Quaternionic Analysis and Elliptic Boundary Value Problems [Reprint 2021 ed.] 9783112576182, 9783112576175

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Mathematica/ Research Quaternionic Analysis and Elliptic Boundary Value Problems

K. Gürlebeck W. Sprößig

Volume 56

AKADEMIE-VERLAG

BERLIN

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K. Gurlebeck • W. SprdBig Quaternionic Analysis and Elliptic Boundary Value Problems

Mathematical Research

• Mathematische Forschung

Wissenschaftliche Beiträge herausgegeben von der Akademie der Wissenschaften der D D R Karl-Weierstraß-Institut für Mathematik

Band 56 Quaternionic Analysis and Elliptic Boundary Value Problems by K. Gurlebeck and W. SproBig

Quaternionic Analysis and Elliptic Boundary Value Problems

by Klaus Gürlebeck and Wolfgang Sprößig

Akademie-Verlag Berlin 1989

Autoren: Dr. Klaus

Gürlebeck

Technische

Universität

Karl-Marx-Stadt

Mathematik

Sektion

Prof. Dr. Wolfgang Bergakademie

Spräßig

Freiberg

Mathematik

Sektion

Die Titel dieser Reihe werden

vom Originalmanuskript

der Autoren

repro-

Berlin,Leipziger

3-4,Berlin,DDR-1086

duziert,

ISBN

3-05-500756-5

ISSN

0138-3019

Erschienen

im Akademie-Verlag

(c) Akademie-Verlag Lizenznummer: Printed

202-100/413/8(1

in the German Democratic

Gesamtherstellung:

VEB Kongreß-

Lektor: Dr, Reinhard LSV

03400

Republic und

Höppner

1035

Bestellnummer:

Str,

Berlin iq8q

764 0 7 2 6

(2182/56)

Werbedruck,Oberlungwitz,DDR-Q273

Prafatorv Mota« It

is well known that complex methods nay be

advantageously

used for the treatment of boundary value problems of differential

equations

in the plane.

Moreover it

partial is

very

important to transfer results of the classical function theoRn.

ry to function theories over domains in description by

the

A comprehensive

of hypercomplex function theories is being

made

research group of R.DELANGHE (Gent) with their

book

"Clifford Analysis" ([BDS]>. The application to solving boundary

value

problems by the help

of

hypercomplex

function

theories is not developed in the same extent. The

main aim of this book consists in the statement of a new

strategy

for . solving linear and

problems

of

nonlinear

boundary

partial differential equations of

value

mathematical

physics by the help of hypercomplex analysis. In our opinion, it is the first summarizing presentation of a complete hypercomplex

solution theory including Analytical

and

numerical

investigations in only one closed theory. Using a special operator calculus and a hypercomplex function theory, the authors study questions of the existence, uniqueness,

representation

and

regularity of solutions of

above

mentioned problems in a unified form. For the sake of simplicity,

the authors restrict their investigations to the

case

of quaternionic calculus. Sometimes, if it seems to be necessary, it is referred to general results in CLIFFORD algebras. Furthermore

suitable

numerical approaches which

are

well-

to the quaternionic calculus are included too.

adapted authors

not only give an insight into

methods

but

also

introduce a

new

boundary

The

collocation

collocation

procedure.

Occurring for the first time, a discrete model of the quaternionic

function

theory

was developed and applied

to

con-

structing and investigating of finite difference methods. The

first

chapter makes the reader familiar

with

knowledge in the field of quaternionic analysis.

a

basic

Most of the

results are also valid in more general algebras. In

Chapter 2 the authors have studied algebraic and functio-

nal

analytical properties of generating operators

and

D,

Fp ,

Tg,

which denote a CAUCHY-type operator, a quaternionic

analogue to the complex T-operator and a generalized

CAUCHY-

5

RIEMANN operator, The of

respectively.

third chapter only contains an orthogonal the space

of quaternionic-valued functions into ¿,n ,* ker Dftl^ H < g ) a n d H^^' T^3

the subspaces sition

decomposition

L 0 „(6)

is an essential methodological

instrument

throughout

the following explanations. In

Chapter 4 a series of linear and nonlinear elliptic boun-

dary value problems of mathematical physics has been investigated

by the help of a unified method in a

rather

complete

manner. Starting

with some fundamental functional analytic theorems,

a quaternionic version of the boundary collocation method

is

treated in Chapter 5. Finally, in Chapter 6 a discrete quaternionic function theory is

introduced.

well-adapted given

These

results are used in order to

numerical

in Chapter 4.

approach

to the

The line of action is

analytical

find

a

theory

demonstrated

by

considering the discrete NAVIER-STOKES problem. The book finishes with an Appendix.

It is intended to give a

short survey about other questions in the hypercomplex theory which have been investigated recently. in

The authors apologize

advance that in this summary not all important ideas

and

papers can be mentioned. The monograph is suitable

for mathematicians, physicists and

engineers

in research institutes.

textbook.

All

It has the character of a

the necessary mathematical

preparations

are

made available. The structure of the method presented is very simple and makes possible a formal use for practical computations. Suitably chosen examples make the reader familiar with the

topics and methods of quaternionic analysis.

Other spe-

cial branches such as approximation theory,

theory of right-

invertible operators,

methods,

boundary collocation

finite

difference methods and equations of mathematical physics will be mentioned in the book.

Knowledge in numerical mathematics

is desirable and facilitates the understanding. We have to thank Prof. Stadt

B.SILBERMANN,Prof. H .JACKEL (Karl-Marx-

University of Technology) for suggesting

of this book

Thanks are also due

(Helsinki University of Technology),

to Prof.

Prof.

the writing P.LOUNESTO

R.DELANGHE (Gent

State University), Dr. V.SOUCEK (Prague Charles University),

6

Prof. B.GOLDSCHMIDT (Halle University), Doz. Dr. H.MALONEK (Pedagogical University of Halle) for stinulating discussions and useful hints for references. He also wish to thank Hiss. BERNHARDT, who looked for mistakes after typing. Furthermore we should like to thank Hr. H.STRAUCH for giving essential advice concerning the English language of the nanuscript. Finally, our thanks go to the Akademie-Verlag, especially to Dr.R.HOPPNER for the realization of this monograph. Karl-Marx-Stadt, Freiberg, January 1989, K.GURLEBECK, H.SPROSSIG

7

CONTENTS 1. Quaternionic Analysis 1.1. 1.2. 1.3. 1.4. 1.5.

Algebra of Real Quaternions H-regular Functions A Generalized LEIBNIZ Rule BOREL-POMPEIU's Fornula Basic Statements of H-regular Functions

2. Operators 2.3. Properties of the T-Operator 2.4. VEKUA's Theorens 2.5. Some Integral Operators on the Manifold

11 11 13 23 26 29 48 48 53 56

3. Orthogonal Decomposition of the Space

64

4. Some Boundary Value Problems of DIRICHLET's Type

67

4.1. 4.2. 4.3. 4.4. 4.5. 4.6. 4.7. 4.8.

