An (R) be hypercomplex differen→ tiable at − a . Then → → → a ) = (f )′ (− a ) + (g)′ (− a ). 1. (f + g)′ (− → → → → → a ) = (f )′ (− a )g(− a ) + f (− a )g ′ (− a ). 2. (f g)′ (− Proof On the basis of Theorem 4.4, Theorem4.6 and Theorem4.7, we can get 1. Moreover from Theorems 4.4, Lemma 4.6 and Theorem 4.8, we can derive 2.

5

Regular and Harmonic Functions in Complex Clifford Analysis

Let e := {e1 , ..., en } be the orthogonal basis of Rn . According to the structure of real Clifford analysis, we use the complex numbers to construct the complex Clifford algebra A(C); its elements possess the form  Z= cA eA , (5.1) A

where cA are the complex numbers and eA is as above. The norm of the element is defined as  |cA |2 . (5.2) |Z| = A

Complex Clifford analysis studies the functions defined on Cn and taking value on a complex Clifford algebra. About Complex Clifford analysis, F. Sommen and J. Ryan have obtained some results. In 1982,

General Regular and Harmonic Functions

31

F. Sommen discussed the relation between functions of several variables and monogenic functions by using the Fourier-Borel transformation (see [72]2)). In 1982, J. Ryan gave a simulation of Cauchy’s theorem, proved the invariant property of the complex left regular function under the action of a Lie group and structured the holomorphic function by complex harmonic functions (see [68]1)). In 1983, J. Ryan also gave a general Cauchy integral formula for a kind of special holomorphic function and used this formula to prove that all these special holomorphic functions form a Fr´echet model (see [68]2).

5.1

Complex regular functions

Now we give the concept of a complex regular function by using the Dirac operator. Definition 5.1. Let U ⊂ Cn be a domain and f : U → An (C) be a holomorphic function. If for every point z ∈ U , we have ∂f =

n 

j=1

ej

∂f (z) = 0, ∂zj

then f (z) is called a complex left regular function in U . If for every point z ∈ U , we have

f∂ =

n  ∂f (z)

j=1

∂zj

ej = 0,

then f (z) is called a complex right regular function in U . In the following, we mainly discuss the complex left regular function, and the complex right regular function can be analogously discussed. Let A = {j1 , ..., jr }, Ak = {j1 , ...jk−1 , jk+1 , ..., jr }, 2 ≤ j1 < j2 < ... < jr ≤ n,

KA = Kj1 ...jr , AK = j1 ...jr K, and denote the sign of permutation (l, j1 , ..., jr ) by sgn(l, A). And by a

32

Chapter I

direct calculation, we get ejk ek

∂fA

k

∂zjk

eA = k

k

∂zjk

(−1)k−1 eA (k = 1, 2, ..., r),

∂fkA ∂fkA 2 ∂fkA ekA = ek eA = − eA ∂zk ∂zk ∂zk

= −sgn(k, A) ek

∂fA

∂fkA eA (k < j1 , k = 1, 2, ..., j1 − 1), ∂zk

∂fAk ∂fAk 2 ∂fAk eAk = (−1)r ek eA = (−1)r+1 eA ∂zk ∂zk ∂zk

Ak = −sgn(k, A) ∂f ∂zk eA (jr < k, k = jr + 1, jr + 2, ..., n),

ek

∂f{k}∪A ∂fj1 ,...,k,...,jr ej1 ,...,k,...,jr = −sgn(k, A) eA , ∂zk ∂zk

/ A), (j1 < k < jr , k ∈ and then n 

n ∂  ∂f (z)  ej = fA eA ∂zj ∂zj A j=1

ej

j=1

=

⎧ r  ⎨ A

⎩

k−1

(−1)



j1 |x|, we can find |y|2 − |x|2 1 1 x. =y + n n−2 |y − x| (y − x)|y − x| (y − x)|y − x|n−2 Putting this into Un (x), we get Un (x) = +

1 ωp r

1 ωp r





1 dSy (y − x)|y − x|n−2

u(y)y

∂Br

u(y)

∂Br

x dSy . (y − x)|y − x|n−2

Moreover we have *

∂ ∂x

+α

Un (x) = 1 + ωp r

1 ωp r





u(y)y

∂Br

*

∂ u(y) ∂x ∂Br

+α

*

∂ ∂x

+α

1 dSy (y − x)|y − x|n−2

x dSy . (y − x)|y − x|n−2

Therefore, *

+

∂ ∂x

+α

p 

1 Un (o) = ωp r

j=1,α≥λj

1 ωp r



∂Br



*

∂ u(y)y ∂x ∂Br

u(y)αj

*

∂ ∂x

+α−λj

+α

(−1)(|α|) dSy y|y|n−2

(−1)|α|−1 ej dSy , y|y|n−2

134

Chapter V

where λj = (δj1 , ..., δjp ) ∈ N0p , j = 1, ..., p. On the basis of Corollary 1.4, Chapter 1 and Theorem 1.3, we can find

=

 * +   ∂ α  Un (o)  ∂x

1 ωp r



+

p 

∂Br

|u(y)| αj

j=1,λj ≥1

4

|(−n + 1)(|α|) | r γ |α|+n−1 ⎫

1)(|α|−1) ⎬

|(−n + γ |α|+n−2

⎭

dSy 6

≤

|(−n+1)(|α|) |+|α||(−n+1)(|α|−1) | Nr ω1p γ |α|+n−p

≤

|(−n + 1)(|α|) | + |α||(−n + 1)(|α|−1) | Nr . γ |α|+n−p

1 ∂Br rp−1 dSy

Theorem 1.7 is an improvement of the corresponding results in [36], [72]3). Corollary 1.8 Let p − 1, q − 1 ∈ N with p ≤ q, α ∈ N0p , and u ∈ C 1 (∂Br , Aq (C)) be harmonic in a neighborhood of B r ⊂ Rp . Then *  +  ∂ α  |(−p + 1)(|α|) + |α|(−p + 1)(|α|−1)   ≤ Nr . U (0) n  ∂x  |γ||α|

Proof By Poisson’s integral formula of harmonic functions (see [36]) U (x) =

r2 − |x|2 ωp r



∂Br

u(y) dSy |y − x|p

and Theorem 1.7, we immediately get the result in Corollary 1.8.

2

Three Kinds of Poincar´ e-Bertrand Transformation Formulas of Singular Integrals with a Cauchy’s Kernel in Real Clifford Analysis

Tongde Zhong and Sheng Gong have studied the Poincar´e-Bertrand (P-B) transformation formulas and singular integrals in several complex variables (see [89], [88]2), [88]3), [88]4), [20]). In 1992, M. R. Kandmamov gave a new proof of P-B transformation formulas in several complex variables functions (see [35]). Under the illumination of [88]2), [35],

135

Singular Integrals and Equations Dependent on Parameters

on the basis of the results of Section 1, Chapter 3, we shall prove three kinds of P-B transformation formulas of singular integrals with Cauchy’s kernel in real Clifford analysis. 2.1

Preparation

Let Ω1 ⊂ Rm , Ω2 ⊂ Rk be differentiable, oriented, bounded, compact manifolds, whose dimensions are m, k respectively, and ∂Ωi be the boundary of Ωi , i = 1, 2. ∂Ωi (i = 1, 2) are oriented Liapunov surfaces, and let the orientation be coordinated with that of Ωi . Let ∂Ω = ∂Ω1 × ∂Ω2 . Denote by ∂Ωη the variable η = (η1 , η2 ) on ∂Ω, and set f (η 1 , η 2 ; ξ 1 , ξ 2 ) ∈ H(∂Ω, β), 0 < β < 1 (see Section 1, Chapter 3 ). For convenience, we give the signs

A=

η1 − ζ 1 η2 − ζ 2 , , B = ωm |η 1 − ζ 1 |m ωk |η 2 − ζ 2 |k

C=

ξ1 − η1 ξ2 − η2 D = , , ωm |ξ 1 − η 1 |m ωk |ξ 2 − η 2 |k E = f (η 1 , η 2 ; ξ 1 , ξ 2 ).

In Section 1, Chapter 3, we have defined the Cauchy principal value of singular integrals with the Cauchy’s kernel on the character manifold, which is called the one time singular integral. Similarly to several dimensional singular integrals (see [88]3)), we must give a precise definition of the two times singular integral as follows. Definition 2.1 

∂Ωη

=



Adση1 dση2 B

∂Ωη

=

Adση1



∂Ω1η1

=



The two times singular integral is defined as

∂Ω1η1



Adση1 Adση1



∂Ωξ

∂Ωξ

¯ ξ2 dσξ1 C E Ddσ



∂Ω2η2

4



¯ Ddσ ¯ ξ2 dση2 B dσξ1 C¯ E

∂Ω2η2

dση2 B 



∂Ω1ξ1

∂Ωξ

¯ Ddσ ¯ ξ2 dσξ1 C¯ E

dσξ1 C(



∂Ω2ξ2

 

5

¯ Ddσ ¯ ξ2 ) dση2 B . E

(2.1) Similarly to several dimensional singular integrals (see [88]3)), we also give the definition of a two times singular integral which has to be

136

Chapter V

changed in integral order as follows 

∂Ωξ

=

dσξ1 dσξ2



∂Ωξ

=





∂Ω1ξ1

∂Ωη



dσξ1 [

∂Ω1ξ1

=



∂Ωη

¯ Ddσ ¯ η2 B Adση1 C¯ E

¯ Ddσ ¯ η2 B]dσξ2 Adση1 C¯ E



dσξ1 [

∂Ω2ξ2

dσξ1 {



∂Ω2ξ2

dσξ2 



(2.2)

∂Ωη

[

∂Ω1η1

6

¯ Ddσ ¯ η2 B] Adση1 C¯ E

Adση1 C(



∂Ωiξi (∂Ωiηi ) expresses the integral i ξ (η i ), and the integrals are all in

where

∂Ω2η2

¯ Ddσ ¯ η2 B)]dσξ2 }, E

over ∂Ωi (i = 1, 2) for the vari-

able the sense of Cauchy principal values (see Chapter I). In general, (2.1), (2.2) are not the same. For discussing their relation, we give the Poincar´e-Bertrand transformation formulas. From Chapters I and III, it is clear that the following results are valid: ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨

f gβ ≤ J1 f β gβ , (J1 a positive constant, f, g ∈ H(∂Ω, β)) 

∂Ω1η1

Adση1 =



∂Ω2η2

1 dση2 B = , ζ ∈ ∂Ω; 2

∂ η1 A = A ∂ η1 = ∂ η2 B = B ∂ η2 = 0, ζ i = η i , i = 1.2;

⎪ ⎪ (η 1 − ζ 1 )∂ η1 = m, (ξ 1 − η 1 )∂ η1 = −m, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ (η 2 − ζ 2 )∂ η2 = k, (ξ 2 − η 2 )∂ η2 = −k; ⎪ ⎪ ⎪ ⎪ ⎪   ⎪ ⎪ ω n εm ω k εk ⎪ 1 2 ⎪ ⎪ , , dη = dη = ⎪ ⎪ m k ⎪ B(ζ 1 ,ε) B(ξ 2 ,ε) ⎪ ⎪ ⎪ ⎪ ⎩ 1 1 1 2 2

here B(ζ , ε) : |η − ζ | ≤ ε, B(ξ , ε) : |η − ξ 2 | ≤ ε.

Theorem 2.1

The following equalities hold:

(2.3)

137

Singular Integrals and Equations Dependent on Parameters 1

1

h1η1 (ζ , ξ ) = 2

2

h2η2 (ξ , ζ ) = 



∂Ω1η1



∂Ω2η2

A Cdση1 = 0,

∂Ω1η1

Adση1 C = 0, Ddση2 B = 0, 

∂Ω2η2

dση2 D B = 0,

where ζ 1 , ξ 1 ∈ ∂Ω1 , ζ 1 = ξ 1 , ζ 2 , ξ 2 ∈ ∂Ω2 , ζ 2 = ξ 2 . Proof By means of the Cauchy principal value, we have 1

1

h1η1 (ζ , ξ ) = lim



ε→0 ∂Ω 1 \(B(ζ 1 ,ε) 1η

)

Adση1 C = B0 .

B(ξ 1 ,ε))

Here ∂ is the boundary, and the orientation of Ω1 , B(ζ 1 , ε) , B(ξ 1 , ε) ) Ω1 \(B(ζ 1 , ε) B(ξ 1 , ε)) on the communal part is harmonious, and the ) orientations of ∂Ω1 , ∂[Ω1 \(B(ζ 1 , ε) B(ξ 1 , ε))] , ∂B(ζ 1 , ε) , ∂B(ξ 1 , ε) ) are the induced orientations of Ω1 , Ω1 \(B(ζ 1 , ε) B(ξ 1 , ε)) , B(ζ 1 , ε) , B(ξ 1 , ε) respectively. Similarly to [36], we rewrite B0 as a sum of the following three terms. B0 = lim



ε→0 ∂[Ω1 \(B(ζ 1 ,ε)

+lim



ε→0 Ω1

(

Adση1 C + lim



ε→0 Ω1

∂B(ζ 1 ,ε)

)

(

Adση1 C

B(ξ 1 ,ε))]

Adση1 C = B1 +B2 +B3 .

∂B(ξ 1 ,ε)

By using the Stokes theorem (see Section 2, Chapter 1 or Theorem 9.2 in [6]) and (2.3), the limit B1 = lim



ε→0 Ω1 \(B(ζ 1 ,ε)

)

B(ξ 1 ,ε))

[A(∂ η1 C) + (A ∂ η1 )C]dση1 = 0

(2.4)

is derived. Noting that |η 1 − ζ 1 | = ε on ∂B(ζ 1 , ε) and using the Stokes theorem and formula (2.3), we get 1 B2 = lim 2 ε→0



1 1 Adση1 C = lim m 2 ε→0 ε ωm ∂B(ζ 1 ,ε)

=

1 1 lim m 2 ε→0 ε ωm

=

1 1 lim m 2 ε→0 ε ωm



B(ζ 1 ,ε)



B(ζ 1 ,ε)



∂B(ζ 1 ,ε)

(η 1 − ζ 1 )dση1 C

[(η 1 − ζ 1 )(∂ η1 C) + ((η 1 − ζ 1 )∂ η1 )C]dση1 mCdση1 .

138

Chapter V

When ε → 0, η 1 → ζ 1 , we can write C = H + OA (ε), here ξ1 − ζ 1 . ωm |ξ 1 − ζ 1 |m

lim OA (ε) = 0, H =

ε→0

Thus B2 =

1 1 lim m (m)H 2 ε→0 ε ωm



B(ζ 1 ,ε)

dη 1 =

mHεm ωm H 1 lim m = . 2 ε→0 ε ωm m 2

(2.5)

Similarly when ε → 0, (ζ 1 − η 1 )∂ η2 = −m, we know −H . 2

B3 =

(2.6)

Combining (2.4), (2.5), (2.6), we get h1η1 (ζ 1 , ξ 1 ) = 0. Moreover we can prove  Ddση2 B = 0, h2η2 (ξ 2 , ζ 2 ) = ∂Ω2η2

and



∂Ω1η1

1

∂Ω1η1

∂Ω2η2

dση2 D B = 0.

Let ϕ(ξ 1 , ξ 2 ) ∈ H(∂Ω, β), 0 < β < 1. Then

Theorem 2.2 



A Cdση1 = 0,

2

ϕ(ξ , ξ )Adση1 C =



∂Ω2η2

Ddση2 Bϕ(ξ 1 , ξ 2 ) = 0,

where ζ 1 , ξ 1 ∈ ∂Ω1 , ζ 1 = ξ 1 , ζ 2 , ξ 2 ∈ ∂Ω2 , ζ 2 = ξ 2 . Proof Similarly to the proof of Theorem 2.1, it suffices to notice that ∂ ηi ϕ(ξ 1 , ξ 2 ) = ϕ(ξ 1 , ξ 2 )∂ ηi = 0, i = 1, 2. Theorem 2.3 lim

The limit 

δ→0 σδ (ζ 1 ,η 1 )

Adση1 = lim



δ→0 σδ (ζ 2 ,η 2 )

dση2 B = 0 (

is valid, in which the orientation of σδ (ζ k , η k ) = ∂Ωkηk B(ζ k , δ), ζ k ∈ ∂Ωk , k = 1, 2, Ωk , B(ζ k , δ) is coordinated, and the orientation of ∂Ωk is the inductive orientation of Ωk .

139

Singular Integrals and Equations Dependent on Parameters

Proof We only prove the first formula. Letting 0 < δ < δ, σδ (ζ 1 , η 1 ) = ( Ω1η1 B(ζ 1 , δ) and using (2.3), we have lim



δ→0 σδ (ζ 1 ,η 1 )



= lim lim



Adση1 = lim lim



δ→0 δ→0 (σδ (ζ 1 ,η 1 ))\(σ (ζ 1 ,η 1 )) δ



δ→0 δ→0 (∂Ω 1 \σ (ζ 1 ,η 1 ))\(∂Ω 1 \σδ (ζ 1 ,η 1 )) 1η 1η δ

= lim



δ→0 (∂Ω1η1 \σδ (ζ 1 ,η 1 ))

Adση1 − lim

Adση1



Adση1





δ→0 ∂Ω 1 \σδ (ζ 1 ,η 1 ) 1η

Adση1

1 1 − = 0. 2 2

=

Similarly we can prove the second formula. 2.2 Some singular integrals whose integral order can be exchanged In the following, ϕ(η i , ξ i ) ∈ H(∂Ωi , β) (i = 1, 2, 0 < β < 1) mean that the function ϕ about η i and ξ i all belong to H(∂Ωi , β), i = 1, 2 (see Section 3, Chapter II, [88]3)). Theorem 2.4 ∂Ω1 . Then 

Adση1

∂Ω1η1

= 



∂Ω1ξ1



∂Ω1ξ1

Θ= =



dσξ1

Adση1

∂Ω1η1

=

Suppose that ϕ(η 1 , ξ 1 ) ∈ H(∂Ω1 , β), 0 < β < 1, ζ 1 ∈



dσξ1

∂Ω1η1

∂Ω1ξ1







∂Ω1η1



∂Ω1η1



Adση1 C[ϕ(η 1 , ξ 1 ) − ϕ(ξ 1 , ξ 1 )]; dσξ1 C[ϕ(ξ 1 , ξ 1 ) − ϕ(η 1 , η 1 )]

∂Ω1ξ1

Adση1

dσξ1

dσξ1 C[ϕ(η 1 , ξ 1 ) − ϕ(ξ 1 , ξ 1 )]

∂Ω1ξ1



Adση1 C[ϕ(ξ 1 , ξ 1 ) − ϕ(η 1 , η 1 )];

∂Ω1ξ1

∂Ω1η1

dσξ1 C[ϕ(η 1 , η 1 ) − ϕ(ζ 1 , ζ 1 )]

Adση1 C[ϕ(η 1 , η 1 ) − ϕ(ζ 1 , ζ 1 )] = Θ′ .

Proof We only prove the third formula, because the other formulas

140

Chapter V

can be similarly proved. Let Θ = Θ0 + Θδ , Θ′ = Θ′0 + Θ′δ , where Θ0 = Θδ = Θ′0

=

Θ′δ =





∂Ω1η1

∂Ω1η1



Adση1



Adση1

σδ (η 1 , ξ 1 )





∂Ω1ξ1 \σδ (η 1 , ξ 1 )

∂Ω1ξ1 \σδ (η 1 , ξ 1 )

σδ (η 1 , ξ 1 )

dσξ1

dσξ1



dσξ1 C[ϕ(η 1 , η 1 ) − ϕ(ζ 1 , ζ 1 )],



∂Ω1η1

dσξ1 C[ϕ(η 1 , η 1 ) − ϕ(ζ 1 , ζ 1 )],

∂Ω1η1

Adση1 C[ϕ(η 1 , η 1 ) − ϕ(ζ 1 , ζ 1 )],

Adση1 C[ϕ(η 1 , η 1 ) − ϕ(ζ 1 , ζ 1 )],

in which the orientations of σδ (η 1 , ξ 1 ) = ∂Ω1ξ1 ∩ B(η 1 , δ), Ω1 and B(η 1 , δ) are all coordinate, and the orientation of ∂Ω1 is the inductive orientation of Ω1 . From Section 1, Chapter III and [29]2), we know that Θ0 , Θ′0 are integrals in the normal sense, by using the Fubini Theorem [7], [6], [19], whose integral order can be exchanged, i.e. Θ0 = Θ′0 , hence |Θ − Θ′ | ≤ |Θδ | + |Θ′δ |. In addition, from Section 1, Chapter III (or see [29]2)) we can get 

∂Ω1η1

    Adση1 [ϕ(η 1 , η 1 ) − ϕ(ζ 1 , ζ 1 )] ≤ N ,

where N is a positive constant. From Theorem 2.3, we know when there exists ε > 0 independent of η 1 such that  δ6 is small enough,    σδ (η1 , ξ 1 ) dσξ 1 C  ≤ 2ε/N , so ε |Θδ | ≤ . 2

(2.7)

Next we consider Θ′δ . By using Theorem 2.1, we can substitute ϕ(ζ 1 , ζ 1 ) by ϕ(ξ 1 , ξ 1 ) in Θ′δ , thus Θ′δ can be written as Θ′δ

=



σδ (η 1 , ξ 1 )

dσξ1



∂Ω1η1

Adση1 C[Ψ1 + Ψ2 + Ψ3 ],

in which Ψ1 = ϕ(η 1 , η 1 ) − ϕ(η 1 , ξ 1 ) − ϕ(ζ 1 , η 1 ) + ϕ(ζ 1 , ξ 1 ), Ψ2 = ϕ(η 1 , ξ 1 ) − ϕ(ξ 1 , ξ 1 ), Ψ3 = ϕ(ζ 1 , η 1 ) − ϕ(ζ 1 , ξ 1 ). Since



Adση1 C[Ψ1 +Ψ2 +Ψ3 ]

∂Ω1η1

141

Singular Integrals and Equations Dependent on Parameters = lim



Adση1 C[Ψ1 + Ψ2 + Ψ3 ],

(2.8)

δ→0 ∂Ω 1 \σδ (ζ 1 ,η 1 ) 1η

where σδ (ζ 1 , η 1 ) = ∂Ω1η1 Θ′′δ

=



σδ (η 1 ,ξ 1 )

(

dσξ1

B(ζ 1 , δ), we consider



∂Ω1η1 \σδ (ζ 1 ,η 1 )

Adση1 C[Ψ1 + Ψ2 +Ψ3 ]

= P1 + P2 + P3 . β/2 β/2

From Section 1, Chapter III or [29]2), we have |Ψ1 | ≤ J2 ρ1 ρ2 , herein J2 is a positive constant, ρ1 = |η 1 − ξ 1 |, ρ2 = |η 1 − ζ 1 |, |dσξ1 | ≤ L1 ρm−2 dρ2 , L1 , L2 are positive constants. Thus dρ1 , |dση1 | ≤ L2 ρm−2 2 1 |P1 | ≤ L1 L2 J2

δ

β

dρ1 ρ−m+1 ρ12 ρm−2 1 1

0

= L1 L2 J2

δ

β −1 2

ρ1

dρ1

0

|P2 | ≤ J4

δ

ρ1β−1 dρ1

0

∂Ω1η1 \σδ (ζ 1 ,η 1 ) β

∂Ω1η1 \σδ (ζ 1 ,η 1 )

ρ22

−1

dρ2 ρ−m+1 ρ22 ρm−2 2 2 β

dρ2 ≤ J3 δ 2 , (2.9)

0

= J4

β



ρ−m+1 ρβ1 ρm−2 dρ1 1 1 δ





∂Ω1η1 \σδ



∂Ω1η1 \σδ (ζ 1 ,η 1 )

(ζ 1 ,η 1 )

ρ−m+1 ρm−2 dρ2 2 2 (2.10)

β ρ−1 2 dρ2 ≤ J5 δ .

Similarly we can get |P3 | ≤ J6 δ β ,

(2.11)

in which Ji (i = 3, 4, 5, 6) are positive constants. From (2.8)−(2.11), we see that if δ is small enough, the inequality |Θ′δ | ≤

ε 2

(2.12)

is derived. From (2.7), (2.12), it follows that |Θ − Θ′ | < ε. Due to the arbitrariness of ε, the equality Θ = Θ′ is obvious. Similarly we can prove Theorem 2.5

Let ϕ(η 2 , ξ 2 ) ∈ H(∂Ω2 , β), 0 < β < 1, ζ 2 ∈ ∂Ω2 . Then

142

Chapter V 

∂Ω2η2

= 

= 

∂Ω2ξ2

∂Ω2ξ2



∂Ω2ξ2



∂Ω2η2

∂Ω2ξ2



∂Ω2η2

∂Ω2ξ2



dσξ2

[ϕ(η 2 , ξ 2 ) − ϕ(ξ 2 , ξ 2 )]Ddση2 B;

[ϕ(ξ 2 , ξ 2 ) − ϕ(η 2 , η 2 )]Ddσξ2



dσξ2

dση2 B

∂Ω2ξ2

[ϕ(η 2 , ξ 2 ) − ϕ(ξ 2 , ξ 2 )]Ddσξ2



dσξ2

dση2 B



∂Ω2η2

=

dση2 B



∂Ω2η2



[ϕ(ξ 2 , ξ 2 ) − ϕ(η 2 , η 2 )]Ddση2 B;

[ϕ(η 2 , η 2 ) − ϕ(ζ 2 , ζ 2 )]Ddσξ2

∂Ω2η2

[ϕ(η 2 , η 2 ) − ϕ(ζ 2 , ζ 2 )]Ddση2 B.

Theorem 2.6 If f (η 1 , η 2 ; ξ 1 , ξ 2 ) ∈ H(∂Ω, β), 0 < β < 1, ζ 1 ∈ ∂Ω1 , ζ 2 ∈ ∂Ω2 , then the following integrals order can be exchanged for the integral order. 

∂Ωη



∂Ωξ

dσξ1 C[f (η 1 , η 2 ; ξ 1 , ξ 2 ) − f (η 1 , η 2 ; η 1 , ξ 2 )

−f (η 1 , η 2 ; ξ 1 , η 2 ) + f (η 1 , η 2 ; η 1 , η 2 )]Ddσξ2 ,



∂Ωη



∂Ωη



Adση1 dση2 B

∂Ωη

Adση1 dση2 B



∂Ωξ

dσξ1 C[f (η 1 , η 2 ; ξ 1 , η 2 ) − f (η 1 , η 2 ; η 1 , η 2 )

−f (η 1 , ζ 2 ; ξ 1 , ζ 2 )+]f (η 1 , ζ 2 ; η 1 , ζ 2 )]Ddσξ2 , Adση1 dση2 B



∂Ωξ

dσξ1 C[f (η 1 , η 2 ; η 1 , ξ 2 ) − f (η 1 , η 2 ; η 1 , η 2 )

−f (ζ 1 , η 2 ; ζ 1 , ξ 2 ) + f (ζ 1 , η 2 ; ζ 1 , η 2 )]Ddσξ2 , Adση1 dση2 B



∂Ωξ

dσξ1 C[f (η 1 , η 2 ; η 1 , η 2 ) − f (ζ 1 , η 2 ; ζ 1 , η 2 )

−f (η 1 , ζ 2 ; η 1 , ζ 2 ) + f (ζ 1 , ζ 2 ; ζ 1 , ζ 2 )]Ddσξ2 . 2.3 Three kinds of Poincar´e-Bertrand transformation formulas of two times singular integrals Theorem 2.7

Let E = f (η 1 , η 2 ; ξ 1 , ξ 2 ) ∈ H(∂Ω, β), 0 < β < 1, ζ 1 ∈

Singular Integrals and Equations Dependent on Parameters ∂Ω1 , ζ 2 ∈ ∂Ω2 . Then 

△1 = =

∂Ωη



∂Ωξ

dσξ1 dσξ2



1 + 4 +

∂Ω1ξ1



∂Ω2ξ2

+

Adση1 dση2 B 

∂Ωη

dσξ1

dσξ2





∂Ωξ

143

¯ Ddσ ¯ ξ2 dσξ1 C E

¯ η1 C¯ E ¯ Ddσ ¯ η2 B Adσ



∂Ω1η1

∂Ω2η2

Adση1 Cf (η 1 , ζ 2 ; ξ 1 , ζ 2 ) 1

2

1

2

f (ζ , η ; ζ , ξ )Ddση2 B



1 f (ζ 1 , ζ 2 ; ζ 1 , ζ 2 ) = △2 . 16

Proof Transform △1 into the form 

△1 =

Adση1 dση2 B

∂Ωη



¯ ξ2 = D1 +· · ·+D11 , dσξ1 C(Φ1 +· · ·+Φ11 )Ddσ

∂Ωξ

where Φ1 =f (η 1 , η 2 ; ξ 1 , ξ 2 )−f (η 1 , η 2 ; η 1 , ξ 2 )−f (η 1 , η 2 ; ξ 1 , η 2 )+f (η 1, η 2 ; η 1 ,η 2 ), Φ2 =f (η 1 , η 2 ; ξ 1 , η 2 )−f (η 1 , η 2 ; η 1 , η 2 )−f (η 1 , ζ 2 ; ξ 1 , ζ 2 )+f (η 1, ζ 2 ; η 1 ,ζ 2 ), Φ3 =f (η 1 , η 2 ; η 1 , ξ 2 )−f (η 1 , η 2 ; η 1 , η 2 )−f (ζ 1 , η 2 ; ζ 1 , ξ 2 )+f (ζ 1, η 2 ; ζ 1 ,η 2 ), Φ4 =f (η 1 , η 2 ; η 1 , η 2 )−f (ζ 1 , η 2 ; ζ 1 , η 2 )−f (η 1 , ζ 2 ; η 1 , ζ 2 )−f (ζ 1, ζ 2 ; ζ 1 ,ζ 2 ), Φ5 = f (η 1 , ζ 2 ; ξ 1 , ζ 2 ) − f (ξ 1 , ζ 2 ; ξ 1 , ζ 2 ), Φ6 = f (ζ 1 , η 2 ; ζ 1 , ξ 2 ) − f (ζ 1 , ξ 2 ; ζ 1 , ξ 2 ), Φ7 = f (ξ 1 , ζ 2 ; ξ 1 , ζ 2 ) − f (η 1 , ζ 2 ; η 1 , ζ 2 ), Φ8 = f (ζ 1 , ξ 2 ; ζ 1 , ξ 2 ) − f (ζ 1 , η 2 ; ζ 1 , η 2 ), Φ9 = f (η 1 , ζ 2 ; η 1 , ζ 2 ) − f (ζ 1 , ζ 2 ; ζ 1 , ζ 2 ), Φ10 = f (ζ 1 , η 2 ; ζ 1 , η 2 ) − f (ζ 1 , ζ 2 ; ζ 1 , ζ 2 ), Φ11 = f (ζ 1 , ζ 2 ; ζ 1 , ζ 2 ). For every Di , i = 1, ..., 4, we exchange their integral orders by using Theorem 2.6 and then merge them, and for every Di , i = 5, ..., 10, we

144

Chapter V

exchange their integral orders by using (2.3), Theorem 2.4, 2.5 and then merge them. Moreover for Φ11 , by using (2.3) we obtain △1 = 

1 + 4

=

∂Ωξ

∂Ω1ξ1



1 + 4 +



∂Ω2ξ2

dσξ1 dσξ2

dσξ1 dσξ2





∂Ωη

∂Ω1η1



∂Ω2η2

¯ η2 B Adση1 C(Φ1 + · · · + Φ4 )Ddσ

¯ 5 + Φ7 + Φ9 ) Adση1 C(Φ ¯ (Φ6 + Φ8 + Φ10 )Ddση2 B

1 f (ζ 1 , ζ 2 ; ζ 1 , ζ 2 ) 16 

∂Ωξ



dσξ1 dσξ2

∂Ωη

2

Adση1 C f (η 1 , η 2 ; ξ 1 , ξ 2 ) − f (η 1 , ζ 2 ; ξ 1 , ζ 2 ) 3

−f (ζ 1 , η 2 ; ζ 1 , ξ 2 ) + f (ζ 1 , ζ 2 ; ζ 1 , ζ 2 ) Ddση2 B + + +

1 4 1 4



∂Ω1ξ1



∂Ω2ξ2

dσξ1 dσξ2



∂Ω1η1



∂Ω2η2

2

3

Adση1 C¯ f (η 1 , ζ 2 ; ξ 1 , ζ 2 ) − f (ζ 1 , ζ 2 ; ζ 1 , ζ 2 ) 2

3

¯ f (ζ 1 , η 2 ; ζ 1 , ξ 2 ) − f (ζ 1 , ζ 2 ; ζ 1 , ζ 2 ) Ddση2 B

1 f (ζ 1 , ζ 2 ; ζ 1 , ζ 2 ). 16

Finally, applying Theorem 2.1 we can get △1 =



dσξ1 dσξ2

∂Ωξ

+ + +

1 4 1 4



dσξ1

∂Ω1ξ1



dσξ2

∂Ω2ξ2



∂Ωη



¯ Adση1 Cf (η 1 , η 2 ; ξ 1 , ξ 2 )Ddση2 B ¯ (η 1 , ζ 2 ; ξ 1 , ζ 2 ) Adση1 Cf

∂Ω1η1



¯ f (ζ 1 , η 2 ; ζ 1 , ξ 2 )Ddση2 B

∂Ω2η2

1 f (ζ 1 , ζ 2 ; ζ 1 , ζ 2 ) = △2 . 16

Theorem 2.8 Let G = b(η 1 , η 2 ), F = f (ξ 1 , ξ 2 ) ∈ H(∂Ω, β), 0 < β < 1, ζ 1 ∈ ∂Ω1 , ζ 2 ∈ ∂Ω2 . Then 

∂Ωη

Adση1 dση2 B



∂Ωξ

¯ ξ2 Gdσξ1 C F¯ Ddσ

Singular Integrals and Equations Dependent on Parameters

=



∂Ωξ



1 + 4 +

∂Ω1ξ1



∂Ω2ξ2

+



dσξ1 dσξ2

dσξ2

¯ η2 B A¯ Gdση1 C¯ F¯ Ddσ

∂Ωη

dσξ1

145



∂Ω1η1



Ab(η 1 , ζ 2 )dση1 Cf (ξ 1 , ζ 2 ) 1

∂Ω2η2

2

1

2

b(ζ , η )f (ζ , ξ )Ddση2 B



1 b(ζ 1 , ζ 2 )f (ζ 1 , ζ 2 ). 16

Similar to the proof of Theorem 2.7, we can prove Theorems 2.8 and 2.9. Theorem 2.9 Let E = f (η 1 , η 2 ; ξ 1 , ξ 2 ) ∈ H(∂Ω, β), 0 < β < 1, ζ 1 ∈ ∂Ω1 , ζ 2 ∈ ∂Ω2 . Then 

∂Ωη

=



∂Ωξ

dσξ1 dσξ2



1 + 4 +

Adση1 dση2 B

∂Ω1ξ1



∂Ω2ξ2

+

3



∂Ωξ

∂Ωη

dσξ1

dσξ2





Cdσξ1 Edσξ2 D

¯ η2 D B A¯ Cdση1 Edσ



∂Ω1η1

∂Ω2η2

A Cdση1 f (η 1 , ζ 2 ; ξ 1 , ζ 2 ) 1

2

1

2

f (ζ , η ; ζ , ξ )dση2 D B



1 f (ζ 1 , ζ 2 ; ζ 1 , ζ 2 ). 16

The Composition Formula and Inverse Formula of Singular Integrals with a Cauchy’s Kernel in Real Clifford Analysis

In this section, we first prove the composition formula and the inverse formula of singular integrals with Cauchy’s kernel by using the Poincar´eBertrand transformation formula, and then we give the second proof method for the composition formula and the inverse formula by using the Plemelj formula for Cauchy’s integrals.

