Boundary Value Problems for Partial Differential Equations and Applications in Electrodynamics

functions

f t n

belong

, F~'F(
the functional

functional number

R

, . . . ,cr ) , x = ( x , . . . ,x ) , d x = d x . . .dx . a n d o"x=cr x + . . .+u x .

h e r e tr=[ir It

(1.16)

a

r

e

arbitrary

2

i s satisfied:

I D % ( x ) l ] , (pes,

I a I ^q w h e r e C,/ Linear

and g a r e c o n s t a n t s depending continuous

generalized

functional

functions.

greater

than

function

f

that

of

over

o n L b u t now d e p e n d i n g

S will

I f t h e growth

rate

a

at

polynomial

c a n be a s s o c i a t e d w i t h

be a l s o r e f e r r e d of \x\^,

then

e v e r y such f u n c t i o n

way:

7

f (x)

function a

on p.

later

as

i s not

generalized

i n the following

f(x)
( f . f H

If

L

i s a

generalized

function,

F o u r i e r t r a n s f o r m F~ (L)

inverse

l

,

(1.18)

PES.

i t s Fourier

transform

are defined as f o l l o w s

(f(L),*)-{L r(f)),

F(L) and

[1] I

^ s ,

f

(i.i9)

,

(F"'(L),<ji)-(L,r* ()»), p e s . t h e f u n c t i o n f(x)

If

(1.16) is

may d i v e r g e .

defined

with

t h e formula

them as g e n e r a l i z e d

t h e Fourier

transformation o f the function ,. . . ,x ) . 1

to

f u n c t i o n U{x,t)eH

every

At

every

Let

fixed

=[

U(t)

U{X,t) {X)dx

defined

by t h e f o r m u l a :

,
V

t t Ot h e f u n c t i o n a l

on t as a

U(x,t)cH

We p u t i n c o r r e s p o n d e n c e

ri

the functional

<[/
depending

function

f u n c t i o n s i n accordance

(1.18).

t o t h e v a r i a b l e x-tx

respect

g r o w t h a t IJTI-*», t h e i n t e g r a l

T h e r e f o r e F o u r i e r t r a n s f o r m a t i o n f o r such

by c o n s i d e r i n g

B e l o w we d e f i n e with

has a polynomial

(1.20)

U(£) i s a

(1.21)

generalized

function

parameter.

U ( x , t ) c M , t h e n f o r (1.21)

l(P(t),f)r*e,(i*tl

implies:

' m*x\(l+\xl')

*

Y

a

iD »>(x> l | ,p«s,tso,

(1,22)

\a\sq where not

c

l

''

l

l

If

l l

a n l z

function

e

constants

we s h a l l

depending

depending

say t h a t

i s not greater

U(x,t)

on f u n c t i o n

but

than

on parameter

t h e growth

that

t satisfies the

rate

o f a polynomial

o f generalized a t t-»+~ { h a s a

growth).

us denote

fixed

r

f u n c t i o n V(t)

(1.22)

L/(£)

polynomial Let

a

^ "J

o n

generalized

inequality

at

''

depending

the Fourier

t * 0 by F ( U ( t ) ) .

transform

From

(F(U(t)),cp)=J

(1.19)

of the generalized and (1.21)

f J ( c f , t ) F ( p ) (cr)dcr ,

ft

f u n c t i o n [/(£)

i t follows
that: (1-23)

The

F(U(t))

functional

function

U(x,t)

will

with

be

named

respect

the

Fourier

transform

r = ( x , . . . ,xj

to

and

i

of

the

denoted

by

function

of

F (D(x,t)). %

It

should

polynomial The

be

noted

derivatives

respect

that

to

the

of

the

variable

C

(

It

is

clear t

variable U(x,t)eK

is

the

be

dt

t j

'*')

=

a

,
at

every

£s0.

defined

the

product

the

of

the

generalized

functional

W(t)

1

is

i t

a

function

follows

with

respect

ift(x)

and

that

at

lxl-*°.

1.3.

i s an

of

us

mm,

the

infinitely

and

the

differentiable

i t s derivatives

consider

are

not

f o r Equation

the

M

o

Applying x=(x

[X),...

g ( x ) = (g

class

equation

condition

the

we

,g

(x))

(cf section

,...,x ) t o

to

the

p.S.

function

,1.25,

^(x)

is

assumed

function

greater

than

(1.26)

which

that

of

growth

polynomial

Cauchy's b o u n d a r y

condition:

(1.1)

(1.1)

Fourier the

both

is

rate

a

with

r j ( x , 0 ) = g ( x ) , XEK",

where

i f

sense:

Cauchy P r o b l e m

Let

of

that

formula:

( V ( X ) V ( C ) , i p ( x ) ) = (» ( t ) , ^ ( x ) p ( x ) ) , peS, where

2

I - *)

ipeS

(1.23)

r r"

The

with

s

(
)=F

by

(U(x,t))

V(t)=F

sense:

• ?* -

fixed

From

U(t

derivative also

function

m?L. U JSi^lrm in

generalized

i n a weak

f ^ i - f i

"5t

half-axis

the

t may

generalized

r i s assumed

(W{Z)

that

on

then

variable

F (t7(x, t ) )

growth.

a

given

(1-27)

vector-function

belonging

to

1.1). transformation sides

of

with

equation

obtain:

9

respect (1.1)

and

to to

the

variable

the

boundary

A (a)

d

£

^< >

+ B(cr)V(t)=0

, t>0,

(1.28)

7(0)-l, where

V ( t l - f r , ( t ) ,. t , ,F ( t j )

transforms lV

(X,t),...,V

m

g

(as

(r,t))

(if)) with

and

generalized and

d-29)

L=(i,

functions) given

x-{x^,

ym-F ip(X t]U The

problem

the

Fourier U( x,t)

solution

at

L=0

=

L

g(x)=(g (x),..., i

.. . ,x^) , L=F(g).

r

(1-28),(1.29)

are

the

vector-functions

respect t o the variable

i

I>J

of

will

(1-30)

be

called

the

homogeneous

problem. (ir) and b (cr) b e t h e e l e m e n t s o f t h e m a t r i c e s A'a) a n d B(cr) Jk Jk ( j f K = l , - - - ,<") - E q u a t i o n ( 1 . 2 8 ) may be d e t a i l e d i n t h e f o l l o w i n g f o r m : Let

a

I J

! " ^ — c i t —

3

+

b

i

i

, ^

)

V

^

t

)

\

=

•

0

t

>

0

'

J *

-

1

m

(

1

-

3

1

)

k=i In

(1.29)

7(0) i s the l i m i t

of the vector-functional

V(t)

a t t-t+0,

i.e. (HO)

(V(t),p)

, pes,

(1.33)

nowhere ( K ( t ) ,p} = ( ( l ^ ( t ) ,p) , . .-, ( ^ ( t ) , p ) ) . Definition belongs fixed

1.6.

The

functional

t o the class M

t = 0 and s a t i s f y J

- d V ^ d t

I

, i f V^{t)

a

If

(t) . ,f> aC ( 1 + t )

—

on

t h e parameter

belong

j

T

1

J

2

max (l+lfl )
t o S' a t

t

every

, then the vector

V(t)

z

_—, ) ID (p(o-} I , , f -1 l a l aq a

(1.34)

peS, t > 0 ,

" d q are constants depending

t h e components o f t h e v e c t o r

class M

and •

the estimates:

j=0,1; where

V ( t ) depending

(1-33)

K(t)=(V

on

belongs t o M .

10

K^ft).

(t),...,V

( t ) ) belong t o the

Since t h e s o l u t i o n U ( x , t ) M,

i t is

Therefore searched

evident, the

It M*,

as a

existence

necessary

[1.28),(1.29)

and

1.5.

( 1 . 2 8 ) , (1.29)

Let

of

the

(1.28)

aim of t h i s

will

problem

equation

i n the class

on

L,

solution

of

Cauchy's

i s to

providing

condition plays

the

homogeneous solution,

is the matrix Proof.

and i n

cauchy's

problem

theorem. be

regular.

i s solvable

i f and

only

i f the

vector

the condition:

V(t)=lV

defined

1

Let with

by

(1.28),

i s an

(1.35)

(1.29)

identity

( a t L=o)

matrix

has

o f order

m

only

the

a n d CJ(IT)

(1.12).

L e t t h e problem

V (t))eM . m

(t)

condition

problem

where E

Necessity.

an

(1.1),(1.7)

u

trivial

the

problem

solution.

(1.1)

M*

i n the class

paragraph

( E - u ( t r ) )L=C, and

be

condition

condition

M* . I n t h e s e q u e l t h i s

i s t h e main

functional L satisfies

H*.

class

o f t h e boundary

i n t h e i n v e s t i q a t i o n o f the problem

construction of this

i n t h e class

the

(1.2a),(l.29)

f o r equation

[ 2 ] . The

sufficient

i n the class

The f o l l o w i n g r e s u l t Theorem

side

to

S'.

uniqueness

role

belongs

problem

the right

i s not correct

and

important

Cauchy

t h a t Cauchy's p r o b l e m

rule,

the

M*,

(1.1) i s searched

= F (<7 (X, t ) )

of

i n the class

i s known

find

the

solution

i n the class

(1.29) b e i n g

of equation V(t)

that

We

(1.28),

shall

prove

(1.29) that

have t h e s o l u t i o n L

satisfies

the

(1.35).

W ( t ) be

t h e Laplace

respect

transformation

to

the

transformation variable

t .

of the vector By

definition

function V(t) of

Laplace

([13],p.494): +D (r(A},f)-

| (V(Z)

,
ReX>0,

(1.36)

o where p i s an a r b i t r a r y Thus

function of the class

t h e f u n c t i o n a l W(X)

i s a generalized

S. f u n c t i o n a t each complex

n u m b e r X, ReXsO. Calculating

t h e Laplace

(1.28) w i t h r e s p e c t

transformation

of

both

sides

of

equation

t o t h e v a r i a b l e t and i n v i e w o f t h e c o n d i t i o n : /(0)=L

11

(1.37)

we

find: (A(o-)X+fl(cr) )W{A)=A(
the

us denote

sphere

the class

lcl
by S

of functions

.By

Re\>0.

(1.38)

belonging t o S with

definition,

carrier

carriers in

of the function

p (cr)

o is

the

closure

( 1 . 3 8 ) we

of

points

s e t o-eR"

f o r which

(J(cr)X+B(i7) )W(A)=A(
p ( c ) «0.

By

virtue

equality

\o-\
ReX>0.

(1-39)

( 1 . 3 9 ) means t h e f o l l o w i n g : ( ( ^ ( o - ) X + B ( c ) ) W ( A ) ,p) = ( . 4 ( c ) L , p ) ,

for

of

have i n p a r t i c u l a r :

any f u n c t i o n

peS^

(1-40)

Re\>0.

, o

If

the vector-functional

^(
L=(L ,...,L ]

i s the vector-functional a n d t h e v e c t o r L.

Since

equation

(section

numbers e

Q

(1-1) i s r e g u l a r ,

6.1)

and p

o

f o r every

so t h a t

s',

and

matrix

vector:

(L^p) ) . then

positive

a c c o r d i n g t o lemmas

number

r

there

o

are

6.4

and

positive

the inequalities:

det(A(
s

a t l
d e t (A(o-) x+B(c))«0 a t l o l s r ^

are

t o the class

the product of the

(L,p) i s t h e f o l l o w i n g ( L , * > ) - ( ( ! . , ,ip)

6.6

) belongs

o b t a i n e d as

0
(1-41)

o

| X | >p , Re\>0 q

(1-42)

satisfied.

Let satisfy From

r

Q

be

a

fixed

positive

the inequalities ( 1 . 3 9 ) we

number

and

and

p

g

are

chosen

to

( 1 . 41) , ( 1 . 4 2 ) .

have: (A(v)\+B((T)

)W(k)=A'
J c r K r ^ , 0
(1-43)

and (A(c)X+B(cT))Ii'(A)^(c)L,

12

|cr|
U I > p , ReX>0. o

(1.44)

Remark.

I f ipeS

and

r

the function

p(er) i s d e f i n e d

i n the

sphere

o |rr|ar

o

t h e p r o d u c t $(a)
sphere,

i s assumed t o be e q u a l t o z e r o b e y o n d o

Multiplying

both

[A(cr)X+B(cr) ) " ' we

sides o f equations

(1.43)

l

f

l

Laplace's

by t h e m a t r i x

( V ( t ) , tp)

( f ( t ) ,
with

i s

inverse transform

(1-45)

ltr|
I X | 2 p , ReX>0.