LAPLACE Equation HELMHOLTZ Equation Equations of Linear Elasticity Tine-independent MAXWELL Equations STOKES Equations NAVIER-STOKES Equations Stream Problems with Free Convection Approximation of STOKES Equations by Boundary Value Problems of Linear Elasticity

5. H-•regular Boundary Collocation Methods 5.1. Complete Systems of H-regular Functions 5.2. Nunerical Properties of H-complete Systems of H-regular Functions 5.3. Foundation of a Collocation Method with H-regular Functions for Several Elliptic Boundary Value Problems 5.4. Numerical Examples 6. Discrete Quaternionic Function Theory 6.1. Fundamental Solutions of the Discrete Laplacian 6.2. Fundamental Solutions of a Discrete Generalized CAUCHY-RIEMANN Operator 6.3. Elements of a Discrete Quaternionic Function Theory 6.4. Main Properties of Discrete Operators 6.5. Numerical Solution of Boundary Value Problems of NAVIER-STOKES Equations 6.6. Concluding Remarks

67 76 85 91 97 106 113 119 122 122 129 138 146 153 153 164 175 188 197 208

Appendix

210

References

241

Subject Index

251

Notations

253

9

I, It

BUATERNIOmC is

natural

ANALYSIS

to look for generalizations

of

functions over the conplex field

The

developnent

proposed

to

to

higher

of physics at the end of the

new questions in nathenatics.

necessary

of the

dimensions.

last

century

Above all

find algebraic possibilities in

advantageously

theory

it

was

order

to

carry out calculations with vector

functions

over 3-dimensional domains.

An algebraic assumption for such

applications

invention of the quaternions

1843.

was HAMILTON'S

This

discovery was published

famous paper Initially

no special class of

1-dimensional in

his

'regular' functions among

all

in 1866.

functions was considered,similarly

to the

This decisive step was made many

years

case.

the

in

in

[Ham] "Elements of Quaternions"

quaternionic-valued later

in a final form

important papers

G.MOISIL/N.TEODORESCU

[MT],

by

which

R.FUETER

[Fuel]

and

may be regarded as

the

starting point for the function theory of quaternions. The aim of the present chapter real quaternion theory further considerations. possible

to

mention

In this context

papers

by

our

it is completely

all the things which are

quaternionic functions. sential

is to give a short survey of a

in such a way as it is needed for known

im-

about

For this purpose we refer to the es-

A.SUDBERY ,

J.RYAN

[Sud] , [Ryj) ,

P.LOUNESTO

J.BURES

[Bu]

and

[Fue3].

We will not deal with generalizations of the classi-

cal function theory in other abstract the

book

of

[SouJ]

,

[Gall]

the lecture notes by

V.SOUCEK

[Lou*

B.GOLDSCHMIDT

R.FUETER

algebras,

F.BRACKX/R.DELANGHE/F.SOHMEN

and

[Fue2] either.

In

[BDS]

the

reader can find a function theory in CLIFFORD algebras,

while

the publications of K.HABETHA mental

[Habl] ,

function-theoretical

theorems

[Hab2] contain in

more

fundageneral

algebras.

Algebra of Real In

Quaternions

this section we will present all the algebraic

properties

of quaternions which are used throughout the whole book. 1.1.1

4

Let R

be the

4-dimensional Euclidean vector

space.

11

W e c h o o s e the o r t h o n o r m a l b a s i s e2=(0,0,1,0),

e3=(0,0,0,1)

can be w r i t t e n

+

a

law is g i v e n

a-b

>

a.e. we

Ùî

obtain

1 1

a n o t h e r v e c t o r , then a m u l t i p l i c a t i o n

- a-b)eg + a x b

and

vector product

in

is n o t c o m m u t a t i v e is

S =

by

ab = ( ^ b g where

(1.1)

S a.e. l l. i=0

I n t r o d u c i n g the a b b r e v i a t i o n 0e0

e^=(0,1,0,0), a=(a0,a1(a2>a3) €

as =

a=a

e0=(1,0,0,0),

. Hence, a vector

a x b

are

+ ab '0

(1.2)

the s c a l a r p r o d u c t

and

R , respectively. Obviously,

this

the

product 4 in g e n e r a l . I n this way the v e c t o r s p a c e R

f u r n i s h e d w i t h the a l g e b r a i c s t r u c t u r e of a

ring,

which

will

be n a m e d a l g e b r a of real q u a t e r n i o n s and d e n o t e d b y

This

letter

quaternions, 1.1.2 real

is

chosen

in honour of the d i s c o v e r e r

of

W.R.HAMILTON.

Q u a t e r n i o n s m a y be i d e n t i f i e d w i t h a s p e c i a l kind 4 x 4 -

m a t r i c e s , w h i c h have the 0

"al

a

l

a

a

2

a

a

3

a

i=0,l,2,3,

Here

H. the

representation

"a2

10 0 0 1

a

2

a

0

"al

a

l

«0

3

"a2

(1.3)

d e n o t e real n u m b e r s .

m a t r i x c a l c u l u s by r

3

3

0

(1.1), a b a s i s

10 0 0 0 10 0 0 0 10

_a

_a

0-l 0 0

10 0 0 0

0

0 0

S o m e t i m e s this n o t a t i o n

0-1

10

of

form

'2' 0 0 -1 0 0 0 1 0 0 10--1 0

S i m i l a r l y to

the

eg is g i v e n in the

0 1 0 0

fa '

is r e p l a c e d b y l,i,j,k

e

.

3

=

0 0 -l 0 0-i 0 0 l 0 0 l 0 0 0

In [BDS]

it

is s h o w n that H is an even s u b a l g e b r a of the well-lmown P A U L I a l g e b r a of q u a n t u m 1.1.3 S

12

=

a

physics.

W e d e f i n e the c o n j u g a t e 0e0

quaternion

F r o m (1.2) it i m m e d i a t e l y

follows

aS = 5a = >

a.^ , and t h e r e f o r e the Euclidean norn |a| = i a s ! 1

i-0

He d e f i n e the r e a l p a r t and the p u r e l y imaginary part of a by Re a = a^en, = JL ( a + 5 ) ,

Im a = >

Z

a.e. 1

i=T

The i n v e r s e of the quaternion a-1 = a

3

J a I-2

i

.

a ( a^0 ) i s obtained as

.

A s t r a i g h t f o r w a r d computation leads t o the (i)

= —( a - i T )

1

identities

SB - B a ,

(ii)

| ab | - | a I| b | ,

(iii)

Re(ab) = Re(ba)

where

(1.4)

a, b are a r b i t r a r y elements of

1.1.4

It

rule

H.

i s easy to show by the use of

(1.2)

the

multiplication

that the b a s i s quaternions f u l f i l the

following

relations: e.2 = -e0 I

, i = 1,2,3,

(i)

* 0 2 =:

(ii)

e. e . + e . e . = 0 , i X j , i,j J i i 0 = e. , i= 0 , 1 , 2 , 3 , e0ei l = eie0

(iii) (iv) 1.2

e

e0

le2 =

'

e3

•'

= ej

e2e3

(1.5)

= 1,2,3

, e3e1 = e2 .

H-rftgnlar Functions

1.2.1 Throughout the whole book l e t 6 be a bounded domain of o R , and "J G a s u f f i c i e n t l y smooth LIAPUNOV surface.Then functions

u

d e f i n e d in G and

values

H

w i l l be

in

on

P ,

respectively,

considered.

The

so-called

with

H-valued

f u n c t i o n s may be w r i t t e n as 3 u(x) = S H u - i U ) ^ , x6G , 1 i —0 1 where the f u n c t i o n s u ^ ( x ) are r e a l - v a l u e d . as c o n t i n u i t y ,

differentiability,

which are ascribed t o ponents

u.(x)

BANACH

spaces

(m=0,1,2,...

, ,

Properties

integrability

such

and so on ,

u have t o be possessed by a l l the com-

i = 0,1,2,3. of

(1.6)

In t h i s manner the

H-valued f u n c t i o n s are denoted by

«£[0,1]

),

L

p H

(p^l),

Hjj

(s>0)

cjj' " and

13

(P*-1' k = 0 , l , 2 , . . . ) . M o r e o v e r ,

*p H

all H - v a l u e d f u n c t i o n s of the s p a c e shing

on the b o u n d a r y t h e

**2

i n n e r

H(G>

s t a n d s for

Wj h^®^'

T .In case of p = 2

w h i c h

we

a r e

vani-

introduce

Product

(U.v)*Jffv dG .