146

Chapter V

We suppose E = F (ξ 1 , ξ 2 ; ζ 1 , ζ 2 ) = f (ξ 1 , ξ 2 ) in Theorem 2.9, and use Theorem 2.1, then the composition formula of singular integrals with Cauchy’s kernel can be obtained. Theorem 3.1 1, 2. Then 

∂Ωη

Let f (ξ 1 , ξ 2 ) ∈ H(∂Ω, β), 0 < β < 1, ζ i ∈ ∂Ωi , i =

Adση1 dση2 B



∂Ωξ

Cdσξ1 f (ξ 1 , ξ 2 )dσξ2 D =

1 f (ζ 1 , ζ 2 ). 16

(3.1)

Denote the operator Wf = 4



∂Ωξ

Cdσξ1 f (ξ 1 , ξ 2 )dσξ2 D,

then the composition formula can written as W 2 f = W (W f ) = f.

(3.2)

By using the composition formula, the inverse formula for singular integrals with Cauchy’s kernel can also be obtained. Theorem 3.2

If f (ξ 1 , ξ 2 ) ∈ H(∂Ω, β), 0 < β < 1, and we write W f = g(η 1 , η 2 ),

(3.3)

W g = f (ζ 1 , ζ 2 ).

(3.4)

then

Proof From (3.3), (3.2), we get (3.4) and W g = W (W f ) = f (ζ 1 , ζ 2 ). Inversely, we have (3.3) from (3.4), and (3.3), (3.4) are called the inverse formulas for singular integrals with Cauchy’s kernel on characteristic manifolds. Obviously it is equivalent to the composition formula. If (3.3) is seen as a singular integral equation with the unknown function f , then from inverse formula (3.4), we can derive that the equation has a unique solution. As a corollary, we can get the inverse formula on the smooth closed manifold as follows: ⎧  ⎪ 1 ⎪ g(η ) = 2 Cdσξ1 f (ξ 1 ), ⎪ ⎪ ⎨ ∂Ω1ξ1  ⎪ ⎪ 1 ⎪ ⎪ Adση1 g(η 1 ). ⎩ f (ζ ) = ∂Ω1η1

(3.5)

Singular Integrals and Equations Dependent on Parameters

147

From Theorem 1.1 in Section 1, Chapter III, we know that Cauchy type integral F (x, y) =



∂Ωξ

ξ1 − x ξ2 − y 1 2 f (ξ , ξ )dσ dσ , 2 1 ξ ωm |ξ 1 − x|m ξ ωk |ξ 2 − y|k / ∂Ω2 x∈ / ∂Ω1 , y ∈

± is a biregular function in Ω± 1 × Ω2 , and F (x, ∞) = F (∞, y) = F (∞, ∞) = 0. Using Theorem 1.1 (the Plemelj formula) in Section 1, Chapter III, it is easy to see that

F ++ (η 1 , η 2 ) + F +− (η 1 , η 2 ) + F −+ (η 1 , η 2 ) + F −− (η 1 , η 2 ) = g(η 1 , η 2 ), (η 1 , η 2 ) ∈ ∂Ω.

(3.6)

From (3.3), we can get g(η 1 , η 2 ) = W f . If we consider another biregular ± function Q(x, y) in Ω± 1 × Ω2 , Q(x, y) =

⎧ + − − ⎨ F (x, y), (x, y) ∈ Ω+ 1 × Ω2 or Ω1 × Ω2

⎩ −F (x, y), (x, y) ∈ Ω+ × Ω− or Ω− × Ω+ , 1 2 1 2

(3.7)

then (3.6) can be written as

Q++ (η 1 , η 2 ) − Q+− (η 1 , η 2 ) − Q−+ (η 1 , η 2 ) + Q−− (η 1 , η 2 ) = g(η 1 , η 2 ),

(3.8)

where (η 1 , η 2 ) ∈ ∂Ω, i.e. Q(x, y) is a solution of (3.8). From Theorem 1.1 (Plemelj formula) of Section 1, Chapter III again, we see that the Cauchy type integral Q1 (x, y) =



∂Ωη

η1 − x η2 − y 1 2 dσ 1 g(η , η )dση 2 η ωm |η 1 − x|m ωk |η 2 − y|k

is a solution of (3.8). Let Q2 (x, y) = Q(x, y) − Q1 (x, y). From condition (3.8), we know that Q2 satisfies the homogeneous boundary condition of (3.8): +− 1 2 −+ 1 2 −− 1 2 1 2 Q++ 2 (η , η ) − Q2 (η , η ) − Q2 (η , η ) + Q2 (η , η )

= 0, (η 1 , η 2 ) ∈ ∂Ω. This shows that Q2 (x, y) possesses the jump degree zero, hence Q±± 2 (x, y) can be extended to a biregular function through each other’s ∂Ω, namely Q2 (x, y) is a biregular function in Rm × Rk and Q2 (x, ∞) =

148

Chapter V

Q2 (∞, y) = Q2 (∞, ∞) = 0. From the Liouville theorem (see [19]), we get Q2 ≡ 0, thus Q(x, y) ≡ Q1 (x, y). From the above Plemelj formula again, we have F ++ (η 1 , η 2 ) − F +− (η 1 , η 2 ) − F −+ (η 1 , η 2 ) + F −− (η 1 , η 2 ) = f (η 1 , η 2 ), (η 1 , η 2 ) ∈ ∂Ω, Q++ (η 1 , η 2 ) + Q+− (η 1 , η 2 ) + Q−+ (η 1 , η 2 ) + Q−− (η 1 , η 2 ) = W g(η 1 , η 2 ), (η 1 , η 2 ) ∈ ∂Ω. From (3.7), the definition of g(η 1 , η 2 ) and two formulas as stated above, we obtain f (η 1 , η 2 ) = W g(η 1 , η 2 ) = W (W f ) = W 2 f, which shows that the inverse formula is true.

4

The Fredholm Theory of a Kind of Singular Integral Equations in Real Clifford Analysis

In this section we deal with a kind of integral equations with quasiCauchy kernel in real Clifford analysis. Firstly we write the condition for an integral equation, which can be reduced to the Fredholm type equation, and find the regularization operator and prove the regularization theorem. Let b(η 1 , η 2 ), c(η 1 , η 2 ), ϕ(η 1 , η 2 ) ∈ H(∂Ω, β), 0 < β < 1, and introduce the singular integral operator with Cauchy kernel and quasi-Cauchy kernel K, L: 1

2

(Kf )(η , η ) = 4



∂Ωξ

(Lf )(η 1 , η 2 ) =



∂Ωξ

dσξ1 Cf (ξ 1 , ξ 2 )Ddσξ2 ,

L1 (ξ 1 , η 1 )dσξ1 f (ξ 1 , ξ 2 )dσξ2 L2 (ξ 2 , η 2 ),

where the Cauchy kernels C, D have been given in Section 2, and the quasi-Cauchy kernel is defined as L1 (ξ 1 , η 1 ) =

l2 (ξ 2 , η 2 ) l1 (ξ 1 , η 1 ) 2 2 2 , L (ξ , η ) = , |ξ 1 − η 1 |m−1−r1 |ξ 2 − η 2 |k−1−r1

where li (ξ i , η i ) ∈ H(∂Ω, αi ), 0 < αi < 1, ri > 0, i = 1, 2. We consider the singular integral equation with quasi-Cauchy kernel in H(∂Ω, β): Sf = f + bKf + cLf = ϕ,

(4.1)

Singular Integrals and Equations Dependent on Parameters

149

in which ∂Ω is the regularizing manifold. The following regularization theorem about (4.1) can be obtained. Theorem 4.1 Let (1 − b2 ) in (4.1) be invertible on ∂Ω, and 1/(1 − b2 ) < Y, Y a positive constant. Then when bβ is small enough, (4.1) can be transformed into a Fredholm type equation. Proof Firstly we give the operator M : M ψ ≡ 1(ψ − bKψ)/(1 − b2 ), ψ ∈ H(∂Ω, β), and prove that M is a regularization operator of (4.1). In fact, when M acts on the two sides of (4.1), the right-hand side of (4.1) becomes a function g such that M ϕ = g and the left-hand side becomes M (Sf ) =

1 [f + cLf − bK(bKf ) − bK(cLf )]. 1 − b2

By applying Theorem 2.8 to K(bKf ) in the above formula and exchanging the integral order of K(cLf ) (from Theorems 2.4, 2.5, 2.6, we see that the order can be exchanged), hence (4.1) possesses the form [M (SF )](ζ 1 , ζ 2 ) = f+

c 16b 4b 4b 4b Lf − Uf − Jf − Qf − Vf 2 2 2 2 1−b 1−b 1−b 1−b 1 − b2

(4.2)

= g(ζ 1 , ζ 2 ), in which 1

2

(U f )(ζ , ζ ) = 1

2

(Jf )(ζ , ζ ) =







∂Ωξ

dσξ1 dσξ2

∂Ω1ξ1

(Qf )(ζ 1 , ζ 2 ) =

dσξ2

∂Ω2ξ2

(V f )(ζ 1 , ζ 2 ) = ×



dσξ1



∂Ωξ







∂Ωη

¯ 1 , η 2 )dση1 Cf ¯ (ξ 1 , ξ 2 )Ddση2 B, Ab(η

Ab(η 1 , ζ 2 )dση1 Cf (ξ 1 , ξ 2 ),

∂Ω1η1

b(ζ 1 , ζ 2 )f (ζ 1 , ξ 2 )Ddση2 B,

∂Ω2η2

dσξ1 dσξ2

¯ 1 , η 2 )L1 (ξ 1 , η 1 )dση1 f (ξ 1 , ξ 2 )dση2 L2 (ξ 2 , η 2 )B. Ac(η

∂Ωη

We write the kernel of U, J, Q, V, L as U1 , J1 , Q1 , V1 , L1 , herein U1 (ζ 1 , ζ 2 ; ξ 1 , ξ 2 ) =



∂Ωη

¯ 1 , η 2 )dση1 CDdσ ¯ Ab(η η 2 B,

150

Chapter V 1

2

1

J1 (ζ , ζ , ξ ) = 1

2



2

Q1 (ζ , ζ , ξ ) =

∂Ω1η1



∂Ω2η2

Ab(η 1 , ζ 2 )dση1 C, b(ζ 1 , η 2 )Ddση2 B,

V1 (ζ 1 , ζ 2 ; ξ 1 , ξ 2 ) =



∂Ωη

¯ 1 , η 2 )L1 (ξ 1 , η 1 )dση1 dση2 L2 (ξ 2 , η 2 )B, Ac(η

L1 (ζ 1 , ζ 2 ; ξ 1 , ξ 2 ) = L1 (ξ 1 , ζ 1 )L2 (ξ 2 , ζ 2 ). Moreover setting v(η 1 , η 2 ) = b(η 1 , η 2 ) − b(η 1 , ξ 2 ) − b(ξ 1 , η 2 ) + b(ξ 1 , ξ 2 ), and using Theorem 2.2, we know that U1 can be transformed into U1 =



∂Ωη

¯ 1 , η 2 )dση1 CDdσ ¯ Av(η η 2 B.

From Section 1, Chapter III or [29]2), we get β

β

|v| ≤ J2 |η 1 − ξ 1 | 2 |η 2 − ξ 2 | 2

(4.3)

where J2 is a positive constant. Hence |U1 | ≤ J3



|dση1 ||dση2 |

, β − ζ 2 |k−1− 2 |η 2 − ζ 2 |k−1 (4.4) in which J3 is a positive constant. Using the Hadamard Theorem (see [19]), we see that when m > β2 + 1, k > β2 + 1, the estimate ∂Ωη

|η 1 − ζ 1 |m−1 |η 1 −

|U1 | ≤

β ξ 1 |m−1− 2 |η 2

J4 β

β

|ξ 1 − ζ 1 |m−1− 2 |ξ 2 − ζ 2 |k−1− 2

,

(4.5)

is derived, where J4 is a positive constant. Similarly, we have |V1 | ≤ J5 ≤



|ξ 1

∂Ωη

−

|dση1 ||dση2 | |η 1 − ζ 1 |m−1 |η 1 − ξ 1 |m−1−r1 |η 2 − ζ 2 |k−1 |η 2 − ξ 2 |k−1−r2

J6 1 m−1−r 1 |ξ 2 ζ |

− ζ 2 |k−1−r2

. (4.6)

Singular Integrals and Equations Dependent on Parameters

151

where J5 , J6 are positive constants. Using Theorem 2.2, we know that J1 can be changed as J1 =



∂Ω1η1

Ab1 (η 1 , ζ 2 )dση1 C,

in which b1 (η 1 , ζ 2 ) = b(η 1 , ζ 2 ) − b(ζ 1 , ζ 2 ), and |b1 | ≤ J7 |η 1 − ζ 1 |β ,

(4.7)

herein J7 is a positive constant. By the Hadamard Theorem and (4.7), when m > β + 1, we have |J1 | ≤ J8



∂Ω1η1

|η 1

−

|dση1 | 1 m−1−β ζ | |η 1

− ξ 1 |m−1

J9 . ≤ 1 |ζ − ξ 1 |m−1−β

(4.8)

Next we discuss Q1 . We can get, when k > β + 1, J10 . |ξ 2 − ζ 2 |k−1−β

(4.9)

J11 , |ξ 1 − ζ 1 |m−1−r1 |ξ 2 − ζ 2 |k−1−r2

(4.10)

|Q1 | ≤ For L1 obviously we have |L1 | ≤

in which J8 , J9 , J10 , J11 are all positive constants. Combining (4.5), (4.6), (4.8), (4.9) and (4.10), it is easy to see that (4.2) is a weak singular equation. By the Hadamard Theorem, when ζ 1 , ξ 1 ∈ ∂Ω1 and µ1 , µ2 > 0, we get 

∂Ω1η1

|η 1

−

|dση1 | 1 m−1−µ 1 |η 1 ζ |

−

ξ 1 |m−1−µ2

≤

|ζ 1

−

J12 , m−1−(µ 1 1 +µ2 ) ξ |

(4.11)

where m > µ1 + µ2 + 1, J12 is a positive constant. In addition for ζ 2 , ξ 2 ∈ ∂Ω2 , the same estimate can be concluded. Similarly by the Hadamard theorem, when ζ, ξ ∈ ∂Ω and µi > 0, 1 ≤ i ≤ 4, we can obtain 

∂Ωη

|dση1 ||dση2 | |η 1 − ζ 1 |m−1−µ1 |η 1 − ξ 1 |m−1−µ2 |η 2 − ζ 2 |k−1−µ3 |η 2 − ξ 2 |m−1−µ4 ≤

|ζ 1

−

J13 m−1−(µ +µ 1 1 2 ) |ζ 2 ξ |

− ξ 2 |k−1−(µ3 +µ4 )

, (4.12)

152

Chapter V

where m > µ1 + µ2 + 1, k > µ3 + µ4 + 1, J13 is a positive constant. Thus (p) from (4.5), (4.11), (4.12), we see that the reiterative kernel U1 of U1 for p times satisfies the inequality (see [46]) (p)

U1

≤

J14 m−1− pβ 2

|ζ 1 − ξ 1 |

|ζ 2 − ξ 2 |k−1−

pβ 2

;

(4.13)

here J14 is a positive constant. Moreover the reiterative kernels of V1 , J1 , Q1 , L1 for p times satisfy the similar inequality. Because of ζ i ∈ ∂Ω (i = 1, 2), the integrals 

∂Ω1η1



∂Ω2η2

|dση1 | , 0 < µ1 < 1, |η 1 − ζ 1 |m−1−µ1 |η 2

|dση2 | , 0 < µ3 < 1 − ζ 2 |k−1−µ3

are uniformly bounded, hence the mixed reiterative kernels of U1 , V1 , J1 , Q1 , L1 for p times satisfy the similar inequality (see [46]). From (4.13) we see that for the positive integer p satisfying k − 1 − (p) ≤ 0, m − 1 − pβ 2 ≤ 0, the reiterative kernel U1 of U1 for p times are all bounded functions, it is sufficient to assume p ≥ max{[2(m − 1)/β] + 1, [2(k −1)/β]+1}. Similarly we can consider the reiterative kernels and mixed reiterative kernels of U1 , V1 , J1 , Q1 , L1 for p times. We know if p is large enough the reiterative kernels and mixed reiterative kernels for p times are all bounded functions. This shows that (4.2) is a Fredholm equation, and the Fredholm theorem holds for it. pβ 2

Finally we prove that there exists an inverse of M . In fact, from Section 1, Chapter III or [29]2), we know that KΨβ ≤ J15 Ψβ , where J15 is a positive constant, hence when bβ is small enough, the operator bKΨ is a compact operator. Moreover from the hypothesis 1/(1 − b2 )β < Y, we see that when bβ is small enough, there exists an inverse operator M −1 (see [46]). This shows that (4.1) is equivalent to the Fredholm equation (4.2). Remark 1 When bβ , cβ , ϕβ are appropriately small, from Section 1, Chapter III (or see [29]2)), we know that (4.1) is solvable Remark 2 When a is invertible, the singular integral equation (see [19]) af + bKf + cLf = ϕ, (4.14) can be rewritten as f + a−1 bKf + a−1 cLf = a−1 ϕ.

Singular Integrals and Equations Dependent on Parameters

153

Hence from Theorem 4.1, we know that under some condition (4.14) is equivalent to a Fredholm equation.

5

Generalized Integrals and Integral Equations in Real Clifford Analysis

By using the method of resolution of the identity, in this section we define the generalized integrals in the sense of M. Spivak (see [73]) on open manifold for unbounded functions in real Clifford analysis, and discuss the solvability and the series expression of solutions for the second kind of generalized integral equations. Finally we give the error estimate for the approximate calculation. Let Ω ⊂ Rn be an n-dimensional bounded manifold. We consider a class of functions belonging to CΩ (A), where the functions are defined in Ω and with values in the real Clifford space An (R). Now we give the definition of resolution of the identity. Let Ω be as stated before, θ be an open covering on Ω, and for every U ⊂ θ, we have U ⊂ Ω. Then we call θ a permissible open covering on Ω. Thus there must exist a group of sets Φ of function ϕ belonging to C ∞ ; ϕ is defined on an open set including Ω, and satisfies 1. For every x ∈ Ω, we have 0 ≤ ϕ(x) ≤ 1. 2. For every x ∈ Ω, there exists an open set V including x such that there exists a finite ϕ ∈ Φ which isn’t equal to 0, V . 3. For every x ∈ Ω, we have




ϕ(x) = 1.

ϕ∈Φ

4. For every ϕ ∈ Φ, there exists an open set U belonging to θ such that ϕ is equal to 0 on a closed subset in U. If Φ ⊂ C ∞ satisfies 1 − 3, then Φ is called a resolution of the identity on Ω. If Φ also satisfies 4, then Φ is said to be a resolution of the identity belonging to θ. If Φ is a resolution of the identity belonging to θ on Ω, f is a function from Ω to R and f is bounded in an open set of each point in Ω, the measure of the set {x : f is discontinuous on x} is 0, then for 6
any ϕ ∈ θ, ϕ|f | on Ω is integrable. If ϕ∈Φ Ω ϕ|f | converges, then f is said to be a generalized integrable function on Ω.

154

Chapter V

We consider the function class CΩ (A), where the functions are defined in Ω and with values in the Clifford space. Definition 5.1 Let f (x) ∈ CΩ (A), θ be a permissible open covering on Ω, Φ(⊂ C ∞ ) be a resolution of the identity belonging to θ on Ω so
6 that for any ϕ ∈ θ, ϕ|f | be integrable on Ω. If Ω ϕ|f | as series is ϕ∈Φ

convergent (see [73]), we say that f is integrable as a generalized integral in the sense of M. Spivak. The sum of the series is called the integral of f on Ω. All the generalized integrable functions in CΩ (A) can be written as IΩ (A).

In the following, the generalized integrable functions are considered such functions in the sense of M. Spivak. Lemma 5.1 If Ω ∈ Rn is as stated above, then f (x) = ΣfA (x)eA ∈ IΩ (A), if and only if each fA (x) is generalized integrable on Ω. Lemma 5.2 If Ω ∈ Rn , IΩ (A) is as stated above, then for any f, g ∈ IΩ (A) and Clifford number λ ∈ An (R) we have f + g ∈ IΩ (A) and λf ∈ IΩ (A). Proof Let f, g ∈ IΩ (A), i.e. there exist a permissible open covering θ1 on Ω and a resolution of the identity Φ1 belonging to θ1 , such that f is bounded in an open set of 6each point in Ω. Hence, for any ϕ ∈
θ1 , ϕ|f | is integrable, and ϕ∈Φ Ω ϕ|f | as a series is convergent. At the same time, there exists another permissible open covering θ2 on Ω and a resolution of the identity Φ2 ∈ θ2 , such that g is bounded in one open set of each point in Ω, and then for any ϕ ∈ θ1 , ϕ|f | is integrable, and 6
ϕ|g| as a series is convergent. Make a permissible open covering ϕ∈Φ ) Ω ) θ = θ1 θ2 = {U |U = u1 u2 , ui ∈ θi , i = 1, 2} on Ω and a resolution 2 of the identity Φ = {ϕ|ϕ = ϕ1 +ϕ , ϕi ∈ Φi , i = 1, 2} on Ω, then 2 

ϕ∈Φ Ω

≤

 

ϕ1 ∈Φ1

Ω

ϕ|f + g| ≤

ϕ1 |f | +



Ω

 

ϕ∈Φ



ϕ1 |g| +

Ω

ϕ|f | +

 

ϕ2 ∈Φ2

Ω





ϕ|g|

Ω

ϕ2 |f | +



Ω



ϕ2 |g| ,

and each integral on the right-hand side of the inequality is convergent. Hence the integral on the left-hand side is also convergent. That is to say that f + g is generalized integrable. In the same way, if λ is a real Clifford number, then λf is generalized integrable too. The proof is completed.

Singular Integrals and Equations Dependent on Parameters

155




Definition 5.2 Let f (x, y) = A fA (x, y)eA , (x, y) ∈ Ω × Ω,where each fA (x, y) : Ω × Ω → R1 is a real function. If fA (x, y) is square integrable for each variable (the other variable is looked on as a constant), i.e. |f |2 looked on as a function of x or y is generalized integrable and 6 6 2 Ω Ω |f (x, y)| dxdy < ∞. Then we define f (x, y) as a square generalized integrable function. All the square generalized integrable functions can be written as IΩ×Ω (A). Lemma 5.3

If f (x, y), g(x, y) ∈ IΩ×Ω (A) and Φ2 ∈ IΩ (A), then

1. About u (here x,y are seen as constants) f (x, u)g(u, y) ∈ IΩ (A). 2. 3.

6

6

Ω f (x, u)g(u, y)du Ω f (x, u)Φ

2 (u)du

∈ IΩ×Ω (A).

∈ IΩ (A).


Definition 5.3 Let K(x, y) = A KA (x, y)eA ∈ IΩ×Ω (A) be a function. We define the corresponding kernel as K 0 (x, y) =



KA (x, y)eA hA ,

A

where hA = heA = her1 ...erh = her1 ...herh , A = {r1 , ..., rh }, and each hei is a transformation hei (ej ) =

⎧ ⎨ ej ,

⎩ −e , j

i = j, i = j,

and define hA as a left exchange factor. Definition 5.4

Define the equation ϕ(x) − λ



K 0 (x, u)ϕ(u)du = f (x)

(5.1)

Ω

as a second kind integral equation, where ϕ(x) is an unknown function, K 0 (x, y) is a function defined in Definition 5.3, f is a known function satisfying f 2 ∈ IΩ , and λ(∈ An (R)) is a real Clifford constant. In the following, we find a solution of equation (5.1). We find a solution of equation (5.1) by using a successive iteration. Let ⎧ ⎨ ϕ0 (x) = f (x),

⎩ ϕ (x) = f (x) + λ 6 K 0 (x, u)ϕ m−1 (u)du, m = 1, 2.... m Ω

(5.2)

156

Chapter V

It is easy to prove that if the sequence of functions {ϕm (x)} on Ω uniformly converges to a function, then the function is a solution of equation (5.1). In order to study the property of {ϕm (x)}, we give the following definition. Definition 5.5

If K(x, y) ∈ IΩ×Ω (A) is as stated before, we define K2 (x, y) =

Km (x, y) =



Ω



K(x, t)K 0 (t, y)dt,

Ω

K(x, t)Km−1 (t, y)dt, m ≥ 3,

as the reiterative kernel of K(x, y) (m ≥ 2) for m times. If K 0 (x, y), K(x, y), λ, f (x) is as stated above, then

Theorem 5.4

⎧ ⎨ K 0 (x, u)f (x) = f (x)K(x, u),

(5.3)

⎩ K 0 (x, u)λ = λK(x, u).

By the definition of the exchange factor, the theorem is easy to prove. Consider the sequence of functions {ϕm (x)} as follows: ϕ1 (x) = f (x) + λ ϕ2 (x) = f (x) + λ = f (x) + λ





Ω



Ω

0

= f (x) + λ

K 0 (x, t)ϕ1 (t)dt 

K (x, t) f (t) + λ

Ω



K 0 (x, t)ϕ0 (t)dt,

0





K (t, u)f (u)du dt

Ω

K (x, u)f (u) + (λ)

2

 

K 0 (x, u)f (u) + (λ)2



Ω

Ω

0



K(x, t)K (t, u)f (u)du dt

Ω

Ω

Ω

= f (x) + λ



0

K2 (x, u)f (u)du,

ϕ3 (x) = .... In general, we have ϕn (x) = f (x)+λ



Ω

K 0 (x, u)f (u)du+

n 

m=2

(λ)m



Ω

Km (x, u)f (u)du,

n = 1, 2, .... (5.4)

157

Singular Integrals and Equations Dependent on Parameters If f (x) ∈ IΩ (A) and

Theorem 5.5



Ω

|f (u)|2 dvu = H 2 , |K(x, u)| ≤ M (u),

which is valid for each x ∈ Ω, u ∈ Ω and M (u) is a generalized square 6 integrable function Ω |M (u)|2 dvu = L2 , where dvu is the volume element of Ω, then |



Ω

Km (x, u)f (u)du| ≤ HLm J1m , m = 1, 2, ....

(5.5)

Proof If m = 1, we denote K(x, u) = K 0 (x, u) and have |



Ω

K1 (x, u)f (u)du| ≤ ≤ J1





|K1 (x, u)f (u)|dvu ≤ J1

Ω

2

|M (u)| dvu

Ω

 1  2

2

Ω

|f (u)| dvu

1 2



Ω

|M (u)| |f (u)|dvu

= J1 HL.