(1.46)

D

o

i s Laplace's

respect

expressed

0
o

(W(X) ,
the vector-function

vector-function then

(1.44)

lo-|
fcr(X) = (,l (cr)X+B(cr) )~ A'
and

obtain:

V ( X ) = (^(cT") X+B (tr) ) ~ A [&) L

peS) ,

this

i . e . ifi (a) tp (cr) = 0 a t l c r | e r .

transform

t o the variable

throuqh

(fif(fc>, (p]

t by

of the

(at

fixed

means

of

((13),p.499):

(K(t),fJ- ^ i j

j

( W ( X ) , p ) e x p ( X C ) d X , a>0

(1.47)

a- Im

* S i n c e 7 ( t ) e M , i t means t h a t tpsS i n c r e a s e n o f a s t e r

fixed may

be a n y a r b i t r a r y L-0.

Let lcl
Then

t h e elements o f t h e vector

positive

number

i n accordance w i t h o

v(t)=o and

((13),p.499). equation

R e X i p . Hence a t a = p , f o r (1.47) Q

since r

o

i s arbitrary,

we

So we h a v e p r o v e d

that

we

have W(X)=0 a t

(1.48)

o

obtain: t>0.

(1-49)

t h e homogeneous p r o b l e m

( 1 . 2 8 ) , ( 1 . 2 9 ) has

det(A(o-)X p

is

then:

p

p

,

+ B { o - ) ) = a ( c ) X + a ( c r ) X " + . . .+ a ( o " ) ,

a

polynomial variable

o

i

non-negative

(1.50)

p

integer,

a (tr) , a (cr) , . . . , 3 ^ (cr) o

j

n

(1.51)

o

(1.50)

and

are

c ~ ( < r , , . . . ,0"^] a n d : a ( u ) * 0 , o-eR .

From

only

solution.

Since e q u a t i o n (1.1) i s r e g u l a r ,

where

(1.46)

implies:

a t t > o , |cr|
V(t)=0,

trivial

C(t),p) at

t h a n a p o l y n o m i a l a t t->+». So i n ( 1 . 4 7 ) a

(1.51)

i t follows

13

that

the

elements

b

jk

(X,cr)

m) o f t h e m a t r i x

(j=l

{A (a) A+B(ir) ) ,

p

|h (A, -)l^Ur k

where the

C is

| c | s r , |X|sp Q

([14],p.16)

0

a

( A ( a ) X+B(cr) ) ' i n

o n r . So t h e m a t r i x

can

g

theestimate:

a t l*l=ff ,lAl*P ,

0

c o n s t a n t depending

o

domain

satisfy

be

represented

by

Cauchy's

integral

a),

(at fixed

fi 1

(J(.r)X+B(cr))~ =S(X,cr) + ^

where

^(o-)*',

ICl^T,,.

£

I*I P -

(1.52)

0

6=m-l-p,

f(*.«^)= - 2S1

j i- ' ")^* ")^ ^!' 1

0

0

1

( 1

-

5 3 )

l£l-P„ j

Kj(
In

(1.54) t h e i n t e g r a l

variable

,

d

M(c)e+B(c))- - 4

being

along t h e

,j=0,...,m-l-p.

r

calculated with

circumference

l£l=P

respect i

n

0

(1.54)

t o

complex

counter clockwise

direction. The

sum

principal (A(o-) at

o ( c ) +0^ (cr) A+-. . + a (IT) k

i n the right

S

o

s

part

of

X+B(cr))with

the

Laurent

series

side

expansion

respect t o the variable

o f (1.52) of

X i n vicinity

the of

i s a

matrix infinity

f i x e d ireR", l o - | ^ r o

Let

pes

. Then e q u a t i o n s o ReX-*+», i f a n d o n l y i f : u^o-lAfo-JL^O It the

( 1 . 4 6)

a t l
and

(1.55)

i s necessary

imply

,9)-*0

j=l,...,m-l-p.

i s known t h a t t h e L a p l a c e t r a n s f o r m condition

(1.52)

at

(1.55)

(1T(A) >p(
f o rt h e s o l v a b i l i t y

o f t h e problem

(1.28) , (1.29) . Let t h e condition

( 1 . 5 5 ) be s a t i s f i e d .

t A(a)\+B(a)

)~ A(O)L=Bi\,O)A{0-)L, 1

L

The at

formula

fixed

(1.53)

function

implies

ipeS

that

the vector

i s analytic o

domain

Then

IXI=P and 0

14

from

( 1 . 5 2 ) we

obtain:

\&\
(1.56)

Q

function

t o t h e complex

(B (cr,\)

variable

A{
i n the

(B{\,v)A(o-)L,tp)-tO So,

i n accordance w i t h

Jordan's

a t j &.)-#*• • lemma

( S ( X , c r ) ^ ( c r ) L , p ) exp(Xt

lim

{[13],p.439):

) d A = 0 , a t ipeS

,

t±0,

[1-57)

lying

i n the

half-plane

i n ( 1 . 5 7 ) we

obtain:


r

i s the part

of circumference

lAI=g

RsX^a-const,a>0. Substituting

f

lim

B{\,cr)A(u)L

from

(1.56)

((.4(0-)\+B(o-) ) " ' a { ( f ) L , f ((F)) e x p { A t ) riA=0, a t p e S , t i O .

(1.58)

q let

r

b e t h e s t r a i g h t - l i n e s e g m e n t EeA=e lying within q 0 I A I & J a n d 5 b e t h e u n i o n o f T a n d y i . e. S = r u T .

q

q

A c c o r d i n g t o Cauchy's theorem f

q

q q

the

circle

q

([13),p.45): 1

^(A(o-) A + B ( c ) ) " A(o-)L,p(a') j e x p f A t ) d A -

5

q (^(o-)A+B(D-) )

A(a)L,
\exp{\t.)d\,

( 1 . 59)

s w h e r e g > p , t^O a n d tpzS^

.

Q

o By

passing

to

the

limit

in

(1.59)

at

q->+
and

using

( l . 58)

we

obtain:

f

( U(o-) A+B(o-) )~ A(o-)L,
exp(At)dA

E - i n

( ^ ( c ) A + B ( c ) ) ~ i ( c ) L , ( p J e j r p ( A t ) d A , ipeS.t^O. ,

J

IS

(1.60)

From e q u a l i t i e s

(1.45),(1.47)(at

a=c ) and

( 1 . 6 0 ) we

Q

| | ( J ( , T ) X + B ( c f ) )- A(
lV(t),p)m

have:

,
1

(1.61)

r

& Let

L^

be

a

differentiable and

complex

number.

And

functional function

variable since

A

belonging

with at

to

respect

s'

to

o

o

L

be

variable

lo-|^r ,p -£s| A I * p + e ,

the functional

a'\,a)

and

real

where

o

E

infinitely

o-=(oi s a

i s l i n e a r and c o n t i n u o u s ,

f

](

. ..

,aj

positive we

have

([1],p.66)1

f

where f

J l y «(A,
(1-62)

j

i s any a r b i t r a r y f u n c t i o n

belonging t o the class S o

Taking (1.61)

the

equality

(1.62)

into

account,

l e t us

write

equation

as: ; r ( t ) ,f)~(u{a,t)L,f)

,

>peS

,tiO,

r

(1.63)

where

w < l r

'

t )

=

2WI j

+

(^(^)^ B(T))"'A(cr)exp(At)dA.

(1.64)

\ Evidently,

the function

"(c,t)=

(1.64)

-^j

J

may

be r e w r i t t e n

as:

l

(A((r)A+B(c))" A( r)exp(At)dA,

(1.65)

(

T(o-) where r(
equality

(1.63)

in

V{t)=u(a-,t)L, From

lemma

6.16

(1.12).

means:

(chapter

6,

|o-|
(1.66)

o

section

16

6.3)

i t follows

that

a{
belongs

t o t h e class

M*.

number, t h e n

from

positive

As

rv,, i n we

( 1 . 6 6 )

V(t)=W(a,t)L, substituting

E=0

obtain

into

(1.67)

(1.29)

we

the equality

(1.35)

f o r the solvability

Sufficiency. that of

t h e problem

and

arbitrary

fixed

(1-67)

r e c a l l i n g t h e boundary

(1.35).

Thus, t h e n e c e s s i t y

of t h e problem

K ( t ) defined

i s an

creR", t^O.

Let the condition

the function

(1.66),

obtain:

(1.28),(1.29),

(1.35)

be

(1.67)

condition

i s proved.

satisfied.

by t h e f o r m u l a

condition

of

We

shall

show

i s the solution

(1.28) ,(1.29) .

The e x p r e s s i o n

(1.65)

3

*(g) "]

r

,

C

>

implies:

+

B(o-)u(o-,t)=

t

^ i j

J A(cr)exp(At)cU=0.

(1.68)

Her) Hence,

( 1 . 3 5 ) we 7(0)

And h e n c e , 7 ( E ) = u ( i T , t } L the

V{t)

the vector-functional

From t h e c o n d i t i o n

class

H

-td(o-, t ) L

- u(cr,0)L

= u(ff)l

i s finite

Cauchy p r o b l e m This proved

1.5,

sides

(1.27)

= L. ( 1 . 28) , ( 1 . 2 9 ) i n

o f c h a r a c t e r i s t i c e q u a t i o n (1.3)

dependent

i n the class

from

on

a,

then

M has o n l y

the formula

(1.48),

the

homogeneous

trivial the

solution.

later

being

a s s u m p t i o n t h a t p (IT) - c o n s t < o .

o f Theorems

1 . 1 , 1.2

transformation of equation

the following

(1.28).

i s proved.

not

follows

under t h e only

Fourier

in

does

(1.1),

statement

1.4. P r o o f

both

and

equation

i s t h e s o l u t i o n of t h e problem

. The t h e o r e m

Remark 1 . 1 . I f t h e number o f r o o t s p(o-)

satisfies

have:

with

and

1.3

respect

to

variable

( 1 . 1 ) and t h e b o u n d a r y

x = ( x , . . . ,xj

condition

[

on

(1.7) r e s u l t

problem: A ' a )

d

V

^ -

+ B(c-)7(£) = 0, P(
17

t>0,

(1.69) (1-70)

w h e r e V(t)

and u a r e F o u r i e r t r a n s f o r m s o f t h e v e c t o r - f u n c t i o n s

£(x)

and

r e s p e c t t o t h e v a r i a b l e x~(X ,

with

U(x,t)

. . . ,X ) i

t

o

l"(t)=F (B[X,tJ) and u = F ( f ( x ) ) . As

t ) eM

f (x) eM,

and

K{ t ) e M *

M

functional

M i s d e f i n e d i n s e c t i o n 1.3.

We s h a l l * * class K .

search

The

and H

then

functions

relation

are defined

the solution

a n d peS'

i n section

The

classes

of

1.1, and t h e cases o f

o f t h e problems

between t h e problems

(1.71)

(1.69),

(1.70)

( 1 . 1 ) , ( 1 . 7 ) and ( 1 . 6 9 ) , ( 1 . 7 0 )

i n the

i s the

following. 1 . 1 . L e t U(x, t )

Lemma

V(t)-F

then f(t)

(U(X,t))

i s the solution

solution

the

problem

solution U(x,t)

the proof

(1.69) , (1.70)

will

be

and

can

U{x,t)

then

i s

I f be the

i somitted.

( 1 . 1 ) , ( 1 . 7 ) c a n be s o l v e d a s f o l l o w s . i n the class

M , then

a s V ( t ) = F {U{x,t)),

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

construct

(1.69),(1.70)

U(x,t)sH,

where

( 1 . 1 ) , (1.7) ,

{ 1 . 69) , ( 1 . 70) .

(1.1),(1.7).

i sobvious,thus

the problem

o f t h e problem

o f t h e problem

t h e problem

{U[x,t)),

of t h eproblem

Lemma l . l So

of

a s V(t)-F

represented

be t h e s o l u t i o n

i sthe solution

the solution

of

First, show

where

t h e problem

a l l s o l u t i o n s o f t h e problem

we

we s o l v e that

this

I7(jr,t)eM.

Then

(1.1) , (1.7) .

(1.1),(1.7)

We

may

i n the class M i n

t h i s way. Proof

o f theorem 1.1.

t h e c l a s s M*, t h e n

L e t K ( t ) be t h e s o l u t i o n

o f equation

b y v i r t u e o f t h e o r e m 1.5 ( s e c t i o n

(1.69) i n

1.3) we h a v e :

(E -u(o-))l'(0)=0,

(1.72)

i

where

w(tr)

problem

i s the matrix

(1.69),(1.70)

(1.69),(1.70),(1.72) Let

t h e problem

( 1 . 1 ) , (1.7)

t o t h e theorem

condition

(1.14).

in the

1.4

By v i r t u e

6.6) t h e s y s t e m

t h e class

by t h e f o r m u l a M

(1.12).

i s equivalent

Hence, t h e

t o t h e problem

i n t h e same c l a s s .