(1.7)

G

1.2-2

Let

vector

a

a ^ f c g . a j . a g . a g ) b e an a r b i t r a r y v e c t o r in R 4 . T h e c o r r e s p o n d s to the q u a t e r n i o n

by

a =

3

D. = jr, i = 1,2,3 , i oxi RIEMANN operator

1

1

is

an a n a l o g u e to the ~

D=(0,D^,D2,Dg)

, where

we o b t a i n the s o - c a l l e d

3 D = X I D.e. = i l 1 1

operator

a, w h i c h is g i v e n

a.e.

i=0

R e p l a c i n g a b y the f o r m a l v e c t o r

which

in

CAUCHY-

,

(1.8)

2-dimensional

CAUCHY-RIEMANN

, z=x+iy. Its a c t i o n on the H - v a l u e d

function

3 0=0 is d e n o t e d b y 1.2. 3

Du

3

uD

from the

right.

Definition

An H - v a l u e d f u n c t i o n u{Cjj(6)

and

Du=0

s o l v e s the e q u a t i o n in

3

from the left a n d b y

u in

is c a l l e d H - l e f t r e g u l a r

in

G. If an H - v a l u e d f u n c t i o n

uD=0,

t h e n it is n a m e d

H-right

G

regular

G. T h r o u g h o u t the w h o l e b o o k we a g r e e u p o n the f a c t

H - l e f t r e g u l a r f u n c t i o n s shall b e c a l l e d

iff

ufiCjJ(G) that

H-regular.

1.2.4 U s i n g ( 1 . 5 ) it is easy to see that Du m a y b e w r i t t e n D u = ( - d i v u )e 0 + g r a d u 0 + rot u

,

(1.9)

where grad u 0

= ( DlU0

div u = D l U l and

14

>e x + ( D 2 u 0

+ D2u2 + D3u3 ,

)e 2 + ( D 3 u 0

as

)e3 ,

e

r o t u =oltt

Therefore

l

e

2

e

D

1

D

2

°3

U

1

U

2

u

3 3

an H - v a l u e d f u n c t i o n

u

is n a m e d H - r e g u l a r

satisfies the so-called MOISIL-TEODORESCU

if

u

system

d i v u =0 , g r a d Ug + r o t u = 0 . In p a r t i c u l a r ,

where

(1.10)

u^ = c o n s t ,

this system describes

i r r o t a t i o n a l fluid w i t h o u t sources or The

consideration

the

development

mous

paper

of s y s t e m

an

sinks.

(1.10) was the starting p o i n t

of t h e h y p e r c o m p l e x

function

of G . M O I S I L a n d N . T E O D O R E S C U

theory. The

[MT]

in fa-

appeared

in

1931. 1.2.5

T h e s e t of a l l H - r e g u l a r f u n c t i o n s A JHJ((GG ) . N o t i c e

noted by DD where then

- A

¿^

equation

in

Au=0

R .

1.2.6

The following simple of two

regular. He

take

Du=0,

while

D(uu)

= Du

Assume

that

2

0 ^ G

1

u = x

is n o t H - r e g u l a r . T h i s

of

in

G

function

is

l2

however

+

0 -

is n o t n e c e s s a r i l y

then we

/J

2x2e2

u(x)/0.

In

x

22

^

H-

have

0

theory the

•

~

consists

m o s t of t h e o t h e r p e c u l i a r i t i e s . of

,

contrast

* 0.

to

the

function

1

+

t h e p r o d u c t of t w o H - v a l u e d

theory

functions

because e

I

a

quaternio-

~e„

x

is

example shows that the

l l

^

U

function

functions.

e

then

u e cfj(G),

in

harnonic

2

(x) = ( - x 1 e 1 + *2e2

The main trouble,

of

H-regular functions

complex

Du-1««)=

H-regular

theory

= [

1-dimensional u

of an

, if

Du=0

the t h e o r y of H - r e g u l a r

here refines the

nic product

Obviously

follows from

function. Therefore

considered

de-

(l.u)

That means that each component harmonic

be

.

is the L a p l a c i a n

the

in G w i l l

that

, in

2 the

functions, It is a l s o

X 0 non-comnutativity which

causes

the r e a s o n w h y

conformal mappings has been developed only

in

a a

15

smaller

extent,

t h a n in the c l a s s i c a l c o m p l e x theory.

r e s u l t s in this f i e l d will be r e f e r r e d in the 1.2.7

In order to o v e r c o m e the a b o v e m e n t i o n e d

R. D E L A N G H E i n t r o d u c e d variables. analytic is

if and only if for e a c h

H-regular. Let

z

d^ € H

is

proved

the

linearly

that the

independent).

identity z

In

with

totally

proved

it in C L I F F O R D

1.2.8

Let

All

variables.

i'

property in

case

to

study are

DELANGHE

R.

T h e n there e x i s t real

numbers

by

0

a

00

a

01

a

02

a

03

e

0

d

l

a

10

a

ll

a

12

a

13

e

l

d

2

a

20

a

21

a

22

a

23

e

2

d

3

a

30

a

31

a

32

a

33j

e

3j

the

J

analy-

H-regular functions which

analytic

dJ£H, i = 0,1,2,3

=

the

i-3 n e c e s s a r y

d^d^

d.d.,

algebras.

a. . w h i c h are d e f i n e d U d

=

the f o l l o w i n g w e intend

some s t r u c t u r a l p r o p e r t i e s of connected

d^

be a totally

tic v a r i a b l e . In [Guel] it is p o i n t e d o u t t h a t of c o m m u t a t i v i t y of the p r o d u c t s functions.

z

by

i , 0 = 1 , 2 , 3 is s u f f i c i e n t in order t h a t

of H - v a l u e d

totally

k - t h p o w e r of

(1.12)

(not n e c e s s a r i l y

it

k€N

analytic

is c a l l e d

l l

i-0

In [Dell]

zeH

be r e p r e s e n t e d =

with

difficulties,

in [Dell] the idea of t o t a l l y

In this s e n s e , a v a r i a b l e

Some

Appendix.

further considerations

in this

=

1I'

• i«flf

subsection

(1.13)

will

be

c o n c e r n e d w i t h the rank of the m a t r i x 01

a

02

a

a

ll

a

12

a

13

a

21

a

22

a

23

31

a

32

a

33^

a

A' =

a

The

exact

03

(1-14)

p r o o f s of the f o l l o w i n g p r o p e r t i e s of

f u n c t i o n s will be o m i t t e d here. The reader can find

H-regular them

in

[Guel]. Property 1 d.d. i J

16

d.d. J

i

i,j = 0 , 1 , 2 , 3 , if and only if

rank A ' < 2 .

Property 2 Let

a = ( ag.a^ag.ftg ) e R

M(a)

=

®0

a

a

a

2

a

l

Further let

u fe

if and only if

H

H

€ Ah(G)

u.(x)d. , rank A' < 2 , then 1

T3S

for all

1

n £ N .

Property 6 £ AAl_h (G) , rank A' < 2 , then If z € With G' = {' x £ R 4 : z(x) t 0 } . 1.2.9 Let

Theorem z

be a

(L u)(x) = n

where

n

€ A h (G') for all n

( T.AGRANQB'n Interpolation Polynomial ^ totally ) €Hn

u =
i=0

3

"a03"a12 a

02"a13

x. a.