Suppose that the estimate (5.5) is true for m − 1; we shall prove that the estimate (5.5) is also true for m, i.e. |



Ω

=|

 

Km (x, u)f (u)du| = |



≤ J1 ≤ J1



K(x, t)[

Ω

Ω



Ω



Ω

K(x, t)K(m−1) (t, u)dtf (u)du|

K(m−1) (t, u)f (u)du]dt| 2

|K(x, t)| dvt 2

Ω Ω

|M (t)| dvt

 1  2

Ω

 1  2

Ω

|



Ω

2

K(m−1) (t, u)f (u)du| dvt

2(m−1) 2 J1 H dvt

1 2

1

2

≤ HLm J1m .

Theorem 5.6 Under the same result as in Theorem 5.2, then for the real Clifford number λ satisfying |λ| < 1/J12 L, equation (5.1) has a unique solution, and the solution is the limit of the sequence of functions: (5.4). Proof From Theorem 5.4, it is easy to see that the above sequence of functions uniformly converges on Ω. Obviously its limit function is a solution of equation (5.1). In the following, we will prove the uniqueness of the solution. If there exists λ ∈ An (R), and |λ| < 1/J12 L, then there exist two solutions ϕ1 (x), ϕ2 (x) of equation (5.1), i.e. ϕ1 (x) − λ ϕ2 (x) − λ



Ω



Ω

K 0 (x, u)ϕ1 (u)du = f (x), K 0 (x, y)ϕ2 (u)du = f (x).

158

Chapter V

Let the first equality be subtracted from the second one, then we get ϕ1 (x) − ϕ2 (x) − λ



Ω

K 0 (x, u)[ϕ1 (u) − ϕ2 (u)]du = 0.

Denote ω(x) = ϕ1 (x) − ϕ2 (x), then we have ω(x) = λ



K 0 (x, u)ω(u)du,

Ω

and then 2

|ω(x)| ≤

J12 |λ|2



Ω

0

2

|K (x, u)| dvu



Ω

|ω(u)|2 dvu .

Making the volume integration go to x on Ω in both sides of the inequality, i.e.   Ω

|ω(x)|2 dvx J12 ≤ L2 |λ|2

we get

(1 − and



J12 L2 |λ|2 )

Ω



Ω

Ω

|ω(u)|2 dvu ,

|ω(u)|2 dvx ≤ 0,

|ω(u)|2 dvx = 0, ϕ1 = ϕ2 .

The proof of Theorem 5.3 is finished. Theorem 5.7 Under the same conditions described in Theorem 5.4, we substitute {ϕn } in (5.4) into the accurate solution of equation (5.1), then the norm error is not greater than |λ|n+1 Ln+1 HJ12n+1 . 1 − |λ|2 L2 J12

CHAPTER VI SEVERAL KINDS OF HIGH ORDER SINGULAR INTEGRALS AND DIFFERENTIAL INTEGRAL EQUATIONS IN REAL CLIFFORD ANALYSIS In the first section of this chapter, we shall introduce six kinds of high order singular integrals of quasi-Bochner-Martinelli type with one singular point, definitions of their Hadamard principal values, recurrence formulas, calculational formulas and differential formulas. In the second section, after proving the lemma of Hile type, we shall discuss the properties of high order singular integral operators and then prove the H¨older continuity of several kinds of high order singular integrals of quasi-Bochner-Martinelli type on the integral path. In the third section, we shall prove the existence and uniqueness of solutions for three kinds of nonlinear differential integral equations with high order singular integrals of quasi-Bochner-Martinelli type by the method of integral equations. In the fourth section, we shall give the definitions of high order singular integrals with two singular points, and prove the Poincar´e-Bertrand permutation formulas for high order singular integrals of quasi-Bochner-Martinelli type in real Clifford analysis by using the differential formulas (see [29]10), [65]).

1

The Hadamard Principal Value and Differential Formulas of High Order Singular Integrals with One Singular Point in Real Clifford Analysis

First of all, we introduce the concept of Hadamard principal value of high order singular integrals with one singular point for functions of one complex variable. Suppose that L is a simple smooth closed curve, f ′ on L is H¨older continuous, and f ′ ∈ H. Let τo ∈ L; it is clear that the integral  f (τ ) dτ (τ − τ0 )2 L

160

Chapter VI

at τ = τ0 possesses a singularity of high order (> 1). In general, it is divergent, even under the sense of Cauchy principal value. For instance, if we define 

L

f (τ ) dτ = lim η→0 (τ − τ0 )2



L−Lη

f (τ ) dτ, (τ − τ0 )2

(1.1)

in which Lη = L ∩ {|τ − τ0 | ≤ η}, then its limit on the right-hand side of the above equality usually doesn’t exist. This is because we know 

L−Lη

f (τ )dτ = (τ − τ0 )2



L−Lη





f ′ (τ ) f (τ2 ) f (τ1 ) dτ + − = I1 + I2 , (τ − τ0 ) τ2 − τ0 τ1 − τ0

by means of integration by parts, where τ1 , τ2 ∈ L ∩ {|τ − τ0 | = η}. Noting f ′ ∈ H, if η → 0, then I1 →



L

f ′ (τ ) dτ, (τ − τ0 )

and lim I2 = lim {

η→0

η→0

f (τ2 )−f (τ0 ) f (τ1 )−f (τ0 ) 1 1 − +f (τ0 )( − )} τ2 −τ0 τ1 −τ0 τ2 −τ0 τ1 −τ0 1 1 = f (τ0 ) lim ( − ). η→0 τ2 − τ0 τ1 − τ0

This shows that this integral generally doesn’t exist (except f (τ0 ) = 0) Hence we cannot define this kind of singular integrals by (1.1). Now we first consider the case: n = 2 > 1 on the left side of (1.1). It is a singular integral whose singularity is higher than one order. Moreover, we conclude that L and n satisfy conditions such that the integral 

L−Lη

f (τ ) dτ (τ − τ0 )n

(1.2)

can converge under the sense of Cauchy principal value. Some authors have discussed the problem and acquired some results. But for applications, we shall discuss the high order singular integral (1.2) from another view. This view was first proposed by J. Hadamard for similar singular integrals on the real axis, namely the idea of the so-called finite part of an integral [70]. In 1957, C. Fox generalized this idea to integral (1.2) with the positive integer n [16]. Afterwards Chuanrong Wang discussed the problem in [78], and then Jianke Lu in [45], generalized n to the case of a general positive real number and an integral with many singular points. Using the idea of Hadamard principal value, we can define 

L

f (τ ) dτ = (τ − τ0 )2



L

f ′ (τ ) dτ, τ − τ0

High Order Singular Integrals and Differential Integral Equations i.e.

 * L

−1 τ − τ0 )

+′

*



161

+

−1 − f (τ )dτ = f ′ (τ )dτ, τ − τ0 L

−1 −1 ′ ) possesses a lower order singularity than ( ) . Thus τ − τ0 τ − τ0 we define a high order singular integral by induction on an integral having a low order singularity. It is also said that this integral is defined by cutting out the divergent part I2 . This idea is easily generalized to a general high order singular integral (1.2). Provided that we do the integration by parts and cut out the terms with divergent part several times, then we can write the definition as desired. In [46], Jianke Lu straight forwardly defines where −(



L

f (τ ) 1 dτ = n+1 (τ − τ0 ) n!



L

f (n) (τ ) dτ , τ − τ0

τ0 ∈ L.

In 1990, Xiaoqin Wang obtained some results about the Hadamard principal value of high order singular integrals for functions of several complex variables (see [79]). In this chapter, we discuss the first kind of function in Clifford analysis
f (x) : Rn → An (R), where the element in An (R) is u = uA eA (uA ∈ A




¯ = x1 e1 − R), the element in Rn is x = nk=1 xk ek , herein e1 = 1, x
n x e , and denote the operator k=2 k k ∂¯x =

n  ∂ ∂ ∂ ek , ∂x = e1 . − ek ∂x ∂x ∂x 1 k k k=2 k=1 n 

Before giving the induction definition of high order singular integrals, we prove several lemmas. Lemma 1.1

Let u(x) =


A

( )xj

uA (x)eA , v(x) =

∂( ) ∂uA = , (vi )xk be continuous. Then . ∂xj ∂xk ∂ x (uv) = (∂ x u)v + u(∂ x v) +

n 

n 

j=2

i=1

vi (x)ei , x ∈ Rn ,

(ej u − uej )vxj ,

(1.3)

(uej − ej u)vxj ,

(1.4)

j=2

∂x (uv) = (∂x u)v + u(∂x v) +

n


162

Chapter VI u(∂ x v + ∂x v) = −(∂ x u)v − (∂x u)v + ∂ x (uv) + ∂x (uv) −

n


j=2

(ej u − uej )v xj −

n


j=2

(1.5)

(uej − ej u)vxj .

Proof It is clear that ∂ x (uv) = = = = =

n


n



(uA vi )xj eA ei A i=1 n n n n





vi ei + ej (uA )xj eA ej (uA )eA (vi )xj ei i=1 j=1 i=1 j=1 A A n n

(∂ x u)v + ej u(vxj ) = (∂ x u)v + ue1 vx1 + ej u(vxj ) j=1 j=2 n n

(∂ x u)v + ue1 vx1 + u ej vxj + (ej u − uej )vxj j=2 j=2 n
(ej u − uej )vxj . (∂ x u)v + u(∂ x v) + j=2 j=1

ej

Similarly, we can prove (1.4). Substitute v¯ into the position of v in (1.3), and add it to (1.4), then we immediately get (1.5). Corollary 1.2

If the conditions in Lemma 1.1 are satisfied, then

(1)

If v is independent of x, then ∂x (uv) = (∂x u)v.

(2)

If u = u1 e1 , then ∂¯x (uv) = (∂¯x u)v + u(∂¯x v); ∂x (uv) = (∂x u)v + u(∂x v), u(∂¯x v¯ + ∂x v) = −(∂¯x u)¯ v ) + ∂x (uv). v − (∂x u)v + ∂ x (u¯

Lemma 1.3

Let α > 0, x, y ∈ Rn , x = y, v1 =

1 . Then (2 − n − α)|x − y|n+α−2

x−y , v2 = (−α)|x − y|n+α

∂v1 ej (n + α)(xj − yj ) = − v1 , n+α ∂xj (−α)|x − y| |x − y|2

(1.6)

∂¯ v1 e¯j (n + α)(xj − yj ) = − v¯1 , n+α ∂xj (−α)|x − y| |x − y|2

(1.7)

High Order Singular Integrals and Differential Integral Equations ∂v2 xj − yj = , 1 ≤ j ≤ n, ∂xj |x − y|n+α

163

(1.8)

where e¯j =

⎧ ⎨ e1 ,

j = 1,

⎩ −e , j

2 ≤ j ≤ n,

x−y =

n 

j=1

(xj − yj )ej .

Proof Noting ∂v1 ∂xj

n −1  [(xi − yi )|x − y|−n−α ]xj ei α i=1

=

−1 ej |x − y|−n−α α n n −n−α−2 −n−α  −1  ( (xi −yi )2 ) 2 · 2(xj −yj ) + (xi −yi )ei α i=1 2 i=1 v1 ej − (n + α)(xj − yj ) , = −α|x − y|n+α |x − y|2

=

and substituting v 1 into the position of v1 in (1.6), we get (1.7), and n −n−α ∂v2 xj −yj −n−α+2  = , ( (xi −yi )2 ) 2 · 2(xj −yj ) = ∂xj 2(2−n−α) i=1 |x−y|n+α

hence (1.8) is valid. Let x, y ∈ Rn , x = y, α > 0. Then

Lemma 1.4

∂ x (¯ x − y¯) = n, ∂x (x − y) = n;

⎧ ⎨ ∂ x |x − y|σ = σ|x − y|σ−2 (x − y),

∂x(

⎩ ∂ |x − y|σ = σ|x − y|σ−2 (¯ x − y), σ > 0; x

x−y 1 x ¯−y ) = ∂x ( )= ; n+α n+α (−α)|x − y| (−α)|x − y| |x − y|n+α ⎧ ⎪ ⎪ ⎪ ⎨ ∂x( ⎪ ⎪ ⎪ ⎩

x−y 1 )= , n+α−2 (2 − n − α)|x − y| |x − y|n+α x ¯ − y¯ 1 )= . ∂x ( n+α−2 (2 − n − α)|x − y| |x − y|n+α

(1.9) (1.10)

(1.11)

(1.12)

164

Chapter VI

Proof It is evident that n 

x − y¯) = ∂¯x (¯

n  ∂ (xi − yi )ei ] [(x1 − y1 )e1 − ∂xj i=2

ej

j=1

= e1 e1 +

n 

ej (−1)ej = n.

j=2

Similarly, we can prove ∂x (x − y) = n. In addition, noting that ∂¯x |x − y|σ = =

n 

ej

j=1 n 

n σ ∂  ( (xi − yi )2 ) 2 ∂xj i=1

n σ σ  −1 ej ( (xi − yi )2 ) 2 · 2(xj − yj ) 2 i=1 j=1

= σ|x − y|σ−2

n 

j=1

(xj − yj )ej

= σ|x − y|σ−2 (x − y), (1.10) is derived. Moreover we can prove ∂x |x−y|σ = σ|x−y|σ−2 (¯ x −y). By means of Corollary 1.2 and (1.10), we get x ¯ − y¯ ) (−α)|x − y|n+α 1 |x − y|−n−α )](¯ x − y¯) + ∂ x (x − y) [∂¯x ( −α (−α)|x − y|n+α n n+α |x − y|−n−α−2 (x − y)(¯ x − y¯) + α −α|x − y|n+α 1 n n ( + 1) + n+α α |x − y| −α|x − y|n+α 1 . |x − y|n+α

∂¯x ( = = = =

1 x−y )= , namely −α|x − y|n+α |x − y|n+α (1.11) is true. On the basis of (1.10), the equality (1.12) can be derived.

Finally, we can similarly prove ∂x (

Lemma 1.5

Let u(x) =


A

uA (x)eA , v(x) =

ditions of Lemma 1.1 be satisfied. Then ∂¯x (vu) = (∂¯x v)u + v(∂¯x u) − 2

n


i=1

n 

i,j=2,i=j

vi (x)ei , and the con-

vi ei ej uxj ,

(1.13)

High Order Singular Integrals and Differential Integral Equations n 

∂x (vu) = (∂x v)u + v(∂x u) + 2

vi ei ej uxj .

165

(1.14)

i,j=2,i=j

Proof Taking into account ∂¯x (vu) = =

n


j=1 n


j=1

ej

n



i=1 A

(vi uA )xj ei eA

ej vxj u +

n


j=1

ej

= (∂¯x v)u + ve1 ux1 +

n


i=1 n


A

(uA )xj eA

ej uxj ) +

j=2 n n



j=2 i=1 n


= (∂¯x v)u + v(∂¯x u) − 2




ej vuxj

j=2 n


= (∂¯x v)u + v(e1 ux1 + = (∂¯x v)u + v(∂¯x u) +

vi ei

n


j=2

ej vuxj − v

vi ej ei uxj −

i,j=2,i=j

n n



j=2 i=1

n


j=2

ej uxj

vi ei ej uxj

vi ei ej uxj ,

(1.13) is derived. Similarly we can prove (1.14). Corollary 1.6 If the conditions in Theorem 1.4 are satisfied, then 1) If u is independent of x, then ∂¯x (vu) = (∂¯x v)u, ∂x (vu) = (∂x v)u. 2) When v = v1 e1 , then ∂¯x (vu) = (∂¯x v)u+v(∂¯x u), ∂x (vu) = (∂x v)u+ v(∂x u). Suppose that D is a connected open set in Rn , Ω is the boundary of D, and ∂¯x f (x, y), ∂x f (x, y) (x, y ∈ Ω) are H¨ older continuous. In (1.4), set 1 v= , u = f (x, y); (2 − n − α)|x − y|n+α−2 by (1.12), (1.8), we have f (x, y) −



x−y 1 = ∂x f (x, y) n+α |x − y| (2 − n − α)|x − y|n+α−2



n  ∂x f (x, y) xj − yj [f (x, y)ej − ej f (x, y)] − , n+α−2 (2 − n − α)|x − y| |x − y|n+α j=2

x−y xj − yj has α (> 0) order singularity, and also n+α |x − y| |x − y|n+α has a high order singularity. By means of the above equality, we know in which

166

Chapter VI

f (x, y) ] also has α(> 0) order singularity. However (2 − n − α)|x − y|n+α−2 x−y 1 possesses a lower order singularity than . Hence |x − y|n+α−2 |x − y|n+α according to the idea of Hadamard’s principal value of integrals and using the lower order singular integrals to inductively define the high order singular integrals, we can give the following definitions.

∂x [

Definition 1.1 

Ω

The α (> 0) order singular integral is defined as

f (x, y)(¯ x − y¯) dσx = |x − y|n+α



Ω

∂x f (x, y)dσx , y ∈ Ω. (n + α − 2)|x − y|n+α−2

1 , u = f (x, y); by (2 − n − α)|x − y|n+α−2 means of (1.12), (1.8), we can give the following definition. Similarly, in (1.3) let v =

Definition 1.2 

Ω

f (x, y)(x−y) dσx = |x−y|n+α



Ω

∂¯x f (x, y)dσx , α > 0, y ∈ Ω. (n+α−2)|x−y|n+α−2

In (1.5), let u = f (x, y), v = (1.7), we have

x−y (−α)|x−y|n+α ;

in view of (1.11), (1.6),

2f (x, y) |x − y|n+α   (∂¯x f (x, y))(¯ x − y¯) x−y + (∂x f (x, y)) = − (−α)|x − y|n+α (−α)|x − y|n+α +∂¯x −



n 

j=2





f (x, y)(¯ x − y¯) f (x, y)(x − y) + ∂x n+α (−α)|x − y| (−α)|x − y|n+α

(ej f (x, y) − f (x, y)ej ) ℵ −

n 

j=2



(f (x, y)ej − ej f (x, y)) ℵ,

where ℵ=

*

+

ej (n + α)(xj − yj ) (x − y) − · . n+α 2 (−α)|x − y| |x − y| (−α)|x − y|n+α

Similarly to the above discussion, we see that in the right-hand side 1 possesses an α + 1 order of the above equality, the term |x − y|n+α x−y x−y , have an singularity and other terms n+α (−α)|x − y| (−α)|x − y|n+α α (> 0) order singularity. So we can inductively give

High Order Singular Integrals and Differential Integral Equations Definition 1.3 

Ω

167

The integral

f (x, y)dσx = |x − y|n+α



Ω

(∂¯x f )(¯ x − y¯) + (∂x f )(x − y) dσx , α > 0, y ∈ Ω 2α|x − y|n+α

is called the first kind of high order singular integral of quasi-BochnerMartinelli type with one singular point. In (1.14), set u = f (x, y), v =

1 ; (2 − n − α)|x − y|n+α−2

by means of the components vi = 0 (2 ≤ i ≤ n) of v and (1.12), we can get (¯ x − y¯)f (x, y) f (x, y) ∂x f (x, y) = ∂x [ ]− . n+α n+α−2 |x−y| (2−n−α)|x−y| (2−n−α)|x−y|n+α−2 Similarly to the above discussion, we see that order singularity than the α order of

1 has a lower |x − y|n+α−2

x ¯ − y¯ , so we can also give |x − y|n+α

Definition 1.4 

Ω

(¯ x − y¯)f (x, y)dσx = |x − y|n+α



Ω

∂x f (x, y)dσx , α > 0, y ∈ Ω. (n + α − 2)|x − y|n+α−2

By means of (1.12), (1.13), we can define other high order singular integrals through a similar method. Definition 1.5 

Ω

(x − y)f (x, y)dσx = |x − y|n+α



Ω

∂¯x f (x, y)dσx , α > 0, y ∈ Ω. (n + α − 2)|x − y|n+α−2

In view of Definitions 1.1, 1.2, 1.4 and 1.5, we get 

Ω



Ω

(¯ x − y¯)f (x, y)dσx = |x − y|n+α (x − y)f (x, y)dσx = |x − y|n+α



Ω



Ω

f (x, y)(¯ x − y¯) dσx , α > 0, y ∈ Ω, |x − y|n+α

(1.15)

f (x, y)(x − y) dσx , α > 0, y ∈ Ω. |x − y|n+α

(1.16)

The above two high order singular integrals are called the second and third kinds of high order singular integrals of quasi-Bochner-Martinelli type with one singular point respectively.

168

Chapter VI

Definition 1.6 If f (x, y), (x, y) ∈ Ω × Ω is still H¨older continuous after the action of the operator ∂¯x , ∂x for p (≤ m) times, and 0 < β < 1 (m) is its H¨older index, then we say that f (x, y) belongs to Hx (β), and (m) (0) write f ∈ Hx (β). When m = 0, denote by f ∈ Hx (β) the H¨older continuity of f on x ∈ Ω, where β is the H¨older index. Similarly we can (m) define f ∈ Hy (β). In the following, we prove the recurrence formulas of the first, second and third kinds of high order singular integrals of quasi-BochnerMartinelli type. Theorem 1.7 Then

(2m+2)

Let n > 1, α > 2m > 0, f (x, y) ∈ Hx

(β), y ∈ Ω.



f (x, y)(¯ x − y¯)dσx µ = n+α |x − y| n + α − 2m − 2



∆m x (∂x f (x, y))dσx , (1.17) |x − y|n+α−2m−2



f (x, y)(x − y) µ dσx = n+α |x − y| n + α − 2m − 2



¯ ∆m x (∂x f (x, y))dσx , (1.18) |x − y|n+α−2m−2

Ω

Ω



Ω

f (x, y)dσx µ = |x − y|n+α α(n + α − 2m − 2)

Ω

Ω



Ω

∆m+1 f (x, y)dσx x , |x − y|n+α−2m−2

(1.19)

where the operator ∆x = ∂x ∂¯x = ∂¯x ∂x , µ=

(α − 2m − 2)!!(n + α − 2m − 2)!! , (α − 2)!!(n + α − 2)!!

and r!! expresses the multiplication of the integers from r to the least integer every time decrease 2, and if −2 < r ≤ 0, denote r!! = 1, then µ=

(n + α − 2m − 2)!! . (n + α − 2)!!(α − 2)(α − 4) · · · (α − 2m)

Proof On the basis of Definitions 1.1, 1.2 and 1.3, we have 



∂x f (x, y)dσx f (x, y)(¯ x − y¯)dσx = n+α n+α−2 |x − y| Ω (n + α − 2)|x − y| Ω  (∂ x ∂x f )(x − y) + (∂x ∂x f )(x − y) = dσx n+α−2 Ω 2(n + α − 2)(α − 2)|x − y|  [∆x ∂x f (x, y)]dσx = (n + α − 2)(n + α − 4)(α − 2)|x − y|n+α−4 Ω

(1.20)

High Order Singular Integrals and Differential Integral Equations

169



[(∂ x ∆x ∂x f )(x − y) + (∂x ∆x ∂x f )(x − y)]dσx n+α−4 Ω 2(n + α − 2)(n + α − 4)(α − 2)(α − 4)|x − y|  [∆2x ∂x f (x, y)]dσx = n+α−6 Ω (n + α − 2)(n + α − 4)(n + α − 6)(α − 2)(α − 4)|x − y|  2 [∆x ∂x f (x, y)]dσx (α − 6)!!(n + α − 6)!! = . (n + α − 6)(α − 2)!!(n + α − 2)!! Ω |x − y|n+α−6 =

Inductively, we can get 

Ω

f (x, y)(¯ x − y¯)dσx µ = n+α |x − y| n + α − 2m − 2



Ω

(∆x m ∂x f (x, y))dσx . |x − y|n+α−2m−2

This shows that (1.17) is valid. Similarly, we can prove (1.18). By means of Definitions 1.1, 1.2 and 1.3, we know 



(∂ x f )(x − y) + (∂x f )(x − y) f (x, y)dσx 1 = dσx n+α 2α Ω |x − y|n+α Ω |x − y|  (∆x f )dσx 1 = α(n + α − 2) Ω |x − y|n+α−2  [(∂ x ∆x f )(x − y) + (∂x ∆x f )(x − y)] 1 dσx = 2α(n + α − 2)(α − 2) Ω |x − y|n+α−2  [∆2x f (x, y)]dσx = n+α−4 Ω α(α − 2)(n + α − 2)(n + α − 4)|x − y|  [∆2x f (x, y)]dσx (α − 4)!!(n + α − 4)!! = . α(n + α − 4)(α − 2)!!(n + α − 2)!! Ω |x − y|n+α−4 Moreover, it is easy to verify (1.19). (2k+2)

(β), y ∈ Ω, n > 1, 0 < r < Let f (x, y) ∈ Hx (n − r − 3)!! . Then there exist the first, sec1, λ = (n + 2k − r − 1)!!(2k − r − 1)!! ond and third kinds of high order singular integrals of quasi-BochnerMartinelli type. Moreover, they can be expressed in the forms Theorem 1.8



Ω



Ω



f (x, y)(¯ x − y¯)dσx =λ n+2k+1−r |x − y| f (x, y)(x − y)dσx =λ |x − y|n+2k+1−r



Ω



Ω

f (x, y)dσx λ = n+2k+1−r |x−y| 2k+1−r Ω

[∆kx ∂x f (x, y)]dσx , |x − y|n−1−r

(1.21)

[∆kx ∂¯x f (x, y)]dσx , |x − y|n−1−r

(1.22)



[∆k+1 x f (x, y)]dσx . n−1−r |x−y| Ω

(1.23)

170

Chapter VI

Proof In Theorem 1.7, setting α = 2k + 1 − r, m = k, and according to (1.20), we get µ (n − r − 1)!! = = λ, n + α − 2m − 2 (n − r − 1)(n + 2k − r − 1)!!(2k − r − 1)!! hence (1.21), (1.22), (1.23) are correct. The integrals on the right-hand side of (1.21), (1.22),(1.23) converge under the general sense as in Section 4, Chapter II. So there exist the above three kinds of high order singular integrals of quasi-Bochner-Martinelli type, and then (1.21), (1.22), (1.23) are their calculational formulas. This completes the proof. The values calculated by (1.21) (1.22) (1.23) are called the Hadamard principal value of the first, second and third kinds of high order singular integrals of quasi-Bochner-Martinelli type. In the following, we derive the differential formulas of three kinds of high order singular integrals of quasi-Bochner-Martinelli type. Theorem 1.9

Let u(y) =


A

v3 =

uA (y)eA , (uA (y))yk be continuous, and

1 , 0 < r < 1, x, y ∈ Rn , x = y. Then |x − y|n−1−r 







n 

(uej − ej u)

(¯ x − y¯)u ∂v3 u(¯ x − y¯) − = (n − r − 1) , (1.24) n+1−r ∂yj |x − y| |x − y|n+1−r

n 

(ej u − uej )

(x − y)u u(x − y) ∂v3 = (n − r − 1) − . (1.25) n+1−r ∂yj |x − y| |x − y|n+1−r

j=2

j=2

Proof In accordance with (1.8), we have ∂v3 (xj − yj ) = (n − r − 1) , ∂yj |x − y|n+1−r

and then n 

j=2

=

(uej − ej u)

∂v3 ∂yj

⎤ n n   (x − y )e u (x − y )e j j j ⎦ j j j − (n − r − 1) ⎣u n+1−r n+1−r ⎡



j=2

|x − y|

j=2

|x − y|

−u(x1 − y1 )e1 (x1 − y1 )e1 u +(n − 1 − r) ( + |x − y|n+1−r |x − y|n+1−r   (¯ x − y¯)u u(¯ x − y¯) − . = (n − r − 1) |x − y|n+1−r |x − y|n+1−r



High Order Singular Integrals and Differential Integral Equations

171

Thus (1.24) is obtained. By means of [(¯ x − y¯) + (x − y)]u − u[(¯ x − y¯) + (x − y)] = 0, we know that (1.25) is true. (m)

Definition 1.7 Let f (x, y) ∈ Hx (β1 ), (x, y) ∈ Ω × Ω, and f (x, y) ∈ (p) Hy (β2 ), 0 < βi < 1, i = 1, 2. Then we say that f (x, y) belongs to H (m,p) (β1 , β2 ), and is denoted by f (x, y) ∈ H (m,p) (β1 , β2 ). Theorem 1.10 Let f (x, y) ∈ H (m+2k+2,m) (β1 , β2 ), 0 < βi < 1, i = 1, 2, 0 < r < 1, λ be as stated in Theorem 1.8. Then ∂ym



f (x, y)(¯ x − y¯)dσx =λ |x − y|n+2k+1−r



[(∂y + ∂x )m ∆kx ∂x f (x, y)] dσx , (1.26) |x − y|n−1−r

∂ym



f (x, y)(x − y)dσx =λ |x − y|n+2k+1−r



[(∂y + ∂x )m ∆kx ∂¯x f (x, y)]dσx , (1.27) |x − y|n−1−r

∂ym

Ω

Ω



Ω

Ω

Ω

f (x, y)dσx λ = |x − y|n+2k+1−r 2k + 1 − r



Ω

[(∂y + ∂x )m ∆k+1 x f (x, y)]dσx . |x − y|n−1−r (1.28)

Proof On the basis of (1.12), we have ∂y

y−x x−y 1 = −(n − 1 − r) = (n − 1 − r) . n−1−r n+1−r |x − y| |x − y| |x − y|n+1−r

By means of (1.21), (1.4), (1.24), (1.15) and Definition 1.1, we obtain 

f (x, y)(¯ x − y¯)dσx n+2k+1−r Ω |x − y|   [∂y ∆kx ∂x f (x, y)]dσx [∆kx ∂x f (x, y)](x − y)dσx =λ + λ(n − 1 − r) |x − y|n−1−r |x − y|n+1−r Ω Ω  (x − y)[∆kx ∂x f (x, y)]dσx +λ(n − 1 − r) |x − y|n+1−r Ω  [∆kx ∂x f (x, y)](x − y)dσx −λ(n − 1 − r) |x − y|n+1−r Ω   [∆kx ∂x f (x, y)](x − y)dσx (∂y ∆kx ∂x )f (x, y)dσx + λ(n − 1 − r) =λ |x − y|n−1−r |x − y|n+1−r Ω Ω   [∂y ∆kx ∂x f (x, y)]dσx [∂x ∆kx ∂x f (x, y)]dσx =λ + λ(n − 1 − r) n−1−r |x − y|n−1−r Ω Ω (n − 1 − r)|x − y|  (∂y + ∂x )(∆kx ∂x f (x, y)) dσx . =λ |x − y|n−1−r Ω ∂y

172

Chapter VI

Inductively, it is easy to see that (1.26) is valid. By means of (1.22), (1.23), we can use the same method to prove (1.27) and (1.28). Theorem 1.11 If f (x, y) ∈ H (m+2k+2,m) (β1 , β2 ), 0 < βi < 1, i = 1, 2, 0 < r < 1, λ is as stated in Theorem 1.8, then ∂¯ym



f (x, y)(¯ x − y¯)dσx =λ n+2k+1−r |x − y|



[(∂¯y + ∂¯x )m ∆kx ∂x f (x, y)]dσx , |x − y|n−1−r

(1.29)

∂¯ym



f (x, y)(x − y)dσx =λ |x − y|n+2k+1−r



[(∂¯y + ∂¯x )m ∆kx ∂¯x f (x, y)]dσx , |x − y|n−1−r

(1.30)

∂¯ym

Ω

Ω



Ω

Ω

Ω

f (x, y)dσx λ = n+2k+1−r |x − y| 2k + 1 − r



Ω

[(∂¯y + ∂¯x )m ∆k+1 x f (x, y)]dσx . |x − y|n−1−r (1.31)

Proof In view of (1.12), we have ∂¯y

1 −(n − 1 − r)(y − x) (n − 1 − r)(x − y) = = . n−1−r n+1−r |x − y| |x − y| |x − y|n+1−r

By means of (1.21),(1.3),(1.25),(1.16) and Definition 1.2, we get 

f (x, y)(¯ x − y¯)dσx n+2k+1−r Ω |x − y|   ¯ k [∆kx ∂x f (x, y)](n − 1 − r)(x − y)dσx [∂y ∆x ∂x f (x, y)]dσx + λ =λ |x − y|n−1−r |x − y|n+1−r Ω Ω  k (n − 1 − r)(x − y)[∆x ∂x f (x, y)]dσx +λ |x − y|n+1−r Ω  (n − 1 − r)[∆kx ∂x f (x, y)](x − y)dσx −λ |x − y|n+1−r Ω  ¯ k  [∂y ∆x ∂x f (x, y)]dσx [∆kx ∂x f (x, y)](x − y)dσx =λ + λ(n − 1 − r) |x − y|n−1−r |x − y|n+1−r Ω Ω   ¯ k [∂¯x ∆kx ∂x f (x, y)]dσx [∂y ∆x ∂x f (x, y)]dσx + λ(n − 1 − r) =λ n−1−r |x − y|n−1−r Ω (n − 1 − r)|x − y| Ω  k (∂¯y + ∂¯x )(∆x f (x, y)) =λ dσx . |x − y|n−1−r Ω

∂¯y

Inductively, it is easy to see that (1.29) is true. By means of (1.22), (1.23), we can use the same method to prove (1.30) and (1.31).