According

section

defined

i n t h e class

satisfies this

o f theorem

(1.70),(1.72)

S' i s u n i q u e l y

Lopatinsky's

condition

6.4 a n d f o r m u l a

with

18

t ot h e

(6.158)(Ch.6,

respect t o t h e vector V{0)

s o l v a b l e and t h e s o l u t i o n

formula:

condition.

i s equivalent

i s d e f i n e d by


K(0)=^

(1.73)

i

J=l 1

where from

( i —P(r

the formula

(1.70)

i s reduced

Since t h e r i g h t to

theorem

(1.69),

r

( x ) ) a n d a(
1.5

(1.73)

(6.158)

.. ,a(
t o cauchy

i s m-dimensional

Chapter

problem

6.

Hence

(1.69),

vector

function

t h e problem

(1.73)

(1.69),

i n the class

M .

s i d e o f (1.73) s a t i s f i e s

e q u a t i o n (1.72) and a c c o r d i n g

and t h e f o r m u l a

section

i s solvable,

(1.67)(

the solution

1 . 3 ) , Cauchy

i s unique

problem

and i s d e t e r m i n e d

by t h e f o r m u l a :

V{t)

- u(o-,t)Y

CUCT))!^ J =

w h e r e u{
(1.74)

1

i s t h e m a t r i x d e t e r m i n e d by ( 1 . 6 5 ) ( s e c t i o n 1 . 3 ) .

a (cr) b e t h e m a t r i x w i t h c o l u m n s c o i n c i d i n g w i t h t h e v e c t o r s r " ( c ) . n ( c ) i s a r e c t a n g u l a r m a t r i x o f d i m e n s i o n (m-r) . T h e n

t h e f o r m u l a (1.74) becomes: 7(t)

= u(cr,t)a(£r) J J ,

(1.75)

where V={» ,...,U )=F(f(X)). 1

How

we s h a l l

represented

show t h a t

as

V(t } = F

lemma 1 . 1 , U(x,t) As

the

vector

(j=i,2,..-,r)

the vector-functional

(t/(x, t ) ) , where

i s the solution f

;

a

Q

q

the

according t o (1.7).

components

f (x)

n

(1.77)

, 0£|al<», x e R ,

a

of

[x)) 6 M ,

Then (1.1),

j

theestimates:

\D fj(X)|£C jl+lxl j

where C

C(e)=u(ff,t)«(ef)|j c a n be

17 ( x , t ) s M .

o f t h e problem

f ( x ) = ( f ( x ) ,. . . , f

satisfy

(1.76)

r

and y a r e a c o n s t a n t , y i s a n o n - n e g a t i v e i n t e g e r

independent

a. L e t q be a p o s i t i v e V

(x,t) "

—

i n t e g e r and: f u(cr,t)a((r) fi+|o-| |

( 2 " ) " ^

2

1

19

'

n

e x p ( - i c x ) dcr, XER ,

(1.78)

17 (x,t)=\W

n

(a,tig

(Jf-o-)do-, x e R ,

t*0,

(1-79)

where

q f(x) J

and

I i s the unit operator.

Since the

f(x)=(f

same

(x),...,f

estimates

vector-function 6.3

" ax

2

g

a n d 6.4) we

with

the

( x ) . From

infer

number q

natural

i

(x)) satisfies

with

same

lemmas

that

the estimates constant

6.16 a n d 6.17

f o r any n a t u r a l

, so t h a t

the derivatives

respect t o the variable

t

y

(1.77), i s

(Chapter

number

6,

for

section

f one c a n f i n d

of vector

a

W (x,t)

function

up t o v - t h o r d e r a r e c o n t i n u o u s

n

domain t = 0 , xeK and these d e r i v a t i v e s

then

true

i n the

s a t i s f y theestimates:

6 (jf.t) | C ( l + e ) 3

2

2

"(l+lxl )

p

, OSftsi',

(1.80)

w h e r e y i s t h e same c o n s t a n t a s i n ( 1 . 7 7 ) . The

estimate

mentioned (1.78)

(1.80)

lemmas.

by p a r t s

The f o l l o w i n g

To

at

|x|sl follows

obtain

this

immediately

estimate

a t |x|al,

2 ( n + l + y ) times and then a p p l y these inequality

from we

t h e above

integrate i n

lemmas.

i s obvious:

l+|o--x| 's2 (1+|0-1 ") ( l + l x l * ) , cr.XeH". From t h e i n e q u a l i t i e s integral

(1.79)

a t q=f

(1.77),

(1.80)

converges

v

and (1.81)

(1.81)

i tf o l l o w s

that the

and t h e i n e q u a l i t y S

ID^rj^ holds,

(x,t)|

where

and

y

(1.79)

i t follows:

sC

i ) ( x

(l+t)

2

1

''(l+lxl ) ',

a r e t h e same

as

Oifrsv,

i n inequalities

(1.82)

(1.77)

and

(1.80). From

F

(1/ ( x , t ) ) = u ( o - , t ) a ( r / } u , v = 0 , 1 v

20

(1.83)

Thus: U

As the

the vector equality

(1.84)

(x,t),

U

function

U (x,t)

function

{x,t)=U

(x, t )

i s true

belongs

V{t)

(1.75) and ( 1 . 8 3 )

i s the solution as

satisfies

f o r any n a t u r a l

K(t)-F

Since

(1-84)

the estimates n u m b e r v,

(1.82)

and

then the vector

t o M.

Comparing t h e e q u a l i t i e s

represented

V=l,2

i n (1.85)

we

obtain:

(17 ( x , t ) ) . o

o f t h e problem

(1.85)

(1.69),

(1,70)

U (x,t) , in o

the vector-function

and c a n be accordance

q

with

lemma

uniqueness

1.1,

i s the solution

of the solution

of

t h e problem

o f t h e problem

(1.1),

(1.1),

(1.7).

The

(1.7)follows

from

(1.74) .

Thus

Lopatinsky's condition

problem Let

(1.1),

provides t h e unique

(1.7) i n t h e class

Lopatinsky's condition

solvability

(1.13)

be v i o l a t e d

a t some

point

rank

We

shall

prove

non-trivial Let

us

that

of the

M. O ES°, (

(1.86)

t h e homogeneous

problem

(1.1),

(1.7) has

a

solution.

denote

a

non-trivial

solution

of

t h e system

of

algebraic

equations: (E_

- u(o" ) ) a = 0 , 0

P(o- )a=0,

(1.88)

o

by a=(a ,...,«_) ]

and l e t : V(Z)

where 6{
(1-87)

= u(a-,t)6(c-
i s Dirac'= delta

function:

( 5 { o - o - ) ,p(o-)) - *>(cy o

21

(1.89)

and

u>(cr,t)

One

i s the matrix of

can e a s i l y

problem

(1.69),

Since

verify

(1.65). V(t)

that

i s the solution

o f t h e homogeneous

(1.70).

u(cr,t)S ( o — f f ) = u { c r , t ) a ( o — c ) , o

o

then

Q

the

V(t)

functional

becomes: V(t) From

( 1 . 9 0 ) we

= u(c ,t)5(o-u )a. o

(1-90)

o

have: V(t)

-

F^'U'x.t)),

(1.91)

where V(x,t) From

lemma

Thus,

by

solution We

6.16

U{x,t)

function

= u(
(Ch. 6,

section

virtue

of

lemma

that

U(x , C ) i O .

is a

t=0

=

a point

o~=o e R . n

1.2.

function

[7(jr,t)

i n

(1.92)

and

nontrivial

L e t n^2

The l a t t e r

us c o n s i d e r t h e system

a

solution

o f t h e homogeneous

and L o p a t i n s k y ' s c o n d i t i o n

be

i s m-dimensional

violated

means:

(1.93)

of equations: (1.94)

f

P(C)K(0)=0,

to

problem

1.1 i s p r o v e d .

(B_-M:(ff)|I'(0)=&

w h e r e V(0)

M.

i s the

recalling

rank

Let

vector

aexp(-ixa ).

( 1 . 7 ) i n t h e c l a s s M. T h e t h e o r e m

Proof o f theorem at

the

obtain:

U(x,a) U(x,t)

that

(1.1),(1.7).

Substituting

o

Hence,

i t follows

1.1, t h e v e c t o r

u ( < 7 , 0 ) = u ( c r ) a n d u ( c r ) a = a we

(1.1),

6.3)

(1.92)

d e f i n e d by t h e f o r m u l a (1.92) belongs t o t h e c l a s s

o f t h e homogeneous p r o b l e m

show

.

o

vector

(1.95)

functional

of the class

S' w h i c h i s

be f o u n d . In

Chapter

6, s e c t i o n

6.6 we h a v e s h o w n

22

( T h e o r e m 6.5) t h a t

under t h e

condition

(1.93) ,

linearly

t h e system

independent

(1.94), l

solutions

(1.95)

has

7(0)(j=l,2,...)

infinite

number

concentrated

of

at the


point

Q

Now

we

shall

consider

the

system

(1.69)

with

Cauchy's

boundary

condition: ]

V{0) It

i s clear

every

that

j

fixed

= 7(0).

the solution

i s also

a

(1-96)

o f Cauchy

solution

problem

(1.69),

o f t h e homogeneous

(1.96)

problem

at

(1.69) ,

[1.70) . ]

Since theorem

7(0) s a t i s f i e s 1.5

(1.70)

and

the condition

formula

i n the class H

(1.94),

(1.67) ( s e c t i o n

then

J

be w r i t t e n

with

(1.69), as:

J

= u(o-,t)7(0) .

(1-97)

) As t h e v e c t o r f u n c t i o n a l s 7 ( 0 ) a r e l i n e a r l y 1 is true f o r 7(t)(j=l,2,...) . J As t h e v e c t o r

problem

h a s t h e s o l u t i o n a n d i t c a n be e x p r e s s e d 7(t)

may

i n accordance

1 - 3 ) , Cauchy's

functionals

independent,

the

latter

7 ( 0 ) a r e c o n c e n t r a t e a t t h e p o i n t cr^, t h e y

i n t h e form: J l V ( 0 ) = F (b(x)exp(-ix(T )

),

o

(1.98)

]

where b

(x)

i s a m-dimensional

polynomials of

theorem

vector

v a r i a b l e s x={x

on t h e r e a l

1.1, one c a n p r o v e t h a t

function,

elements

^,. . . ,x ^) .

from

Similar

(1.97) and

(1.98)

o f which

are

t o the proof i t follows:

I J V(t)=F {t7(x,t)),

(1.99)

>

J

w h e r e U(x,t)

i s a m-dimensional

According solutions

to of

functionals

lemma

the

1.1,

vector

the

homogeneous

J V ( t ) are

problem

linearly

function

vector

o f c l a s s M. J U(x,t)

functions

(1.1),

independent,

(1.7), equation

and

are

the

since

the

(1.99)

implies

1 that

the

independent.

vector-functions Theorem

1.2

U(x,t)

(j-1,2,...)

i s proved.

23

are

also

linearly

P r o o f of theorem at

infinite

1.3.

L e t n = l and L o p a t i n s k y ' s c o n d i t i o n be

numbers o f p o i n t s f ( J - 1 , 2 , . . . ) .

rank

We

shall

prove

that

.

homogeneous

latter

means:

j=l,2

< m,

fit,)

The

violated

(1.100)

(1.1),

problem

(1.7)

i n the class

h a s i n f i n i t e number o f l i n e a r l y i n d e p e n d e n t s o l u t i o n s . ) l l L e t a=(oc ,...,cc ) b e a n o n t r i v i a l s o l u t i o n o f a l g e b r a i c

K

system:

i

J (S -W(:|,))« B

) P(?j)« •

0.

(1.101)

0-

( 1 . 102)

J

o f v e c t o r a a r e complex numbers.

The e l e m e n t s While proving

J U(x,t) are

nontrivial

(1.103) under the

J = u(F ,t) expl-ixe ), i

i t follows

that

these

n = l and

i£

the class

M

problem

(1.69),

(1.69),

(1.70),

b

system

y

. As

we

(1.70) (1.72)

virtue

solvable

number

of

have

be

shall

shown

when p r o v i n g

i n t h e same

M*

with

independent

(1.70),

(1.72)

solutions.

at finite

number

(1.69),

t h e theorem

i s equivalent

So in

of

(1.70)

1.1,

t o the

the

problem

class.

i s violated,

(1.72)

(1.7)

(l.l),

independent

violated

From

independent.

solve t h e problem

i n the class

(1.103)

( c f (1.92)).

problem

6.6

at finite

and

6.7

r e s p e c t t o V(0)

solutions.

number

(chapter

and t h e c o r r e s p o n d i n g homogeneous

linearly

t h e system

we

o f theorems

(1.70),

always

of

number o f l i n e a r l y

functions:

.