[Dell], o = 1,2,3,

0

a

3l"a13

a

12~a21

a

ll+a33

a

0l"a32

0l"a23

a

ll+a22,

_a

and

2 Since

z

£ AJJ(G) , we get

w h e r e

3

0=0

As it has conclude (

a

been

J

3

o=0

0 1

true for any

S( a ji> a j2 , a o3^'''

jl'a02'aj3

Su=0

ker

S

=

®

by consideration of (1.17) , 3

3

o=0

3 i

x = ( j

3

0

x0,x^,x2>xg =

T )

,

we

0»1>2,3 , w h i c h m e a n s

"

Now we distinguish different cases with respect to dim ker S: (i)

dim ker S = 0 : for

i = 1,2,3

That can only be realized if ;

which would imply

o = 0,1,2,3 , dim ker S = 3.

therefore This

is

a^

= 0

S=0 in

contradiction to our assumption. (ii)

d i m ker S = 1 : belong to

20

Because all vectors ( a . - , a . 0 , a . ~ ) 01 J£ jo ker S, it clearly follows rank A'

R

is harmonic in A

u(x) = u 0 ( x ) - Im is

H-regular in

R.

FUETER

method

in his important paper

constructing

functions

this

1 .2. IB

C

Proposition

H

let

a

complex

in the classical plane function

theory.

[Dea]

the

and also in

proof,

[Sud]

rFue41

yl (x+iy) = x +

P

is an analytic function in

y

l]mp[

and f(p) = Re p + i|Im p|

into the upper half-plane of u(p) =

1936

from

assertion will be formulated without

p 6 H , in

[Fue4] of

H-regular functions

which is given in detail in

For

G , then

I t2 (Du 0 )(tx) x dt o G, and it holds Re u = u 0 .

found

for

analytic Here

[Sudl

denote a mapping from

C. Then

, if

the open set

7 p (u( ?(p))) , that

be an embedding of

is

H

u:C --> C

UcC

and

H-regular in the open

?_1(U) £ H .

set

The previous propositions have given us a certain about the

size of the set of

feeling

H-regular functions.

1-2.19 Preparing

a numerical approach for the

solution of certain

partial differential equations we shall introduce the wing system of H-regular functions { ^

¿

with

follo-

3

) in a domain G C R

til

k

-_lf^p

where the points

e


]ekeiV

'k=l

Setting 3 3 u = . Z Z < D k u i ) v j°k e i e j a n d S 2 i 3-B k=l k=l 3 3 3 we obtain for S. = > / (D. u 4 )e. e. ]> v.e. = (Du)v 1 K 1 K 1 i=0 k=l j=0 3 3 The treatnent of the tern Sj requires some algebraic calculations whioh will be carried out here; S

u

2

i

< D

k W i

k,i=l a. i,k=l

24

8

u

i

0 u e ^ + + (rot v ) x u - ( d i v v ) u + (div u ) v + (v-rot u ) O g - v x rot u By

adding

(v-grad)u

both

i d e n t i t i e s the r e l a t i o n

s u b t r a c t i n g leads to (iv) 1.3.5

and

(v)

aroses,

(vi).

^

Bamrk

Especially Corollary

1.3.4 e m p h a s i z e s the e f f i c i e n c y of

q u a t e r n i o n i c a n a l y s i s d e v e l o p e d h e r e in c o n n e c t i o n r e p r e s e n t a t i o n of f o r m u l a s in p h y s i c s and v e c t o r 1.3.B

papers A.SUDBERY

respectively,

obtained

[Sud]

and

p r o d u c t rules,

side w a s c h o s e n s i m i l a r l y to ( 1 . 2 5 ) in m o r e d e t a i l in

the

analysis.

1.4.1

W.SPROESSIG w h e r e the

.

[Sp 1],

right-hand

T h e s e p a p e r s are

cited

[GS 3].

BOREL-POHPEIUb

nic

the

with

Remark

In e a r l i e r

1.4

while

Formula

T h i s s e c t i o n d e a l s w i t h the d e r i v a t i o n of a q u a t e r n i o version

domain

G

described

of a g e n e r a l i z e d and the b o u n d a r y

in

1.2.1

BOREL-POMPEIU "i G

P =

fulfil

formula. the

The

conditions

. Let

3

o( i (y)e i 1

where

ot ^ is the

the p o i n t y

i - t h c o m p o n e n t of the outer n o r m a l on T

. Assume

at

I of | = 1 .

Setting e ( x ) = — — ,3 *Tin

, where

x = > x.e. . frj- i i

In [BDS, 8 . 8 ] it is p r o v e d that nal sense tor

e(x)

is , in a d i s t r i b u t i o -

, a f u n d a m e n t a l s o l u t i o n of the d i f f e r e n t i a l

D. I n t r o d u c e the

operators

j e(x-y) o U y ) u ( y ) d T y

(Fp u)(x) =

and

r (TgU)(x) = -

j

e(x-y)u(y)dGy .

G If

there

instead of

26

are no c o n f u s i o n s , Tr

, Fp

we s h a l l

, respectively.

write

T,

F

opera-

1.4.2 Proposition ( GAUSS' Formula Let u e C*(G) 0 Cjj(G) . Then we have L u dT - J Du dG (1.28) r d Proof Using GAUSS' formula for real-valued functions we obtain u dG = S ¿ D.u. e. e. dG = f D.u. dG1e3 e, = G i=l i=l G 3-0 j-0 j=0

r G

= £= r1 iuj d r eiej = 1 * u d r . r J=0 „ 1-4.3 Proposition aa Let u £ C '(G) , 0 < p

both

and

+

v = u. LEIBNIZ formula now

wfeiiDu-

sides over the domain

G^ = G \ B

(x)

we

obtain the identity 1 ¡4-1 - r u ] « 5 ' yV ri-x-vi

1

+

f J

T Z

«G

g

A

I*"'*!

dG

c

1, ]u(y> * rii-ji

v = - wrg' I

dG

, r>y v +

(Du)(y) dG, £ y ' "

(1.31)

Applying GAUSS'formula it follows that -1- ( _ J — d. (y)u(y) dr UT J lx-^1 y

- ~ fire

1 f = TfiT 1 r r gJ

TT w

where

y-x =

jfi r i

=

k=l

dG

A

c£ >y „+

=

f 1 -1 T~ i*-^ dG y,

(y. - x. )e. . Hence K

r^7Tc' 0. i Obviously ,

of

31

3 (x) = C

J

Further we have

2

for all

x e G .

3

Uj(x)D^Uj = 0

, i=l,2,3 .

0=0 A second differentiation yields 3

3

H

^TjD^)

2

= 0

,

i = l, 2, 3 .

3=0

j=0

Since

all components of an

H-regular function are harnonic,

we conclude 3 for all

x € G and

j = 0,1,2,3

i=l Finally,

we

means that

have u

D-u^ = 0 ,

is constant in

i = l,2,3 , j=0,1,2,3,

which

G.

^

We have the immediate

1-5.6 If

Corollary

u £ Ajj(G) n C H ( G ) , then it follows

sup_|u(x)| = sup |u(x)|. x eG xcT

1.5.7 Let

Proposition u £ Ah(R3>.

is a constant

( I.TOUVXLLB'b Theorem If I u(x) I < N for all

Ì x €. R 3 , then

u(x)

H-valued function.