High Order Singular Integrals and Differential Integral Equations

173

Corollary 1.12 Let f (x, y) ∈ H (m+p+2k+2,m+p) (β1 , β2 ), 0 < βi < 1, i = 1, 2, 0 < r < 1, λ be as stated in Theorem 1.8. Then 



[(∂¯y + ∂¯x )m (∂y +∂x )p ∆kx ∂x f (x, y)]dσx , |x−y|n−1−r Ω (1.32)  ¯  m p k ¯ ¯ (∂y + ∂x ) (∂y +∂x ) ∆x ∂x f (x, y)]dσx f (x, y)(x−y)dσx ∂¯ym ∂yp =λ , n+2k+1−r |x−y|n−1−r Ω Ω |x−y| (1.33)  f (x, y)dσx ∂¯ym ∂yp |x − y|n+2k+1−r Ω (1.34)  [(∂¯y + ∂¯x )m (∂y + ∂x )p ∆k+1 λ x f (x, y)]dσx = . 2k + 1 − r Ω |x − y|n−1−r

∂¯ym ∂yp

f (x, y)(¯ x − y¯)dσx =λ n+2k+1−r Ω |x−y|

Proof It is easy to prove this corollary by using Theorems 1.10 and 1.11. Theorem 1.13 If f (x, y) ∈ H (m+2k+2,p) (β1 , β2 ), 0 < βi < 1, i = 1, 2, t1 , t2 ∈ Ω, 0 < r < 1, λ is as stated in Theorem 1.8, then ∂¯tm1 ∂tp2



 [(∂¯xm ∂tp2 ∆kx ∂x f (x, t2 )]dσx f (x, t2 )(¯ x − t¯1 )dσx =λ , (1.35) |x − t1 |n+2k+1−r |x − t1 |n−1−r Ω

∂¯tm1 ∂tp2



f (x, t2 )(x − t1 )dσx =λ |x − t1 |n+2k+1−r

∂¯tm1 ∂tp2



f (x, t2 )dσx λ = n+2k+1−r |x − t1 | 2k + 1 − r

Ω

Ω

Ω



Ω

[(∂¯xm ∂tp2 ∆kx ∂¯x f (x, t2 )]dσx , (1.36) |x − t1 |n−1−r 

Ω

[(∂¯xm ∂tp2 ∆k+1 x f (x, t2 )]dσx . |x − t1 |n−1−r (1.37)

Proof According to (1.21), Corollaries 1.2, 1.6, (1.12), (1.16) and Definition 1.2, we have 

f (x, t2 )(¯ x − t¯1 )dσx n+2k+1−r Ω |x − t1 |  [∆kx ∂x f (x, t2 )]dσx = ∂¯tm1 ∂tp2 λ |x − t1 |n−1−r Ω  p k [∂t2 ∆x ∂x f (x, t2 )]dσx = λ∂¯tm1 |x − t1 |n−1−r Ω ∂¯tm1 ∂tp2

174

Chapter VI 

(n − 1 − r)(x − t1 )[∂tp2 ∆kx ∂x f (x, t2 )]dσx |x − t1 |n+1−r Ω  ∂tp2 ∆kx ∂x f (x, t2 )](x − t1 )dσx = λ(n − 1 − r)∂¯tm−1 1 |x − t1 |n+1−r Ω  [(∂¯x ∂tp2 ∆kx ∂x f (x, t2 )]dσx = λ∂¯tm−1 1 |x − t1 |n−1−r Ω  [(∂¯xm ∂tp2 ∆kx ∂x f (x, t2 )]dσx . = λ |x − t1 |n−1−r Ω

= λ∂¯tm−1 1

This shows that (1.35) is correct. Similarly, By means of (1.22), Corollaries 1.2 and 1.6, (1.12), (1.15) and Definition 1.2, we get (1.36). At last, by using (1.23), Corollaries 1.2, 1.6, (1.12), (1.16) and Definition 1.2, we can get (1.37). In the following, we discuss the fourth, fifth and sixth kinds of high order singular integrals of quasi-Bochner-Martinelli type. Theorem 1.14

Suppose that u(x) =

are as those in Lemma 1.1. Then


A

uA (x)eA , v(x) =

n


i=1

vi (x)ei

[(∂x − ∂¯x )(v + v¯)]u = (v + v¯)[(∂¯x − ∂x )u] + (∂x − ∂¯x )[(v + v¯)u] , (1.38) [∂x (v + v¯)]u = −(v + v¯)∂x u + ∂x [(v + v¯)u] .

(1.39)

Proof We substitute v by v¯ in Lemma 1.5 and notice when j ≥ 2, v¯j = −vj , then ∂¯x (¯ v u) = (∂¯x v¯)u + v¯(∂¯x u) + 2

n 

vi ei ej uxj ,

(1.40)

n 

vi ei ej uxj .

(1.41)

i,j=2,i=j

and v u) = (∂x v¯)u + v¯(∂x u) − 2 ∂x (¯

i,j=2,i=j

In view of (1.12), (1.14), (1.40), (1.41), we get (1.38). By means of (1.14), (1.41), it is easy to derive (1.39). The proof is finished.

High Order Singular Integrals and Differential Integral Equations

175

x ¯−¯ y Suppose α > 0, and let v = (−α)|x−y| n+α , u = f (x, y) in (1.38) of Theorem 1.11. By using Lemma 1.3, we can get

x − y¯)2 n + α (¯ (x − y)2 [(∂x − ∂¯x )(v+¯ [ v )]u = − ]f (x, y), (1.42) α |x − y|n+α+2 |x − y|n+α+2 x ¯ − y¯ x−y 1 + ][(∂x − ∂¯x )f (x, y)]. (v + v¯)[(∂¯x − ∂x )u] = [ n+α α |x − y| |x − y|n+α (1.43) The right-hand side of (1.43) has a lower order singularity than the right-hand side of (1.42). Similarly to the discussion of Definition 1.1, due to (1.38), we shall use the integral on the right-hand side of (1.43) to define the following integral. Definition 1.8   Ω

=



(¯ x − y¯)2 (x − y)2 f (x, y)dσx − |x − y|n+α−2 |x − y|n+α+2

1 n+α



Ω

(¯ x − y¯) + (x − y) [(∂x − ∂¯x )f (x, y)]dσx , |x − y|n+α

(1.44)

where α > 0, y ∈ Ω. The singular integral on the left side of (1.44) is called the fourth kind of high order singular integral of quasi-Bochner-Martinelli type with one singular point. Similarly, let v =

x ¯ − y¯ , u = f (x, y) in (1.39), then we get |x − y|n+α





2−n−α (¯ x − y¯)2 (n + α) [∂x (v + v¯)]u = − f (x, y), |x − y|n+α |x − y|n+α+2 −(v + v¯)∂x u =

−[(¯ x − y¯) + (x − y)] ∂x f (x, y). |x − y|n+α

(1.45)

(1.46)

In the terms on the right-hand side of (1.45), (1.46), it is only the term: (¯ x − y¯)2 (n + α) f (x, y), whose high order singular integral has not been |x − y|n+α+2 defined. Similarly, by means of (1.39), we can inductively define the fifth kind of high order singular integrals of quasi-Bochner-Martinelli type with one singular point.

176

Chapter VI

Definition 1.9 

Ω



f (x, y)dσx (¯ x − y¯)2 f (x, y)dσx −(n + α − 2) = n+α+2 n+α |x − y| n+α Ω |x − y|  [(¯ x − y¯) + (x − y)]∂x f (x, y) 1 dσx , + n+α Ω |x − y|n+α

(1.47)

if y ∈ Ω, α > 0. Finally according to (1.44), (1.47), it is easy to give the definition of the sixth kind of high order singular integrals of quasi-BochnerMartinelli type with one singular point. Definition 1.10 

Ω



(x − y)2 f (x, y)dσx f (x, y)dσx (n + α − 2) = |x − y|n+α+2 −(n + α) Ω |x − y|n+α  [(¯ x − y¯) + (x − y)] ¯ 1 + ∂x f (x, y)dσx , n+α Ω |x − y|n+α

(1.48)

for y ∈ Ω, α > 0. As a supplement to Lemma 1.4, it is easy to prove the following equalities: ∂¯x (

x−y 2−n (n + α)(x − y)2 )= − , α > 0, n+α+2 n+α |x − y| |x − y| |x − y|n+α+2

(1.49)

2−n (n + α)(¯ x − y¯)2 x ¯ − y¯ ) = − , α > 0. |x − y|n+α |x − y|n+α |x − y|n+α+2

(1.50)

∂x (

From the second term on the right side of (1.49), (1.50), we see that for investigating high order singular integrals in real Clifford analysis, it is necessary to discuss the fourth, fifth and sixth kinds of high order singular integrals of quasi-Bochner-Martinelli type with one singular point. By means of Definitions 1.8, 1.9, 1.10 and recurrence formulas, calculational formulas, and differential formulas for the first, second, and third kinds of high order singular integrals of quasi-Bochner-Martinelli type with one singular point, we can get the recurrence formulas, calculational formulas, and differential formulas for the fourth, fifth and sixth kinds of high order singular integrals of quasi-Bochner-Martinelli type. Because of page limitation, we don’t prove all formulas, and only prove one of every kind of high order singular integrals; the other proofs are left to readers.

High Order Singular Integrals and Differential Integral Equations

177

Theorem 1.15 If n > 1, α > 2m > 0, f (x, y) ∈ Hx2m+2 (β), y ∈ Ω, µ is as stated in Lemma 1.6, then we have the recurrence formula   Ω

=



(¯ x − y¯)2 (x − y)2 − f (x, y)dσx |x − y|n+α+2 |x − y|n+α+2 7 8  ∆m (∂ 2 − ∂¯2 )f (x, y) µ x

x

(n + α)(n + α − 2m − 2)

Ω

|x −

x y|n+α−2m−2

(1.51) dσx .

Proof By Definition 1.8, (1.15), (1.16), (1.17) and (1.18), we have   Ω

= = =



(¯ x − y¯)2 (x − y)2 f (x, y)dσx − |x − y|n+α+2 |x − y|n+α+2 

(¯ x − y¯) + (x − y) 7 ¯x )f (x, y)8 dσx (∂ − ∂ x |x − y|n+α Ω  ¯ ¯ ∆m µ x (∂x + ∂x )(∂x − ∂x )f (x, y) dσx , n+α−2m−2 (n + α)(n + α − 2m − 2) Ω |x − y| 7 2 8  ¯2 ∆m µ x (∂x − ∂x )f (x, y) dσx . (n + α)(n + α − 2m − 2) Ω |x − y|n+α−2m−2 1 n+α

(2k+2)

(β), y ∈ Ω, λ is as Theorem 1.16 Suppose that f (x, y) ∈ Hx stated in Theorem 1.8, and n > 1, 0 < r < 1. Then 

(¯ x − y¯)2 f (x, y)dσx n+2k+3−r Ω |x − y|  ∆k+1 λ(2 − n) x f (x, y) dσx = (n + 2k + 1 − r)(2k + 1 − r) Ω |x − y|n−1−r 7 8  ∆kx ∂x2 f (x, y) λ + dσx . n + 2k + 1 − r Ω |x − y|n−1−r Proof By Definition 1.9, (1.15), and Theorem 1.8, we get 

(¯ x − y¯)2 f (x, y)dσx n+2k+3−r Ω |x − y|  f (x, y)dσx −(n + 2k − 1 − r) = n+2k+1−r n + 2k + 1 − r Ω |x − y|  [(¯ x − y¯) + (x − y)] ∂x f (x, y)dσx 1 + n + 2k + 1 − r Ω |x − y|n+2k+1−r

(1.52)

178

Chapter VI 

2

3

∆k+1 x f (x, y) dσx

−λ(n + 2k − 1 − r) (n + 2k + 1 − r)(2k + 1 − r) Ω |x − y|n−1−r 7 8  ∆kx (∂x + ∂¯x )∂x f (x, y) dσx λ + n + 2k + 1 − r Ω |x − y|n−1−r  ∆k+1 λ(2 − n) x f (x, y) dσx = (n + 2k + 1 − r)(2k + 1 − r) Ω |x − y|n−1−r 7 8  ∆kx ∂x2 f (x, y) dσx λ + . n + 2k + 1 − r Ω |x − y|n−1−r =

Theorem 1.17 Let f (x, y) ∈ H (m+2k+2,m) (β1 , β2 ), 0 < βi < 1, 0 < r < 1, λ be the same as that in Theorem 1.8. Then ∂ym =



Ω

(x − y)2 f (x, y)dσx |x − y|n+2k+3−r

λ(2 − n) (n + 2k + 1 − r)(2k + 1 − r) +

λ n + 2k + 1 − r



Ω

2



Ω

2

3

(∂y + ∂x )m ∆k+1 x f (x, y) dσx |x − y|n−1−r 3

(∂y + ∂x )m ∆kx ∂¯x2 f (x, y) dσx |x − y|n−1−r

. (1.53)

Proof By Definitions 1.5, 1.10, and Theorem 1.10, we can get 

(x − y)2 f (x, y)dσx n+2k+3−r Ω |x − y|  f (x, y)dσx (n + 2k − 1 − r) m ∂y = n+2k+1−r −(n + 2k + 1 − r) Ω |x − y|  [(¯ x − y¯) + (x − y)] ∂¯x f (x, y)dσx 1 ∂ym + n + 2k + 1 − r |x − y|n+2k+1−r Ω  (∂y + ∂x )m ∆k+1 (n + 2k − 1 − r)λ x f (x, y)dσx = −(n + 2k + 1 − r)(2k + 1 − r) Ω |x − y|n−1−r 2 3  {(∂ + ∂ )m (∆k+1 + ∆k ∂¯2 )f (x, y) } y x x x x λ dσx + n + 2k + 1 − r Ω |x − y|n−1−r ∂ym

=

λ(2 − n) (n + 2k + 1 − r)(2k + 1 − r) +

λ n + 2k + 1 − r



Ω

2



Ω

2

3

(∂y + ∂x )m ∆k+1 x f (x, y) dσx |x − y|n−1−r 3

(∂y + ∂x )m ∆kx ∂¯x2 f (x, y) dσx |x − y|n−1−r

.

High Order Singular Integrals and Differential Integral Equations

2

179

The H¨ older Continuity of High Order Singular Integrals in Real Clifford Analysis By the known inequality |σ1µ − σ2µ | ≤ |σ1 − σ2 |µ ,

(2.1)

where 0 ≤ µ ≤ 1, σi > 0, i = 1, 2 [87], we can prove the following theorem similar to the Hile type lemma (see Section 2, Chapter II). Theorem 2.1 Let m be a positive integer, α > 0 and [α] be the integral part of α; x, y, yˆ, t, tˆ ∈ Rn , t, tˆ = 0, x = y, x = yˆ. Then        1   tˆk  t − tˆ 1  m−1      m − m ≤  |tˆ|−m ,  |t| |tˆ|  k=0  t   t 

and

   αk [α]    x − yˆ  [α]+1   1 1   ≤  −  |x − y|α |x − yˆ|α  k=0  x − y 

  α  y − yˆ  [α]+1   |x − yˆ|−α . x − y 

(2.2)

(2.3)

Proof Noting that

   ˆm    |t| − |t|m   1  1    |t|m − ˆm  = |t|m |tˆ|m t  m−1  
ˆ m−1−k k ˆ     |t| |t| |t| − |t| m−1   tˆk  t − tˆ k=0 ≤ =     |tˆ|−m , t  t  |t|m |tˆ|m k=0

it is clear that (2.2) is true. By means of (2.1), (2.2), we have     1 1    |x − y|α − |x − yˆ|α      1 1   =  −  α α  (|x − y| [α]+1 )[α]+1  [α]+1 [α]+1 ) (|x − yˆ|   α α  αk [α]   α '−[α]−1  x − yˆ  [α]+1  |x − y| [α]+1 − |x − yˆ| [α]+1  &   ≤   |x − yˆ| [α]+1 α x − y    |x − y| [α]+1 k=0   α  [α]+1  αk  [α]    x − yˆ  [α]+1 |x − y| − |x − yˆ|   |x − yˆ|−α ≤ α x − y  [α]+1

≤

k=0 [α] 

αk   x − yˆ  [α]+1   x − y 

k=0

|x − y|

  α  y − yˆ  [α]+1   |x − yˆ|−α . x − y 

180

Chapter VI

From this section to the last section in this chapter, we suppose that the boundary Ω of the domain D is a smooth, oriented, compact, Liapunov surface, and the orientation of Ω is the induced orientation of D. In view of the definition of Liapunov surface, there exists a positive number d > 0, such that for any point N0 ∈ Ω, the sphere with the center at N0 and radius d (or less then d) can be divided Ω into two parts, and the interior part of the sphere is denoted by Ω′ . We choose N0 as the origin of the local generalized sphere coordinate system, such that xn -axis and the outwards normal direction of Ω at N0 are identical. If d is small enough and the outward normal line of Ω′ at any point N (ξ1 , · · · , ξn ) is n0 . Let r0 = |N0 N |. Denote by ρ0 the length of the project of r0 on the tangent plane through N0 and by (ρ0 , φ1 , · · · , φn−2 ) the local generalized  D(ξ1 ,...,ξn−1 )  sphere coordinate of N , and let J =  D(ρ  be the Jacobian 0 ,ϕ1 ,...,ϕn−2 ) determinant of coordinate transformation. In Section 2, Chapter I, we have obtained 1 (2.4) cos(n0 , xn ) ≥ , |J| ≤ ρ0 n−2 . 2 For any Φ ∈ H (¯n+2k+2,¯n) (β1 , β2 ), 0 < βi < 1, i = 1, 2, the norm of Φ is defined as Φβ =



C(∂¯xm+2k+2 ∂y p Φ,Ω × Ω)+

0≤m+p≤¯ n



H(∂¯xm+2k+2 ∂y p Φ,Ω × Ω).

0≤m+p≤¯ n

ˆ = ∂¯m+2k+2 ∂ p Φ, where For convenience, we denote Φ x y ˆ Ω × Ω) = C(Φ, ˆ Ω × Ω) = H(Φ,

max

(x,y)∈Ω×Ω

ˆ |Φ(x, y)|,

ˆ 1 , y1 ) − Φ(x ˆ 2 , y2 )| |Φ(x , β β2 1 (x1 ,y1 ),(x2 ,y2 )∈Ω×Ω |x1 − x2 | + |y1 − y2 | sup

in which (x1 , y1 ) = (x2 , y2 ). This definition is different from the definition of H(f, ∂Ω × ∂Ωβ) in Section 4, Chapter II, but it is the same as that in Tongde Zhong’s paper in 1980 [88]2), and it is easy to prove H (¯n+2k+2, n¯ ) (β1 , β2 ) is a compact Banach space, and its norm possesses the property F + Gβ ≤ F β + Gβ , F Gβ ≤ 2n−1 F β Gβ , where F, G ∈ H (¯n+2k+2,¯n) (β1 , β2 ). Theorem 2.2

If the operator on Ω is defined as (P ϕ)(y) =



Ω

ϕ(x, y)dσx , |x − y|n−1−r

181

High Order Singular Integrals and Differential Integral Equations

here 0 < r < 1, ϕ(x, y) ∈ H (0,0) (β1 , β2 ), 0 < βi < 1, i = 1, 2, ω = (0) min(β1 , β2 ), r > β1 , β1 < n−1−r n−1 , y ∈ Ω, then P ϕ ∈ Hy (ω) on Ω. Proof We introduce the operator (θ1 ϕ)(y) =



Ω

ϕ(x, y) − ϕ(y, y) dσx , |x − y|n−1−r

(θ2 ϕ)(y) =



Ω

ϕ(y, y)dσx . |x − y|n−1−r

Obviously (P ϕ)(y) = (θ1 ϕ)(y) + (θ2 ϕ)(y). Firstly, we discuss the H¨older continuity of (θ1 ϕ)(y). From Section 2, Chapter 1, we know dσx = m(x)dsx , m(x) =

n


j=1

ej cos(m, ej ) is the outward normal direction of

x on Ω. For any y1 , y2 ∈ Ω, we denote δ = |y1 − y2 |. Let 3δ < d, here d is as stated before. We make a sphere with the center at y1 and radius 3δ. Denote by Ω1 the interior part of Ω and by Ω2 the left part. It is easy to see that |(θ1 ϕ)(y1 ) − (θ1 ϕ)(y2 )|





  ϕ(x, y1 ) − ϕ(y1 , y1 ) ϕ(x, y2 ) − ϕ(y2 , y2 )   dσ | + dσ ≤ | Ω1 x x   |x − y1 |n−1−r |x − y2 |n−1−r Ω1    6 1 1  − )(ϕ(x, y1 ) − ϕ(y1 , y1 ))dσx  +  Ω2 ( n−1−r n−1−r |x − y1 | |x − y2 |   6 [(ϕ(x, y1 ) − ϕ(x, y2 )) + (ϕ(y2 , y2 ) − ϕ(y1 , y1 ))]   dσx  +  Ω2 n−1−r |x − y2 | 6

= L1 + L2 + L3 + L4 .

The formula (2.4) about N0 ∈ Ω discussed at the beginning of the section is used as y1 , and the projective domain of Ω1 on the tangent plane of y1 is denoted by π1 , from which we can get 

dsx n−1−r−β1 Ω1 |x − y1 | 6 cos(n0 , xn )dsx = G1 ϕβ Ω1 |x − y1 |n−1−r−β1 cos(n0 , xn )  dξ1 · · · dξn−1 ≤ G1 ϕβ 1 π1 (|x − y1 |n−1−r−β1 ) 2 L1 ≤ G1 ϕβ

≤ G2 ϕβ

6 3δ (ρ0 0

ρ0

)n−2 dρ

0 n−1−r−β1

(2.5)

≤ G3 ϕβ δ r δ β1

≤ G4 ϕβ δ β1 = G4 ϕβ δ ω δ β1 −ω ≤ G5 ϕβ δ ω = G5 ϕβ |y1 − y2 |ω . Similarly, we have L2 ≤ G6 ϕβ |y1 − y2 |ω ,

(2.6)

182

Chapter VI

where Gi (i = 1, ..., 6) are positive constants independent of y1 , y2 . In the following, Gi (i ≥ 1) are denoted as positive constants with the similar property. Next, we estimate L3 . By means of Theorem 2.1, the inequality     1 1    |x − y |n−1−r − |x − y |n−1−r  1 2   k(n−1−r)   n−1−r n−2
 x − y2  n−1  y1 − y2  n−1    ≤ | x − y2 |−n+1−r x − y  x−y  k=0

1

1

is derived. For any x ∈ Ω2 , we have |x − y1 | ≥ 3δ = 3|y1 − y2 |, |x − y2 | ≥  x−y2  2δ, hence 12 ≤  x−y  ≤ 2, |x − y2 | ≥ 12 |x − y1 |. Moreover, noting that 1 (n − 1 − r)/(n − 1) > β1 , we have  n−1−r     y1 − y2  n−1  y 1 − y 2  β1 δ β1     ≤ x−y   x − y  = ( |x − y | ) , 1 1 1

and then 

δ β1 ϕβ |x − y1 |β1 dsx = G7 |y1 − y2 |β1 |x − y1 |n−1−r+β1 Ω2 ≤ G8 ϕβ |y1 − y2 |β1 ≤ G9 ϕβ |y1 − y2 |ω .

L3 ≤ G7



Ω2

ϕβ dsx |x − y1 |n−1−r (2.7)

By virtue of |(ϕ(x, y1 ) − ϕ(x, y2 )) + (ϕ(y2 , y2 ) − ϕ(y1 , y1 ))| 2

≤ ϕβ |y1 − y2 |β2 + ϕβ |y1 − y2 |β1 + |y1 − y2 |β2 ≤ G10 ϕβ |y1 − y2 |ω ,

3

we know L4 ≤ G11 ϕβ |y1 − y2 |ω .

(2.8)

Next we discuss the H¨older continuity of (θ2 ϕ)(y). It is not difficult to see that |(θ2 ϕ)(y1 ) − (θ2 ϕ)(y2 )|

     dσx , ≤ (ϕ(y1 , y1 ) − ϕ(y2 , y2 ))  n−1−r Ω |x − y1 |   * +   1 1  − dσ + |ϕ(y2 , y2 )|  x n−1−r n−1−r |x − y | |x − y | Ω

1

2

High Order Singular Integrals and Differential Integral Equations

≤ G12 ϕβ (|y1 − y2

|β1

+ |y1 − y2

|β2 )

  + ϕβ 

Ω

183 

 dσx   n−1−r |x − y1 |

1     dσ x  + ϕβ  n−1−r  Ω1 |x − y2 |  *  +   1 1  + ϕβ  − dσx  n−1−r n−1−r |x − y2 | Ω2 |x − y1 |

= L5 + L6 + L7 + L8 .

Firstly, it is easy to see that L5 ≤ G13 ϕβ |y1 − y2 |ω .

(2.9)

Secondly, similarly to the deduction of (2.5), and according to the condition of r > β1 ≥ ω, we get L6 ≤ G14 ϕβ δ r ≤ G15 ϕβ δ ω = G15 ϕβ |y1 − y2 |ω .

(2.10)

Moreover, we have L7 ≤ G16 ϕβ |y1 − y2 |ω .

(2.11)

Finally, similarly to the deduction of (2.7), and noting that r − β1 > 0, we get 

dσx ≤ G18 ϕβ |y1 − y2 |ω . n−1−(r−β ) 1 Ω2 |x − y1 | (2.12) In view of (2.5) − (2.12), the inequality β1

L8 ≤ G17 ϕβ |y1 − y2 |

|(P ϕ)(y1 ) − (P ϕ)(y2 )| ≤ G19 ϕβ |y1 − y2 |ω ≤ G20 |y1 − y2 |ω ,

(2.13)

is derived, namely P ϕ ∈ Hy0 (ω). The proof of this theorem is finished. Now, we verify the h¨older continuity for every kind of high order singular integrals of quasi-Bochner-Martinelli type on an integral path. Theorem 2.3 Let f (x, y) ∈ H (2k+2,0) (β1 , β2 ), 0 < r < 1, r > β1 , n − 1 > r/(1 − β1 ), ω = min(β1 , β2 ), y ∈ Ω, 0 < βi < 1, i = 1, 2. Then the six kinds of high order singular integrals of quasi-Bochner-Martinelli type as stated before all belong to Hy0 (ω) on Ω. Proof On the basis of Theorem 2.2 and the calculation formulas of every kind of high order singular integrals of quasi-Bochner-Martinelli type, we can prove this theorem.

184

3

Chapter VI

Nonlinear Differential Integral Equations including Three Kinds of High Order Singular Integrals of QuasiBochner-Martinelli Type in Real Clifford Analysis

From the Definitions 1.8, 1.9, 1.10, we know that the fourth, fifth, and sixth kinds of high order singular integrals of quasi-Bochner-Martinelli type with one singular point can be expressed by the first, second, and third kinds of high order singular integrals of quasi-Bochner-Martinelli type with one singular point. So when we consider the nonlinear differential equations including high order singular integrals of quasi-BochnerMartinelli type in real Clifford analysis, it is sufficient to discuss the equations with the first, second, and third kinds of high order singular integrals of quasi-Bochner-Martinelli type. In the field of differential integral equations, Y. Hino investigated the linear Volterra differential integral equation in 1990 (see [24]). Due to the enlightenment from the Volterra differential integral equation, in this section we shall prove the existence and uniqueness of solutions for some nonlinear differential integral equations with the first, second, and third kinds of high order singular integrals of quasi-BochnerMartinelli type by using the results in Sections 1 and 2, the method of integral equations and the Schauder fixed-point theorem. In this section, Ω is the same as that in the above section. Now we introduce three high order singular integral operators over Ω × Ω, i.e. 

ϕ(x, t2 )(¯ x − t¯1 )dσx , n+2k+1−r 1 Ω |x − t1 |  ϕ(x, t2 )(x − t1 )dσx (S2 ϕ)(t1 , t2 ) = , n+2k+1−r2 Ω |x − t1 |  ϕ(x, t2 )dσx (S3 ϕ)(t1 , t2 ) = , n+2k+1−r3 Ω |x − t1 |

(S1 ϕ)(t1 , t2 ) =

where 0 < ri < 1, i = 1, 2, 3, (t1 , t2 ) ∈ Ω × Ω, and the nonlinear differential integral equation including Si ϕ (1 ≤ i ≤ 3) is Wϕ =

3


i=1

(ai (t1 , t2 )




0≤ki +mi ≤ni

∂¯tk1i ∂tm2 i Si ϕ)

(3.1) +g(t1 , t2 )f [t1 , t2 , ∂¯tp11 ∂tq21 S1 ϕ, ∂¯tp12 ∂tq22 S2 ϕ, ∂¯tp13 ∂tq23 S3 ϕ] = ϕ(t1 , t2 ), (t1 , t2 ) ∈ Ω × Ω, here ki , mi , ni , pi , qi are non-negative integers.