(1.7)

are l i n e a r l y

t h e homogeneous

. First,

As L o p a t i n s k y c o n d i t i o n •--

(1.1),

solutions

Lopatinsky's condition

F

the vector

j-1,2,...

i

(1.100)

the condition

points

the

a

s o l u t i o n s of t h e problem

c l a s s M h a s an i n f i n i t e

Let

in

1.1 we h a v e shown t h a t

theorem

Hence

of points 6,

6.6)

section

i n the class

S'

system

finite

has

the general

a

is

solution

i s d e t e r m i n e d by t h e f o r m u l a : N 0

^

H0)=L+) k =1

24

k C^L,

(1.104)

where

L i s a

particular

solution

o f non-homogeneous

problem

[1.70),

k [1.72)

and

L

homogeneous complex Thus, (1.69),

( f t = l , . . . ,N)

problem

t h e problem

then

(1.69),

i n t h e class

Since t h e r i g h t

1.3),

(1.72),

independent

and

C

j f

solutions

C ,.•.,C

a

t

e

of the

arbitrary

constants.

(1.104)

(1.72),

are linearly

(1.70),

side

(1.70)

i s reduced

of the condition

according

t o theorem

Cauchy's p r o b l e m

t o Cauchy's

problem

M*.

(1.69),

1.5

(1.104) s a t i s f i e s

and t h e f o r m u l a

the equation

(1.66)

(section

(1.104) has t h e s o l u t i o n : N

7(t)-w({r,t) where (o(c,t} Let

i s the matrix

( i = F ( f (x) ) ,

where

of

(1.105)

(1.65).

f ( x ) sM .

I n Chapter

6,

section

6.6,

i t

i s

]

shown the

(theorem

system

6.S)

(1.70),

that

i n this

L=F(

where

case

the solution L

(j=0,...,W)

of

(1.72) can be r e p r e s e n t e d a s :

g j

(x)),

j=0,...,K,

(1.106)

g^fxJeM^.

Hence, t h e v e c t o r

f u n c t i o n u(cr,t)Z. ( j = 0 , . . . , N ) can be e x p r e s s e d a s :

] ) <ji(o-,t)L=F 'U(x,t))

,

x

j=0,

. . . ,K,

(1.107)

J

w h e r e U ( x , C ) a r e some m - d i m e n s i o n a l Thus,

the general

solution

vector

(1.105)

functions

o f t h e problem

o f the class (1.69),

M.

(1.104)

is:

7(t)

By v i r t u e

= F I u<x,t)

+\

o f lemma 1 . 1 , t h e v e c t o r

cu'x.t)

(1.108)

function: N

U(x,t)

= Btx t)*T

C U{X,t).

t

k

25

( 1 . 109)

is

the solution Thus,

of

points

(1.7)

F ,F , . . . t h e

complex

prove

linearly

now

(1.1),

(1.7).

Lopatinsky's condition

i s d e t e r m i n e d by

arbitrary We

of t h e problem

i f n = l and

the

general formula

is violated

solution

of

at finite

the

( 1 . 1 0 9 ) , where

number

problem

(1.1),

(ft=l,...,N)

are

numbers.

that

the vector

J U(x,t)

functions

independent, obviously,

N)

(j=0

from formula

(1.107)

we

are

have:

J J u ( o - , 0 ) L = F (U(JC,0)) .

(1.110) J 1

The

latter

and

the

equalities

u ( i r , 0) = u ( c )

and

k>(o~) L=L(

j = l , . . . ,N)

imply: J ) L • F^([7{x,0)),

j=l,

(1.111)

J

Since then

the vector

the

formula

(j=l,...,W) (1.104)

finite of

(1.111)

are also

1.6.

implies

linearly

in particular

Theorem

result

I f n=l

and

number o f p o i n t s ,

At

(1.72)

t h e end

independent class

s'.

that

are

the

linearly

vector

i n d e p e n d e n t . The

independent, ) U{x,t)

functions

latter

and

the formula

in Lopatinsky's condition

t h e number o f

t h e homogeneous p r o b l e m

number o f t h e l i n e a r l y (1.70),

L(j-1,...,N)

functionals

(1.1),

linearly

is

violated

independent

(1.7) i n t h e c l a s s M

at

a

solutions

i s equal t o the

i n d e p e n d e n t s o l u t i o n s o f t h e homogeneous

system

i n t h e c l a s s S' .

of Chapter solutions

H e r e we

6,

section

6.6,

we

find

o f t h e homogeneous s y s t e m

mean l i n e a r

independence

t h e number o f (1.70),

(1.72)

over t h e f i e l d s

of

linearly i n the complex

numbers. 1.5.

Examples

Example

1. L e t us

c o n s i d e r t h e system au — 1

at

dV + a — 'ex 3

of equations:

+ b U =0,jf<ER',t>0, 1

26

1

(1.112)

a v

au b U

2

2

Sx where to

U={0

a ,b

,b sf!

can

easily

J

2

2

1

solution

are given

be

found,

which

belongs

numbers.

that

t h e system

of equations

i s r e g u l a r i f and

i f : bjSO , b > 0 ,

a a ib

;

Let

t h e c o n d i t i o n (1.114)

of

the

we

must

impose one

Me

take

i t as:

system

(1.112),

be

boundary

«

and

(

function

from

Applying

i s uniquely

problem

(1.112),

solution. (1.115)

I f 0^=0,

has

We

0^*0,

w h e r e C>0 class

H.

We

take

and

lR

f(x)

is a

and

1.6

we

given

(1.7)

(1.113).

is a

given

obtain:

(1.112),

(1.115)

(1.113),

a ro ;

the

i n the class H

an

+

elliptic

one

^£

-

a

1

2 C

U

i s always

solvable

independent

(1.112),(1.113), solutions.

in half-plane:

xex',t>0,

= 0,

i n the

(1.116)

2 x

t o be

condition i n the

3

(1.115)

non-homogeneous

linearly

independent

equation

i s the solution

t h e boundary

f (x)

2

£

consider

fi(x,t)

to

(1.115)

a =0 t h e homogeneous p r o b l e m

4 * where

1.3

number o f l i n e a r l y

at

regularity

(1.112),

a +a^«0, a n d

homogeneous p r o b l e m has

S

of

according

t h e system

constants,

the problem

(1.113),

infinite

E x a m p l e 2.

Then t h e o r d e r

Therefore,

s o l v a b l e . I f 11^-0,0^*0,

the corresponding

(1.114)

o

1.1,

class M

.

M-

I f a^O

1.1.

i

0 ) = f (x) ,XER' ,

2

real

the class

theorems

Statement

i s one.

+ aU (x,

are given

;

c o n d i t i o n on

i

where

]

satisfied.

(1.113)

a U (x,0)

and

to

3

2

verify

(1.113)

flx

i s the

;

1

One

2

( x , t) , t ? ( x , T.))

(

t h e c l a s s M,

only

2

+ a — '=0,Jt6R', t > 0 ,

found,

which

form:

" ^ ° ' + pf7(X,0) = f (x) ,

function

from

belongs t o the

the

class

«R*, M,

a

(1.117, and

8

are

real

constants. The

equation

(1.116)

may

be

rewritten

27

as

t h e system

(1.1) u s i n g

the

following notation: Bjt t)-V(X,t)

,

r

The e q u a t i o n Applying

a

U (x,t)=

U

(1.116) i s r e g u l a r and i t s o r d e r

theorems

l.l,

1.3

and

1.6

t

^ '

2

)

•

of regularity

t o t h e problem

(1.116),

i s 1. (1.117}

we o b t a i n : Statement class

M

1.2.

The

i s always

solutions

of

non-homogeneous p r o b l e m

solvable

the

corresponding

(1.116),

(1.117)

N of linearly

a n d t h e number

homogeneous

problem

i n the

independent

i s defined

as

follows: N=0 a t a*Q,

B*C

B
o r a=0,

N = l a t a*0, B=C,

(1.118)

N=2 a t a = 0 , S i C . E x a m p l e 3. He c o n s i d e r t ^ ^ 2 u

Shi

at

ay

ax

w h e r e 00

a n d U(x,y,t)

the class

M.

The b o u n d a r y

f(x,y)

c

8t''

i s a

equation

*

U

,

t

i s the solution

c o n d i t i o n i s taken d U t ,

where

an e l l i p t i c

+

given

B t 7

i

y

)

^

i

t

>

0

(1.119)

i

t o be f o u n d ,

which

belongs t o

i n t h e form:

x

<
function

X

i n half-space:

= f

2

f*»7) '

from

(f.yjEK ,

the class

M

o

(1.120)

and

p

i s a

real

constant. Applying

theorems

1. l

and

1.2

t o t h e problem

(1.119),

(1.120)

we

obtain: Statement is

1.3.

uniquely

has

infinite

Examples x={x

if

. . . ,xj

I f e
solvable.

t h e problem

number o f l i n e a r l y 2

and

3

(1.119),

(1.120)

I f SiC, t h e c o r r e s p o n d i n q

show

has c r u c i a l

that

independent dimension

28

(1.1),

H

problem

solutions. of

spatial

i n f l u e n c e on t h e f i n i t e n e s s

s o l u t i o n s o f t h e homogeneous p r o b l e m

i n the class

homogeneous

(1.7).

variables

o f t h e number o f

CHAPTER 2

THE SYSTEM OF SINGULAR INTEGRAL EQUATIONS I N THE CLASS OF ANALYTIC FUNCTIONS.

L e t D* b e a f i n i t e sufficiency the

simply connected

domain i n t h e complex p l a n e

smooth boundary I " . The p o s i t i v e

one w h i c h

leaves

t h e domain

a l o n g r . F u r t h e r we i n t e g r a t e Definition

D* o n t h e l e f t

be an a n a l y t i c

is

an a n a l y t i c

vector-function

plz)-(p

function

i nthat

on contour

side

over T i n p o s i t i v e

when

with T i s

o n e moves

direction. (z) , . . . ,tp ( z ) )

i s said

i n domain D , i f i t ' s every

element

2.1. T h e v e c t o r - f u n c t i o n

to

direction

domain.

We c o n s i d e r t h e s y s t e m o f i n t e g r a l

equations:

r w h e r e K ( z , t ) i s a s g u a r e m a t r i x o f o r d e r n, t h e e l e m e n t s o f w h i c h a r e analytic

with

respect

domain

D*

( z ) ) i s an

t o be

vector-functions We s u p p o s e condition

Jp

We

also

t^t^eT,

from

i

;

t o t h e v a r i a b l e s z and t

£=£+11)),

given

analytic

that

i

2.3

a t zeD'ur,

Holder's tel",

i.e.

the inequality:

C

and a

a r e some

positive

constants,

and t . g

f (z)

a n d tp ( z )

satisfy

i f i t i snot specified

vector-functions w i l l

integral

fixed

i s a

n

where

z ,z ,t

suppose

section

f(z)=(f(z),...,f (z))

(z,C) s a t i s f y

D*ur. F u r t h e r m o r e ,

In

( a t every

vector-functions i nthe

t h e e l e m e n t s o f t h e m a t r i x JC(z,t) s a t i s f y

respect

z ,z eD'uI";

independent

found,

z

analytic

i n t h e d o m a i n D*.

that

with

elements K

the

at

t o complex v a r i a b l e

¥>(z) = (u> (a),

and

be c o n s i d e r e d

of t h i s

chapter

as

Holder's

condition i n

otherwise, a l l n-dimensional

vector-columns.

we

also

consider

t h e system

of

equations:

.

w

+

^

j M

m

+

^

r

j r c t i p y * r

29

=

f

(

z

)

,

( 2

.

2 )

where K ( z , t ) and W ( z , t ) of

which

areanalytic


the

n,

o f order

r e s p e c t t o complex v a r i a b l e

t h e elements

z i ndomain D ,

i s t h e v e c t o r - f u n c t i o n s t o be f o u n d ,

which i s

n

i n d o m a i n D*,

analytic p(z),

a r e square m a t r i c e s

with

f(z)=(f


(z),...,f

i s a complex

(z)) i sa given

conjugated

analytic

with

respect t o

vector-functions

i n

d o m a i n D'. Here

we

suppose

H(z,t),and

that

t h e elements

Holder's

matrices

K(z,t),

o f v e c t o r - f u n c t i o n s f ( z ) a n d p [z)

both

satisfy

c o n d i t i o n a t zeO u T;

The e q u a t i o n s

2.1. R e d u c t i o n

According

tel".

chapter w i l l

forelliptic

be a p p l i e d i n s t u d y o f t h e boundary

equations

o f Equation

Equation

of

( 2 . 1 ) and ( 2 . 2 ) a t f = 0 a r e s a i d t o be homogeneous.

The r e s u l t s o f t h i s value problem

t h e elements

(Chapter 5 ) .