Exjaaf. From

where

32

CAUCHY's integral formula we obtain the

R is

representation

an arbitrarily chosen positive real number. Then

differentiation by

D^

C

3

SR(x)

j=l

leads to

Using the relations

D

1x

~bis 3(x,-a,)t«-ix-aiz

» L J l - j

t

. j

we get the eatinate

I

3_

x

I H I Di x r 1 1 ^

XItF^ j=l

1

r^Ti-

whenoe ' V

f J'

•

3 D

i.x

J I J=0

SR 9 M fT R 1 i»T R 3

,

_ ~

9H T ~

Hoc(y)||u(y)|

1

'

, , "1'2'3 •

Since

R is arbitrarily chosen, it follows for

D^u=0

for

1.5.8

BfiH&xk

1.5.9 Let

for all

Propoaition u £ A|j(G) and

valued

R —-* oo

i=l,2,3 , and the assertion is checked.

If in Proposition 1.5.7 u(x) = 0

dSR

*

function

u(x) — »

0

for

|x) — >

ao

then

f TAYLOR Sarioa Bxpannion > x be an arbitrary point of G. Then the u permits,

in a sufficiently snail

bourhood of a, a TAYLOR series expansion of the forn OO a (a u(x) = k ' 00} ,x_a,k k-0

,

x £ R3 .

H-

neigh-

where

0

Take

¿ — i= l

I x-a I

£ > 0 such that

e

i

B £ ( a ) c. G .

^

CAUCHY's integral

formula

(1.37)

(y)u(y ) dS f .

Figure 2: In

[Muej it is proved that w i t h

i=l 0' 0i

-

t =

'^

, where

i=l

"r^l. la-yi

the series 1 f^TT

Pn0. Here

of

derivatives

in

P (3,t) d e n o t e s the n 3 n in R , w h i c h can be

formula

< 1 '.S0.d..e. Z 0J U i j=l

n, 1 | u _ a i 13

=

1

where 0

f «0J d

1

0i '

**

ij UJi-1

.

i = j.

It is easy to see that now d p k k k=0

Du^^ — >

Du^^ = 0

that

there

exist

u^, k = 0,1,2,3, such that on

con

P a c t subsets of G,

Du .

for i = 1,2,...

in G , we finally obtain

u C Ajj(G). The last step is possible as the domain exhausted by compact sets.

38

.

u^*^, k = 0,1,2,3, are harmonic functions ,

by

V'u^^ZZi

i

G

can be M

1.5.14 Let u

(l

Corollary (i

). _ „

be a sequence of

1 € n

\

( 2. WBIERSTRASS' Thaoren ^

i = 1,2,3,..., belong

i = 1,2,3,... . If the series

to 0 there exists a natural number N( £ ) such that

1 k=l

u

N+k(x)l ™ *

< E

holds for xe P As the sum ^

1

and any natural number 1 u N + ^(x)

AJJ(G) n Cjj(G) it immediately follows

1 from the maximum modulus theorem | y u„ it(x) I < £ for x £ G , • k= 1 which yields the desired result. „ 9 1.5.15 Proposition ( Extension Theore» ^ 3 Let G^ and Gg be bounded domains in R , = r,

, i = 1,2 , r

x

n r2 = r

0

be smooth LIAPUNOV surfaces . If the ( k •) uv y (x) belong to A H < G k ) fl C H for on P g , then the

u(x) =

is

,iet r

k

,k = 0,1,2 ,

H-valued

functions

k = 1,2 and coincide

H-valued function

u

(1)

(x)

, x £ Gx

u

(2)

(x)

, x 6 G2

u

(1)

(x) = u(2)(x)

H-regular in

G^ f] Gg = 9 >

, x € P0

G^ U Gg U I

This extension is unique .

Proof Let a € F g be a fixed point , £ > 0 an arbitrary real number, such

that

denotes For

T 0 D B^(a) f) ( f j ij

>

an open ball with the radius L

3 .

If for any smooth

y c G

G.

Proof Let

{ G^ }jtc u be a regular sequence of domains ,

contracting to the point

40

x C G .

Denote

3G^ -

which P^

is Ob-

viously , we have

x £ fl G^. k>J0

Using

LEBESGUE's

function

w G L

theorer. then we obtain for each

lia t4~, J w ( y ) dG k n-* 3 , that there oxists r,„(6) n

extensive

The reader can find it in

than

[BMl

and

[BDS] .

To

continue,

we

recall

some

facts

about

GEGENBAUER

polynomials. For this reason we formulate 1.5.18

Lentia [Lcn]

Suppose

e [0,1) ,

function

(1-2

coefficients

t e[-1,1] ,

t+ «. ^ ) ^

v e R , then we expand the

in a power

series

are the GEGENBAUER polynomials

,

where

^(t) »

the

namely

V5V k = 2 _ C. (t) d 0

J T i T

Sj

K

The limit for R — # Co(ti [ ^ [u(co ) d 0 W

r { X C R 3

42

( I.AllRBNT Sorlmi Expansion > is H-regular in the open

annulus

: r < | x-a | < R , a e R 3 , r >. 0 }

and continuous on its closure

B r ^(a).

Then

u(x) allows the

following series expansion: u(x) = Y H

Ia-X|n

an(a,

n=- oo

where

r

,

,

v,

_

1

R ' is arbitrarily chosen in (r,R) and t„ = \ » Furthermore we put

C^2(t0)=0

and

t

x

Q)

f ,— 1 —r . la-x| -a| 0

( t

0

)

.

Proof Let B

(a) = { x e B D (a): r < r. < |a-x) < R, _2

^

of the H-valued function

then there exists a sufficiently

ball B (a) such that I x- a I"1 Proof The theorem of the LAURENT series expansion yields

46

small

CO u

=

n

0 a0 ) l x - a | n + ¿ 1

a n ( a , 6 _0 ) | x - a | " n "

F r o n D e f i n i t i o n 1.5.24 it follows lin | x - a | B u ( x ) = a x—»a x^a For any

£

with

(a, 0 0 > ?

£. < Bin I a S,

0.

(a,

0„)l

w e find a real n u n b e r

0. £ for all p o i n t s

x

with

2. OPERATORS 2.1

Above

all

in this section we intend to

discuss

functional analytic properties of the operators Fp

introduced in the foregoing Section 1.

considerations

to

To functions

such

ensure

A^(G)

topics that

which

the

D,

We

are

class

some

T^

and

restrict

our

needed

later.

all

H-regular

of

is a subspace in certain functional

spaces

we prove the

The classes

A H ( G ) o C H " ' ^ ( G ) , 0k(x) = where

r

SUP

Fron the nean value

is

i [4»n(y) - 4 > k ( y ) ] d B r

chosen sufficiently

14» (x> - 4 k < x ) l

< C | + B -t k |l

X* Cjr

with

G r = { x t G:dist(x, i G)>r}.

uniformly theorem

theorem

1.5.3) we find

function 0.

1.5.13)

Consequently,

we

,

It

follows that

rm

Hence the sequence converges

for any compact subset of (Theorem

snail.

y

get

we have

G. to

Using an

0,

WEIERSTRASS'

H-regular

complete.

^

2.3 Properties of the T-Qperator 2.3.1 Theorem Let

u fLj

'

(i)

the integral

Then

(TGu)(x)

and tends to zero for 3 " belongs to A„(R \G),

48

limit

and the proof is

exists for all |x|

besides

x €

RAG

(T_u)(x)

B .

y

^

3

g

In an a n a l o g o u s

manner

we nay

deduce

2.3.4 Corollary Let

u € CH(0),

then

(i)

I(TQU)(X)I

(ii)

|)(x')li, L

^

if

}|x-x'||ul

2

conpact.