High Order Singular Integrals and Differential Integral Equations Theorem 3.1

185

If the operator

(P Ψ)(t1 , t2 ) =



Ω

Ψ(x, t2 )dσx , 0 < r < 1, |x − t1 |n−1−r (0,0)

is given, then for any Ψ(x, y) ∈ H(β1 ,β2 ) , 0 < βi < 1, i = 1, 2, n − 1 > r/(1 − β1 ), there exists a positive constant J1 independent of Ψ, such that P Ψβ ≤ JΨβ . (3.2) Proof From Section 2, Chapter II, it is clear that dσx = m(x)dsx . Now we introduce the singular integral operator 

(Ψ(t1 , t2 ) − Ψ(x, t2 ))dσx |x − t1 |n−1−r Ω  dσx = Ψ(t1 , t2 ) − P Ψ. n−1−r Ω |x − t1 |

(QΨ)(t1 , t2 ) =

(3.3)

Firstly, we estimate |(QΨ)(t1 , t2 )| ≤ G1 Ψβ ≤ G1 Ψβ



Ω



Ω

|t1 − x|β1 |m(x)|dsx |x − t1 |n−1−r

dsx ≤ G2 Ψβ , |x − t1 |n−1−r−β1

(3.4)

where G1 , G2 are positive constants independent of Ψ.


= Ω × Ω, and for In order to consider H(QΨ, Ω × Ω, β), denote
ˆ ˆ ˆ any (t1 , t2 ), (t1 , t2 ) ∈ , set δ = |t1 − t1 |. Let 3δ < d, and d be as stated in Section 2. We make a sphere with the center at t1 and radius 3δ. Denote by Ω1 the interior part and by Ω2 the left part, hence |(QΨ)(t1 , t2 ) − (QΨ)(t1 , t2 )|

    1 , t2 ) − Ψ(x, t2 ))dσx     (Ψ( t (Ψ(t , t ) − Ψ(x, t ))dσ 1 2 2 x + ≤      |x − t1 |n−1−r |x − t1 |n−1−r Ω1 Ω1      1 1   − (Ψ(t , t ) − Ψ(x, t ))dσ +   1 2 2 x n−1−r    Ω2 |x − t1 |n−1−r |x − t1 |     [(Ψ(x, t2 ) − Ψ(x, t2 )) + (Ψ(t1 , t2 ) − Ψ(t1 , t2 ))]   dσ +  x   Ω2 |x − t1 |n−1−r

= L1 + L2 + L3 + L4 .

(3.5)

186

Chapter VI

The formula (2.4) about N0 ∈ Ω in Section 2 is used to t1 , and the projective domain of Ω on the tangent plane of t1 is denoted by π1 , thus we can obtain  dsx L1 ≤ G3 Ψβ n−1−r−β1 Ω1 |x − t1 |  cos(n0 , xn )dsx = G3 Ψβ n−1−r−β1 cos(n , x ) Ω1 |x − t1 | 0 n  dξ1 ...dξn−1 ≤ G3 Ψβ 1 π1 |x − t1 |n−1−r−β1 2 (3.6) ≤ G4 Ψβ

 3δ n−2 ρ0 dρ0 0

1 ρn−1−r−β 0

β1

≤ G5 Ψβ δ r δ β1 ≤ G6 Ψβ |t1 − tˆ1 |

β β ≤ G6 Ψβ [|t1 − tˆ1 | 1 + |t2 − tˆ2 | 2 ],

where Gi (i = 3, ..., 6) are positive constants independent of ti , tˆi , i = 1, 2. Similarly, we have β β L2 ≤ G7 Ψβ [|t1 − tˆ1 | 1 + |t2 − tˆ2 | 2 ].

(3.7)

Secondly, we estimate L3 . By means of Theorem 2.1, the inequality

≤

    1 1   −   n−1−r ˆ   |x − t1 |n−1−r |x − t1 |

n−2  k=0

|

t1 − tˆ1 n−1−r x − tˆ1 k(n−1−r) | n−1 | | n−1 |x − tˆ1 |−n+1+r x − t1 x − t1

is derived. For any x ∈ Ω2 , we get |x − t1 | ≥ 3δ, |x − tˆ1 | ≥ 2δ, hence 



1  x − tˆ1  1 ≤  ≤ 2, |x − tˆ1 | ≥ |x − t1 |. 2  x − t1  2

Moreover we have

and then L3



  n−1−r    t − tˆ  n−1  t − tˆ β1 δ 1 1  1  1 ≤ ) β1 ,    =(  x − t1   x − t1  |x − t1 |

δ β1 |x − t1 |β1 Ψβ dsx = G8 |t1 − tˆ1 |β1 ≤ G8 n−1−r+β1 |x − t | Ω2 1 ˆ ≤ G9 Ψβ [|t1 − t1 |β1 + |t2 − tˆ2 |β2 ].



Ω2

Ψβ dsx |x − t1 |n−1−r (3.8)

187

High Order Singular Integrals and Differential Integral Equations In view of |(Ψ(x, tˆ2 ) − Ψ(x, t2 )) + (Ψ(t1 , t2 ) − Ψ(tˆ1 , tˆ2 ))| ≤ [G10 |t2 − tˆ2 |β2 + G11 (|t1 − tˆ1 |β1 + |t2 − tˆ2 |β2 )]Ψβ ≤ G12 Ψβ [|t1 − tˆ1 |β1 + |t2 − tˆ2 |β2 ], it is easy to see that L4 ≤ G13 Ψβ [|t1 − tˆ1 |β1 + |t2 − tˆ2 |β2 ].

(3.9)

In view of (3.5) − (3.9), we know when 3|t1 − tˆ1 | < d, |(QΨ)(t1 , t2 ) − (QΨ)(tˆ1 , tˆ2 )| ≤ G14 Ψβ [|t1 − tˆ1 |β1 + |t2 − tˆ2 |β2 ]. (3.10) It is clear that when 3|t1 − tˆ1 | ≥ d, the above estimation is correct. From (3.4), (3.10), it follows that QΨβ ≤ G15 Ψβ . So in view of (3.3), we have P Ψβ ≤ QΨβ + G16 Ψβ ≤ J1 Ψβ , here Gi (7 ≤ i ≤ 16) are positive constants independent of ti , tˆi , (i = 6 dσx 1, 2), G16 = sup Ω , J1 is a positive constant independent |x − t1 |n−1−r t1 ∈Ω of Ψ. This completes the proof of Theorem 3.1.

Corollary 3.2 Suppose ϕ(t1 , t2 ) ∈ H (¯n+2k+2,¯n) (β1 , β2 ) ⊂ H (0,0) (β1 , β2 ), 0 ≤ ki +mi ≤ ni ≤ n ¯ , n−1 > ri /(1−β1 ), i = 1, 2, 3, where n ¯ , k, ki , mi , ni are non-negative integers. Then ∂¯tk1i ∂tm2 i Si ϕ(t1 , t2 )β ≤ J2 Ψβ , where the norm is the norm of the element in space H (0, 0) (β1 , β2 ). Proof It is easy to prove this corollary by means of Theorems 1.13 and 3.1. The set of functions which have arbitrary order generalized derivative ¯ ∂t1 , ∂t2 on Ω×Ω is denoted by D∞ (Ω×Ω). It is clear that D∞ (Ω×Ω) ⊂ H (0,0) (β1 , β2 ). Theorem 3.3 Let g(t1 , t2 ), f (t1 , t2 , Φ1 (t1 , t2 ), Φ2 (t1 , t2 ), Φ3 (t1 , t2 )) and ai (t1 , t2 ) (i = 1, 2, 3) in the nonlinear differential integral equations (3.1) belong to D∞ (Ω×Ω), where Φi (t1 , t2 ) = (∂¯tp1i ∂tq2i Si ϕ)(t1 , t2 ), (t1 , t2 ) ∈ Ω× Ω, pi + qi ≤ ni ≤ n ¯ , si φ, 0 < ri < 1, n − 1 > ri /(1 − β1 ) (i = 1, 2, 3) are as stated in Corollary 3.2, and f (0, 0, 0, 0, 0) = 0. Then when 0 < r = 2n−1 J2 (¯ n + 1)2

n


i=1

ai β < 1, gβ < σ, 0 < σ ≤

M (1−r) , 2n−1 (G19 +G20 M )

the

nonlinear differential integral equation (3.1) has a solution ϕ ∈ D∞ (Ω × Ω); and when f ≡ 1, 0 < r < 1, the solution is unique, where M is a

188

Chapter VI

given positive number such that ϕβ ≤ M, and G19 , G20 are positive constants. Proof According to the condition f ∈ D∞ (Ω × Ω), for any (t1 , t2 ), (tˆ1 , tˆ2 ) ∈ Ω × Ω and any Φji ∈ An (R), (i = 1, 2), 1 ≤ j ≤ 3, we have (1) (2) (3) (1) (2) (3) |f (t1 , t2 , Φ1 , Φ1 , Φ1 ) − f (tˆ1 , tˆ2 , Φ2 , Φ2 , Φ2 )| β

≤ G17 [|t1 − t2 |β1 + |tˆ1 − tˆ2 | 2 ] + G18

3


(j)

j=1

(j)

(3.11)

|Φ1 − Φ2 |,

(j) in which G17 , G18 are positive constants independent of ti , tˆi , Φi , i = 1, 2, 1 ≤ j ≤ 3. Next, we consider the subset

⎫   ϕ ∈ D (Ω × Ω) ⊂ H (0,0) (β , β ), ⎬ ∞ 1 2  T = ϕ(t1 , t2 )  ⎩  (t1 , t2 ) ∈ Ω × Ω, ϕβ ≤ M, M > 0 ⎭ ⎧ ⎨

of H (0,0) (β1 , β2 ), in which the norm is defined in the space H (0,0) (β1 , β2 ). For any ϕ ∈ T , by means of (3.1), Theorem 1.13 and the above conditions, we know Wϕ ∈ D∞ (Ω × Ω). In view of (3.1) and Corollary 3.1, the estimate W ϕβ ≤ 2n−1

3


i=1

ai β




0≤ki +mi ≤ni

∂¯tk1i ∂tm2 i Si ϕβ + 2n−1 gβ f β

3
≤ 2n−1 ai β (¯ n + 1)2 J2 ϕβ + 2n−1 gβ f β i=1

(3.12) is concluded. In accordance with f (0, 0, 0, 0, 0) = 0, (3.11) and Corollary 3.2, we have |f | ≤ G17 [|t1 − 0|β1 + |t2 − 0|β2 ] + G18 ≤ G19 + 3G18 J2 ϕβ .

3
|∂¯pi ∂ qi Si ϕ|

i=1

t1 t2

(3.13)

Noting (3.11), we have |f (t1 , t2 , ∂¯tp11 ∂tq21 S1 ϕ(t1 , t2 ), ..., ∂¯tp13 ∂tq23 S3 ϕ(t1 , t2 )) −f (tˆ1 , tˆ2 , ∂¯tp11 ∂tq21 S1 ϕ(tˆ1 , tˆ2 ), ..., ∂¯tp13 ∂tq23 S3 ϕ(tˆ1 , tˆ2 ))| ≤ G17 [|t1 − tˆ1 |β1 +|t2 − tˆ2 |β2 ]+3G18 J2 ϕβ [|t1 − tˆ1 |β1 +|t2 − tˆ2 |β2 ] ≤ [G17 + 3G18 J2 ϕβ ] [|t1 − tˆ1 |β1 + |t2 − tˆ2 |β2 ].

(3.14)

High Order Singular Integrals and Differential Integral Equations

189

From (3.13),(3.14), it follows that f β ≤ G19 + G20 ϕβ .

(3.15)

By means of (3.12), (3.13) and the conditions, we get W ϕβ ≤ 2n−1

3


i=1

ai β (¯ n + 1)2 J2 M + 2n−1 δ(G19 + G20 M )

≤ M r + 2n−1 δ(G19 + G20 M ) ≤ M. This shows that W maps the set T into itself. Next, we prove that W is a continuous mapping. we are free to choose a sequence ϕ(n) ∈ T (n = 1, 2, ...), such that {ϕ(n) (t1 , t2 )} uniformly converges to ϕ(t1 , t2 ) ∈ T, (t1 , t2 ) ∈ Ω × Ω. For arbitrarily given positive number ε, when n is large enough, ϕ(n) − ϕβ can be small enough, hence by (3.11), (3.12) and Corollary 3.2, we see that when n is large enough, the following inequality holds: W ϕ(n) (t1 , t2 ) − W ϕ(t1 , t2 )β ≤ 2n−1

3


i=1

ai β (¯ n +1)2 J2 ϕ(n) − ϕβ

+2n−1 gβ (3G18 J2 ϕ(n) − ϕβ ) < ε. This shows that W is a continuous mapping, which maps T into itself. According to the Ascoli-Arzela theorem, we see that T is a compact set of space H 0, 0 (β1 , β2 ). Hence the continuous mapping W maps the closed convex set T in H (0,0) (β1 , β2 ) onto itself. Moreover W (T ) is also a compact set in H (0,0) (β1 , β2 ). By the Schauder fixed-point theorem, there exists a function ϕ0 ∈ D∞ (Ω×Ω) satisfying equation (3.1); here we mention that though the conditions added to ai , g, f are stronger, the solution found still satisfies the above condition. At last when f ≡ 1, similarly to Section 4, Chapter II, by using the contraction mapping theorem, we can verify the uniqueness of the solution for equation (3.1).

4

A Kind of High Order Singular Integrals of QuasiBochner-Martinelli Type With two Singular Points and Poincar´ e-Bertrand Permutation Formulas in Real Clifford Analysis

From the enlightenment of Sheng Gong’s paper about singular integrals of several complex variables [20], in this section, we first discuss

190

Chapter VI

high order singular integrals of quasi-Bochner-Martinelli type with two singular points. Denote still by Ω the boundary of the connected open set D in Rn . If x = t (x, t ∈ Ω), by means of the separability of D, we assume that Dx , Dt satisfy the conditions Dx ∩ Dt = ∅, Dx ∪ Dt = D, Ωx , Ωt are the boundaries of Dx , Dt respectively, x ∈ Ωx , t ∈ Ωt , and the orientations of Ωx , Ωt are harmonious with the orientation of Ω. Moreover, the orientations of Ω, Ωx , Ωt are induced orientations of D, Dx , Dt re( spectively. In addition, suppose that Dx Dt = Σ, and for any y ∈ Σ, |y − x| = |y − t| holds. The integral kernels with singular points x, t are denoted by K x (x, y), K t (x, y) respectively. By the additive property of general integrals, the singular integral with two singular points can be defined as follows. Definition 4.1 Let ϕ(x, y) ∈ H (m, p) (β1 , β2 ), 0 < βi < 1, i = 1, 2; the singular integral with two singular points on Ω is defined as 

Ω

=



K t (y, t)ϕ(x, y)K x (x, y)dσy t

x

K (y, t)ϕ(x, y)K (x, y)dσy +

Ωt



Ωx

K t (y, t)ϕ(x, y)K x (x, y)dσy ,

where m, p are determined by the orders of singularity of kernels K x (x, y), K t (y, t) respectively. In the following, we first discuss the singular integral which can exchange the integral order. Theorem 4.1 Let ϕ(x, y) ∈ H (0,0) (β1 , β2 ), 0 < βi < 1, r > 0, h > 0, n − 1 > r + h, x, t ∈ Ω. Then  

Ω

Ω

=

  Ω



ϕ(x, y) dσy dσx n−1−r |y − t| |x − y|n−1−h

1 |y − t|n−1−r



Ω



ϕ(x, y)dσx dσy . |x − y|n−1−h

(4.1)

Proof According to the Hadamard theorem [19], when (n − 1) > r + h, we have     ϕ(x, y)dσy G21  ≤ .  n−1−r |x − y|n−1−h  (n−1)−(r+h) |y − t| |x − t| Ω

This shows that the above integral with two singular points has a weak singularity, and then the left side of (4.1) is the integral in a general

191

High Order Singular Integrals and Differential Integral Equations

sense. By means of the Fubini theorem [19], we know that this integral order can be exchanged. Hence (4.1) holds. Now, we deduce the Poincar´e-Bertrand permutation formula for the first and second kinds of high order singular integrals of quasi-BochnerMartinelli type. Theorem 4.2. Let ϕ(x, y) ∈ H (1,1) (β1 , β2 ), 0 < βi < 1, r , h > βi , i = 1, 2, 0 < r < 1, n − 1 > h + r, x, t ∈ Ω. Then  

  y¯ − t¯ ϕ(x, y)dσx dσy n+1−r n−1−h Ω |y − t| Ω |x − y|    (¯ y − t¯)ϕ(x, y)dσy dσx = n+1−r |x − y|n−1−h Ω |y − t| Ω  

+

Ωt

Ω

 

+

Ω

*

+



1 ϕ(x, y) ∂x dσy dσx n−1−r (n − 1 − r)|y − t| |x − y|n−1−h *

1 ϕ(x, y) (∂x +∂y ) n−1−h |y−t|n−1−r Ωx (n−1−r)|x−y|

+



dσy dσx . (4.2)

Proof Under the conditions of Definition 1.1, Theorem 1.8, Corollary 1.2, Lemma 1.4 and Theorem 4.1, we can get 

Ω

=

y¯ − t¯ |y − t|n+1−r

  Ω

=

  Ω

 

+

Ω

=

Ω

ϕ(x, y)dσx dσy |x − y|n−1−h

1 (n − 1 − r)|y − t|n−1−r 1 (n − 1 − r)|y − t|n−1−r

1 (n−1−r)|y−t|n−1−r

  Ω



Ω





∂y

Ω

*

Ω

*

+

∂y ϕ(x, y)dσx |x − y|n−1−h

+

dσy 

ϕ(x, y)(¯ x − y¯)(n−1−h) dσx dσy |x−y|n+1−h Ω 

∂y ϕ(x, y)dσy dσx (n − 1 − r)|y − t|n−1−r |x − y|n−1−h  



ϕ(x, y) dσx dσy |x − y|n−1−h



(4.3)



1 ∂x ϕ(x, y)dσx n−1−h dσy + n−1−r n−1−h n − 1 − r Ω |y − t| Ω (n − 1 − h)|x − y|    (∂y + ∂x )ϕ(x, y)dσy 1 = dσx . n − 1 − r Ω Ω |y − t|n−1−r |x − y|n−1−h On the basis of Definitions 4.1 and 1.1, Corollary 1.2, Lemma 1.4 and

192

Chapter VI

(4.3), we can derive  

= =

=

=

=



(¯ y − t¯)ϕ(x, y)dσy dσx n+1−r |x − y|n−1−h Ω |y − t| Ω + *     (¯ y − t¯)ϕ(x, y)dσy dσx ) + ( |y − t|n+1−r |x − y|n−1−h Ωx Ω Ωt * +    1 ϕ(x, y) ∂ dσy dσx n−1−r y |x − y|n−1−h Ωt (n − 1 − r)|y − t| Ω    (¯ y − t¯)ϕ(x, y)dσy dσx + n−1−h |y − t|n+1−r Ωx |x − y| Ω    ∂y ϕ(x, y)dσy dσx n−1−r |x − y|n−1−h Ωt (n − 1 − r)|y − t| Ω    ϕ(x, y)(n − 1 − h)(¯ x − y¯)dσy + dσx n−1−r |x − y|n+1−h Ωt (n − 1 − r)|y − t| Ω * +    1 1 ∂ + ϕ(x, y)dσ y dσx n−1−h (1+r−n) y |y−t|n−1−r Ω Ωx |x−y|    ∂y ϕ(x, y)dσy dσx + n−1−r |x − y|n−1−h Ωx (n − 1 − r)|y − t| Ω    ∂y ϕ(x, y)dσy dσx − n−1−r |x − y|n−1−h Ωx (n − 1 − r)|y − t| Ω    ∂y ϕ(x, y)dσy dσx n−1−r |x − y|n−1−h Ω (n − 1 − r)|y − t| Ω    ∂x ϕ(x, y)dσy dσx + n−1−r |x − y|n−1−h Ω (n − 1 − r)|y − t| Ω    ∂x ϕ(x, y)dσy − dσx n−1−r |x − y|n−1−h Ωt (n − 1 − r)|y − t| Ω    ϕ(x, y)(n − 1 − h)(¯ x − y¯)dσy + dσx n−1−r |x − y|n+1−h Ωt (n − 1 − r)|y − t| Ω + *    1 1 ∂ ϕ(x, y)dσ + y dσx n−1−h y |y−t|n−1−r Ωx (1+r−n)|x−y| Ω    ∂x ϕ(x, y)dσy − dσx n−1−r |x − y|n−1−h Ωx (n − 1 − r)|y − t| Ω    ∂y ϕ(x, y)dσy dσx − n−1−r |x−y|n−1−h Ωx (n−1−r)|y−t| Ω    (∂y + ∂x )ϕ(x, y)dσy dσx n−1−r |x − y|n−1−h Ω (n − 1 − r)|y − t| Ω    ∂x ϕ(x, y)dσy dσx − n−1−r |x−y|n−1−h Ωt (n−1−r)|y−t| Ω + * +    * ϕ(x, y) · (−1) 1 ∂ dσ + x y dσx n−1−r |x − y|n−1−h Ω Ωt (n−1−r)|y−t|

High Order Singular Integrals and Differential Integral Equations

  *

+

*

+



193

1 1 ∂y ϕ(x, y)dσy dσx − n−1−h x |y−t|n−1−r Ω Ω (n−1−r)|x−y|    (∂x + ∂y )ϕ(x, y)dσy dσx − n−1−r |x − y|n−1−h Ωx (n − 1 − r)|y − t| Ω   (∂x + ∂y )ϕ(x, y)dσy [ = ]dσx (n − 1 − r)|y − t|n−1−r |x − y|n−1−h Ω Ω   1 ϕ(x, y) ∂x ( )dσy ]dσx − [ n−1−r |x − y|n−1−h Ω Ωt (n − 1 − r)|y − t|   1 1 − [ ∂y ( )ϕ(x, y)dσy ]dσx n−1−h |y − t|n−1−r Ω Ωx (n − 1 − r)|x − y|   (∂x + ∂y )ϕ(x, y)dσy − [ ]dσx x (n − 1 − r)|y − t|n−1−r |x − y|n−1−h Ω Ω   ϕ(x, y)dσx y¯ − t¯ ]dσy [ = n+1−r n−1−h Ω |x − y| Ω |y − t|   1 ϕ(x, y) ∂ ( )dσy ]dσx − [ n−1−r x |x − y|n−1−h t (n − 1 − r)|y − t| Ω Ω   1 1 ( ∂ ( )ϕ(x, y)dσy ]dσx − [ n−1−h y |y − t|n−1−r x (n − 1 − r)|x − y| Ω Ω   (∂y + ∂x )ϕ(x, y)dσy − [ ]dσx . n−1−r |x − y|n−1−h Ω Ωx (n − 1 − r)|y − t| (4.4) In view of (4.4) and Corollary 1.2, we have  

  y¯ − t¯ ϕ(x, y)dσx dσy n+1−r n−1−h Ω |y − t| Ω |x − y|    (¯ y − t¯)ϕ(x, y)dσy dσx = n+1−r |x − y|n−1−h Ω |y − t| Ω + *    1 ϕ(x, y) + ∂ dσy dσx n−1−r x |x − y|n−1−h Ωt (n − 1 − r)|y − t| Ω    1 ϕ(x, y) ∂ ( )dσ + y dσx n−1−h y |y − t|n−1−r Ωx (n − 1 − r)|x − y| Ω    ∂x ϕ(x, y)dσy dσx + n−1−h |y − t|n−1−r Ωx (n − 1 − r)|x − y| Ω    (¯ y − t¯)ϕ(x, y)dσy = dσx n+1−r |x − y|n−1−h Ω |y − t| Ω * +    ϕ(x, y) 1 ∂ dσ + y dσx n−1−r x |x − y|n−1−h Ωt (n − 1 − r)|y − t| Ω

194

+

Chapter VI   Ω

Ωx

Ω

Ωx

 

+

*

+



1 ϕ(x, y) ∂y dσy dσx n−1−h (n − 1 − r)|x − y| |y − t|n−1−r + *  ϕ(x, y) 1 ∂x dσy dσx . (n − 1 − r)|x − y|n−1−h |y − t|n−1−r

(4.5)

Merge the last two terms in (4.5), and we get (4.2). The proof of this theorem is completed. In short, the method of the proof in Theorem 4.2 is as follows: Firstly, by using calculational formulas, differential formulas in Section 1 and the Fubini theorem (see [6],[7][19]), we establish the relation between the singular integral before permutation, the singular integral after permutation with high order quasi-Bochner-Martinelli type kernels and the singular integrals with two singular points to be exchanged in integral order respectively. Next we can get the Poincar´e-Bertrand permutation formulas. The above differential formulas play an important pole in the proof. This method is different from that used to prove the Poincar´eBertrand permutation formulas of non-high order singular integrals in Section 2, Chapter V. If W (ki , ri ) (1 ≤ i ≤ 6) are expressed the general first−sixth kinds of high order quasi-Bochner-Martinelli kernels in Theorems 1.8 and 1.16, then we can use the same method in the proof of Theorem 4.2 to prove the Poincar´e-Bertrand permutation formulas of the general high order singular integrals with the kernels W (ki , ri ) and W (kj , rj ) (1 ≤ i, j ≤ 6). Because of page limitation we do not prove them one by one. The proofs are left as exercises. Concluding Remark It is well known that the representation in circular cylinder domains for several complex variables is worse than the representation in the unit disk for functions of one complex variable. Hence in theory of several complex variables, the regular functions determined by Cauchy integral formulas in several domains possess different representations, which were an important research subject by many scholars. In 1961, F. Norguet enumerated 66 main papers about Cauchy integral formulas of several complex variables in the appendix of his paper [56]. In [56], the main Cauchy integral formulas are classified as five kinds, for example, the Cauchy integral formulas in four kinds of canonical domains, Cauchy integral formulas of Bochner-Martinelli type and so on. These Cauchy integral formulas are generalized to the Cauchy integral formula of a complex variable from different angles and views. We are enlightened by the variability of Cauchy integral formulas in

High Order Singular Integrals and Differential Integral Equations

195

several complex variables. Because of the non-commutativity property in Clifford algebra, the expressions of integral operators adopted in this chapter are different from the Cauchy integral expression 1 ωn



Ω

x−y dσx f (x), Ω = ∂D ⊂ Rn , |x − y|n

used by other scholars (see [6],[19]). In this chapter, we use integral expressions, that be got in Chapter I, such that many calculational formulas, recurrence formulas and differential formulas of singular integrals as stated in Section 1 are succinct and regular. Furthermore, we can give another method to prove the Poincar´e-Bertrand permutation formula of high order singular integrals by using differential formulas. It is also interesting to investigate various integral expressions of generalized or doubly regular functions in Clifford analysis.

CHAPTER VII RELATION BETWEEN CLIFFORD ANALYSIS AND ELLIPTIC EQUATIONS In this chapter, we first introduce the solvability of some oblique derivative problems for uniformly and degenerate elliptic equations of second order, and then discuss the existence of solutions of some boundary value problems for some degenerate elliptic systems of first order in Clifford analysis by using the above results for elliptic equations.

1

Oblique Derivative Problems for Uniformly Elliptic Equations of Second Order

1.1 Formulation of oblique derivative problem for nonlinear elliptic equations Let Q be a bounded domain in RN and the boundary ∂Q ∈ Cµ2 (0 < µ < 1), herein N (> 1) is a positive integer. We consider the nonlinear elliptic equation of second order F (x, u, Dx u, Dx2 u) = 0 in Q,

(1.1)

N 

(1.2)

namely Lu =

N 

aij uxi xj +

i=1

i,j=1

where Dx u = aij = c=

(uxi ), Dx2 u  1 0

 1 0

bi uxi + cu = f in Q,

= (uxi xj ), and

Fτ rij (x, u, p, τ r)dτ, bi =

 1 0

Fτ pi (x, u, τ p, 0)dτ,

Fτ u (x, τ u, 0, 0)dτ, f = −F (x, 0, 0, 0),

r = Dx2 u, p = Dx u, rij =

∂u ∂u , pi = . ∂xi ∂xj ∂xi

198

Chapter VII

Suppose that (1.1) (or (1.2)) satisfies Condition C, i.e. for arbitrary functions u1 (x), u2 (x) ∈ Cβ1 (Q) ∩ W22 (Q), F (x, u, Dx u, Dx2 u) satisfy the following conditions: F (x, u1 , Dx u1 , Dx2 u1 ) − F (x, u2 , Dx u2 , Dx2 u2 ) N 

=

a ˜ij uxi xj +

N 

˜bi ux + c˜u, i

(1.3)

i=1

i,j=1

where 0 < β < 1, u = u1 − u2 and a ˜ij = c˜ =

 1 0

 1 0

Fuxi xj (x, u ˜, p˜, r˜)dτ, ˜bi =

 1 0

Fuxi (x, u ˜, p˜, r˜)dτ,

Fu (x, u ˜, p˜, r˜)dτ, u ˜ = u2 + τ (u1 − u2 ),

p˜ = Dx [u2 + τ (u1 − u2 )], r˜ = Dx2 [u2 + τ (u1 − u2 )], and a ˜ij , ˜bi , c˜, f satisfy the conditions q0

N 

j=1

sup

N 

|ξj |2 ≤ N 

a ˜2ij /inf [

Q i,j=1

Q

i=1

N 

i,j=1

a ˜ij ξi ξj ≤ q0−1

a ˜ii ]2 ≤ q1
N + 2) are non-negative constants. Moreover, for almost every point x ∈ Q and Dx2 u ∈ RN (N +1)/2 , the functions a ˜ij (x, u, Dx u, Dx2 u), ˜bi (x, u, Dx u), c˜(x, u) are continuous in u ∈ R and D x u ∈ RN . The so-called oblique derivative problem(Problem O) is to find a continuously differentiable solution u = u(x) ∈ Cβ1 (Q) ∩ W22 (Q) satisfying the boundary conditions: lu = d lu =

∂u + bu = g(x), x ∈ ∂Q, i.e. ∂ν

N 

j=1

dj

∂u + bu = g(x), x ∈ ∂Q, ∂xj

(1.6)

Clifford Analysis and Elliptic Equations

199

in which dj (x), b(x), g(x) satisfy the conditions Cα1 [b(x), ∂Q] ≤ k0 , Cα1 [g(x), Q] ≤ k2 , cos(ν, n) ≥ q0 > 0, b(x) ≥ 0 on ∂Q, N 

j=1

(1.7)

dj cos(ν, xj ) ≥ q0 > 0, Cα1 [dj (x), ∂Q] ≤ k0 , j = 1, ..., ,N,

in which n is the unit outward q0 (0 < q0 < 1) are non-negative has the conditions d = 1, ν = n, is the Neumann boundary value N.

normal on ∂Q, α (0 < α < 1), k0 , k2 , constants. In particular, if Problem O b = 0 on ∂Q in (1.6), then Problem O problem, which will be called Problem

In the following, we give a priori estimates of solutions for Problem O. Then, by using the method of parameter extension and the LeraySchauder theorem, we prove the existence and uniqueness of solutions for Problem O.

