Transmutation and operator differential equations

TRANSMUTATION AND OPERATOR DIFFERENTIAL EQUATIONS

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NORTH-HOLLAND MATHEMATICS STUDIES

37

Notas de Matematica (67) Editor: Leopoldo Nachbin Universidade Federal do Rio de Janeiro and University of Rochester

Transmutation . and Operator Differential Equations R. W. Carroll Mathematics Department University of Illinois

1979

NORTH-HOLLAND PUBLISHING COMPANY - AMSTERDAM

NEW YORK

OXFORD

t

North-Hollunrl Publishing Company, I079

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ISBN: 0444853286

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Lihrary of Congress Calaloging in Publication Data

C a r r o l l , Robert Wayne, 1930Transmutation and o p e r a t o r d i f f e r e n t i a l equations. (Notas de matem&icas ; 67) (North-Holland mathematics s t u d i e s j 37) 1. D i f f e r e n t i a l equations, P a r t i a l . 2. Operator equations. I. T i t l e . 11. S e r i e s . QAl.N86 no. 67 [ Q A j 7 4 ] 5 l O l . 8 ~ [515'.353] 79-12341 ISBN 0-444 -85328 -6

PRINTED IN THE NETHERLANDS

PREFACE

With the advent of modern functional analysis and the theory of distributions it became possible in the early 1 9 5 0 ' s to systematically explore and develop the theory of partial differential equations (PDE) to a degree previously unimaginable. One important feature of this work was the determination of "natural" spaces of functions of distributions in which to pose various differential problems.

Then

linear differential operators were treated as abstract linear operators A

mapping

a domain of definition D(A) vector spaces F

and

G.

C

F

into G

for suitable locally convex topological

The abstract operational properties of

A

were studied

and existence-uniqueness theorems for example were then deduced as results in operator theory. This kind of approach was extensively developed also for certain types of nonlinear problems beginning in the early 1960's.

In theoretical and

applied work in other aspects of PDE one frequently has recourse to these now well established operational methods and distribution techniques.

It is no accident

that this material interacts naturally with various geometrical and variational points of view for example and enriches studies i n these contexts. Much of this material already appears in book form but research continues and we will report on some recent work as indicated below. One fascinating area of study which took form in these years involves abstract evolution equations of the form du/dt an

F valued function of

t

with

=

A(t)u

with

u(t) E D(A(t))

u ( 0 ) = uo.

TIere

in

gations of such problems played off operator properties of the A(t) d/dt

F.

Investi-

against

to produce a variety o f powerful results and in this spirit

one is led to study ordinary differential equations cients of the form

is

where we shall confine our

attention to families of densely defined linear operators A(t)

properties of

t+u(t)

(Dt = d/dt; u ' j )

= Diu)

(ODE)

with operator coeffi-

R. W. CARROLL

vi

where the A.(t) J

n

could be thought of as arising from differential expressions in

...,x

"space" variables

(xl,

) = x

but could of course also be more general

operators (e.g. pseudodifferential operators). type problems with Cauchy data incomplete Cauchy data usually take A (t) = I m (1.2)

d< = dt

A(t):

u(j'(0)

~ ( " ( 0 ) = uj

(0


S

E

CN(R)'

since taking

E ' C F'

CY a this asserts that V



+

=

CN(lR)

E

and

= lim a

=

V

lim Cf U

C

equicontinuous, one knows that



uniformly for

F, to what we define a s

is a closed convex disced neighborhood

(nbh)

of

0 in

V).

for

Y

E F

=

5


I f' E

B'.

C (L; F)

of

0 in F

,h> E 2 V

for

n

=

v

(S,h)

-+



u n B'

one has the quantity n > n

( s o that

n

a

F

<S,h>

for

=

Sn E A ' )

while

can be put in any closed convex = Vo,

n

2 no,

and

a 2 uo(V).

ll arbitrarily small for n u

are

I : CN(L)'

space CN(L)'

[ll)

+

U)

and

C

so

~ ~ ( F8) ; as

that t

t to

+

G(t,*)

+

or

t

+



=

+

€

=

Co(Ls($))

.G(t,*) E h

h

<w,vt $>

G(t,s)

Let u s note here that if

f

E

Lz(O; F)

;6

and

and applying the Parseval formula one obtains h

h

+-+

< IP(-is)T,@>

= 271


2

=

T

2



am

F))).

h



for

as a multiplier.

distribution pairings yield

(2.25)

1J

A-

h

Similarly one looks at

g..(t,s)

then G(t,s)c(s)

Ls(&; F)

E

.

= h

G(to,s)

c ( s ) = T(')uok(s)

=

then natural

t

+

E

GENERAL SYSTEMS

21

is any matrix of the form IP(t,D )

(here P ( D x )

notation i(t,s)

=

E(t,-is) A

=

FE(t,Dx)

=

above).

FE(t,x)

Similarly using the

we have

h

++ <E(t,-is)T,p = 2n<E(t,D )?,$(-x)>

(2.26)

h

2n