(2.1) t o t h e Singular

Integral

i nt h e Class o f Holder

t o Sokhotzhy-Plemel's f o r m u l a ( [ 1 8 ] , p . 6 6 ) :

f

f zeD*, t s r . o

In the

equation

(2.3) and f u r t h e r

sense o f p r i n c i p a l

value

a l l singular integrals

([18],

p.50).

Proceeding

z-*t =£ +iii <=r, (zeD*) i n t h e b o t h s i d e s o f e q u a t i o n o

the

o

Q

formula

a r e assumed i n to thelimit a t

(2.1) and applying

( 2 . 3 ) we o b t a i n . f K(C , t ) < p ( t ) d t

(a+ f i c r v v M V

+

~

^

f

t

c

= < „' - V

r

-

2

4

! - '

t where E i s i d e n t i t y m a t r i x o f o r d e r n. Equation

(2.4) w i l l

be c o n s i d e r e d

i na wider

class o f functions:

, K ( t ,e)*(t)dt (E

+

^ivv)*cv

+

sir — ^

f t ), t T, (

o

o

(2.5,

r where found,

|K(t) = (^ ( t ) ,. .. , ^ ( 1 ) ) l

defined

on

i s n-dimensional

t h e contour

T,

30

v e c t o r - f u n c t i o n t o be

t h e elements

o f which

satisfy

Holder's

condition.

boundary

value

of

The d i f f e r e n c e i n equation

from

the class

f ( t

side

of

) of

Q

equation

equation

(2.5) i s t h e

(2.1)(of

analytic

(2.4) and

(2.5) t h e v e c t o r - f u n c t i o n of Holder,

o f any a n a l y t i c

( 2 . 5 ) i s due t o t h e f a c t

p

i s an a r b i t r a r y

i . e . i t i s not, i n general,

function

t h e boundary

vector-function.

have t h e f o l l o w i n g :

Theorem sense. value

2.1. Equations

( 2 . 1 ) and

I f p(z) i s the solution
of

analytic

( i n D')

equation Proof.

equation

(2.5),

(2.5) are e q u i v a l e n t

o f equation

(ter) i s the solution

solution

How

side

between equations

that

We

right right

f(t)).

vector-function

value

The the

(2.1),

of equation

then

boundary

which

is

in

following

i t s boundary

i f tli(t)

(2.5) and

i t i s the

v e c t o r - f u n c t i o n s
then

the

i s the

value

of

an

solution

of

(2.1). The f i r s t

l e t (fr(t)

be

part

o f t h e o r e m 2.1 f o l l o w s

the

solution

of

equation

from

the equality

(2.5).

We

(2.4).

consider

the

vector-function:

r and

denote t h e boundary value

Proceeding applying

to

the

the formula

limit

o f $ ( z ) a t z->t6r(zeD*) in

equation

(2.3) and e q u a t i o n *(t)=tf(t)

Substituting the

solution

Theorem if

the

2.1

implies:

equation

independent general

(H(t)=*(t)

of equation

0(t)=¥>(t), w h e r e 4)

the general

is

(2.7),

p ( z ) i s an

solution

and

(2.7)

i n ( 2 . 6 ) we

solvable,

equation

z->ter(zeD')

obtain:

obtain that

*(z) i s

Theorem 2.1 i s p r o v e d .

of equations

of

( 2 . 5 ) we

by * ( £ ) . at

at ter.

1) t h e e q u a t i o n

(2.5)

solutions

solution

from (2.1).

(2.6)

( 2 . 1 ) i s s o l v a b l e i f and 2)

the

(2.1) and

(2.5)

arbitrary

of equation

is

numbers

of

(2.5) a r e equal,

defined

solution

by

the

of equation

only

linearly 3) t h e formula (2.1) and

( 2 . 1 ) i s d e f i n e d by t h e f o r m u l a :

r w h e r e # ( t ) i s an a r b i t r a r y F u r t h e r , we w i l l

solution o f equation

see t h a t t h i s

equivalence

31

(2.5).

i s important not only f o r

analysis

o f equation

when t h e r i g h t class

2.2.

(2-1),

but also

side of the latter

( s a t i s f y i n q Holder's

f o r analysis

o f e q u a t i o n (2.5)

i s any v e c t o r - f u n c t i o n

of

Holder's

condition).

Analysis of Equation (2.1)

Together

with

equation

( 2 . 5 ) we c o n s i d e r t h e e q u a t i o n : cj(t)K(t,t )dt o

-o,

t «r,

(2.9)

o

r where

u ( t ) - ( u ( t ) .... , W ( t ) )

In

Holder's

equation

condition

(2.9) i s c a l l e d

respect t o equation us

on

the

system

(2.5) are

called

Q

should

condition From

be

noted

f o requation

(2.10)

equations

B(t )=

of

and

o

that

normal

[2.11)

i tfollows

(or satisfying

the

when

of function t

t^T.

o

(2.11)

i s also

(2-11)

the

normality

that

the normality

condition f o r

i s :

i t i s assumed t h a t

index

once, d i v i d e d

x-

type

o

condition

t er.

o

continuous

a(t)

on r w i l l

i s path-tracing

(2.12)

Q

the condition

H ( C ) be a c o m p l e x - v a l u e d The

o

(2.9).

(2.5) and (2.9)

Herein a f t e r

arga(t),

o

det((A(t )-B(t ))»0,

o

Let

with

(2.10)

|K(t ,t ).

o

d e t ( E + K ( t , t ))'0,

by

vector-line.

condition)([IB],p.507),if: det((A(t )+B(t ))*0,

tel".

found,

denote:

normality

Let

be

(2.5)([18],p.507).

0

It

to

a a s s o c i a t e d homogeneous e q u a t i o n

A(t l=E+ f K ( W , The

vector-function

r.

( 2 . 9 ) u ( t ) i s c o n s i d e r e d as a

The e q u a t i o n

Let

is

n

satisfying

(2.12)

function

issatisfied. on T and a ( t ) * 0 a t

be c a l l e d

t h e contour

r

t h e increment of

i n positive

direction

b y 2n.

us d e n o t e

t h e index

Normal-type

of function

equations

det(E+K(t,t))

(2.5) a r e c o m p l e t e l y

32

on t h e c o n t o u r T analyzed

i n [18]

(p.510), where t h e f o l l o w i n g 1)

The

number

homogeneous

of

equation

two

statements are

linearly

(2.5)

and

proved:

independent

the

number

solutions

of

linearly

s o l u t i o n s o f t h e homogeneous e q u a t i o n ( 2 . 9 ) a r e f i n i t e

of

and:

m -m =-x. o

2) N o n - h o m o q e n e o u s

equation

vector-function f ( t )

|

(2.13)

a

(2.5)

satisfies

the

independent

is

solvable

the following

i f and

only

i f the

condition:

j - 1 , 2 , . .. ,m' ,

cj(t)f (t)dt=0;

(2-14)

a

r I where

U ( t ) ( j = l , 2 , . . . ,ra' )

In

this

section,

linear

equivalence of

above m e n t i o n e d T h e o r e m 2.Z.

The

We u[z)

shall = (u (z) i

Proof. is

2.3.

The

i f vector

Lemma

equations

number m

(2.1)

and

of l i n e a r l y

o

(2.1)

and

and

linear

(2.5)

i n two

need t h e f o l l o w i n g the

independence

(theorem

independent m'

of

a

equation

satisfies respect

t o Cauchy's theorem

1 2m

and

solutions

linearly

(2.1)

the

of the

independent

and

i s solvable

the condition

(in

domain

the conditions to

2.1)

i f and

(2.14).

£>')

vector-function

e q u a t i o n (2.9) , t h e n cj(z)=0.

n

with

are

simple

analytic

, . . . ,ej (z) ) s a t i s f i e s Let u(z)

the

theorems.

t h e number

non-homogeneous

I f

of

numbers.

function f ( z ) satisfies

2,1,

analytic

accordinq

solutions

o f t h e homogeneous e q u a t i o n (2.9) a r e f i n i t e

Theorem only

independent

dependence

o f complex

statements r e s u l t

homogeneous e q u a t i o n solutions

linearly

(2.9).

assumed o v e r t h e f i e l d The

are

g

homogeneous e q u a t i o n

complex

of

lemma 2 . 1 .

variable

z

since

i n domain

K(z,t)

D' ,

then

([13],p.45):

M(t)K(t,tJcK J

t-z

- =0,

VzeD ,

(2.16)

r w h e r e D~ i s t h e c o m p l e m e n t o f D*u Proceeding

to

the

limit

at

T t o t h e whole

z-»t

(zeD~)

33

complex p l a n e .

and

applying

Sokhotzky-

Plemelj's

formula ([18],p.66)

we

obtain

D(t)K(t,t ) d t +

- ^ ' . i f t ' o - V

lib-

-

t

0

;

t

r

2

V -

' -

1 7

'

r Comparing

(2.9)

and

(2.17),

we

find

t d ( t ) =0, t e r . Q

Lemma

o

2.1

is

proved. Let

us c o n s i d e r t h e f u n c t i o n a l l (f)=

(see

(2.14)):

| o(t)f(t)dt,

j

j = l

m;.

(2.18)

r The

functionals

I

Holder's class.

We

as f u n c t i o n a l s

over

are

shall

linearly

prove t h a t

the class

independent

as

functionals

they are also l i n e a r l y

of analytic

boundary v a l u e s b e l o n g i n g t o t h e Holder

over

independent

vector-functions

i n D*

with

class.

Let

C

i V

f

)

+

C

A

(

f

)

+

---+

5

W ^

Substituting

from

(2.IB) i n t o

|

0

"

G

'

V f

-

(2-19)

0

( 2 . 1 9 ) , we

have:

U>(C)f(t)d
(2.20)

r where l - S j i U t t ) +...+

u(t) 1

As

(2.20)

(2.21)

o

k

k

u ( t ) ,. . . , u ( t )

also

h C u ( t ) , H=m' .

the

are the solutions

solution

of

this

of equation

equation.

In

(2.9),

particular,

then tj(t) i s from

equation

and

applying

i t follows:

r Proceeding

to

the

limit

at

S o k h o t z k y - P l e m e l j ' s f o r m u l a , we

z-it^r

(zcD~)

in

(2.22)

obtain:

" * V •lb J

34

"

S

3

?

t

a

.

»

i

We

( i n d o m a i n D')

consider the analytic

vector-function:

r Let

us

denote

Proceeding

t h e boundary

to

the

values

limit

at

Sokhotzky-Plemelj's formula,we

of

t ( z ) a t z->t er,zeD*

by

o

z^t (zeD*)

in

o

(2.24)

and

i(t ). o

applying

obtain:

r Comparing

(2.23)

and

(2.25)

one

*(t )

- u(t )

o

Hence, ( J ( t )

finds: , t er.

Q

i s t h e boundary

(2.26)

o

value of a n a l y t i c vector-function * ( z ) . 1

According

t o lemma

independent, condition

then

(2.14)

2.1 u ( t ) = 0

C =...-C^-0

(k=m^).

i

are linearly

k

a n d s i n c e d>(t) , . . . , u ( t ) Thus,

we

are

proved

linearly that

the

independent.

2.3. A n a l y s i s o f t h e E q u a t i o n ( 2 . 2 )

Equation here of

(2-2) should

we w i l l

be

analyzed

formulate the results

similarly

(2.1),

so

moments

proofs f o r the statements. Here,

linear

(2,2)

and t h a t

(2.2)

will

(2.2)

independence

t o the limit

and a p p l y i n g

E»

of the solutions

of solvability

conditions

0

of real

(zeD*)

i n both

S o k h o t z k y - P l e m e l j ' s f o r m u l a , we

1 tK(t .t ))Kt ) 0

a t z^tt^r

0

1 + £j

f K(t ,t)V»(t)dt —

[

suppose

2T7I |

equation

numbers. sides

of

equation

obtain:

0

c

r +

o f homogeneous

f o r non-homogeneous e q u a t i o n

be a s s u m e d t o be o v e r t h e f i e l d

Proceeding

We

t o equation

and p o i n t o u t t h e e s s e n t i a l

M(t.E)V(t)dt -t^l

= < >' f

that:

35

C

0

+ i«(t ,t ) (t ) o

o

P

0

+

det (E+K(t ,C >)*0, o

This (Cf.

condition

C er.

o

i s the condition

of

(2.28}

o

normality

f o r equation

(2.27)

[ 1 4 ] , p.275).

Equation f(t )

(2.27)

will

i s the right

o

The

be

solved

i n Holder's

side of equation

normal-type

equations

classes,

assuming

that

(2.2).