1.2

A priori estimates of solutions for Problem O for (1.2)

We first prove the following theorem. Theorem 1.1 If the equation (1.2) satisfies Condition C, then the solution of Problem O is unique. Proof Let u1 (x), u2 (x) be two solutions of Problem O, it is easily seen that u = u1 − u2 is a solution of the boundary value problem N 

i,j=1

a ˜ij uxi xj +

N 

˜bi ux + c˜u = 0 in Q, i

(1.8)

∂u + bu = 0, x ∈ ∂Q, ∂ν

(1.9)

i=1

lu(x) = 0 i.e. d

where a ˜ij , ˜bi , c˜ are as stated in (1.3). By the maximum principle of solutions for (1.8), u(x) attains its maximum in Q at a point P0 ∈ ∂Q, and lu|P0 > 0; this contradicts (1.9), hence u(x) = 0 in Q, i.e. u1 (x) = u2 (x), x ∈ Q. ¯ and C 1 (Q) of In the following, we shall give the estimates of C 1 (Q) β solutions u(x) of Problem O.

200

Chapter VII

Theorem 1.2 Under the same condition as in Theorem 1.1, any solution u(x) of Problem O for (1.2) satisfies the estimate ||u||C 1 (Q) ¯ = ||u||C(Q) ¯ +

N  i=1

||uxi ||C(Q) ≤ M1 ,

(1.10)

where M1 is a non-negative constant only dependent on q, p, α, k, Q, i.e. M1 = M1 (q, p, α, k, Q), q = (q0 , q1 ), k = (k0 , k1 , k2 ). Proof Suppose that (1.10) is not true, there exist sequences of funcm m m m m m tions {am ij }, {bi }, {c }, {f } and {a (x)}, {b (x)}, {g (x)}, which m satisfy Condition C and the conditions in (1.7), and {am ij }, {bi }, 0 0 m m m 0 m m 0 {c }, {f } weakly converge to aij , bi , c , f , and {a }, {b }, {g } uniformly converge to a0 , b0 , g 0 on ∂Q respectively. Furthermore the boundary value problem N 

i,j=1

am ij uxi xj +

N 

m m bm in Q, i uxi + c u = f

(1.11)

i=1

∂um + bm um = g m (x), x ∈ ∂Q (1.12) ∂ν has a solution um (x), such that ||um ||C 1 (Q) ¯ = Am (m = 1, 2, ...) is unbounded (there is no harm in assuming that Am ≥ 1, and limm→∞ Am = +∞). It is easy to see that U m = um /Am is a solution of the initialboundary value problem lum (x) = g m (x), i.e. am

˜ m= LU

N 

i,j=1

m m m am ij Uxi xj = B , B = −

lU m (x) =

N  i=1

m m m m bm i Uxi −c U +f /Am , (1.13)

∂U m gm gm + bm U m = , i.e. am , x ∈ ∂Q. Am ∂n Am

(1.14)




m m m m Noting that N i=1 bi Uxi +c U in (1.13) is bounded, by using the result in Theorem 1.3 below, we can obtain the estimate N 

||Uxmi ||Cβ (Q) ¯ ≤ M2 ,

(1.15)

||U m ||W 2 (Q) ≤ M3 = M3 (q, p, α, k, Q), m = 1, 2, ....

(1.16)

m ||U m ||C 1 (Q) ¯ + ¯ = ||U ||Cβ (Q) β

2

i=1

where β(0 < β ≤ α), Mj = Mj (q, p, α, k, Q)(j = 2, 3) are non-negative constants. Hence from {U m }, {Uxmi }, we can choose a subsequence {U mk } such that {U mk }, {Uxmi k } uniformly converge to U 0 , Ux0i in Q

201

Clifford Analysis and Elliptic Equations

respectively, {Uxmi xkj } weak converges to Ux0i xj in Q, and U 0 is a solution of the boundary value problem: N 

˜0 a0ij U xi xj

+

N 

˜ 0 + c0 U 0 = 0, b0i U xi

(1.17)

∂U 0 + bU 0 = 0, x ∈ ∂Q. ∂ν

(1.18)

i=1

i,j=1

lU 0 (x) = 0, i.e. a

According to Theorem 1.1, we know that U 0 (x) = 0, x ∈ Q. However, ∗ ¯ from ||U m ||C 1 (Q) ¯ = 1, we can derive that there exists a point x ∈ Q,
∗ 0 such that |U 0 (x∗ )| + N i=1 | Uxi (x )| > 0. This contradiction proves that (1.10) is true. Theorem 1.3 Under the same condition as in Theorem 1.1, any solution u(x) of Problem O satisfies the estimates ||u||C 1 (Q) ¯ ≤ M4 = M4 (q, p, α, k, Q),

(1.19)

||u||W 2 (Q) ≤ M5 = M5 (q, p, α, k, Q),

(1.20)

β

2

where β(0 < β ≤ α), M4 , M5 are non-negative constants. Proof

First of all, we find a solution u ˆ(x) of the equation ∆ˆ u−u ˆ=0

(1.21)

with the boundary condition (1.6), which satisfies the estimate ||ˆ u||C 2 (Q) ¯ ≤ M6 = M6 (q, p, α, k, Q).

(1.22)

(see [38]). Thus the function u ˜(x) = u(x) − u ˆ(x)

(1.23)

is a solution of the equation L˜ u=

N 

aij u ˜ xi x j +

i,j=1

N 

bi u ˜xi + c˜ u = f˜,

(1.24)

i=1

l˜ u(x) = 0, x ∈ ∂Q,

(1.25)

where f˜ = f − Lˆ u. Introduce a local coordinate system on the neighborhood G of a surface S1 ∈ ∂Q: xi = hi (ξ1 , ..., ξN −1 )ξN + gi (ξ1 , ..., ξN −1 ), i = 1, ..., N,

(1.26)

202

Chapter VII

where ξN = 0 is just the surface S1 : xi = gi (ξ1 , ..., ξN −1 )(i = 1, ..., N ), and  N  di (x)  2 hi (ξ) = d2i (x). , i = 1, ..., N, d (x) = d(x) xi =gi (ξ) i=1

Then the boundary condition (1.25) can be reduced to the form ∂u ˜ + ˜b = 0 on ξN = 0, ∂ξN

(1.27)

where u ˜ = u ˜[x(ξ)], ˜b = b[x(ξ)]. Secondly, we find a solution v(x) of Problem N for the equation (1.21) with the boundary condition ∂v = ˜b on ξN = 0, ∂ξN

(1.28)

which satisfies the estimate ||v||C 2 (Q) ¯ ≤ M7 = M7 (q, p, α, k, Q) < ∞,

(1.29)

(see [38]) and the function V (x) = u ˜(x)ev(x)

(1.30)

is a solution of the boundary value problem in the form N 

a ˜ij Vξi ξj +

N 

˜bi Vx + c˜V = f˜, i

(1.31)

i=1

i,j=1

∂V = 0, ξN = 0. ∂ξN

(1.32)

On the basis of Theorem 1.4 below, we can derive the estimates of V (ξ), i.e. (1.33) ||V ||C 1 (Q) ¯ ≤ M8 = M8 (q, p, α, k, Q), β

||V ||W 2 (Q) ≤ M9 = M9 (q, p, α, k, Q), 2

(1.34)

where β (0 < β ≤ α), M8 , M9 are non-negative constants. Combining (1.22), (1.29), (1.33) and (1.34), the estimates (1.19) and (1.20) are obtained. Now, we shall give some estimates of solutions of Problem N for (1.2). Theorem 1.4 Suppose that the equation (1.2) satisfies Condition C. Then any solution u(x) of Problem N satisfies the estimates ¯ ≤ M10 = M10 (q, p, α, k, Q), Cβ1 [u, Q]

(1.35)

203

Clifford Analysis and Elliptic Equations ||u||W 2 (Q) ≤ M11 = M11 (q, p, α, k, Q),

(1.36)

2

where M10 , M11 are non-negative constants, and β is a constant as stated in (1.19). Proof First of all, choosing that x∗ is an arbitrary point in Q and ε ¯ such is a small positive number, we construct a function J(x) ∈ Cα2 (Q) that J(x) =

⎧ ⎨ 1,

⎩ 0,

x ∈ Ωε , x ∈ Q\∂Q2ε ,

0 ≤ J(x) ≤ 1, x ∈ ∂Q2ε \Qε ,

in which Qε = {|x − x∗ | ≤ ε} and J(x) satisfies the estimate C 2 [J(x), Q] ≤ M12 = M12 (ε, Q).

(1.37)

Denote U (x) = J(x)u(x); obviously U = U (x) is a solution of the boundary value problem N 

aij Uxi xj +

N  i=1

i,j=1

¯ bi Uxi + cU = f ∗ , x ∈ Q,

lU (x) = 0, x ∈ ∂Q,


N

in which d∗ = JLu + i,j=1 aij [Jxi uxj + Jxi xj u] + method of inner estimate in [1],[11], we can obtain

(1.38) (1.39)


N

i=1 bi Jxi u.

¯ ≤ M13 , C 1 [u, Qε ] ≤ M14 , Cβ1 [U, Q] β

By the (1.40)

where Mj = Mj (q, p, α, k, Qε ), j = 13, 14. Combining (1.37) and (1.40), we obtain the estimate Cβ1 [u, Qε ] ≤ M15 = M15 (q, α, k, Q, p).

(1.41)

Next, we choose any point x∗ ∈ S2 = ∂Q and a small positive number d such that S3 = S2 ∩ {|x − x∗ | ≤ d}. Then we can find a solution u ˆ(x) ˆ(x) satisfies the boundary condition of (1.21) on Q, such that u ∂u ˆ = g(x), x ∈ S3 , ∂n which satisfies the estimate (1.22). Thus u ˜(x) = u(x) − u ˆ(x) is a solution of the equation as stated in (1.24), and u˜(x) satisfies the boundary condition ∂u ˜ (1.42) = 0, x ∈ S3 . ∂n

204

Chapter VII

We can define a non-singular transformation of a second order continuously differentiable function ζ = ζ(x), such that S3 maps onto S4 on the plane ζN = 0, the domain Q onto the domain Q1 in the half space ζN < 0, and the equation (1.24) and boundary condition (1.42) are reduced to the equation and boundary condition as follows: N 

Aij u ˜ ζi ζ j +

N 

˜ = D in Q1 Bi u ˜ζi + C u

(1.43)

i=1

i,j=1

∂u ˜ = 0 on S4 . ∂ξN

(1.44)

Now, we extend the function u ˜ to a symmetric domain Q2 of Q1 about S4 , i.e. we define a function U=

⎧ ⎨ u ˜(ζ),

⎩ u ˜(ζ ∗ ),

ζ ∈ Q1 , ζ ∗ ∈ Q2 ,

(1.45)

where ζ ∗ = (ζ1 , ..., ζN −1 , −ζN ), and U (ζ) is a solution of the equation N 

A˜ij Uζi ζj +

N  i=1

i,j=1

˜ =D ˜ in Q1 ∪ Q2 , ˜i Uζ + CU B i

(1.46)

where A˜ij =

˜i = B

C˜ =

⎧ ⎨ Aij (ζ),

⎩ (−1)k A (ζ ∗ ), ij ⎧ ⎨ Bi (ζ),

k=

k=

⎧ ⎨ 1, i = j, i or j = N,

⎩ 0, other cases, ⎧ ⎨ 1, i = N,

⎩ 0, i = ⎩ (−1)k B (ζ ∗ ),  N, i ⎧ ⎧ ⎨ D(ζ), ζ ∈ Q1 , ⎨ C(ζ), ⎩ C(ζ ∗ ),

˜= D

⎩ D(ζ ∗ ),

ζ ∈ Q2 .

By using a similar method in the proof of (1.41), we can derive that U (ζ) and u(x) satisfy the estimate Cβ1 [U, Q1 ∪ Q2 ] ≤ M16 , Cβ1 [u, Q1 ] ≤ M17 ,

(1.47)

where Mj = Mj (q, p, α, k, Q), j = 16, 17. Combining (1.41) and (1.47), the estimates (1.35) and (1.36) are obtained.

205

Clifford Analysis and Elliptic Equations

1.3 Solvability of oblique derivative problem for elliptic equations We first consider a special equation of (1.2), namely ∆u = gm (x, u, Dx u, Dx2 u), gm = ∆u−

N 

i,j=1




aijm uxi xj −

N 

bim u−cm u+fm in Q,

i=1

2 2 where ∆u = N i=1 ∂ u/∂xi , Λ = (2N − 1) inf Q and the coefficients

aijm =

⎧ ⎨ aij /Λ, ⎩ δ /Λ, ij ⎧ ⎨ c/Λ,

cm =

⎩ 0,

(1.48)

bim =

fm =

⎧ ⎨ bi /Λ,


N

i=1 aii /(2N

2 − 2N

− 1),

i, j = 1, ..., N,

⎩ 0, ⎧ ⎨ f /Λ in Qm , ⎩0

in RN \Qm ,

where Qm = {(x) ∈ Q | dist(x, ∂Q) ≥ 1/m}, m is a positive integer, δii = 1, δij = 0 (i = j, i, j = 1, ..., N ). In particular, the linear case of equation (1.48) can be written as ∆u = gm (x, u, Dx u, Dx2 u), gm =

N 

[δij −aijm (x)]uxi xj

i,j=1

−

N  i=1

(1.49)

bim (x)uxi −cm (x)u+fm (x) in Q.

In the following, we will give a representation of solutions of Problem O for equation (1.48). Theorem 1.5 Under the same condition as in Theorem 1.1, if u(x) is any solution of Problem O for equation (1.48), then u(x) can be expressed in the form u(x) = U (x) + V (x) = U (x) + v0 (x) + v(x), ˜ = v(x) = Hρ



Q0

G=

G(x − ζ)ρ(ζ)dζ,

⎧ ⎨ |x−ζ|2−N /(N (2−N )ωN ), N > 2, ⎩ log |x − ζ|/2π, N = 2,

(1.50)

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Chapter VII

where ωN = 2π N/2 /(N Γ(N/2)) is the volume of a unit ball in RN , ρ(x) = ∆u = gm and V (x) is a solution of Problem D0 for (1.48) in Q0 = {|x| < R} with the boundary condition V (x) = 0 on ∂Q0 ; here R is an appropriately large number, such that Q0 ⊃ Q, and U (x) is a ˜ for ∆U = 0 in Q with the boundary condition solution of Problem O (1.58) below, which satisfy the estimates Cβ1 [U, Q] + ||U ||W 2 (Q) ≤ M18 , 2

Cβ1 [V, Q0 ] + ||V ||W 2 (Q0 ) ≤ M19 ,

(1.51)

2

where β(0 < β ≤ α), Mj = Mj (q, p, α, k, Qm ) (j =18,19) are nonnegative constants, q = (q0 , q1 ), k = (k0 , k1 , k2 ). Proof It is easy to see that the solution u(x) of Problem O for equation (1.48) can be expressed by the form (1.50). Noting that aijm = 0 (i = j), bim = 0, cm = 0, fm (x) = 0 in RN \Qm and V (x) is a solution of ˆ 2m = Q\Q2m Problem D0 for (1.48) in Q0 , we can obtain that V (x) in Q satisfies the estimate ˆ 2m ] ≤ M20 = M20 (q, p, α, k, Qm ). C 2 [V (x), Q On the basis of Theorem 1.3, we can see that U (x) satisfies the first estimate in (1.51), and then V (x) satisfies the second estimate in (1.51). Theorem 1.6 If equation (1.2) satisfies Condition C, then Problem O for (1.48) has a solution u(x). Proof In order to prove the existence of solutions of Problem O for the nonlinear equation (1.48) by using the Leray-Schauder theorem, we introduce an equation with the parameter h ∈ [0, 1], ∆u = hgm (x, u, Dx u, Dx2 u) in Q.

(1.52)

ˆ 2 (Q) = Denote by BM a bounded open set in the Banach space B = W 2 1 2 Cβ (Q) ∩ W2 (Q)(0 < β ≤ α), the elements of which are real functions V (x) satisfying the inequalities 1 ¯ ||V ||W ˆ 2 (Q) = Cβ [V, Q] + ||V ||W 2 (Q) < M21 = M19 + 1, 2

2

(1.53)

in which M19 is a non-negative constant as stated in (1.51). We choose any function V˜ (x) ∈ BM and substitute it into the appropriate positions ˜ in the right-hand side of (1.52), and then we define an integral v˜(x) = Hρ as ˜ ρ˜, ρ˜(x) = ∆V˜ . v˜(x) = H (1.54)

207

Clifford Analysis and Elliptic Equations Next we find a solution v˜0 (x) of the boundary value problem in Q0 : ∆˜ v0 = 0 on Q0 ,

(1.55)

v (x) on ∂Q0 , v˜0 (x) = −˜

(1.56)

˜ = 0 on Q, ∆U

(1.57)

and denote the solution Vˆ (x) = v˜(x) + v˜0 (x) of the corresponding Problem D0 in Q0 . Moreover on the basis of the result in [38], we can find a ˜ (x) of the corresponding Problem O ˜ in Q, solution U

˜ ˆ ∂U ˜ = g(x) − ∂ V + b(x)Vˆ on ∂Q. + b(x)U ∂ν ∂ν Now we discuss the equation ˜ + Dx2 V ), 0 ≤ h ≤ 1, ˜, Dx u ˜, Dx2 U ∆V = hgm (x, u

(1.58)

(1.59)

˜ + Vˆ . By Condition C, applying the principle of contracting where u ˜=U mapping, we can find a unique solution V (x) of Problem D0 for equation (1.59) in Q0 satisfying the boundary condition V (x) = 0 on ∂Q0 .

(1.60)

Denote u(x) = U (x) + V (x), where the relation between U and V is the ˜ and V˜ , and by V = S(V˜ , h), u = S1 (V˜ , h) (0 ≤ same as that between U h ≤ 1) the mappings from V˜ onto V and u respectively. Furthermore, if V (x) is a solution of Problem D0 in Q0 for the equation ∆V = hgm (x, u, Dx u, Dx2 U + Dx2 V )), 0 ≤ h ≤ 1,

(1.61)

where u = S1 (V, h), then from Theorem 1.3, the solution V (x) of Problem D0 for (1.61) satisfies the estimate (1.53), consequently V (x) ∈ BM . Set B0 = BM × [0, 1]. In the following, we shall verify that the mapping V = S(V˜ , h) satisfies the three conditions of the Leray-Schauder theorem: 1) For every h ∈ [0, 1], V = S(V˜ , h) continuously maps the Banach space B into itself, and is completely continuous on BM . Besides, for every function V˜ (x) ∈ BM , S(V˜ , h) is uniformly continuous with respect to h ∈ [0, 1]. In fact, we arbitrarily choose V˜l (x) ∈ BM (l = 1, 2, ...); it is clear that from {V˜l (x)} there exists a subsequence {V˜lk (x)} such that ˜l (x)}, {V˜lk (x)}, {V˜lk xi (x)} (i = 1, ..., N ) and corresponding functions {U k

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Chapter VII

˜l x (x)}, {˜ ulk (x)}, {˜ ulk xi (x)} (i = 1, ..., N ) uniformly converge to {U k i ˜0 (x), U ˜0x (x), u V˜0 (x), V˜0xi (x), U ˜0 (x), u ˜0xi (x) (i = 1, ..., N ) in Q0 , Q i ˜ respectively, in which u ˜lk = S1 (Vlk , h), u ˜0 = S1 (V˜0 , h). We can find a solution V0 (x) of Problem D0 for the equation ˜0 + D2 V0 ), 0 ≤ h ≤ 1 in Q0 . ˜0 , Dx u ˜0 , Dx2 U ∆V0 = hgm (x, u x

(1.62)

From Vlk = S(V˜lk , h) and V0 = S(V˜0 , h), we have ˜l + Dx2 Vl ) ˜lk , Dx u ˜lk , Dx2 U ∆(Vlk − V0 ) = h[gm (x, u k k ˜l + Dx2 V0 ) + Cl (x)], 0 ≤ h ≤ 1, ˜lk , Dx u ˜lk , Dx2 U −gm (x, u k k where

˜l + Dx2 V0 ) Clk (x) = gm (x, u ˜lk , Dx u ˜lk , Dx2 U k ˜0 + Dx2 V0 ), x ∈ Q0 . ˜0 , Dx u ˜0 , Dx2 U −gm (x, u

According to a similar method to deriving (2.43), Chapter II in [81], we can prove that (1.63) L2 [Clk (x), Q0 ] → 0 as k → ∞. Moreover according to Theorem 1.3, we can derive that ||Vlk − V0 ||W ˆ 2 (Q0 ) ≤ M22 L2 [Clk , Q0 ], 2

where M22 = M22 (q, p, α, k0 , Qm ) is a non-negative constant, hence Thus from {Vlk (x) − V0 (x)}, ||Vlk − V0 ||W ˆ 2 (Q0 ) → 0 as k → ∞. 2 there exists a subsequence (for convenience we denote the subsequence again by {Vlk (x) − V0 (x)}) such that ||Vlk (x) − V0 (x)||W ˆ 2 (Q0 ) 2

= Cβ1 [Vlk (x) − V0 (x, t), Q0 ] + ||Vlk (x) − V0 (x)||W 2 (Q0 ) → 0 as k → 2 ∞. From this we can obtain that the corresponding subsequence {ulk (x) − u0 (x)} = {S1 (Vlk , h) − S1 (V0 , h)} possesses the property: ||ulk (x) − u0 (x)||W ˆ 2 (Q) → 0 as k → ∞. This shows the complete conti2 ˜ nuity of V = S(V , h) (0 ≤ h ≤ 1) in BM . By using a similar method, we can prove that V = S(V˜ , h) (0 ≤ h ≤ 1) continuously maps BM into B, and V = S(V˜ , h) is uniformly continuous with respect to h ∈ [0, 1] for V˜ ∈ BM . 2) For h = 0, from (1.53) and (1.59), it is clear that V = S(V˜ , 0) ∈ BM . 3) From Theorem 1.3 and (1.53), we see that V = S(V˜ , h)(0 ≤ h ≤ 1) does not have a solution u(x) on the boundary ∂BM = BM \BM .

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Clifford Analysis and Elliptic Equations

Hence by the Leray-Schauder theorem (see [40]), we know that Problem D0 for equation (1.59) with h = 1 has a solution V (x) ∈ BM , and then Problem P of equation (1.52) with h = 1, i.e. (1.48) has a solution u(x) = S1 (V, h) = U (x)+V (x) = U (x)+v0 (x)+v(x) ∈ B. Theorem 1.7 Under the same conditions as in Theorem 1.1, Problem O for the equation (1.2) has a solution. Proof By Theorem 1.3 and Theorem 1.6, Problem O for equation (1.48) possesses a solution um (x), and the solution um (x) of Problem O for (1.48) satisfies the estimates (1.19) and (1.20), where m = 1, 2, .... Thus, we can choose a subsequence {umk (x)}, such that {umk (x)}, {umk xi (x)} (i = 1, ..., N ) in Q uniformly converge to u0 (x), u0xi (x) (i = 1, ..., N ) respectively. Obviously, u0 (x) satisfies the boundary conditions of Problem O. On the basis of principle of compactness of solutions for equation (1.48), we can see that u0 (x) is a solution of Problem O for (1.2).

2

Boundary Value Problems of Degenerate Elliptic Equations of Second Order

2.1 Formulation of the Oblique Derivative Problem for Degenerate Elliptic Equations Let G ∈ Cα2 (0 < α < 1) be a bounded domain in the upper-half space xN > 0, whose boundary is ∂G = S1 ∪ S2 , in which S1 is located on xN = 0 and S2 is located in xN > 0. Denote Γ = S2 ∩ {xN = 0}. We consider the degenerate elliptic equation of second order Lu =

N 

i,j=1

aij (x)uxi xj +

N 

bi (x)uxi +

i=1

h(x) uxN +c(x)u = f (x) in G. (2.1) xkN

Suppose that the equation (2.1) satisfies the following conditions, i.e. Condition C: 1) There exists a positive number q0 (< 1), such that for x = (x1 , ..., xN ) ∈ G, the following equality holds: N 

i,j=1

aij (x)ξi ξj ≥ q0

N 

ξi2 ;

(2.2)

i=1

there is no harm in assuming that aN N = 1 and (aij (x)) is symmetrical.

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Chapter VII

2)

Coefficients of (2.1) satisfy the conditions

Cα [aij (x), G], Cα [bi (x), G], Cα [c(x), G] ≤ k0 , i, j = 1, ..., N, Cα [f (x), G] ≤ k1 , C 2 [h(x), G] ≤ k0 , c(x) ≤ 0, k > 0,

(2.3)

where α(0 < α < 1), kj (j = 0, 1) are non-negative constants. Problem Q Find a bounded solution u(x) of equation (2.1) in G satisfying the oblique derivative boundary condition lu =

∂u + b(x)u = g(x) on S2 , ∂ν

(2.4)

where ν is a vector on every point of S2 , and b(x), g(x) satisfy the conditions Cα1 [cos(ν, n), S2 ], Cα1 [b(x), S2 ] ≤ k0 , Cα1 [g(x), S2 ] ≤ k2 ,

(2.5)

cos(ν, n) > 0, b(x) < 0 on S2 , in which n is the inner normal vector on S2 . The contents of this section are mainly chosen from [42].

2.2 Unique Solvability of the Oblique Derivative Problem for Degenerate Elliptic Equations Lemma 2.1 the operator

If the boundary ∂Q of the domain Q belongs to Cα2 and

Lu =

N 

aij (x)uxi xj +

N 

bi (x)uxi + c(x)u

i=1

i,j=1

satisfies the uniformly elliptic condition, aij (x), bi (x), c(x) ∈ Cα (Q), and the coefficients of boundary operator lu =

N 

di (x)uxi + b(x)u

i=1

satisfy the conditions N  i=1

di (x) cos(n, xi )|∂Q ≥ q0 > 0, Cα1 [di (x),∂Q] ≤k0 ,Cα1 [b(x),∂Q] ≤k2 , (2.6)

211

Clifford Analysis and Elliptic Equations then any function u(x) (∈ Cα2 (Q)) satisfies the estimate Cα2 [u, Q] ≤ M1 {Cα [Lu, Q] + Cα1 [Bu, ∂Q] + c[u, Q]},

where the non-negative constant M1 is only dependent on the coefficients of L, l and ∂Q (see [38]). We use the parameter expression of S2 as follows: xi = xi (ξ1 , ..., ξN −1 ), i = 1, ..., N, in which xi (ξ1 , ..., ξN −1 ) ∈ Cα2 and define by Xi =

1 ∂(xi+1 , ..., xN , x1 , ..., xi−1 ) , Ω ∂(ξ1 , ..., ξN −1 )

the direct cosine of an inner normal line of S2 , where Ω=

N * 1/2  ∂(xi+1 , ..., xN , x1 , ..., xi−1 ) +2 i=1

∂(ξ1 , ..., ξN −1 )

> 0.

Theorem 2.2 Suppose that equation (2.1) satisfies Condition C and Cα1 [aij , G] ≤ k0 , c < 0 in G. Then Problem Q of (2.1) has a bounded solution u(x) ∈ Cα2 (G ∪ S2 ). Proof Choose a decreasing sequence of positive numbers {εm } such that εm = 0 as m → ∞, and define a sequence of domains {Gm } satisfying the conditions: 1) G1 ⊂ G2 ⊂ ...; 2) ∪Gm = G; 3) Gm and its boundary are in xN > 0, and Gm ∩ {xN ≥ εm } = G ∩ {xN ≥ εm }; 4) The boundary Sm of Gm belongs to Cα2 . Introduce the boundary condition ∂um + bm (x)um = gm (x) on Sm , ∂νm where we choose νm , bm (x), gm (x), such that when xm > 2εm , cos(νm , n) > 0; herein n is the inner normal vector of Sm and νm = ν, bm (x) = b(x), gm (x) = g(x). Next for Sm ∩ {xN ≤ εm }, set ⎡

⎤1/2

N N N   ∂um 1 ∂um ∂um  = aij (x)Xj aij (x)Xj )2 ⎦ ( = , B =⎣ ∂νm ∂l B ∂x i i,j=1 i,j=1 i,j=1

,

212

Chapter VII

in which l is the secondary normal vector and its directional cosine is
Yi = N j=1 aij (x)Xj /B. It is clear that cos(n, l) > 0, and bm (x) < −1, 1 Cα [bm , Sm ∩ {xN ≤ εm }], Cα1 [gm , Sm ∩ {xN ≤ εm }] ≤ k3 , where k3 is a positive constant. Moreover for {3εm /2 < xN ≤ 2εm }, let ν = ν, and Cα1 [bm , Sm ∩ {3εm /2 < xN ≤ 2εm }], Cα1 [gm , Sm ∩ {3εm /2 < xN ≤ 2εm }] ≤ k3 . Finally for Sm ∩ {εm < xN ≤ 3εm /2}, ∂um /∂νm is defined to be the relation −1 ∂um ∂um N ∂um = k(ξ1 , ..., ξN −1 ) hi (ξ1 , ..., ξN −1 ) + , ∂l ∂νm ∂ξi i=1

where k(ξ1 , ..., ξN −1 ) > 0, hi (ξ1 , ..., ξN −1 ) ∈ Cα1 , bm < −1, and bm (x), gm (x) satisfy the condition as before. Noting that N N   cos(ν, n) ∂ ∂ ∂ = cos(l, xi ) cos(ν, xi ) = ∂ν ∂xi cos(n, l) ∂xi i=1 i=1

+

N   i=1



∂ cos(ν, n) 3εm cos(l, xi ) < xN < 2εm , cos(ν, xi )− on cos(n, l) ∂xi 2

we have

∂ cos(n, l) ∂ = ∂l cos(ν, n) ∂ν +

N  

cos(l, xi )−

i=1



∂ cos(n, l) 3εm cos(ν, xi ) < xN < 2εm . on cos(ν, n) ∂xi 2

(2.7)

Denote αi = cos(l, xi ) − cos(n, l) cos(ν, xi )/ cos(ν, n), then it is obvious N 

αi Xi = 0.

i=1

Besides for any differentiable function v(x1 , ..., xN ), we have N N   ∂v ∂xj ∂v ∂xj ∂v ∂v = , i = 1, ..., N − 1, i.e. − = 0. ∂ξi j=1 ∂xj ∂ξi ∂x ∂ξ ∂ξ j i i j=1

There is no harm in assuming that ∂(x1 , ..., xN −1 )/∂(ξ1 , ..., ξN −1 ) = 0, we obtain ∂v ∂v ∂(xj+1 , ..., xN , x1 , ..., xj−1 ) = ∂xj ∂xN ∂(ξ1 , ..., ξN −1 ) 9 ∂(xj+1 , ..., xN −1 , −v, x1 , ..., xj−1 ) ∂(x1 , ..., xN −1 ) + , ∂(ξ1 , ..., ξN −1 ) ∂(ξ1 , ..., ξN −1 ) j = 1, ..., N − 1,

213

Clifford Analysis and Elliptic Equations thus 9

−1 ∂(xi+1 , ..., xN , x1 , ..., xi−1 ) ∂v ∂(x1 , ..., xN −1 ) ∂v N αi = αi ∂xi i=1 ∂(ξ1 , ..., ξN −1 ) ∂xN ∂(ξ1 , ..., ξN −1 ) i=1

N 

9

N −1 

∂(xi+1 , ..., xN −1 , −v, x1 , ..., xi−1 ) ∂(x1 , ..., xN −1 ) + αi ∂(ξ1 , ..., ξN −1 ) ∂(ξ1 , ..., ξN −1 ) i=1 +αN

−1 ∂(xi+1 , ..., xN −1 , −v, x1 , ..., xN −1 ) ∂v 1 N = αi ∂xN ΩXN i=1 ∂(ξ1 , ..., ξN −1 )

+

N −1  αi Xi i=1

XN

+ αN



∂v . ∂xN

˜ 1 , ..., ξN −1 ) ∈ C 1 , such that Besides we can find N − 1 functions h(ξ α N  i=1

αi

−1 ∂v N ˜ 1 , ..., ξN −1 ) ∂v , = h(ξ ∂xi i=1 ∂ξi

thus (2.7) can be written as −1 ∂ ∂ ∂ N = k ′ (ξ1 , ..., ξN −1 ) + h′i (ξ1 , ..., ξN −1 ) , ∂l ∂ν i=1 ∂ξi

where k ′ (ξ1 , ..., ξN −1 ) > 0, h′i (ξ1 , ..., ξN −1 ) ∈ Cα1 . Especially k ′ (ξ1 , ..., ξN −1 ) = 1, h′i (ξ1 , ..., ξN −1 ) = 0 (i = 1, ..., N − 1) in xn ≤ εN . Hence we have −1 ∂ ∂ ∂ N = K(ξ1 , ..., ξN −1 ) Hi (ξ1 , ..., ξN −1 ) + , i = 1, ..., N −1. ∂l ∂νm i=1 ∂ξi

Now we consider the boundary value problem (Problem Qm ) Lum = f (x) in Gm ,

∂um + bm um = gm (x) on Sm . ∂νm

(2.8)

By the result in [38] and Lemma 2.1, we know that there exists a solution um (x) of Problem Qm , and the solution um (x) satisfies the estimate C[um (x), Gm ] ≤ M1 , Cα2 [um (x), Gm ] ≤ M2 ,

(2.9)

where M1 = M1 (q0 , α, k, G), M2 = M2 (q0 , α, k, Gm ) are two positive constants, k = (k0 , k1 , k2 ). In the following, we shall prove that um (x),

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Chapter VII

umxi (x), umxi xj (i, j = 1, ..., N ) uniformly converge to the functions u0 (x), u0xi (x), u0xi xj (i, j = 1, ..., N ) in G ∩ {xN ≥ 3η > 0} respectively. In fact, for arbitrary small positive number η, we can assume that ψ(x) ∈ Cα2 (G), ψ(x) = 1 for xN ≥ 2η, ψ(x) = 0 for xN ≤ η, and ∂ψ/∂ν = 0 on S2 . Let vm = um (x)ψ(x), then we have Lvm = ψ(x)f (x) + um (x)Lψ +2

N 

aij (x)

i,j=1

∂um ∂ψ −c(x)ψ(x)um (x) = Fm (x). ∂xi ∂xj

(2.10)

Denote by Gη a bounded domain in G with the boundary Sη ∈ Cα2 , and Sη ∩ {xN ≥ η} = S2 ∩ {xN ≥ η}, Gη ⊂ {xN > 0}. It is clear that there exists a positive integer m such that Gη ⊂ Gm , when 2εm ≤ η, and ∂vm ∂ψ(x) + b(x)vm (x) = ψ(x)gm (x) + b(x)um (x) ∂ν ∂ν

(2.11)

= ψ(x)g(x) for xN ≥ η; moreover for xN < η, we have ψ(x) ≡ 0, vm (x) ≡ 0, hence ∂vm + b(x)vm (x) = ψ(x)g(x) for xN < η. ∂ν Thus

∂vm + b(x)vm (x) = ψ(x)g(x) on Sη . (2.12) ∂ν By the results in [38], Section 1 and Lemma 2.1, the boundary value problem (2.10),(2.12) has a bounded solution vm (x), and the solution satisfies the estimate lvm =

Cα2 [um (x), Gη ] ≤ M3 = M3 (q0 , α, k, Gη ).