S >0

suoh

that

0

CJJ '!* ( D

our assertion in

W

„ ( D , l0 P111

Lg H^r")'

Sp

is shown.

o £ C h 0 ' ^ ( P ) , 0 ( D , 0< |i (6) holds. Proof With u£C® , ' J (T) we get P p u€cjj' ^ ( D , whence follows that FpU=FpPpU is the harmonic extension of PpU into the donain G. Now. the assertion nay be proved by using a result of SMOLITZKI [Sno] on the smoothness of harnonic extensions.g 2.5.17 Remark Proposition 2.5.12 and its oorollaries are also valid for continuous extensions of the operators Sp, Pp and Q p onto the sface Lg 2.5.IB Theorem Let v.w £ A H (G)n »¿(G), ^ e C ^ ' ^ C G ) , 0 .

and Theoren 2.5.10 we

obtain

the

integral equation (iv). Obviously, we have

Out of this follows (iii). The function

u

_

has to be snooth in such a way that

exists, for instance

2

"iu jTT

u c N ^ JJ-

2.5.20 Rwnark Using the preceding assertions we can deduce a generalization of well-known fornula. Let the uecjj'P •

By the help of

BOREL-

POMPEIU's formula T

G

Q T

follows,

F L

F

=

T

Q

D U

=

u

and therefore

u € W ^ h 'l0°(G) •

Finally,

3.3 implies -Au=DDu=DQT Q f=f. 4.1.2 Corollary Let

+

g€w£ ^ Au

/2

= 0

u = g

Corollary #

( P ) , k>J3. DIRICHLET's problem in

G,

(4.3)

on

P

(4.4)

67

u€wj^H'l0°

has a s o l u t i o n

of

tho

f o r n

u = Fpg + TG«Dh,

(4.5)

k+1 where

h

is a ^

^ ( O ) - e x t e n s i o n of

g.

Proof g€»2+H/2(r,)

As

tr h=g.

Put

transforned Av

u=v+h.

v 4

The

problen

w i t h

(4.3)-(4.4)

will

= A h

in

G,

on

P.

the v a l i d i t y of T h e o r e n 4 . 1 . 1 there e x i s t s the „k+l,loo._. ... ( 2 H * w v =

be

into

v = 0 For

h

there exists a « ^ ¿ - e x t e n s i o n

solution

TQQTGAh.

Using BOREL-POMPEIU'e fornula, P=I-Q

and

DD=-A

we

find

v = -T f l QDh + T g Q F p D h = - T g D h + T g P D h = - h + T Q P D h + F p h . With

u=v+h

w e g e t (4.5).

#

4 . 1 . 3 Thftflrfli ( E x i s t e n c e 1 Let

( G )

*2+H

F>'

k

~0-

D I R I C H L E T

s

H

-Au

= f

in

G,

(4.6)

u = g

on

T

(4.7)

has the

'

/ 2 (

F€»2

Problen

solution

u = F p g + TgPDh + T G « T G f e H ^ '

l o c

(G).

(4.6)

k+2 Here

h

d e n o t e s a Wg ^ - e x t e n s i o n of

g.

Proof Let

u^, Ug

be s o l u t i o n s of the p r o b l e n s

(4.1)-(4.2),

( 4 . 3 ) - ( 4 . 4 ) , r e s p e c t i v e l y , then u=u^+ug s o l v e s the b o u n d a r y v a l u e p r o b l e n ( 4 . 6 ) - ( 4 . 7 ) in the s p a c e W^il'l o c ( G ) . * 4.1.4 Let

u

Theoren F€»2

H

( G )

'

G £ , I

2

+

H

/ 2 ( R )

'

"

of ( 4 . 8 ) is the only s o l u t i o n of

problen

T h e

the

H

"

v a l u e d

function

boundary

value

(4.6)-(4.7).

EiOfflf C o n s i d e r the b o u n d a r y v a l u e p r o b l e n P O M P E I U *s solution

68

fornula u

and the fact that

of t h i s p r o b l e n

(Au=0.

tr u = 0 } .

D u € in Q

lead

k

BORELto

a

u = TgDu = TqQDU . On the other hand, beoause of

Du£Aj|(Q),

u = TQDU = TgPDa follows. Therefore PDu = QDu, whence Du = 0 and finally u = 0.

9

4.1.5 Propoaitlgn Let , k C M . Then the operator tr

V r

:W

2ti /2(r >fl

iB p

r

— >

w

2ti/2nar

Is an Isonorphlsn. Erciii By the aid of Theorem 2.3.6, Theorem 2.5.5 and trace theorems we obtain (tr V r ) ( " 2 J / 2 ( r ) ) C *2tH / 2 < r ) Let

vCw£+j|/2 Q = I - Fp(tr T G F p ) _ 1 tr T Q .

69

For

o t « 2 „(S)

w e have

Pu, « u t ^

„(6).

Proof Put

P*=Fp(tr T 6 F p ) _ 1 t r T g .

Theorem

2.3.6

ue*£H(G>,

kil.

and u s i n g a t r a c e t h e o r e m we

/2

Iq *2~H tr T f iUu £ Wg

Let

B y

D •-

Proposition 4.1.5

By

P

Proposition 2.5.12

Following

have

tr T g U € im Q .

yields

(tr T ( J F r ) " 1 t r T Q u € w£~jj / 2 ( H 0 im P p . Consequently, by using Theorem 2.5.5 we P'u £ W 2 , H

(G)f1

obtain

k S r

It is easy to v e r i f y P ' 2 u = ( F r ( t r T G F r , ) " 1 t r T Q ) ( F r < t r T 0 F p ) _ 1 t r T fi u = =Fp(tr T G F p ) _ 1 t r

TGu=P'.

It r e m a i n s to show that space

D({j

I-P'

(G) H(6))n»2 h

Furthermore we

subholds.

TGFr)_1tr

TQu).

that

tr [ T g U - T g F p ( t r T G F p ) _ 1 t r For

the

and

DCWg h ( G ) )

4.1.7

-

u = F p g + T Q PDh + T 0 QT Q f , which belongs to

k+ W P.H H" 1 / 2 '

be an eigenfunction of -.A. corresponding to the Xj and let w=Du. Then

and our assertion (i) is proved. The statement (ii) is tained fron the following estimations: |T»'

=

=

Sap

Taking w=Du fied (ii).

sup IkiiEki . _ uJmQ lul*

^

sup J L M L h

+ sup

ob-

JMviL,; „ ID^IJ t,H

l

in

v = Ppg

on

G (li)

T '

Dw = 0

in

G

tr TgW = Q p g

on

P

Proof First iff

we show that problem (i) has a solution +

ge W2 H

/2(f1)n in p

solution of (i),

r "

Inde

d

« >

let

v e

+

*2 H

v € Sd.

valued functions in this way that it acts on each

component.

Formal factorization of H E L H H O L T Z ' operator is given by ( A + X 2 )I = - D ^ , where

Dv\ = X. e0 +D. 3 U= Writing X Z uiei' i

¡

f=

«

J X T ^iei

j

an
0.

Thnoren

3

x,y e R , x/y, Ai R

and

e x ( x ) = e(x)[cos X |x( + |x| X sin X |x|] +

e^ .

Then D^

U)(x) =

j ex(x-y) «(y)u(y)dry

r

76

, x € R3, ,x£R3\P,

the

(iii)

(S^uXx) = 2 J e ^ x - y ) .B. P •

Finally,

using Corollary 4.2.13 we

2 > A u + X u = -DxD_yu = D x Q T y f = f.

#

4.2.15 Corollary Let g e W 2 + ^ / 2 ( D , ku0. The first boundary value problem A u + X2u = 0 u = g has a solution

in on

G , P

utlf2^,loo(G)

(4.19) (4.20) of the form

u = Fy g + T ^P^D^h , k+2 where h is a V^ jj(G)-extension of g. Proof As g e W2+jj/2(r), there exists a «^(GJ-extenBion h with tr h=g. With u=v+h the boundary value problem (4.19)-(4.20) will be transformed into Z\.V + X2v = - A h - x 2 h in G, on r.