(2.27)

i s completely

analyzed

i n

[ 14 ]

(pp.273-278). As f o r e q u a t i o n ( 2 . 1 ) , are

one c a n p r o v e t h a t

e q u i v a l e n t a n d t h e o r e m 2.1 i s s a t i s f i e d

(2.1)

being

replaced

by

the equation

equations

(2.2) and

(2.2) and

t h e e q u a t i o n (2.5)

b e i n g r e p l a c e d b y t h e e q u a t i o n ( 2 . 2 7 ) when f o r m u l a t i n g We

consider

together

with

the

(2.27)

f o r t h e them, t h e e q u a t i o n

equation

this

(2.27)

theorem.

the

following

equation: ,

,

i

f u(t)K(t,t )dt 0

f a(t )

[ u(t)«(t, t j d t

Q

2ni

I

= ° .' t„eT, a

t-t

(2.29)

r where

a ( r ) = l i m ^-^—, t«=r i-iZ

and

u ( t ) = (u ( t } ,...,u ( t ) ) t

(vector-line),

The e q u a t i o n ( 2 . 2 9 ) equation If

is

n

satisfying

(2.27)

([14],

vector-function

Holder's condition

i s called

(2.30)

t-T

to

be

found

on r .

associated equation t o the

p.275).

the normality condition

(2.28)

i s satisfied,

i t i s shown i n ( [ 1 4 ] ,

p.276) t h a t one can p r o v e t h e f o l l o w i n g t w o s t a t e m e n t s : 1)

The

equation of

number

k

(2.27)

and t h e number

o

of linearly

independent k'

of

o

solutions

linearly

t h e a s s o c i a t e d e q u a t i o n (2.29) a r e f i n i t e

The

non-homogeneous

equation

(2.27)

36

homogeneous solutions

and

where x i s t h e i n d e x o f t h e f u n c t i o n d e t (f?+K(t, t ) ) 2)

of

independent

o n F.

i s solvable

i f and

only i f

the vector

function

f(t)

satisfies

the condition:

j

Be

fc)(t)/(t)dt-0,

(2.31)

r i where

u ( t ) (J=l,...,k' )

are

g

associated The

equation

linearly

independent

solutions

of

the

(2.29).

equivalence

of

equations

(2.2)

and

(2.27) ,

and

the

above-mentioned statements r e s u l t i n : Theorem

2.4.

The

number

homogeneous e q u a t i o n solutions

of

the

non-homogeneous satisfies

2.4.

Let

equation

equation

the condition

Efficient

D'

be

containing

h

of

Q

(2.2) and

linearly

independent

t h e n u m b e r k'

(2.27)

(2.2)

are

is

of

finite,

solvable

solutions

linearly

k -k'=-2x-

and

i f and

of

independent

only

The

i ff ( z )

(2.31).

Method f o r S o l v i n g E q u a t i o n ( 2 . 1 )

a

simply

the origin

connected

domain

with

o f c o - o r d i n a t e s . L e t us

smooth

boundary

consider the

T

following

integral equation:

. 2Si

z

*< >-

f K ( z , t ) (p{£)dt t-«(z)

- i, hi J K (Z,t)«(t)l (l- f ) d t = f ( z ) , 2

r where

K (z,t)

complex

function

with

logarithmic

We

JC (z,t) ?

z

and

respect

t

are at

Holder's function,

a t every f i x e d

(2.32)

condition which

and

functions teD

variables

and on

D*ur,

t o be f o u n d , a n a l y t i c on

D*ur,

i s analytic

In(1-2) with

with

respect

satisfying

is

to

Holder's

f(z) is a

given

i n d o m a i n D* a n d that

respect t o z

branch

of

( i n domain

t e r and e q u a l t o z e r o a t z=0.

impose f o l l o w i n g

D* a n d s a t i s f i e s

analytic zeD

t o both

and (>(z) i s a f u n c t i o n

satisfying

D*)

and

variables

condition

n

r

restrictions

Holder's condition

on c t ( z ) :

ot(z) i s a n a l y t i c

i n domain

on D*u f , a n d t h e v a l u e s a ( z ) a r e :

a(z)eD'

at

zeD*u T,

(2.33)

[^|^-ja7
at

Z S D \ J T,

(2.34)

37

where g i s a It

equation

From

2.1

Since

class.

condition

K^fz, t ) p ( t )

complex

variable

formula

([13],

i t is a

(2.1) t o s i n g u l a r

inteqral

t o zero.

indicate

(2.32),

which

solvability

another,

allows a in

a

more

necessary

terms

of

the

are analytic

2

a t every

fixed

zeD*,

i n D* w i t h

then

according

respect t o to

Cauchy's

p.54):

Substituting

J — r

t=ct(z)

Integrating

both

f f

sides

|

e

D

( 2 . 3 5 ) , we

(«(z»- 35! J r of

0 a n d z a t j - 2 , we

Z

|Z£

i n t o formula

V*,«(2))

(2.35)

g-

with

'

C

e

D

' J

= 1

2

' -

2

< -

3 5

)

obtain:

.

a ( z )

respect

to

(2-36)

C

between

the

find:

K (z,C)»>(t)dt= g i j J K ^ t ^ t J p f t J I n f l -

|)dt.

(2.37)

| K ( z , t ) i o ( t ) d t - f ( z ) , z«D*,

(2.38)

2

o Hence,

case of

that

shall

unique

particular

obtained.

and K ( z , t ) * ( t )

t

K^z,t)p(t)=

limits

we

equation of

is a

i t follows

equation

Here

c o e f f i c i e n t s t o be

(2.32) (2.33)

i s equal

have reduced

method f o r s o l v i n g

sufficient

equation

equation

and i t s i n d e x

we

i n Holder's

efficient

that

the condition

equation

section

equation

and

to verify

(2.1).

normal-type In

constant.

i s easy

r

equation

(2.32)

may

be r e w r i t t e n

as:

i

ip(z)-B(z)p(a(Z))-z"'

2

0

where S[z) It

i s clear

The s o l u t i o n

= R (z,«(Z)).

t h a t |3(z) i s a n a n a l y t i c of equation

(2.38)

(2.39,

(

function

i s sought

38

i n domain

D*.

i n the form o f :

E

C J

jfz

+ z"u>(z) ,

(2.40)

] =o w h e r e C ,.-.,C o

satisfying chosen

ui

are constants,

Holder' s

u ( z ) i s an a n a l y t i c o n D*<J I " , m i s a

condition

( i n D*)

natural

function,

number

t o be

later.

Substituting

tp(z)

from

(2.40)

i n t o equation

(2.33)

we

obtain:

z z ( J ( z ) - S ( z ) (n(z))° (a(z) ) - z"' JjC ( Z , t ) c'u (t) d t = t l ( z ) , N

tl)

z

(2.41)

where

£![z)=f(z)-^

j f z ' + f3(z)^

J -O

In

equation

C ,...,C

-jllatzD'+z-'^

J-0

(2.41)

a r e t o be

From t h e c o n d i t i o n

J =0

the analytic

Performing

(2.34)

(2.42)

0

function

u ( z ) and

the

constants

i t follows: | '<0)|sg.

d i f f e r e n t i a t i o n on

both

( J

(2.42)

sides

of

equation

a n d s u b s t i t u t i n g z = o , we

f! '(0)=0, fi(z) from

(2.43)

Q

t o z up t o ( m - l ) - t h order

Substitutinq

( z , £.) F d£.

found.

a(0)=0,

respect

J

j {|

j-0,1, into

,10-1.

( 2 . 4 4 ) , we

obtain: (2.45)

z

Cjjl-eiOJ^'IO))

1

-

K

^

C

)

) +

^d

i

p

with

obtain: (2-44)

C ( l - f i ( 0 ) - K ( 0 , 0 ) ) = f (0) , o

(2.41)

C=f

,

J

,

(0),

(2.46)

j=l,...,m-i, where d Let

a r e some w e l l - k n o w n c o n s t a n t s . ip the following conditions: 1 - 6 ( 0 ) (n- ( 0 ) )

j

K (0,0) j ^ — *0, j = 0 , l , . . .

39

(2.47)

be s a t i s f i e d . Then the

t h e system

solutions

function From

of

(2.45), the

(2.46)

system

i s uniquely

into

(2.42)

we

solvable.

shall

Substituting

find

the

analytic

fl(z) .

(2.44)

i t follows: £l{2)=z"£l (Z) ,

(2.48)

o

where

fi (z)

i s analytic

0

condition

function

i n d o m a i n D*

which

satisfies

Holder's

on D'<J r .

Substituting

tl(z) from

sides of equation

(2.48)

(2.41)

i n t o e q u a t i o n (2.41)

b y z", we

obtain:

<j(z)=L(u)

+ Q (z),

and d i v i d i n g

both

(2-4°)

where z 5

L(u)=B(z) f ^ ^ - ]

"(f(z))

+ z~"~' | K ( Z , t ) t " u ( t ) d t .

(2.50)

a

o If

the solution

domain

Dur,

u(z) of equation

then

satisfies

Holder's

satisfies

Holder's

equation in

(2.49)

recallinq

(2.50)

the

on

D-'uP,

condition condition

on

D*ur.

i s continuous

riqht we So

side

find

of

that

i n the closed this

equation

this

solution

i t i s sufficient to

i n the class of analytic

c l o s e d d o m a i n D*ur From

(2.49)

that

i n domain

D*

and

solve

continuous

functions.

and t h e c o n d i t i o n

(2.34),

i t follows:

II

I L ( u i ) |s

z

I

m+1

(2.51)

where ||= max l f i ( z ) l , ZeD ur

|(j||=

+

max l w ( z ) I , zi=D*vr

(2.52)

and ||K || = max z.teD'ur We

choose t h e number m

IK ( z , £) | .

satisfy the inequality:

40

(2.53)

(2.54)

This and

choice provides

thus

equation

iteration

t h e norm o f t h e o p e r a t o r L ( u ) b e i n g

(2.49)

i s uniquely

solvable

a n d may

less than 1

be

solved

by

method([19],p.47): z

c j ( Z ) = n ( z ) + L ( f i ) + L < C y + ..., 0

(2.55)

o

w h e r e L ' ( D ) means t h a t t h e o p e r a t o r L i s a p p l i e d t o £i k t i m e s . d

The If

o

s e r i e s (2.55) m

satisfies

satisfied

Substituting

we

solvability It

(2.54) , t h e n

t h e s o l u t i o n s C , . . . ,C

of equation

have

shown

of equation

progression.

t h e c o n d i t i o n (2.47)

is

only f o r j=0,1,...,m-l.

of

the algebraic

and t h e s o l u t i o n u ( z ) o f e q u a t i o n

the solution

Hence,

the condition

geometric

f o r jwtt, so i t needs t o be v e r i f i e d

(2.45),(2.46) find

converges as a decreasing

(2.49)

into

equations ( 2 . 4 0 ) , we

(2.38).

that

the condition

(2.47)

provides

unique

(2.38).

i s easy t o see t h a t t h e c o n d i t i o n (2.47)

i s satisfied, i f :

|JC ( 0 , 0 ) l+IK ( 0 , 0) < 1 ,

(2.56)

]

or t c ' ( 0 ) = 0 , K ( 0 , O ) + K ( 0 , 0 ) * l , K (0,0)«2,3 t

Hence,

From

(2.32)

described

o r (2.57)

i s satisfied,

then

i s uniquely solvable.

method

t h e c o n d i t i o n (2.47)

homogeneous e q u a t i o n

(2.57)

2

i f one o f t h e c o n d i t i o n s (2.56)

the equation

if

2

f o r solving i sviolated

(2.32)

equation a t least

has n o n - t r i v i a l

(2.32)

i tfollows that,

f o r one i n d e x j , t h e n t h e solution.

So, we o b t a i n t h e f o l l o w i n g : Theorem conditions

2.5.

2.5. E q u a t i o n (2.47)

Reduction to

i s u n i q u e l y s o l v a b l e i f and o n l y i f t h e

o f Some C l a s s e s o f S i n g u l a r

Equations

Integral

Equations

( 2 . 1 ) and ( 2 . 2 )

L e t D* be a f i n i t e sufficiently

(2.32)

are satisfied.

smooth

simply

connected

boundary T

domain i n t h e complex plane

a n d D~ b e t h e c o m p l e m e n t

41

o f D*u

with T i n

the

complex

plane.

Let us c o n s i d e r t h e f o l l o w i n g

integral

equations:

K(t„,t)»»(t)dt (

E +

3

*

(

*

„

+

.