(2.13)

Hence vm (x), vmxi (x), vmxi xj (i, j = 1, ..., N ) are uniformly bounded and equicontinuous. According to the Ascoli-Arzela theorem, from {vm (x)} we can choose a subsequence {vmk (x)}, such that {vmk }, {vmk xi }, {vmk xi xj } uniformly converge to u(x), uxi , uxi xj in Gη and u(x) ∈ Cα2 (Gη ). Hence u(x) satisfies the equation and boundary condition Lu = f (x) in xN ≥ 3η,

∂u + b(x)u = g(x) on S2 ∩ {xN ≥ 3η}. ∂ν

This prove that there exists a bounded solution of the above problem in G ∩ {xN ≥ 3η}. Noting the arbitrariness of η, we verify that the

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Clifford Analysis and Elliptic Equations

boundary value problem (2.1),(2.4), i.e. Problem Q, has a bounded solution u(x) ∈ Cα2 (G ∪ S2 ). Corollary 2.3 If the coefficient c(x) ≤ 0 of equation (2.1), and the boundary condition lu =

N  ∂u + b(x)u = di (x)uxi + b(x)u = g(x) on S2 ∂ν i=1

(2.14)

satisfies Cα1 [di (x), S2 ] ≤ k0 ,

N  i=1

di (x) cos(n, xi )|S2 ≥ q0 > 0,

cos(ν, n) > 0, b(x) ≤ 0 on S2 , and one of bi (x)(i = 1, ..., N ) for instance bj (x) > 0(1 ≤ j ≤ N ), then the result in Theorem 2.2 is still valid. Proof

Introduce a transformation u(x) = (C − eµxj )v(x), 1 ≤ j ≤ N ;

equation (2.1) is reduced to N 

aij (x)vxi xj +

N −1  i=1

i,j=1

˜bi (x)vx + h(x) vx + c˜(x)v = f˜(x) in G, i xkN N

where c˜(x) = c(x)−(a11 (x)µ2 +bj (x)µ)eµxj /(C −eµxj ). Noting a11 > q0 , we can choose positive constants C and µ such that c˜(x) < 0 in G and (C − eµxj ) > 1, then the boundary condition (2.14) is transformed into N 

d˜i (x)vxi + ˜b(x)v = g˜(x) on S2 ,

i=1

in which ˜b(x) = b(x) − µbj (x)eµxj /(C − eµxj ) < 0. Theorem 2.4 If the conditions in Theorem 2.2 and b(x) ≤ 0 on S2 hold, then when g(x) ≡ 0, equation (2.1) has a bounded solution u(x) ∈ Cα2 (G ∪ S2 ) satisfying the boundary condition (2.4) on S2 . The proof is the same with that of Theorem 2.2. Theorem 2.5 Let the conditions in Theorem 2.2 hold, and cos(ν, xN ) > 0 on Γ. Moreover one of the following conditions holds:

216

Chapter VII

a) k = 1, h(x1 , ..., xN −1 , 0) ≥ 1, and h(x) about xN is even; b) k > 1, h(x1 , ..., xN −1 , 0) > 0. Then Problem Q of (2.1) has at most a bounded solution u(x) ∈ Cα2 (G ∪ S2 ). Proof It is sufficient to prove the bounded solution u(x) ≡ 0 of the boundary value problem: Lu = 0 and ∂u/∂ν + bu = 0 on S2 . For this we construct a barrier function W (x) as follows: 1) W (x) ≥ c0 > 0 in G, here c0 is a constant; 2) When xN → 0, W (x) → ∞ uniformly holds; 3) LW < 0 in G; 4)

∂W + bW < M12 on S2 . ∂ν

If there exists the function W (x), by Theorem 2.2 we can establish a bounded function w(x) satisfying Lw = 0 and ∂w/∂ν + bw > ∂W/∂l + bW. Let V (x) = W (x) − w(x). Then LV < 0, ∂V /∂ν + bV < 0, and from the boundedness of w(x), it follows that limXN →0 V (x) = +∞. On the basis of the result in [58], we can derive εV ± u0 ≥ 0, where ε is a positive number. In fact, L(εV ± u0 ) = εLV < 0 in G, ∂(εV ± u0 )/∂ν + b(εV ± u0 ) ≤ 0 on S2 , and V → ∞ as XN → 0, hence εV ± u0 cannot take the negative minimum in G. Due to the arbitrariness of ε, we obtain u0 ≡ 0. Now we make the barrier function in case a) as follows:

˜)J + K, W (x) = ln xN − (x1 − a

(2.15)

˜ > 1 for x = (x1 , ..., xN ) ∈ G, and where a ˜ is chosen such that x1 − a J, K are undetermined constants. For a sufficiently small xN , we can choose that a positive number A is large enough, such that 1 − h(x) (xN )2 = [1−h(x1 , ..., xN −1 , 0)−

∂h(x1 ,...,xN −1 ,0)xN ∂xN

+O(x2xN )]x−2 N ≤ A;

the above inequality is also valid for any xN in G. From (2.2), a11 ≥ q0 ,

217

Clifford Analysis and Elliptic Equations we can select a J large enough such that ˜)J−2 LW = −a11 (x)J(J − 1)(x1 − a −b1 (x)J(x1 −˜ a)J−1 +[1−h(x)]x−2 N +c(x)W < 0,

and then choose a sufficiently large K such that W (x) ≥ c0 > 0 in G. Moreover ∂W/∂ν = −(∂xN /∂ν)/xN + O(1). According to the hypothesis: cos(ν, xN ) > 0 near to Γ, hence ∂W/∂ν + bW < M12 on S2 ; this shows that the function W (x) in (2.15) satisfies all conditions of the barrier function. Finally we consider the case b); we choose the function ˜)J + K, 0 < β < 1, W (x) = x−β N − (x1 − a

(2.16)

and for a sufficiently small xN , it is clear that β −k+1 −β−2 β[β + 1 − h(x)xN < 0, ]x < − h(x1 , ..., xN −1 , 0)x−k−β−1 N 2 where J, K are chosen as stated in case a). Moreover from cos(ν, xN ) > 0 on Γ, we have ∂W/∂ν + b(x)W < M12 . Thus the function W (x) in (2.16) satisfies all conditions of the barrier function. This completes the proof. Theorem 2.6 Under the same conditions as in Theorems 2.4 and 2.5, there exists at most a bounded solution u(x) ∈ Cα2 (G ∪ S2 ) of (2.1) satisfying the boundary condition ∂u + bu = 0 on S2 . ∂l

3

The Schwarz Formulas and Dirichlet Problem in Halfspace and in a Ball

First of all, we introduce the regular function in real Clifford analysis. The Clifford algebra An (R) over the space Rn is defined as follows: Let e1 = 1, e2 , · · · , en be the standard orthogonal basis in Rn , and denote by es the general basis element of An (R), where s is any subset of {1, 2, 3, ..., n}, i.e. ⎧ ⎪ e1 = 1, e2j = −1, 2 ≤ j ≤ n, ⎪ ⎪ ⎪ ⎪ ⎨ ej ek = −ek ej , 2 ≤ j < k ≤ n, ⎪ ⎪ es = ej1 ej2 · · · ejs , 2 ≤ j1 < j2 < · · · < js ≤ n, ⎪ ⎪ ⎪ ⎩

s = {j1 , j2 , ..., js }.

218

Chapter VII

It is not difficult to see that An (R) is a 2n−1 -dimensional Clifford algebra space. An arbitrary element of the Clifford algebra An (R) can be written as 

x = x1 e1 +· · ·+xn en +· · ·+xxj1 ...γsej1 ...eγs +· · ·+x2...n e2 ...en =

xs es ,

s

in which x1 , ..., xn , ..., xj1 ...js , ..., x2...n ∈ R.





xs es , y = ys es ∈ An (R), where xs , ys ∈ R. Then Let x =
x y e e , in which es ek is a basis element of An (R) and xy = s s k k s,k xs yk ∈ R. For x = x1 e1 + x2 e2 + · · · + xn en ∈ Rn , its conjugate element is defined by x ¯ = x1 e1 − x2 e2 − · · · − xn en . It is obvious that x¯ x=x ¯x = |x|2 . If x = 0, then x is invertible, and x−1 = x ¯/|x|2 . Denote by D a connected open set in R, and by (r)

FD = {f |f : D → An (R), f (x) =

 s

fs (x)es , fs ∈ C r (D)}

the set of continuously differentiable functions up to degree r, the values of which belong to An (R). Define the differential operators ∂ ∂ ∂ ∂ ∂ ∂¯ = e1 + e2 + · · · + en , ∂ = e1 − · · · − en ; ∂x1 ∂x2 ∂xn ∂x1 ∂xn then we have ¯ = ∂ ∂¯ = ∂∂

n  ∂2

j=1

∂x2j

= ∆.

(r) ¯ = 0, then f (x) is called a (left-)regular If f (x) ∈ FD (r ≥ 1) and ∂f function, and then fs (x) is harmonic in D. In particular if n = 2, the regular function f (x) in a domain D in R2 is an analytic function, and
if n = 3, the regular function f (x) = s fs (x)es = f1 (x) + f2 (x)e2 + f3 (x)e3 + f23 (x)e2 e3 in a domain in R3 is a solution (f1 , f2 , f3 , f23 ) of the system of partial differential equations of first order

⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝

∂ ∂x1 ∂ ∂x2 ∂ ∂x3

− ∂x∂ 2

0

− ∂x∂ 3

∂ ∂x1

0

− ∂x∂ 3 0 ∂ ∂x1 ∂ ∂x2

0 ∂ ∂x3 − ∂x∂ 2 ∂ ∂x1

⎞⎛

f1

⎟⎜ ⎟⎜ ⎟ ⎜ f2 ⎟⎜ ⎟⎜ ⎟ ⎝ f3 ⎠

f23

⎞

⎛

0

⎞

⎟ ⎟ ⎜ ⎟ ⎜ 0 ⎟ ⎟ ⎟ ⎜ ⎟. ⎟=⎜ ⎟ ⎜ 0 ⎟ ⎠ ⎠ ⎝

0

(3.0)

Clifford Analysis and Elliptic Equations

219

We can see that it is an analytic function in the 3-dimensional domain. Hence, the general regular functions possess some properties similar to those of analytic functions in the planar domain. In the following, we first discuss the 3-dimensional space R3 and the corresponding Clifford algebra A3 (R) over R3 . In this case, x = x1 + x2 e2 + x3 e3 + x23 e23 ∈ A3 (R), and we define by Re x = x1 + x2 e2 , Im x = x3 − x23 e2 ,

(3.1)

the real part and imaginary part of x. It is evident that x−x ˜ 1 ˜), Im x = −e3 , x = Re x + e3 Im x, Re x = (x + x 2 2

(3.2)

where x ˜ = Re x − e3 Im x. Let y = Re y + e3 Im y ∈ A3 (R), then we can : =x verify that xy ˜y˜.

3.1

Schwarz Formula and Dirichlet Problem for the Halfspace

Theorem 3.1 Suppose that u(y) is a H¨ older continuous function in the plane E = {x3 = 0}, and u(y) = 0 if |y| is large enough. Then there exists a regular function f (x) in the upper halfspace D, such that Ref + (y) = u(y), y ∈ E.

(3.3)

Then the function f (x) can be expressed as f (x) =

1 2π



E

x3 + e3 [(x1 − y1 ) + e2 (x2 − y2 )] u(y)dSy + e3 g, |y − x|3

(3.4)

where we assume that lim|x|→∞ f (x) = f (∞) = e3 (a3 − a23 e2 ), a3 , a23 are real constants, and then e3 g = f (∞). This is also a representation of the solution of the Dirichlet problem for regular functions in the halfspace D. Proof We assume that f (x) is a desired regular function, where x is any point in D. Let ΣR be the upper half sphere with the center at the origin and radius R, and denote by DR a domain with the boundary ΣR and ER = {|x| < R, x ∈ E}. We choose R so large that x ∈ DR and |y − x| ≥ R/2, y ∈ ER . By the Cauchy integral formula we have 

y¯ − x ¯ 1 n(y)f (y)dSy f (x) = 4π ΣR |y − x|3  y¯ − x ¯ 1 n(y)f (y)dSy = I1 + I2 . + 4π ER |y − x|3

(3.5)

220

Chapter VII

Moreover from *  y¯ − x ¯ y¯   − ≤ |x|   3 3

|y − x|

|y|

1 1 + 2 |y − x||y| |y − x|2 |y|

+

2 4 6|x| ≤ |x|( 3 + 3 ) = 3 , R R R

(3.6)

it follows that    1   4π

ΣR







6|x| M |x| |f (y)|dSy ≤ 3 R 4πR3

≤

1 4π

=

M |x| → 0 as R → ∞. 2R

ΣR



 y¯ − x ¯ y¯ − 3 n(y)f (y)dSy  3 |y − x| |y| 

ΣR

(3.7)

dSy

For arbitrarily given positive constant ε, there exists a large positive constant R0 , such that |f (y) − f (∞)| < ε if |y| ≥ R0 . Thus    

ΣR

which implies that 

ΣR



 y¯ 4 n(y)[f (y) − f (∞)]dSy  ≤ ε, 3 |y| π

y¯ n(y)[f (y) − f (∞)]dSy → 0 as R → ∞. |y|3

(3.8)

Noting that 







y¯− x ¯ y¯− x ¯ 1 1 y¯ n(y)f (y)dSy = − 3 n(y)f (y)dSy 3 3 4π ΣR |y−x| 4π ΣR |y−x| |y|   y¯ y¯ 1 1 n(y)[f (y)−f (∞)]dSy + n(y)f (∞)dSy , + 3 4π ΣR |y| 4π ΣR |y|3

I1 =

and

1 4π



ΣR

y¯ 1 n(y)f (∞)dSy = f (∞), |y|3 2

and applying (3.7)and (3.8), 1 lim I1 = f (∞) 2

R→∞

follows.

(3.9)

221

Clifford Analysis and Elliptic Equations

As for the integral I2 , taking n(y) = −e3 , y = y1 + e2 y2 on ER into account, we have 1 4π

I2 = =

1 4π

1 = 4π





ER

ER



ER

y¯ − x ¯ n(y)f (y)dSy |y − x|3

(y1 − x1 ) − e2 (x2 − y2 ) + e3 x3 (−e3 )f (y)dSy |y − x|3

(3.10)

x3 + e3 [(x1 − y1 ) + e2 (x2 − y2 )] n(y)f (y)dSy . |y − x|3

Letting R tend to ∞, 

x3 +e3 [(x1 −y1 )+e2 (x2 −y2 )] 1 f (y)dSy + f (∞) 3 |y−x| 2 ER

1 f (x) = 4π

(3.11)

can be derived. If Im x < 0, then according to Cauchy’s theorem for regular functions, we can similarly obtain 0=

1 4π



ER

−x3 + e3 [(x1 − y1 ) + e2 (x2 − y2 )] 1 f (y)dSy + f (∞). |y − x|3 2

From the above formula, it is easy to derive 

1 0= 4π

E

x3 + e3 [(x1 − y1 ) + e2 (x2 − y2 ) ˜ 1 f (y)dSy − f˜(∞). |y − x|3 2

(3.12)

Adding (3.11) and (3.12), and noting that f˜(∞) = −f (∞), we get 1 4π

f (x) = =

1 2π



E



x3 +e3 [(x1 −y1 )+e2 (x2 −y2 )] [f (y)+ f˜(y)]dSy +f (∞) |y−x|3 E

x3 + e3 [(x1 − y1 ) + e2 (x2 − y2 )] u(y)dSy + f (∞). |y − x|3

It remains to verify that f (x) is just the desired function. It is sufficient to prove that Re f + (x0 ) = u(x0 ), where x0 is any point on E. We rewrite (3.4) in the form 1 f (x) = 2π +

1 2π





x3 +e3 [(x1 −y1 )+e2 (x2 −y2 )] u(y)dSy |y − x|3 ER

x3 +e3 [(x1 −y1 )+e2 (x2 −y2 )] u(y)dSy +f (∞), |y − x|3 E\ER

(3.13)

222

Chapter VII

where ER = { |y| < R, y ∈ E} and ΣR are as stated before. Due to u(y) = 0 if |y| is sufficiently large, and for piecewise smooth surface ΣR ∪ ER and sectionally H¨ older-continuous function u(y), the Plemelj formula is still true. Hence g + (x0 ) − g − (x0 ) = 2u(x0 ),

(3.14)

where g + (x) =



1 2π

ER ∪ΣR

x3 + e3 [(x1 − y1 ) + e2 (x2 − y2 )] u(y)dSy , x ∈ DR . |y − x|3

¯ R, Setting ζ˜ = x, ζ ∈ D g˜+ (x) =

1 2π



ER ∪ΣR

˜ = 1 = g˜+ (ζ) 2π



x3 − e3 [(x1 − y1 ) + e2 (x2 − y2 )] u(y)dSy |y − x|3

ER ∪ΣR

−ζ3 − e3 [(x1 − y1 ) + e2 (x2 − y2 )] u(y)dSy |y − ζ|3

can be obtained. Letting x tend to x0 , we know that g˜+ (x0 ) = −g − (x0 ). From (3.14), it follows that g + (x0 ) − g − (x0 ) = g + (x0 ) + g˜+ (x0 ) = 2u(x0 ), i.e.

(3.15)

Re g + (x0 ) = u(x0 ). Noting that lim

x→x0

=

1 2π 1 2π



E\ER



E\ER

x3 + e3 [(x1 − y1 ) + e2 (x2 − y2 )] u(y)dSy + f (∞) |y − x|3 e3 [(x01 − y1 ) + e2 (x02 − y2 )] u(y)dSy + f (∞), |y − x0 |3

it is easily seen that 

1 Re 2π



E\ER



e3 [(x01 − y1 ) + e2 (x02 − y2 )] u(y)dSy + f (∞) = 0. |y − x0 |3

Thus we obtain (3.13). Finally, on account of u(y) = 0 if |y| is large enough, lim f (x) = f (∞)

|x|→∞

can be derived. This shows that the above regular function f (x) is unique.

Clifford Analysis and Elliptic Equations

223

Remark By using a similar method, we can also get an expression for the solution of the Dirichlet problem for regular functions in the halfspace D with weaker conditions, namely we suppose u(y) is a bounded continuous function in the plane E = {x3 = 0}, and the solution f (x) of the Dirichlet problem satisfies the condition | Ref (x) | = O(1/|x|) as |x| tends to ∞. Then f (x) may be expressed in the form (3.4), where g(x1 , x2 ) = g1 (x1 , x2 ) + e2 g2 (x1 , x2 ) is an arbitrary function satisfying gx1 = e2 gx2 (see [3]). 3.2

Schwarz Formula and Dirichlet Problem for a Ball

Next, we shall give the Schwarz formula for regular functions in a ball G = {|x| < R, 0 < R < ∞} in R3 . We need the following lemmas. Lemma 3.2 A function f (x) with values in the Clifford algebra A3 (R) is regular in G if and if only Re f (x) and Im f (x) satisfy the system of first order equations 2

∂ ∂ Re f = Im f, x ∈ G, ∂x∗ ∂x3

∂ ∂ Re f = −2 ∗ Im f, x ∈ G, ∂x3 ∂x where x∗ = x1 + x2 e2 , x∗ = x1 − x2 e2 , and

(3.16) (3.17)

∂ ∂ ∂ 1 ∂ ∂ 1 ∂ = ( − e2 ), = ( + e2 ). ∗ ∂x 2 ∂x1 ∂x2 ∂x∗ 2 ∂x1 ∂x2 Proof

It is clear that 2

∂ ∂ (e3 Im f ) = 2e3 Im f. ∂x∗ ∂x∗

Hence

¯ = (2 ∂ + e3 ∂ )(Re f + e3 Im f ) ∂f ∂x∗ ∂x3 ∂ ∂ ∂ ∂ = (2 Re f − Im f ) + e3 ( Re f + 2 ∗ Im f ). (3.18) ∂x∗ ∂x3 ∂x3 ∂x ¯ = 0, from (3.18) follows (3.16) and (3.17). If f (x) is regular in G, i.e. ∂f The inverse statement is also true. Suppose that f (x) is a regular function and f (x) ∈ C 2 (G). Then ¯ Re f = ∂ ∂¯ Re f = ∆Re f = 0, ∆Im f = 0. ∂∂

224

Chapter VII

So Re f and Im f are called harmonic functions in Clifford analysis, and Im f is called the conjugate harmonic function of Re f. Lemma 3.3 Let f (x) = u1 (x) + e2 u2 (x), u1 (x) and u2 (x) be real harmonic functions in G and u1 (x), u2 (x) ∈ C 1 (G). Then the conjugate harmonic function v(x) of u(x) is given by the formula v(x) =

 x3

2

0

∂ ∂ 1 udx3 − T˜( u(x1 , x2 , 0)) + g(x1 , x2 ), ∂x∗ 2 ∂x3

(3.19)

where g(x1 , x2 ) = g1 (x1 , x2 ) + e2 g2 (x1 , x2 ) is an arbitrary function satisfying ∂g/∂x∗ = 0, and 1 T˜u(x) = − π



ER

u(y ∗ ) dσy∗ , y∗ − x∗

(3.20)

where ER = {(x1 , x2 , 0) | x21 + x22 | < R2 }. Proof

From (3.16), it follows that v(x) =

 x3 0

2

∂ udx3 + w(x1 , x2 ). ∂x∗

Substituting the above expression into (3.15) and letting x3 = 0, we obtain ∂ ∂ u(x1 , x2 , x3 )|x3 =0 + 2 ∗ w(x1 , x2 ) = 0, ∂x3 ∂x and then 1 ∂ u(x1 , x2 , x3 )|x3 =0 ) + g(x1 , x2 ) w(x1 , x2 ) = − T˜( 2 ∂x3 can be obtained, in which g(x1 , x2 ) satisfies ∂g/∂x∗ = 0. Thus formula (3.19) holds. Moreover, if v(x) is given by (3.19), we can see that u(x) and v(x) satisfy (3.16). Since u(x) is a harmonic function and 4

∂2 ∂2 ∂2 ∂ ∂ + )u = − u, u = ( ∂x∗ ∂x∗ ∂x21 ∂x22 ∂x23

it is easy to verify that u(x) and v(x) satisfy (3.17). Lemma 3.4 Suppose that u(x) = u1 (x) + e2 u2 (x) is a harmonic function in G = {|x| < R} in Clifford analysis and u(x) is continuous on G. Then u(x) can be expressed as u(x) = −



|y|=R

∂ 1 1 1 R ( − )u(y)dSy , ∂n 4π |x − y| 4π |x||˜ x − y|

(3.21)

225

Clifford Analysis and Elliptic Equations

˜ = R2 x/|x|2 , ∂/∂n where x = x1 + x2 e2 + x3 e3 , y = y1 + y2 e2 + y3 e3 , x denotes the exterior normal derivative with respect to y on the sphere |y| = R, and dSy is the area element of |y| = R. Proof Due to ∆u = 0 in G and because u(x) is continuous on G, from the Poisson formula for harmonic functions in a ball, it follows that the formula (3.21) holds. Now, we find the conjugate harmonic functions of 1/|x − y| and R/|x||˜ x − y| with respect to x. Noting that

=−

 x3

1 ∂ dx3 = 2 ∂x∗ |x − y|

0 ∗ − y )(x3 − y3 ) ∗ |x − y ∗ |2 (x − y)

(x∗

−

 x3 * 0

+

x∗ − y ∗ − dx3 |x − y|3

(x∗ − y ∗ )y3 , |x∗ − y ∗ |2 [(x1 − y1 )2 + (x2 − y2 )2 + y32 ]1/2

and 1  1 y3 1 ∂ =−  2 x =0 2 ∂x3 |x − y| 3 2 [(x1 − y1 ) + (x2 − y2 )2 + y32 ]3/2 (x∗ − y ∗ )y2 ∂ = , ∂x∗ |x∗ − y ∗ |2 [(x1 − y1 )2 + (x2 − y2 )2 + y32 ]3/2

−

we obtain

*

+

∂ 1  1 − T˜  2 ∂x3 |x − y| x3 =0 (x∗ − y ∗ )y2 = ∗ + w(x1 , x2 ), |x − y ∗ |2 [(x1 − y1 )2 + (x2 − y2 )2 + y32 ]1/2 where w(x1 , x2 ) is a function satisfying ∂x∂ ∗ w = 0. On the basis of Lemma 3.3, we know that the conjugate harmonic function of 1/|x − y|(x = y) with respect to x possesses the form −

(x∗ − y ∗ )(x3 − y3 ) + c(x1 , x2 ) + w(x1 , x2 ). |x∗ − y ∗ |2 |x − y|

In particular, choosing c(x1 , x2 ) = −w(x1 , x2 ), it is easy to see that −(x∗ − y ∗ )(x3 − y3 )/|x∗ − y ∗ |2 |x − y| is a conjugate harmonic function of 1/|x − y|. Similarly, we can find that a conjugate function of R/|x||˜ x − y| with respect to y possesses the form −

˜∗ )(y3 − x ˜3 ) R(y ∗ − x R (y ∗ − R2 x∗ /|x|2 )(y3 − R2 |x3 |/|x|2 ) = − |x||y ∗ − x ˜∗ |2 |y − x ˜| |x| |y − R2 x∗ /|x|2 |2 |y − R2 x/|x|2 | =−

R|x|(|x|2 y ∗ − R2 x∗ )(|x|2 y3 − R2 x3 ) , |x21 y ∗ − R2 x∗ |2 ||x|2 y − R2 x|

226

Chapter VII

and a conjugate function of R/|x||˜ x − y| with respect to x possesses the form R|y|(|y|2 x∗ − R2 y ∗ )(|y|2 x3 − R2 y3 ) . − ||y|2 x∗ − R2 y ∗ |2 ||y|2 x − R2 y| Denoting 1 g(x, y) = 4π h(x, y) = −

*

1 4π

+

1 R 1 , − |x − y| |x| |˜ x − y| *

(x∗ − y ∗ )(x2 − y2 ) |x∗ − y ∗ |2 |x − y|



R|y|(|y|2 x∗ − R2 y ∗ )(|y|2 x3 − R2 y3 ) , − ||y|2 x∗ − R2 y ∗ |2 ||y|2 x − R2 y| ¯ =0 then for x = y, the function s(x, y) = g(x, y) + e3 h(x, y) satisfies ∂s with respect to x. Since ∂ ∂ y1 ∂ y2 ∂ y3 = + + , ∂n ∂y1 |y| ∂y2 |y| ∂y3 |y| by calculation, we obtain 1 ∂ − s(x, y)||y|=R= ∂n 4πR 

+e3 −

4

R2 −|x|2 |x−y|3

2(x∗ (x3 −y3 )+x3 (x∗ −y ∗ ) |x∗ −y ∗ |2 |x−y|

4(x∗ −y ∗ )(x3 −y3 )(|x∗ |2 −(x1 y1 +x2 y2 )) + |x∗ − y ∗ |4 |x − y|

(x∗ − y ∗ )(x3 − y3 )(|x|2 − R2 ) + |x∗ − y ∗ |2 |x − y|3

5

(3.22)

.