81

Therefore

it is c l e a r

that X2h).

T_ a\ (Ah + " ' - Xv " F u r t h e r n o r e we g e t v

v = - T ^ D ^ h = -h u=v+h

With

+

Q ^ D y h

= -T_yDvh + T

P ^ h

=

T

-xpxDxh + we g a i n the a s s e r t i o n .

4.2.IB Theorem Let

+ T

^

(Exigtono»^

f e Wg H

n

operator

in

* *£!H/2(r)n

**

iB

Q

X

2 is an i s o m o r p h i s m

if

X

is not an e i g e n v a l u e of

{-A.tr}.

Proof By the aid of P r o p o s i t i o n 4 . 2 . 9 and a trace theorem w e (tr T^F X ) ( W ^ Let

v € »2 H

Theorem

82

+

/2(r) n

4.2.17

/ 2 iB

( D ) C »2*ij P

>-

yield

a n d

tr

TxFKv=0

/2 T

obtain

(D •

xFXv=aand

Theorem 4.2.18

T ^ v

6 ker(A+

and \2>,

whence follows How

let

v=0, as for any

« ( i l Q^.

v e i n Py , F x v=v.

Making use of Corollary 4.2.7 (ii)

we

have* Fyw=0. Then there exists an H-valued function ucker ( A +

>.2)

with

w=tr u.

position 4.2.3) gives us v = D > u i k e r D^ . we find

BOREL-POMPEIU 's formula (Pro-

u=TkD)ku. It is clear that

Applying Proposition 4.2.3 and Remark

v=F)i tr v, whence

u=T)iF> tr v

4.2.8

and so

w = tr T^FyCtr v)c ii tr

^

4.2.19 Corollary u ç H* h ( G )

Let

k>_l. T h e n

for

and

the orthoprojections

we

P*

h a v e

and

f V i ^ n J

Qx

H

( G )

allow the representa-

tions P* - F,(tr T . F ^ ^ t r T, , Q * = I - F x (tr T x F x ) _ 1 t r T k . Proof Let

u€

k

k>.l.

For the sake of brevity we put

P '=Fjj (tr T > F ) l ) _ 1 tr Ty theorem we get

Using Proposition 4.2.8 and a trace

tr T„u e " î ^ ^ D •

Preposition 4.2.16 yields

tr T x u c in Qj, whence (tr T„F x ) _ 1 tr T ^ u c H ^ " ¿ ^ ( D C im Px . Consequently, now using k P.'u € W^

Proposition 4.2.9,we obtain to see that

2

P' =P'.

ker

D

X-

is

eas

y

2

Obviously, (1-P') =I-P". Furthermore we

have (I-P')u=u-P'u=D x (T > u-T x F x (tr T x F x ) - 1 t r T x u)=D )i w with and

tr w=0.

Owing to the uniqueness of the projections

Q > , we obtain

P X =P'

and

Q'=I-P'=c\

P ^ «

4.2.2H Corollary The solutions

'

Let

(4.28) with

f=0

kz0

u = Fpg + T ^ D h where

h

function

iB i ^

'

The boundary value problem

has a solution

'loc(G)

u£

(4.28)-

of the form

,

k+2 jj(G)-extension of the K-valued

g.

Exflflf. As

ii

tr h=g.

h

there exists a K ¡^-extension Put

u=v+h.

The problem (4.28)-(4.28)

with

with

f=0

shall be transformed into the boundary value problen -DM _ 1 Dv = DM _ 1 Dh v = 0

in

G,

on

r .

Applying Theoren 4.3.1 we find the solution v = -T-Q MT p DM _ 1 Dh . Using BOREL-POMFEIU's formula we obtain v = -T-Q Dh + T P Q MF M _ 1 D h = U B b B = -h + F_h - T„P Dh + T„MF M _ 1 D h - T-P MF M _ 1 D h . r un u (j • As

P MF K - 1 Dh=KF„K - 1 Dh m r

our statement follows from

u=v+h. * 9

4.3.3 Theorem (Existence) f W g c W Let fCC W 2, ^H ^ (GG)> ', g c W22/+ H / 2 ( r ) ' (4.28>-(4.29) has the solution

The boundary value

u = F^g + T G P B D h + T G ö B M T G f €w!^jj' loc (G), where

h

k+2 denotes a #2 ^(G)-extension of

problem

(4.31)

g.

Proof The sum of the solutions which are given by Theorem 4.3.1 and its Corollary yields the solution (4.31).

^

4.3.4 Theorem (Uniqueness) Let

f t «2 r ( G ) '

valued

function

k.>0 u

In (4.31) expressed H-

is the only solution

of

the

boundary

vaj ue problem (4 . 28}-Proof The proof immediately follows from Corollary 4.1.6.

^

4.3.8 Propoaltion Let

l0,

0< (iil.

acting within the spaces Proof

The operators

k W „(G) p, n

and

P^

and

are

0ftC U '"(G). n

The proof can be carried out similarly to Proposition 4.1.8.^ 4.3.8 Corollary (Regularity^ Let

fetf£ h (G>

.0,

k£H.

^ The solution

uewi^jjCG)

of

the first b o u n d a r y value problem may be represented by u = v + TQMw , k+2 v £ H^

where

k+1 H^^

are

un

iE + In (bH). First

let

£

and yu be constants.

Then

systen

(4.34)

is

reduced to DE = f , DH = y E . F i n a l l y , we have to consider the b o u n d a r y v a l u e p r o b l e n D y_1DH = f

in

G ,

H = g

on

P,

where

the

stant.

Let

y(x)>0

for

is not n e c e s s a r i l y

,

xeG.

f tlij JJ(G), y e C ^ G ) .

H = 0 has a s o l u t iion on

GV

The first b o u n d a r y value

problem

in 6 ,

(4.37)

on r

(4.38)

H e wi^jj' * o c ( G ) , w h i o h nay be r e p r e s e n t e d b y

V

For T Thh e o r e n 3.1 there exists an H - v a l u e d f u n c t i o n with

con-

Theorea

D y_1DH = f

Proof

(4.36) y = y(x)

By reason of non-vanishing e l e c t r i c c o n d u c t i v i t y

we have 4.4.1

real function

(4.35)

QyyTQf=DH,

where

R

°1 H € J J ( ® )

is identified w i t h the o p e r a t o r

91

of

multiplication by the scalar funotion TGf€W2ji(G)

yields c

y €.

A n

R-

a n d

1

/ V

y .

Theorem

"^H'l0°(G)'

2.3.6

a o

a p p l i c a t i o n of B O R E L - P O M P E I U ' s f o r m u l a

leads

to V y i V

T

=

and therefore

GDH

H£*2^

D y-^H

l

= D r

H

=

, l o c

a

> (G).

y V

y

= f - D with

w e ker D. D y_1DH

4.4.2 Let

=

Finally, C o r o l l a r y 3.3 D

V

"

D

implies =

* V

" .

Therefore

= f.

*

Corollary g £ w£+jj/2(D, D y-1DH

= 0

in

G ,

(4.39)

on

T

(4.40)

H e

H = Fpg + h

problem

H = g has a solution

where

k>J3. T h e f i r s t b o u n d a r y v a l u e

l

o

c

( G )

of t h e

form

TGPyDh\

Is a ^

k+2 jj(G)-extension of

g.