2^T

w(t +

&

t)^(t)dt

— ^ r ^ t -

r where

g ( t > = (£/, ( t )

(tfi ( t ) , . . . , | t t condition

(t))

matrices

(2.1);

(2.27)

i s due

arbitrary

a

T.

a

given

I n equation

t o the fact

5 9 )

function

and

satisfying

(2.58), K(z,t)

0(E) = Holder's

i s the matrix

( 2 . 5 9 ) , K ( z , t ) and H ( z , t ) a r e t h e

(2.58),

that

(2.59)

the right

s a t i s f y i n g Holder's

H e r e we s u p p o s e t h a t

sides

and e q u a t i o n

(2.5),

of t h e former

are

c o n d i t i o n o n T.

the normality condition: d e t [E + K ( t , E ) )»0, t <Er n

is

-

(2.2).

between e q u a t i o n s

functions,

f 2

« V <

vector-function

and i n e q u a t i o n

from equation

The d i f f e r e n c e

is

n

known

on t h e c o n t o u r

from equation

= * t V * 0

,g (t)) is

= g(£ ) , t e r , ( 2 . 5 8 ) =" o " 0

t-t

0

(2.60)

Q

fulfilled. We

shall

equations

show

that

( 2 . 1 ) and

equations

because

some

previous

s e c t i o n , may b e s o l v e d

First, !&(t)

classes

l e t us

of equation

(2.58)

of

consider (2.58)

equation

analytic and

equation

as

(2.58)

and

and

respectively,

reduced

shown

represent

the

i n the

solution

(2.61)

I

X (z) )

are

I0(t) from

a n d D~ r e s p e c t i v e l y , ( p ( t )

o f p ( z ) a n d X ( z ) a t z ^ t e r , zeD* a n d

a n d X(«)= l i n t X ( z ) . (2.61)

the

n

I I I -Mo

Substitutinq

to

i s important

i t was

X(z)=(X,(z)

v e c t o r - f u n c t i o n s i n t h e domains D

zeD

be

, ter,

n

X ( t ) are t h e boundary values

z-ittF,

(2.1),

may

reduction

a s ( [ 1 8 ] ,p. 136) :


(2.59)

This

efficiently.

*(t)=(p(t)-X(t) where

and

(2.2) r e s p e c t i v e l y .

into equation

42

(2.58)

we o b t a i n :

(E

|K(t ,t })( (t )-X(t »

+

0

o

V

0

o

j i j

+

— ?

^

=g{E ) o

(2.62)

r Let

us

denote:

j

, *(z)=¥>(z) +

, K(z,t){(p(t)-Jt(t))dC

JJJJ

, zeD

J - J

.

(2.63)

r Let the

t h e b o u n d a r y v a l u e o f
# ( t ) be limit

at

z-*t e r ,

ZED*

Q

Plemelj's

formula (2.3),

#(t )-»(t

o ) +

0

we

in

a t z-»ter, zeD*. P r o c e e d i n g

(2.63)

obtain

and

applying

to

Sokhotzky-

: f K(t — 2

x t t ^ t ^ ^ g - K g ) ^

t ) (
(2.64)

r H e n c e , e q u a t i o n ( 2 . 6 2 ) may

be

rewritten

as:

0(t )-X(t )=g(t ), n

T h u s , we analytic the

have a p r o b l e m

0(z)

t er

o

of conjuqation

vector-functions

formula

n

and

(2.65)

o

(2.65)

Jt(z).

The

f o r the d e f i n i t i o n

solution

i s defined

of by

([IS],p.136): 0(z)

= G ( z ) , zeD",

X(z)

-

G(z),

(2.66)

z<=D~,

(2.67)

where

r Proceeding

to

the

limit

at

z-+t er(zen )

in

o

formula

(2.67),

we

obtain: X(t , 0

fat,H.

- 0

&

j

,2.69,

f Substituting

*(Z)

and

X(C)

from

(2.66)

and

(2.69)

into

(2.63) ,

we

have: p ( z )

^

[

^

W

^

=

43

f

(

z

l

,

, * l f ,

(2.70,

where

T h u s we

obtain

Theorem

2.6.

variable defined

z

I f the

i n domain

by t h e f o r m u l a

matrix

D*,

fC(z, t )

the general

is

analytic

solution

with

respect

to

(2.58)

i s

of equation

: *(t)-*>(t)-G"(t),

where ^ i ( z ) i s any s o l u t i o n o f e q u a t i o n vector-function

i n domain

D°

and

(2.72)

(2.70)

G~(t)

i n the class of

i s defined

by

analytic

the

formula

(2.69) . Hence,

the solution

equation How I/I(Z)

of equation

(2.58)

i s reduced t o t h e s o l u t i o n

of

(2.1) .

we

shall

from

reduce equation

(2.61)

into

(2.59)

equation

j (£+ i ( t , £ ) ) ( H t ) - X ( t ) ) K

p

o

0

t o equation

( 2 . 5 9 ) , we

o

+

( Kit — 2

^

(2.2).

Substituting

obtain : , t ) «p(C>-*(C))dt —

+

r +

3H(t ,t )(p(t )-X(t )) o

o

o

0

+

f w(t ,t) ( (t)-JC(t))dt P

+

2WT

—

E=E

- f < V -

r

2

V "

< -™>

IL e t us

* ( z ) ^ ( z

denote:

)

+

I

;

F

I

, K(z,t) (?(t)-X(t))dt j ^ —

j

+

r Proceeding

, M(z,t) ( ( j ( t ) - X ( t ) ) d t ^ (2.74)

r

to

the

Sokhotzky-Plemelj's 1

limit

at

z^t sr, o

zeD*

in

(2,74)

and

applying

f o r m u l a , we o b t a i n : ,

i

44

f K(t

t) ( (t)-X(t))dt V

!

,

.

S ^ W ("'V-^^D

+

Hence, e q u a t i o n

(2.73)

Similarly,

+

c

5ffi J_ —

f=£;

2

as :

*(t )-Jt(t )-tr(t ),

t

o

(
f «lt ,t)

may be r e w r i t t e n

0

#(Z)

,

o

f l 6

•

r.

(2.76)

we o b t a i n t h a t X ( t ) i s d e f i n e d b y t h e f o r m u l a

(2.69)

o

i s the solution

1

l

i

of equation

J

2ni J

(2.2),

t-z

(^-")

and

where:

2nT J

T

t-z

r T h u s we

obtain:

Theorem

2.7.

I f the matrices

respect

t o variable

(2.59)

i s defined

solution

of

formulae

(2.69)

2.6.

Let

D°

be

by

formula

K(z,t) D',

and W ( z , t )

the general

(2.72),

( 2 . 2 ) and

where

f ( z ) and

of Functions

C o n n e c t e d Domains t h r o u g h i n the Unit

a

simply

|tl»l o n t h e c o n t o u r

i s an

with

equation arbitrary by t h e

which

are Analytic

Functions

which a r e

circle

connected

domain

with

sufficiently

of the unit

smooth

circumference

r , preserving the orientation.

denote:

a-,

It

p(z)

of

G~(t) are defined

b o u n d a r y T. L e t z = a ( t ) be o n e - t o - o n e m a p p i n g

Let us

are analytic

solution

and ( 2 . 7 7 ) .

Representation

Simply

Analytic

i n domain

equation

Integral in

z

i s supposed

that

t )

=lim T-tl

SLi|ilSLlii,

a(t)satisfies

|t|-|t|-i.

Holder's

condition

a n d a'(t)«0 a t

I t 1 = 1. Let circle

t=R(z) IEI<1.

be t h e c o n f o r m a l Then,

mapping

obviously, every

45

o f t h e domain

analytic

D°

function

onto

the unit

p ( z ) i n domain

D*

represented

as: (2.78)

w h e r e iC(t) i s an In

the

analytic

representation

involved

which

indicate

another

in

the

unit

z-a(t) will

of

be

We

used

Theorem

then

the

function,

curve

[".

i n the

of

be

circle.

expressed

analytic

only

These

unit

conformal mapping

cannot

representation

i n Chapter

function

involving

the

representations

function

explicitly.

of

(p(z)

by

S (z)

is

Here

we

analytic

parametric

equation

analytic

functions

5.

following:

2.8.

D*and

i t may

general,

circle

the

have t h e

domain

in

function (2.78)

I f

(P(z)

is

satisfies be

function

Holder's

represented

analytic

condition

in

in

the

a

simply

closed

connected

domain

D*u

r

as:

(2.79)

where

Z=Q (t)

function the

closed

Proof. i/i(t), is

is

the

analytic

unit circle

Let

which

parametric

i n the

us is

circle

Itl^l

solve

and

equation

in

the

the

curve

s a t i s f y i n g Holder's

#(t)

i s d e f i n e d by

equation

analytic

of

ltl
(2.79) unit

with

circle

(o(z)

respect |tl
T,

0 (t)

is

condition

a in

uniquely

to

the

assuming

function z

that (J( )

known. By

substituting

£=a(t) i n

(2.79),

we

obtain:

(2.80)

r w h e r e 0(£) According

i s the

inverse

function

t o Cauchy's f o r m u l a

of o ( t ) .

( [ I B ] , p . 5 4 ) , we

have: (2.81)

r Substituting

p(z)

from

(2.81)

into

(2.80),

we

obtain (2.82)

f In

[18] (p. 133),

i t is

shown

that

the

equality

(2.82)

is

true

for

every

point

zeD*

i f and o n l y

if: (*<«)-*, ( 5 ) .

where 0 ( z ) i s an a n a l y t i c ]

Let

z=u(C)

be

t h e d o m a i n D*. represented

mapping

( z ) i s an a n a l y t i c i*(t)

from

i s clear,

that

o n e - t o - o n e mapper In

l&(t)

(2.85)

every

( 2 . 7 9 ) . Theorem More

onto 111<1

the

general

(2.84)

function.

i n ( 2 . 8 3 ) we

obtain:

t.CW3»{5)>1 "

e r

- ? -

a

function

z=w(S ( O )

maps

(2.85)

T

onto

itself

in

and p r e s e r v i n g o r i e n t a t i o n .

has unique

from t h i s

formulated

ltl
0 ( t ) i n domain

2

i n D* d o m a i n

"

[ 1 8 ) ( p . 5 7 3 ) , i t was

Hence,

l

= « (u(t) ) ,

(2.84)

MO

shift

function

4 {*)=0. circle

as:

Substituting

It

of the unit

Then, o b v i o u s l y , a n a l y t i c

iMt) where

i n d o m a i n D~ a n d

function

the conformal

(2.83)

fact

proved solution.

We

the problem obtain

of conjugation with

unknown

analytic

function

and t h e r e p r e s e n t a t i o n ( 2 . 8 4 ) .

function 2.8

that

p (z)

i s represented

uniquely

by

the

formula

i s proved.

integral

representations

i n Ref.[20].

47

of

the

form

(2.79)

are

CHAPTER 3

ASYMPTOTIC FORMULAS FOR SOLUTION OF MAXWELL'S EQUATIONS AND THE LAWS OF PROPAGATION OF ELECTROMAGNETIC ENERGY AT GREAT DISTANCES

3.1.

Cauchy P r o b l e m

Let

us w r i t e

complete partial

f o r Maxwell's

t h e system

system

E q u a t i o n s System

o f mathematical statements representing t h e

of electromagnetic

differential

theory

equations, i . e . well-known

equations o f Maxwell's

([15],

pp.289-293, [ 1 6 ] ,

p.27) : rotH=|?+J, dicD=p,

rotE=-|5, and

algebraic

(3.1)

div-B=0,

(3-2)

relations: D-ee E, B=UU H a n d J = c E , o

where E = E ( r , t ) i s i n t e n s i t y of

point

(x,y,z),

i s a magnetic

H=H(?,t)

i sintensity

D=D(r,t)

i s an e l e c t r i c

c (i

i stheelectric

o

o f an e l e c t r i c

field,

i s t h e speed

induction, o f a magnetic f i e l d

constant,

of light

,

induction, e -10 o

/ 4 n c = 8 . 8 5 1 -10"', 2

7

i n vacuum,

i s t h e m a g n e t i c c o n s t a n t , |i =4rr-10 '=1.257*10

o

o

J

i s a conductance

p

i s a charge

c

i sa dielectric

u

i sa magnetic

current

constant,

permeability, electric

medium i s i s o t o p i c

The

main

goal

a t great

Henry/meter

density,

conductivity.

V a l u e s H,D,J,E a n d B a r e v e c t o r s

fields

6

density,

cr i s a s p e c i f i c

that

? i sa radius-vector

t i s time,

B-B(r,t)

c

(3.3)

Q

of this

chapter

values of t

energy over a g r e a t

i n 3 - d i m e n s i o n a l space.

We

suppose

a n d h o m o g e n e o u s , t h u s JJ, c a n d tr a r e s c a l a r . i s t o study

and t o o b t a i n

distance.

49

t h e behavior

o f these

t h e law o f propagation o f

Taking

equation

(3.3) i n t o

account,

t h e system

(3.1),

( 3 . 2 ) may b e

w r i t t e n as: 3

| | = -roCE, d i r B = 0 , ^

_

a

r o t

t i O , (x,y,Z)eR ,

BE, t s O , ( x , y , z ) e R

B

(3--1)

,

(3.5)

p«ee d i r E ,

(3.6)

where EE UU 0 O To to

obtain thecharacteristics

consider Let

M

t h e system

and M

be

Q

(3.4),

and

8 =

EC

(3.7) 0

o f electromagnetic

field

i t i s enough

(3.5) o n l y .

t h e classes

of functions

defined

i n Chapter

1

(section 1.1)i The s o l u t i o n class Let The

o f Maxwell's equation

B-(B (r,t),B (?,t),B

(?,t))

B

initial

i n the

(r\t)).