Theorem 3.5 Let u(x) = u1 (x) + e2 u2 (x) be a continuous function on the sphere |x| = R. Then there exists a continuous function f (x) on G, which satisfies the system of first order equations ∂¯ f = 0 in G = {|x| < R}

(3.23)

and the boundary condition Re f = u(x) on ∂G = {|x| = R},

(3.24)

227

Clifford Analysis and Elliptic Equations and f (x) can be expressed as 1 f (x) = 4πR +

4(x∗

−



4

∂G

y ∗ )(x3



R2 − |x|2 2(x∗ (x3 − y3 ) + x3 (x∗ − y ∗ )) + e − 3 |x − y|3 |x∗ − y ∗ ||x − y|

− y3 )(|x∗ |2 − (x1 y1 + x2 y2 )) |x∗ − y ∗ |4 |x − y|

(x∗ −y ∗ )(x3 −y3 )(|x|2 −R2 ) + |x∗ −y ∗ |2 |x−y|3

5

u(y)dSy +e3 c(x1 , y2 ),

(3.25) where c(x1 , x2 ) = c1 (x1 , x2 ) + e2 c2 (x1 , x2 ) is an arbitrary function satisfying ∂ c(x1 , x2 ) = 0, ∂x∗ where x∗ = x1 + x2 e2 . Proof From the above discussion, we see that the function expressed by (3.25) is a solution of the Dirichlet boundary value problem (3.23) and (3.24). Conversely, if the Dirichlet problem (3.23) and (3.24) has a solution f (x), we denote the integral on the right-hand side of (3.25) by F (x), i.e.

F (x) =

2 4πR +



∂G

4



R2 − |x|2 2x∗ (x3 − y3 ) + x3 (x∗ − y ∗ ) + e − 3 |x − y|3 |x∗ − y ∗ ||x − y|

4(x∗ − y ∗ )(x3 − y3 )(|x∗ |2 − (x1 y1 x2 y2 )) |x∗ − y ∗ |4 |x − y|

(x∗ − y ∗ )(x3 − y3 )(|x|2 − R2 ) + |x∗ − y ∗ |2 |x − y|3

5

u(y)dSy .

¯ (x) = 0 in G and Re F ||x|=R = u(x). Hence ∆(Re f − Re F ) = 0 Then ∂F in G, and (Re f − Re F )||x|=R = 0. This implies that Re f = Re F on G. According to Lemma 3.2, we obtain ∂ ∂ (Im f − Im F ) = 0, (Im f − Im F ) = 0. ∂x∗ ∂x3 Thus Im f − Im F = c(x1 , x2 ) and ∂c(x1 , x2 )/∂x∗ = 0. The theorem is proved (see [4]).

228

4

Chapter VII

Oblique Derivative Problems for Regular Functions and Elliptic Systems in Clifford Analysis

First of all, we consider the case of the Clifford algebra A3 over the
space R3 . Let G be a ball, i.e. G = { 3j=1 |xj |2 < R2 (< ∞)}, and for convenience, let e2 e3 be denoted by e4 . Any point x in G may be written
as x = 3j=1 xj ej , and any function f (x) in G with values in the Clifford
algebra may be denoted by w(x) = 4j=1 wj (x)ej . 4.1 Oblique Derivative Problems for Generalized Regular Functions in R3 A generalized regular function w(x) in G is defined as a solution
w(x) = 4j=1 wj (x)ej (∈ C 2 (G)) for the elliptic system of first order equations in the form ¯ = aw + bw ∂w ¯ + c in G,

(4.1)

where a(x) =

4 

j=1

c(x) =

4 

j=1

aj (x)ej , b(x) =

4 

bj (x)ej ,

j=1

cj (x)ej ∈ Cα1 (G), 0 < α < 1.

Problem P The oblique derivative problem for system (4.1) is to ¯ ∩ C 2 (G) of (4.1) satisfying the boundary find a solution w(x) ∈ Cα1 (G) condition ∂wj + σj (x)wj (x) = τj (x) + hj , x ∈ ∂G, wj (R) = uj , j = 1, 2, ∂νj

(4.2)

where σj (x), τj (x) ∈ Cα1 (∂G), σj (x) ≥ 0 on ∂G, j = 1, 2, hj (j = 1, 2) are unknown real constants to be determined appropriately, uj (j = 1, 2) are real constants, νj (j = 1, 2) are vectors at the point x ∈ ∂G, cos(νj , n) ≥ 0, j = 1, 2, n is the outward normal at x ∈ ∂G, and cos(νj , n) ∈ Cα1 (∂G). If w(x) is a solution of Problem P for system (4.1), then we can verify

229

Clifford Analysis and Elliptic Equations that [w1 (x), w2 (x)] is a solution of the system of second order ⎧ 3  ⎪ ⎪ ⎪ (Aj + Bj )w1xj + B5 w1 + B6 , ∆w = ⎪ 1 ⎪ ⎪ ⎪ j=1 ⎪ ⎪ ⎨ 3  ⎪ Bj w2xj + B7 w2 + (A2 w1 )x1 − (A1 w1 )x2 ⎪ ⎪ ∆w2 = 2 ⎪ ⎪ j=1 ⎪ ⎪ ⎪ ⎪ ⎩

(4.3)

−(A4 w1 )x3 − B2 w1x1 + B1 w1x2 + B4 w1x3 + B8 w1 + B9

satisfying the boundary condition (4.2), where Aj = aj + bj , Bj = aj − bj , j = 1, ..., 4, Bj is only a function of xj , j = 1, 2, 3, B4 is a real constant, and B5 = − B7 =

4 

Aj Bj +

j=1

3 

j=1

Bjxj −

3 

j=1 4 

j=1

Ajxj , B6 = −

4 

Bj Cj +

j=1

3 

Cjxj ,

j=1

Bj2 , B8 = A1 B2 − A2 B1 − A3 B4 + A4 B3 ,

B9 = −C1x2 + C2x1 − C4x3 − B1 C2 + B2 C1 + B3 C4 − B4 C3 . In fact, it follows from (4.1) that ⎧ ⎪ w1x1 − w2x2 − w3x3 = A1 w1 − B2 w2 − B3 w3 − B4 w4 + C1 , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ w2x1 + w1x2 + w4x3 = A2 w1 + B1 w2 − B4 w3 + B3 w4 + C2 , ⎪ ⎪ w3x1 − w4x2 + w1x3 = A3 w1 + B4 w2 + B1 w3 − B2 w4 + C3 , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ w 4x1 + w3x2 − w2x3 = A4 w1 − B3 w2 + B2 w3 + B1 w4 + C4 .

(4.4)

When Bj = Bj (xj ), j = 1, 2, 3, and B4 is a real constant, from (4.4) we can derive the first equation in (4.3), and then the second equation in (4.3) can be obtained. Conversely, if the following conditions hold: ¯ B5 ≥ 0, B7 ≥ 0 on G,

(4.5)

then according to Theorem 1.6, the boundary value problem (4.3),(4.2) has a solution [w1 (x), w2 (x)]. Afterwards, if ¯ B3 = B4 = 0 on G,

(4.6)

230

Chapter VII

then we can find w3 (x), w4 (x) by the following integrals:

 x3 ⎧ ⎪ ⎪ [w1x1 −w2x2 −A1 w1 +B2 w2 −C1 ]dx3 +φ3 (x1 , x2 ), w (x) = 3 ⎨ 0  x3 ⎪ ⎪ ⎩ w4 (x) = [−w2x1 −w1x2 +A2 w1 +B1 w2 +C2 ]dx3 +φ4 (x1 , x2 ),

(4.7)

0

where φ3 (x1 , x2 ), φ4 (x1 , x2 ) satisfy the conditions

⎧ ⎨ φ3x1 − φ4x2 − B1 φ3 + B2 φ4 − C3 = ψ3 (x1 , x2 ),

in which

⎩ φ 3x2 + φ4x1 − B2 φ3 − B1 φ4 − C4 = ψ4 (x1 , x2 ),

(4.8)

ψ3 (x1 , x2 ) = [−w1x3 +A3 w1 ]|x3 =0 , ψ4 (x1 , x2 ) = [w2x3 +A4 w1 ]|x3 =0 , and Aj , Bj , Cj satisfy some conditions such that the integrals in (4.7) are single-valued. Suppose that φ = φ3 + φ4 e2 = φ3 + φ4 i satisfies the Riemann-Hilbert boundary conditions: ⎧ ⎨ Re[λ(t)φ(t)] = r(t)+h(t), t = t1 +it2 ∈ Γ = ∂G∩{x3 = 0}, ⎩

(4.9)

Im[λ(dj )φ(dj )] = gj , j = 1, ..., 2K + 1 for K ≥ 0;

where |λ(t)| = 1, λ(t), r(t) ∈ Cα2 (Γ), dj (j = 1, ..., 2K + 1 for K ≥ 0) are distinct points on Γ, gj (j = 1, ..., 2K + 1 for K ≥ 0) are known real constants, and ⎧ 1 ⎪ ⎪ 0 on Γ for K = ∆Γ argλ(t) ≥ 0, ⎪ ⎨ 2π h(t) = −K−1  ⎪ − m ⎪ ⎪ (h+ ⎩ h0 +Re m +ihm )t on Γ for K < 0; m=1

h0 , h± m (m

= 1, ..., −K − 1) are undetermined real constants. herein According to Theorems 4.1 and 4.6, Chapter 2 in [81], the functions φ3 (x3 , x4 ), φ4 (x1 , x2 ) may be uniquely determined, and then w3 (x), w4 (x) may be also uniquely determined. Then we obtain Theorem 4.1 If the coefficients of the system (4.1) satisfy the conditions (4.5),(4.6), and W (t) = w3 (t) + iw4 (t) satisfies the boundary conditions Re[λ(t)W (t)] = r(t) + h(t), t ∈ Γ = ∂G ∩ {x3 = 0},

(4.10)

Im[λ(dj )W (dj )] = gj , j = 1, ..., 2K + 1 for K ≥ 0,

(4.11)

where λ(t), r(t), h(t), dj , gj and K are as stated in (4.9), then Problem
¯ ∩ C 2 (G), 0 < P has a unique solution w(x) = 4j=1 wj (x)ej ∈ Cα1 (G) α < 1 (see [3]).

231

Clifford Analysis and Elliptic Equations

4.2 Oblique Derivative Problem for a Degenerate Elliptic System of First Order in R3 Now, we discuss the degenerate elliptic system of first order equations ⎧ ⎪ w1x1 −w2x2 −xµ3 w3x3 = A1 w1 −B2 w2 −B3 x1+µ w3 −B4 w4 +C1 , ⎪ 3 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨w + w + w = A w + B w − B xµ w + B w +C , 2x1

1x2

2x3

2

1

1

2

4 3

3

3

4

2

⎪ ⎪ xµ3 w3x1 − w4x2 +w1x3 = A3 w1 +B2 w2 +B1 xµ3 w3 −B2 w4 +C3 , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ µ µ ⎩

(4.12)

w4x1 + x3 w3x2 − w2x3 = A4 w1 −B3 w2 +B2 x3 w3 +B1 w4 +C4 ,

where Aj (x), Bj (x), Cj (x)(j = 1, ..., 4) are known functions as stated c1 (x), Cj (x) = xµ3 cj (x), j = 2, 3, 4, cj (x) ∈ in (4.4), and C1 (x) = x1+µ 3 Cα1 (G), j = 1, ..., 4, µ is a non-negative constant, and G is a domain in the upper halfspace x3 > 0 with the boundary ∂G = D1 ∪ D2 ∈ Cα2 , where D1 is in x3 > 0 and D2 in x3 = 0, Γ = D1 ∩ {x3 = 0}. The Riemann-Hilbert boundary value problem (Problem A) for (4.12) is to
find a bounded solution w(x) = 4j=1 wj (x)ej ∈ Cα1 (G)∩C 2 (G) of (4.12) satisfying the boundary condition ⎧ ∂w3 ⎪ ⎪ ⎪ + σ3 (x)w3 (x) = τ3 (x), x ∈ D1 , ⎨

∂ν3

⎪ ∂w4 ⎪ ⎪ + σ4 (x)w4 (x) = τ4 (x), x ∈ ∂G, ⎩

(4.13)

∂ν4

⎧ ⎨ Re[λ(t)(w1 (t) + iw2 (t)] = r(t) + h[ζ(t)], t = x1 + ix2 ∈ Γ, ⎩

(4.14)

Im[λ(dj )(w1 (dj )+iw2 (dj ))] = gj , j = 1, ..., 2K +1 for K ≥ 0,

where cos(νj , n) > 0, j = 3, 4, n is the outward normal at x ∈ ∂G, and σj (x) > 0, τj (x) ∈ Cα1 (∂D2 ), j = 3, 4, λ(t), r(t), h(t), dj , gj and K are similar as in (4.9), ζ(t) is a conformal mapping from the unit disk {t| < 1} onto D2 . Similarly to before, if Bj is only a function of xj , j = 1, 2, 3, B4 is a real constant, and Aj = Bj , j = 1, ..., 4, then the
solution w(x) = 4j=1 wj (x)ej of Problem O for (4.12) is also a solution of the following boundary value problem for the degenerate system of second order equations: ⎧ ⎪ ∆w3 = 2B1 w3x1 +2B2 w3x2 +A5 x−1 ⎪ 3 w3x3 +A6 w3 +A7 , ⎪ ⎨ 3  ⎪ ⎪ (Aj + Bj )w4xj + A8 w4 + A9 w3 + A10 = ∆w 4 ⎪ ⎩ j=1

(4.15)

232

Chapter VII

with the boundary condition (4.13), in which A5 = −µ + B3 x3 (1 + x3 ), C5 ∈ C 2 (G), A6 = (1 + µ)B3 −

4 

j=1

Bj2 +

3 

j=1

Bjxj + (B32 − B3x2 )(1 − x3 ),

A7 = (−B1 C3 −B2 C4 +B3 C1 +B4 C2−]C1x3 +C3x1 +C4x2 )x−µ 3 , A8 = −

4 

j=1

Bj2 +

3 

j=1

Bjxj , A9 = A4 x3µ−1 (B4 µ + A3 x3 −B3 x23 ),

A10 = −A1 C4 + A2 C3 − A3 C2 + A4 C1 + C2x3 − C3x2 +C4x1 . Conversely, if the following conditions hold: A6 ≥ 0, A8 ≥ 0 in G,

(4.16)

then by means of Corollary 2.3, the boundary value problem (4.15),(4.13) is solvable. Moreover, if A1 = A2 = 0 in G,

(4.17)

then [w1 (x), w2 (x)] can be found by the following integrals:  x3 ⎧ ⎪ w (x) = [−xµ3 w3x1 +w4x2 +B1 xµ3 w3 −B2 w4 +C3 ]dx3 +φ1 (x1 , x2 ), ⎪ ⎨ 1 0

 x3 ⎪ ⎪ ⎩ w2 (x) = [xµ3 w3x2 +w4x1 −B2 xµ3 w3 −B1 w4 −C4 ]dx3 +φ2 (x1 , x2 ), 0

(4.18)

in which φ1 (x1 , x2 ), φ2 (x1 , x2 ) satisfy the conditions

⎧ ⎨ φ1x1 − φ2x2 − A1 φ1 + B2 φ2 = ψ1 (x1 , x2 ), ⎩

where

φ2x1 + φ1x2 − A2 φ1 − B1 φ2 = ψ2 (x1 , x2 ),

ψ1 (x1 , x2 ) = [xµ3 µw3x3 +C1 ]|x3 =0 , ψ2 (x1 , x2 ) = [−w4x3 +C2 ]|x3 =0 , (4.19) and Aj , Bj , Cj satisfy some conditions. Besides, we require that φ = φ1 + φ2 e2 = φ1 + φ2 i satisfies the boundary condition (4.14), i.e. ⎧ ⎨ Re[λ(t)φ(t)] = r(t) + h(t), t ∈ Γ, ⎩

Im[λ(dj )φ(dj )] = gj , j = 1, ..., 2K +1 for K ≥ 0.

(4.20)

233

Clifford Analysis and Elliptic Equations

Similarly to (4.10),(4.11), the function [φ1 (x1 , x2 ), φ2 (x1 , x2 )] is uniquely determined, and hence [w1 (x), w2 (x)] is also uniquely determined. Thus we have Theorem 4.2 Suppose that the coefficients of system (4.12) satisfy conditions (4.16),(4.17) etc. stated as before. Then Problem A for de
generate elliptic system (4.12) has a solution w(x) = 4j=1 wj (x)ej ∈ Cα1 (G ∪ D1 ) ∩ C 2 (G), 0 < α < 1 (see [80]6)). We mention that when the boundary condition (4.13) is replaced by w3 (x) = τ3 (x), w4 (x) = τ4 (x), x ∈ ∂G,

(4.21)

we can similarly discuss the solvability of the boundary value problem (4.12) and (4.21). 4.3 Oblique Derivative Problem for an Elliptic System of First Order in R4 Now, we discuss the elliptic system of first order equations in a do
main Q ⊂ R4 . The so-called regular function w(x) = 8j=1 wj (x)ej is indicated as a solution of the system of first order equations ⎧ ⎪ w1x1 −w2x2 −w3x3 −w4x4 = 0, w1x2 +w2x1 +w5x3 +w6x4 = 0, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ w1x3 +w3x1 −w5x2 +w7x4 = 0, w1x4 +w4x1 −w6x2 −w7x3 = 0,

⎪ ⎪ −w2x3 +w3x2 +w5x1 −w8x4 =0, −w2x4 +w4x2 +w6x1 +w8x3 =0, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ −w +w +w −w = 0, w −w +w +w = 0, 3x4

4x3

7x1

8x2

5x4

6x3

7x2

(4.22)

8x1

in Q, in which for convenience we denote e5 = e2 e3 , e6 = e2 e4 , e7 =
e3 e4 , e8 = e2 e3 e4 . The generalized regular function w(x) = 8j=1 wj (x) is indicated as a solution of the system of first order equations ∂w = Aw + Bw + C in Q,
8


8

(4.23)
8

where A(x) = j=1 Aj (x), B(x) = j=1 Bj (x), C(x) = j=1 Cj (x),
w(x) = w1 (x) − 8j=2 wj (x). Let Q = {|xj | < R (j = 1, 2, 3, 4), 0 < R < ∞}. The oblique derivative problem (Problem P ) of (4.22) in Q is
to find a bounded solution w(x) = 8j=1 wj (x)ej ∈ Cα1 (Q) ∩ C 2 (Q) of (4.22) satisfying the boundary conditions ⎧ ∂wj ⎪ ⎪ ⎨ + σj (x)wj (x) = τj (x) + hj , x ∈ ∂Q,

∂νj

⎪ ⎪ ⎩ w (R) = u , j = 1, 2, 3, 5, j j

(4.24)

234

Chapter VII

where σj (x), τj (x) ∈ Cα1 (∂Q), σj (x) ≥ 0 (j = 1, 2, 3, 5), hj (j = 1, 2, 3, 5) are undetermined constants, and uj (j = 1, 2, 3, 5), α are real constants, νj (j = 1, 2, 3, 5) are vectors at every point x ∈ ∂Q and cos(νj , n) > 0, n is the outward normal at the point x ∈ ∂Q, and cos(νj , n) ∈ Cα1 (∂Q), j = 1, 2, 3, 5. We first find the partial derivatives for the first four equations in (4.22) with respect to x1 , x2 , x3 , x4 , and then add the equations, thus ∆w1 = w1x2 + · · · + w1x2 = 0. 4

1

(4.25)

Similarly we can obtain ∆wj = wjx2 + · · · + wjx2 = 0, j = 2, 3, 5. 1

4

(4.26)

On the basis of Theorem 1.6, there exists a solution [w1 (x), w2 (x), w3 (x), w5 (x)] of the boundary value problem (4.25),(4.26),(4.24), where wj (x) ∈ Cα1 (Q) ∩ C 2 (Q), j = 1, 2, 3, 5. Substituting the functions into (4.22), we have ⎧ ⎨ w4x4 = w1x1 − w2x2 − w3x3 , w6x4 = −w1x2 − w2x1 − w5x3 , ⎩

(4.27)

w7x4 = −w1x3 − w3x1 + w5x2 , w8x4 = −w2x3 + w3x2 + w5x1 .

From the above system, it follows that w4 = w6 = w7 = w8 =

 x4 0

[w1x1 − w2x2 − w3x3 ]dx4 + g4 (x1 , x2 , x3 ),

0

[−w1x2 − w2x1 − w5x3 ]dx4 + g6 (x1 , x2 , x3 ),

0

[−w1x3 − w3x1 + w5x2 ]dx4 + g7 (x1 , x2 , x3 ),

 x4

 x4

 x4 0

(4.28)

[−w2x3 + w3x2 + w5x1 ]dx4 + g8 (x1 , x2 , x3 ),

and gj (x1 , x2 , x3 ) (j = 4, 6, 7, 8) satisfy the system of first order equations ⎧ g4x1 − g6x2 − g7x3 = −w1x4 |x4 =0 = a1 , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ g4x2 + g6x1 + g8x3 = w2x4 |x4 =0 = a2 , ⎪ ⎪ g4x3 + g7x1 − g8x2 = w3x4 |x4 =0 = a3 , ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

g6x3 − g7x2 − g8x1 = w5x4 |x4 =0 = a4 .

(4.29)

235

Clifford Analysis and Elliptic Equations

Let g4 , g6 , g7 , g8 be replaced by w1 , w2 , w3 , w4 , then the system (4.29) can be rewritten as ⎧ ⎨ w1x1 − w2x2 − w3x3 = a1 , w1x2 + w2x1 + w4x3 = a2 , ⎩ w 1x3 + w3x1 − w4x2 = a3 , w2x3 − w3x2 − w4x1 = a4 ,
4

i.e. w(x) =

j=1 wj (xj )ej

(4.30)

satisfies the equation

∂w = e1 wx1 + e2 wx2 + e3 wx3 = a, a =

4 

aj ej ,

j=1

and (4.30) is just a special case of (4.1) with a(x) = b(x) = 0, c(x) = a(x). By Theorem 4.1 and the result in [38], the system (4.30) has
a solution w(x) = 4j=1 wj (x)ej ∈ Cα1 (D) ∩ C 2 (D); herein D = Q ∩ {x4 = 0}. Thus the functions g4 (x), g6 (x), g7 (x), g8 (x) are found. Let these functions be substituted into (4.28). Then wj (x)(j = 4, 6, 7, 8) are
determined, and the function w(x) = 8j=1 wj (x)ej is just the solution of Problem P for (4.22). Hence we have the following theorem. Theorem 4.3 Problem P of the elliptic system of first order equations (4.22), i.e. the regular functions in the domain Q, has a solution w(x) =
8 1 2 j=1 wj (x)ej ∈ Cα (Q) ∪ C (Q), where 0 < α < 1 (see [80]6)).

As for the generalized regular functions, under certain conditions, we can also prove the existence of solutions of Problem P for system (4.23) in Q. By a similar method as stated in Subsection 4.2, we can prove the solvability of Problem P for the corresponding degenerate elliptic systems of first order equations with some conditions in the domain G. Finally, we consider the case of arbitrary dimension n. Let An (R) be a real Clifford algebra and Q be a polycylinder Q1 × · · · × Qn in the space Rn , Qj = {|xj | ≤ Rj }, 0 < Rj < ∞, j = 1, ..., n. Problem P ′ The oblique derivative problem for generalized regular functions in Q ⊂ Rn is defined to be the problem of finding a generalized regular function w(x) =

n−1 2

wj (x)ej =

j=1



wA (x)eA in G,

A

A = {j1 , ..., jk } ⊂ {1, ..., n}, eA = ej1 ...ejk , 1 ≤ j1 < · · · < jk ≤ n, satisfying the boundary condition ∂wj +σj (x)wj (x) = τj +hj , x ∈ ∂Q, wj (R) = uj , j = 1, ..., 2n−2 . (4.31) ∂νj

236

Chapter VII

Here j = A if j ≤ n and if A includes at least two integers greater than 1, then j(n < j ≤ 2n−1 ) denotes one of integers system A such that j and A possess a one to one relation, νj is a vector at the point x ∈ ∂Q, cos(νj , n) ≥ 0, σj (x) ≥ 0, x ∈ ∂Q, hj (j = 1, ..., 2n−1 ) are undetermined constants. Theorem 4.4

A function w(x) =



wA (x)eA in Q

A

is a generalized regular function if and only if the wA (x) satisfy the real elliptic system of first order equations n 

δkB wBxk =

n 

(ad + bd )wM δdM +

d=1

k=1

n 

d=1

(ad − bd )wN δdN + cA , (4.32)

where kB = A, dM = A, dN = A and δbB etc. are proper signs (see [80]10)). Proof Since w(x) is a generalized regular function, it is clear that w(x) satisfies the elliptic system of first order equations ¯ = aw + bw ∂w ¯ + c.

(4.33)

Moreover we have ¯ = ∂w

n 

δkB wBxk eA =

A k=1

Thus




d,A (ad

δkB wBxk eA .

k,A

aw + bw ¯+c= +






d,A (ad

+ bd )wM δdM eA

− bd )wN δdN eA +




A cA eA ,

where kB = A, dM = A and dN = A, which shows that wA (x) satisfies (4.32). Under certain conditions, by introducing various quasi-permutations as stated in Chapter I, from (4.32) we can obtain ∆wk =

n 

dmk wkxm + fk wk + wk , k = 1, ..., 2n−2 .

(4.34)

m=1

On the basis of the result in Section 1, a solution [w1 (x), ..., w2n−2 (x)] of (4.34) satisfying the boundary condition (4.31) can be found. Moreover,

237

Clifford Analysis and Elliptic Equations we rewrite (4.32) in the form Wkxk = Fkl (z1 , ..., zm , W2n−3 +1 , ..., W2n−2 ), k = 2n−3 + 1, ..., 2n−2 , l = 1, ..., m,

(4.35)

where n = 2m, x2k−1 + ix2k = zk , w2k−1 + iw2k = Wk , k = 1, ..., 2n−2 . If Q is a polycylinder, under some conditions, we can obtain [W2n−3 +1 (x), ..., W2n−2 (x)] satisfying Re[z1 K1 ...zm Km Wj (z1 , ..., zm )] = rj (z1 , ..., zm ) + hj , z(z1 , ..., zm ) ∈ ∂Q1 × · · · × ∂Qm , j = 2n−3 + 1, ..., 2n−2 .

(4.36)

The above result can be written as Theorem 4.5 Under some conditions, Problem P ′ for (4.14) has a solution [w1 (x), ..., w2n−1 (x)], where Q is a polycylinder (see [31]2)).

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Index a priori estimates 199 boundary value problem Dirichlet problem 41, 47, 49, 50, 110, 116, 117, 119, 124, 219, 222, 227 Haseman problem 60 initial-boundary value problem 200 linear boundary value problem 75 mixed boundary value problem (Problem P -R-H) 49, 59 Neumann problem 50, 199 nonlinear boundary value problem 60, 68, 75, 95, 99 oblique derivative problem 50, 51, 197, 198, 205, 228, 233, 235 Problem O 199, 200, 206 pseudo-modified problem 115, 122, 123 Riemann-Hilbert boundary value problem 230, 231 Cauchy Cauchy kernel 108, 125, 126, 134, 145, 146, 148 Cauchy integral formula 30, 75, 105, 107, 108, 109, 132, 194, 219 Cauchy-Pompjeu formula 9, 10

Cauchy principal value 9, 10, 75, 76, 77, 99, 135, 137, 160 Cauchy-Riemann condition 43 Cauchy -Riemann equations 24 Cauchy-Riemann system 68 Cauchy type integral 1, 4, 8, 75, 76, 78, 91, 98, 99, 147 Cauchy’s estimates 125, 126, 131 quasi-Cauchy’s kernel 148 Cauchy’s integral 11, 14 Cauchy’s theorem 7, 30, 221 characteristic manifold 95, 98, 107, 108, 109, 114 classical domain 105, 106, 107, 108, 109, 111, 113, 114, 115, 124 classical group 106 Clifford algebra 1, 42, 44, 111, 217, 218, 223, 227, 235 Condition C 198, 199, 200, 202, 206, 207, 209, 211 conjugate 2, 3, 67, 218, 223, 225 equation Cauchy-Riemann equations 24 elliptic equation 197, 205 degenerate elliptic equation 197, 209

250 degenerate elliptic system 197, 230, 232, 235 hyperbolic Laplace equation 70 integral equation 50, 60, 62, 67, 87 M –T equation 5 nonlinear elliptic equation 197 nonlinear equation 206 partial differential equation 218 singular integral equation 82, 92, 100, 103 extremum principle 110 formula calculational formula 159, 176, 183, 194, 195 composition formula 125, 145, 146 differential formula 159, 170, 176, 194, 195 inverse formula 125, 145, 146, 148 permutation formula 159, 191, 194, 195 Plemelj formula 1, 8, 14, 61, 75, 78, 80, 88, 91, 98, 99, 147, 148, 221 Poisson formula 105, 108, 109, 134, 224 recurrence formula 159, 168, 176, 195 Schwarz formula 223 transformation formula 125, 134, 136, 142, 145 function analytic function 48, 49, 105, 108, 113, 114, 116, 117, 122, 218 biregular function 75, 76,

Index 91, 95, 147 complex harmonic function 32, 33, 36 complex regular function 30, 32, 33, 36, 40 differentiable function 204, 212, 218 function vector 95, 96, 97, 98, 99, 100, 101 generalized biregular function 87, 88, 91, 95 generalized holomorphic function 23, 87 generalized integrable function 153, 154 generalized regular function 19, 20, 21, 22, 49, 51, 59, 87, 90, 218, 228, 233, 235, 236 harmonic function 5, 49, 105, 110, 123, 126, 134, 223, 224, 225 holomorphic function 32, 60, 68, 70, 73, 87 hyperbolical harmonic function 68, 70, 71, 72, 73 hypercomplex differentiable function 24, 25 regular function 1, 5, 6, 7, 10, 15, 19, 20, 41, 60, 68, 111, 115, 118, 122, 194, 217, 218, 219, 221, 222, 223, 235 Hadamard principal values 159, 170 Hile lemma 83, 179 hypercomplex structure 1, 24 hyperball 41, 44, 45, 49, 61, 105, 118, 122

Index

251

index A-type index 20 A-type index 20

the second suffix 44, 111, 112 symmetric domain 105, 106, 204

Liapunov surface 7, 60, 62, 76, 97, 135, 180 Lipschitz condition 64, 65, 102

theorem Ascoli-Arzela theorem 67, 87, 103, 189, 214 Cauchy’s theorem 7, 30, 221 contraction mapping theorem 199 fixed point theorem 60, 67, 184 Fredholm theorem 152 Fubini theorem 194 Hartogs theorem 75 Hadamard theorem 190 Leray-Schauder theorem 50, 199, 206, 207, 209 Liouville theorem 148 mean value theorem 107 regularization theorem 148 Stokes theorem 6, 10, 137

maximum principle 199 operator compact operator 152 differential operator 41, 218 Dirac operator 4, 76, 87 harmonic operator 109 inverse operator 152 Laplace operator 5 regularization operator 125, 148, 149 singular integral operator 78, 93, 148, 159, 184, 185 permissible open covering 153, 154 Poisson kernel 45, 108, 110, 119 quasi-permutation 1, 15, 16, 17, 19, 42, 43, 53, 105, 112, 115, 117, 124, 236 resolution of the identity 153 Riemann ball 106 singular integral 125, 134, 135, 139, 142, 145, 146, 159, 160, 161, 166, 167, 168, 169, 170, 174, 175, 176, 183, 184, 189, 190, 191, 194, 195 shift 60, 61, 68, 87, 88, 92, 95 suffix the first suffix 44, 111, 112