4.4.1

analogous

Proof Using

Theorem

4.1.2. 4.4.3 Theorem Let

is

g£«2+H

(4.35)-(4.36)

h

Corollary #

/ 2

(D.

h a s the

denotes a

The first

boundary

value

solution

H = Fpg + TGP^Dh + T G Q y y T G f where

to

(Existence)

f € wji h < G > .

problem

the p r o o f

loc

e

k+2 (j(G)-extension of

(G),

(4.41)

g.

Proof The

sum

of

the

(4.39)-{4.40)

s o l u t i o n s of

problems

(4.37)-(4.38)

s o l v e s ( 4 . 3 5 ) - ( 4 . 3 6 ) a n d a l l o w s the

tation (4.41). 4.4.4 Theorem fUniguangaa) Let

F C « 2

expressed

H


+ 3 2 K 3 C 1 y n|lhO p /(^(4j«-n) 2 ) .

L n < l - £ , £ >0

the additional condition

(ii) oust be fulfilled.

„

*

4.7.5 Ran»rk Similarly

to Section 4.6,

sharper statenents of

regularity

nay be obtained. 4.7-B Ranarh At

S.BERNSTEIH (Freiberg) considers boundary

present

problems

over unbounded domains

ciently

smooth

boundary

G

"3G=P •

with a The

compact,

value suffi-

following formula of

BOREL-POMPEIU's type.

-F_u + T-Du = P G

in

G,

in

R3\G

is also valid. If

G=R ,

examples

then

we

we obtain

TQDu=u.

By the

will characterize the different

help

of

three

situations

in

case of unbounded domains. Example 1:

^

3

Let

G=R ,

Then

u

z

Y^t

2 _3/4 x.(l+|x| ) e .1 . 1

is a continuous and bounded H-valued function, 2

TGu=TGD(l+|x| )

1/4

2

=(l+|x| )

but

1/4

is an unbounded H-valued function. Bxanpln 2; } G=R3, u = - y ~ x i ( l + | x » 2 ) " 3 / 2 e i . i»1 Then a £ L j h^®^' On the other hand it is easy to see that Let

Tqu = TGD(1+|X|2)"1/2

= |Dva 11?

= ( " A Ta ,va ),

= c'a,

and

Finally, we get " V a « ^

= oa 2

»L zh

«

for

a — >

- .

Hence the operator T Q , as a mapping from the subspace ®1 jj(G)CI,2 ^(G) into Lg is not continuous in the norn of L 2 H (G). To

preserve great parts of our theory it is necessary to use

weighted spaces. Let W

2 ; H ( G )= { u e ® H ( G ) : p f u

eL

p^= (1+|x| 2 )

where

2,H(G)

and

D

, i e R . W^'flCG)

iu£L2,H1/2

"P1l2>r ,

and our proof is finished.

^

4 . 8 . 3 Remark Using r e p r e s e n t a t i o n (4.60) the n o r n of the p r e s s u r e can

be estimated b y the

side

f.

4.8.4

Raaarh

The

assumption

L^ ^ ( G ) - n o r n of the

of T h e o r e n 4.8.2 is also important

d e r i v a t i o n of n u n e r i c a l methods.

llp||T

right-hand

for

the

In the proof of P r o p o s i t i o n

4 . 3 . 5 a p o s s i b i l i t y w a s explained to construct a solution the b o u n d a r y v a l u e p r o b l e n of

the

tr u=g}

s o l u t i o n of D I R I C H L E T ' s p r o b l e n

Besides,

we

Theorem be

{DM ^Du=0,

find that

DM-1Du

tends to

{-A.u=0, -A.u

if

tr

u=g}. n

>0.

4 . 8 . 2 s t a t e s that a solution of STOKES e q u a t i o n s may

approximated by solutions of b o u n d a r y value

linear

of

on the b a s i s

e l a s t i c i t y (with exact error b o u n d s ) .

principle

for

problems The

of

iteration

solving H A V I E R - S T O K E S e q u a t i o n s b a s e d on

fixed-point theorem allows that STOKES' p r o b l e m s may b e

the sol-

ved at each step even w i t h a certain error (stability) if the domain known).

of c o n v e r g e n c e

is not left (this d o m a i n is e x p l i c i t l y

If taken in the iteration p r o c e d u r e as s o l u t i o n s

of

STOKES p r o b l e m s , the corresponding s o l u t i o n s of the e q u a t i o n s

120

of

linear

weak

e l a s t i c i t y n e v e r t h e l e s s c o n v e r g e n c e or

convergence

can be p r o v e d and in b o t h cases

estinate can be obtained. Therefore work

at

least

an

error

it could be s u f f i c i e n t

out an e f f e c t i v e n u m e r i c a l a l g o r i t h m to s o l v e the

boundary

v a l u e p r o b l e m of linear e l a s t i c i t y (or the

to

first

LAPLACE

e q u a t i o n ) to m a s t e r the c l a s s i c a l b o u n d a r y v a l u e p r o b l e n s

of

mathematical

We

physics

f r o n the n u n e r i c a l p o i n t of

view.

shall d e a l w i t h a d d i t i o n a l r e q u i r e m e n t s of r e a l i z i n g a rical n e t h o d in C h a p t e r 4.8.!S The

nune-

5.

Ramrk considerations

pointed

Theorem

8.4.2

out that it is r e a l l y p o s s i b l e to c o m p u t e the

velo-

9 u= - l i m i 0 u "I m — > ¿ n Qp=-}Du+ 3 TgQTgf)

city

separating

leading

and p r e s s u r e

p

(with

in the s o l u t i o n of

calculations.

¿ÍS»2 %

to the p r o o f of

= iíS»2

STOKES'

problen

The q u e s t i o n of u s i n g the

W

by

linit

V

to o b t a i n the r e s u l t s of T h e o r e m 4 . 5 . 7 arises. I n d e e d , w e u

«= | =

\ V "

V

m

have

=

- \ V

It is clear t h a t

n

v

tr u=0

t

r

and

V "

F

r>"ltr V

V

•


0, n

mes( B G )—»mes n

i) G

for n—* oo .

Renewing

the

previous considerations then it follows H(f)=0 Because

, of

finally have

124

U Lp>H(Gi)n

A^G,).

the star-shapedness of the domains G and

G&

we

H< If ) = 0 and s o

, if e L p

U

p,n

BfinoxJt

nay

by

(G)

of

Lp>H(G)0

5.1.6 Let The

of

U

L

star-shapedness if

it

(G.) p,M i

is

n

U

of

ensured

A„(G.) H x

t h e domains

that

the

coincides

G

and

closure

in

with

Theorem {x

}.

( l )

^

, UC

Pbe adense

1£H

subset

on

H, .tfr., l

°

"

'

(

V= 1

. e>.

| rf.'!_

* •

system

with

t

is

N

the

(u,v)r

H-complete

scalar

product

= J Gv d T

, u,v

in

£ L

12>H( P )-clos[im ( D

2 H

Pp n C ® ' P ( p >J

.

r

Proof Define

the

function

(U, i f i >

=0

u e L2jH( T ) Let

the

omitted eo

AH(G).

{tr i?i>i

(i)

Ah(G> #

The s u p p o s i t i o n L

(G) n

H=0.

5.1.7 Ga

H

(U,

f

i

)

-

clos[im

= 0 ,

and s o

U- n u .

First

if

, i CN ,

of

P r n cjJ'P

ieN

,

whence

j

d

P

We

get

(F

lim

prove

.

rTu

^f .d P

= 0 ,

i£N

,

T

= J I Z y g ^ j

f o r m u l a we

we

if

( T )]

j

j f

all

and o n l y

u)(x)

- 0

eknu dT

=