B (?,0)=f

,

E (?,0)=f (x),

(3.9)

,

E (?,0)=g (x),

(3.10)

i

(X)

^ (x) ,

f (x) j

]

a n d g^ ( x )

t=0 into

the conditions

(3.8)

0 (x)=const

y

f o r Maxwell's

2

are given

t o (3.10)

equations

functions

t o theclass

equation

i s necessary

]

2

o f t h e considered

t h e second

we

depending

M. Q

Cauchy's

p r o b l e m . The

o f (3.4) and t a k i n g obtain

—

—

f o rsolvability system

(3.8)

2

i

( j = l , 2)

B a n d E be t h e s o l u t i o n

substituting

0)-* {x),

t

o n l y on t h e v a r i a b l e x and b e l o n g i n g

problem

(?,t),E

E tf,

I

condition

E = ( ( E (?,£),E

and

,

L

B (?,0)=g (x)

of

be s o u g h t

B (?,0)=« (X)

! f

Let

(3.5) w i l l

C a u c h y ' s c o n d i t i o n s a r e t o be t a k e n a s :

) [

where

(3.4),

o f f u n c t i o n s H.

account

= 0 . Hence, t h e o f t h e Cauchy's

(3.4),{3.5)

t o be s o l v a b l e .

Thus, t h e c o n d i t i o n (3.8) must be t a k e n a s : B (?.0)= r

where

i sa given

Cauchy's zero

problem

initial

data

real

C , o

B^(?,aJ- * ( x ) ,

(3.11)

2

constant.

f o rMaxwell's ( a t C =0, q

equations

0.,(X)=O,

50

system

f (x)=0, }

(3.4) ,

g (X)*0,

(3.5) with

j=l,2)

will

be

called

a homogeneous

problem.

One h a s t h e f o l l o w i n g : Theorem the

3 . 1 . Cauchy p r o b l e m

initial

d e p e n d s on x a n d t Proof. Remark seek

the

system

(3.4),

1 . 1 ) . Now

(3.5) with

and t h e s o l u t i o n

only.

The u n i q u e n e s s we

(3.4),

i s established

shall

i t i n the class

equations

f o r Maxwell's

data o f (3.9)-(3.11) i s uniquely solvable

prove

of functions

(3.5) and

i n Chapter

l (section

1.3)

existence of the solution. on x

dependinq

t h e boundary

and

conditions

shall

t only.

Then

(3.6)—(3.10) i n

C a r t e s i a n c o - o r d i n a t e s take t h e form: BE ^

= -BEJx.t)

,

(3.12)

E (x,O)=0 (x),

(3.13)

B

8B

2

(x,tj

SB

—

{X,t)

=°<

—

3

=°<

f -

(x,0)=C , SB

BE {x,t)

(x,t)

3E ( x , t )

ax

at

'

BJx,Q)=f

aB (x,t) at

The

solution

of

defined

(3-17)

a

, E J x , 0)=gjx)

t h e problem

. The s o l u t i o n

(3.12)-(3.19)

o f t h e problem

'

(3.16)

,

aE ( X , t ) SB ( x , t •" a t " ~ —ax

BE ( x , t )

—bx

BJx,0)=gJx)

-»<x<+»

BEJx.t),

ax

(x) , EJx,0)=£Jx)

i

4

(x,t)

SB =

at

1

(3 . 15)

0

is

(cf.

We

eejx.t), B^

(3.is)

y

.

(3.19)

i s sought

at

£>0

( 3 . 1 2 ) , (3.13) and ( 3 . 1 4 ) ,

and

(3.15)

by: B (X,t)-C x

,

0

(3.20)

EJx,t)=
(3.21)

Bt

Substituting

in

equation

(3.18)

and

in

the

boundary

conditions

(3-19) : B (x,t)=V<-x,t),

(X,t)=W(-x,t),

;

we

obtain:

51

(3.22)

wptX

SU*^L,

=

ax

PL

=

ot

-(W(r.t) ,

ox

(3.23)

1

7 ( x , 0 ) = g ( - x ) , XER ,

(3.24)

i

W(X,0)=g (-X),

XER'.

2

Thus, Cauchy's problem The

Cauchy's

problem

problem

(3.16),

(3.16),

(3.19)

always

further

we

be

reduced

shall

study

to the

equation

( 1 . 3 ) ( c f . Ch.1) , c o r r e s p o n d i n g t o t h e

(3.16), i s : \ +B>.-KxF'=O 2

The r o o t s

Here

can

Hence,

(3.17).

characteristic

system

(3.18), (3.17) .

(3.25)

1

gej? .

r

o f e q u a t i o n (3.26) a r e d e f i n e d

the

•/ 6 - 4 a F 2

radical

2

means

(3.26)

by:

the

branch

satisfying

the

conditions: / B -iaf 2

I m / B*-4af From e q u a t i o n s ( 3 . 2 7 ) ,

>0 a t 6 > 4 a e ,

(3.29)

>0 a t e <4a£. .

(3.30)

2

2

z

2

(3.28)

i t follows

Re\ (e)<0, i

l

Hence, t h e s y s t e m section

Cauchy's p r o b l e m (Ch.l, problem

section (3.16),

(3.16)

1.1,

i s reqular

Definition

(3.16),

(3.17)

1 . 1 ) . By v i r t u e (3.17)

2

with the order of regularity I t is

satisfies o f Theorem solvable

52

that:

ReA (0)=0.

1.2).

i s uniquely

Z

£ER',

Re^fFJ-eO, 5«R , £*0,

(Ch.1,

2

easy

to

verify

Lopatinsky's condition 1.1

(Ch.l,

section

and Theorem

3.1

r=2 that (1.13)

1.1) t h e

i s proved.

3,2.

Formula

f o r S o l v i n g Cauchy Problem f o r

Maxwell's

Definition is

Equations

System

3.1. The f u n c t i o n f ( x ) belongs t o t h e c l a s s C^(R'), i f i t

infinitely

differentiable

in R

and i s equal t o zero outside

o fa

segment. We r e p r e s e n t t h e f o r m u l a f o r s o l v i n g and

(3.18) , (3.19)

C™(R') .

These

studying

fjx)

with

formulae

asymptotic

Cauchy's p r o b l e m

a n d g^x)

will

(j=l,2)

be u s e d

behavior

(3.16),

belong

i n sections

(3.17)

t o t h e class

3.7 a n d 3.8 when

o f electromagnetic f i e l d

a t t h e great

values o f t . At

t h e construction

of solution

we

n o t deduce

transformations.

it

t h e s o l u t i o n o f t h e c o n s i d e r e d problem under

is really Calculating

(3.17) w i t h

the Fourier

However,

shall

mathematical

transform

r e s p e c t t o v a r i a b l e x, dU —

du

(3.16)

study. and c o n d i t i o n

we o b t a i n :

= -iCtye.t),

t > 0 , $«=»',

(3.31)

= -«ie.u ( s . t j - p t y e . t ] , t > o , e a r , y (?,0)=ll. (e), 2

,t) , WJx,t)=EJX,r.),

CO

(3.32)

e*R\

2

(3.33)

and CO

!*,(€)= [ fAx)explixF)dx, The

i nequation

(F,t)

^M.P

system

rewritten

rigorously

show a f t e r w a r d s t h a t

(e,t j

—

w h e r e WJx ,t)=BJx

we s h a l l

U (e,t)= )

(3.31),

i n matrix ^m

J W (x,t)exp{ix£.)dfi,

j=l,2.

j

(3.32) and t h e boundary

conditions

(3.34)

( 3 . 3 3 ) may b e

form as f o l l o w s : ,t)

+ - 4 ( e ) t 7 ( € , t ) = 0 , t > 0 , feR',

(3.35)

C(€,0)=*t€i .

(3.36)

where W(?,t) = ( U ( i t , t ) , [ J ( C , t ) ) ]

2

53

, *(e) =(

l

*

,

.

(3.37)

« H S * Solving (1.67),

Cauchy's

problem

(3.35),

(3.38)

?)• (3.36)

we

formula

obtain ( c f

Ch.l, section 1.3): W(€.tl=ffl(€,t)#(gJ,

(3.39)

where "IS'')

3

(E^+A'f)

zli

)-*«rp(at)dX,

(3.40)

H?) is plane

a unit X,

equation

matrix

o f order

2, T ( f ; ) i s a c l o s e d c o n t o u r \ ( £ ) a n d k (?)

enclosing the roots

i n complex

of the characteristic

z

f

{ c f . (3.27) and ( 3 . 2 8 ) ) .

Performing

t h e inverse

Fourier

transformation

on f o r m u l a

(3.39)

we

obtain:

| »(5,t)>(5)exp(-:ijr£)d?,

w(x,t)= ^

(3.4i)

where V(x,t)=(B^(x,t),£ (x,tj).

(3.42)

i

Now

l e t us p r o v e

Cauchy's p r o b l e m From

that

t h evector-function

V(X,t)

i st h esolution of

(3 . 1 6 ) , ( 3 . 1 7 ) .

( 3 . 3 8 ) we h a v e :

( E W ( O )

A+0 - I ?

= X +BA+af

Calculating ([13],p.85)

the

integral

(3.40)

l-aii;

by

(3.43]

A

Cauchy's

residue

theorem

we o b t a i n :

w(e,t)=

*,(C)-A (e)

X^ijJ+S -cie

- i f . 1 A e)J P(\^)t)e

i

x

(

2

fX

(O+fi - i f . exp(A (,t)t)

(3.44)

2

A (?)

From

formula

(3.44)

we o b t a i n

w(£,0)=E

54

Hence,

from

formula

(3.41)

we h a v e :

V{x,0)-

- 7 J $?&mp(-£$x}a&-f(x)i

(3.45)

where f ( X ) - ( f j ( x ) , f ( a r ) ) . a

Thus,

t h e vector

initial

condition

function

(3.17).

f (x)eC™(R )

to differentiate Equation

under t h e i n t e g r a l

+ A

at

where V ( x , t ) = ( B

Substituting into

solution From

(x,t),

Taking

E

W(x,t)

account

of equation

formulae

the class

then

( 3 . 1 6 ) m a y be r e w r i t t e n

ay Ifx, ^Sl

(3.40)

formula

(j=l,2),

1

functions

(x,t))

(B ( x , t ) , E

From

satisfies

(3.44)

i t follows

i n formula

(3.41)

Cauchy's that

i f the

i ti s possible

sign.

i n m a t r i x f o r m as f o l l o w s :

m*£L

AV>x,t)=0.

+

(3.46)

(x,t)).

from

(3.41)

we show

that

into

(3.46)

and t a k i n g

equation

W(x,t)

the vector-function

i s

the

(3.46).

(3.41)

and (3.44),

i t follows

that

W{x,t)

belongs t o

M. formula

(3.44)

into

account,

formula

( 3 . 4 1 ) may b e r e w r i t t e n

as: 3 Bjx.fj-

»n

1 2

Y

a

| (s -4«f )

2

(*,(€) ( * j ( € ) + f l ) -

-iS^mlexpt^Ot-iSxldf;, 2

, *™

J = 1

(3.47)

- I

-oa

+A (f)0 (sT))ex (X (i:)t-iex)d :J

where

2

a

0 (?) and ^ I C )

respectively Therefore,

r

e

2

P

t r i e

J

(3.48)

1

Fourier

transforms

/ ( x ) and t

f (x) 2

( c f . (3.34)). i f f (x)ec"(R ) ( j - 1 , 2 ) ,

55

thesolution

o f Cauchy's

problem

(3.16),

In

(3.17)

(3.16),

(3.17).

(3.23), is

i s defined

(3.24)

defined

Applying

we

functions g

0

and

(-6)

Let

the

(£)

T

and g

functions

(3.18),

(3.19)

0

and

2

(EJ

are the Fourier

j

f (x) ]

(J-1,2)

g {x)

and

}

i n R' and s a t i s f y

boundary may

be

belonging

Harmonic

In

this

data

solved

t o the class

C™(R

we

shall

construct

]

i

and

t h e boundary

A^,

u,

are

be c o n s i d e r e d shall

consider

(3.13) given

and

of

and these

case t h e f o r m u l a e f o r

Waves a l o n g

the

the

solution and

x-axis

of

(3.19)

Maxwell's

o f t h e form : ,

(3.49)

0 ) = j ) C o s (ux+(p ) ,

(3.50)

2

real

(3.17)

i n t h e case

( x , 0 ) =A cas{ux+