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
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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 <
uj
=
These could represent evolution
for some j
this is not necessary.
+ 7;
+
u(0)
+ can be studied where u is an
linear operator entries Aij(t).
=
1< m - 1) prescribed (or ( 0 , m-11); although we will
Similarly systems of the type
+ uo
Fm valued
m
= ((A
.(t))) with 13 To formulate guidelines under which problems of
the form (1.1) - (1.2) will be tractable let
vector and
us
A(t)
think of the A (t) j
as arising
a an where Dx = Dll...D
from differential expressions of the form Za. (x,t)DE la with Dk = a/axk. Now the Dk commute,in an obvious sense and generate groups
Tk(E) = exp ED,
in say' Co(Rn)
where
Tk(E)f(x)
= f(xl
surprisingly these features interact naturally with
d/dt
,...,xk + 5 ,...,x,).
Not
and corresponding to
the case where the a (x,t) = a (t) are independent of x one considers in ja ja general spaces F a family Ak of densely defined commuting operators generating in F with
one parameter groups Tk(C) = exp EAk
A.(t)
(t)Aa = ja Ca. (t)Ay '...Ann. Such situations as well as systems with A (t) = EaijaAa la ij to a variety of interesting solvable problems which will be discussed below.
a
J
Some siutations involving noncommutative families Ak related tractable situation occurs when may be employed. For example let F where the Riesz operators
\
=
F
= Ca
are also discussed.
L2 (Eln)
i a/axk = (-A)
and recall that
are continuous in F
and
A
=
1
+
is a
and a commuting
Ca^
(t,R)Ak with continuous jk Similar remarks gjk(t ,R) = xajka(t)Ra.
of bounded operators we can write A.(t) J
+
(-A)'
4Rk
a (x,t) = a (t) ja ja (t)RaAk where
again we have Ca (t)D: = Ca. (t)(-i)l"IR'(A - 1)la1 = 1, jka c1 ja Ja a R" = RI1.. .Rnn. Thus in terms of one nice unbounded operator A
polynomial valued operator functions t
Another
is a Hilbert space and spectral methods
densely defined positive selfadjoint operator. Assuming that
family Rk
lead
=
a apply to systems. We will present some recent results f o r ODE with operator coefficients of the types indicated and relate some of the results to formulas in
PREFACE
operational calculus.
vii
Some v a r i a b l e domain o p e r a t o r problems w i t h
Ak = \ ( t )
are
a l s o t r e a t e d as w e l l a s v a r i o u s o t h e r t e c h n i q u e s f o r h i g h e r o r d e r e q u a t i o n s .
We wish t o emphasize h e r e t h a t t h e u s e of o p e r a t o r methods t o s o l v e d i f f e r e n t i a l
problems r e p r e s e n t s a g r e a t achievement of mathematics i n t h e l a s t 25 y e a r s o r s o . I t i s n o t a n a r t i f i c a l c o n s t r u c t i o n of e f f e t e snobs t o avoid messy a n a l y s i s i n
Lp
(and g e n e r a t e p u b l i c a t i o n s ) b u t r a t h e r a s y n t h e s i s of such c o n t e x t s which i s o l a t e s t h e i m p o r t a n t n a t u r a l f e a t u r e s and produces e l e g a n t g e n e r a l theorems.
Anyone who
s t r u g g l e d w i t h d i f f e r e n t i a l problems i n t h e e a r l y 1950's s u r e l y must b e impressed and g r a t i f i e d by t h e power and economy of t h e s e methods.
Some of t h a t messy a n a l y s i s
was misguided, r e p e t i t i v e , o r unnecessary and came about because t h e wrong problem was b e i n g s t u d i e d o r t h e i m p o r t a n t f e a t u r e s were o b s c u r e .
One d i f f i c u l t y was t o
f i n d t h e r i g h t s p a c e s i n which t h e d i f f e r e n t i a l e x p r e s s i o n s would have good propert i e s when r e a l i z e d a s o p e r a t o r s i n t h e s e s p a c e s .
Today some messy a n a l y s i s i s s t i l l
u s u a l l y n e c e s s a r y (and h e a l t h y ) b u t i t can be b e t t e r f o c u s e d .
Another a s p e c t of t h e m a t e r i a l d i s c u s s e d i n t h i s book can b e i l l u s t r a t e d by t h e f o l l o w i n g examples of " r e l a t e d " d i f f e r e n t i a l e q u a t i o n s .
(1.3)
u
(1.4)
w
(1.5)
z
where
A
t
-
tt tt
2 A u = 0;
Consider t h e problems
u ( 0 ) = uo
- A'w
=
0;
w(0)
+ A2 z
=
0;
z(0) = u
=
uo;
wt(0) = 0
0
i s an a p p r o p r i a t e l i n e a r o p e r a t o r i n a s e p a r a t e d , complete, l o c a l l y
convex t o p o l o g i c a l v e c t o r s p a c e v a l u e s ) and
u
2
0
E D(A ) .
Schwartz t o p o l o g y ) , o r
F
(wherein
For example
D'
=
F
o'(TR) , w i t h
u ( t ) , w ( t ) , and
could be
E = E(R)
z(t)
take their
(= Cm(lR)
with the
A = d / d x , i n which c a s e we would be
d e a l i n g w i t h c e r t a i n Cauchy problems f o r t h e h e a t and wave e q u a t i o n s and a h a l f p l a n e D i r i c h l e t problem f o r t h e L a p l a c e e q u a t i o n .
A s w i l l be proved l a t e r , under
s u i t a b l e h y p o t h e s e s i t i s p o s s i b l e t o connect t h e s o l u t i o n s of ( 1 . 3 ) i n t e g r a l formulas of t h e form
-
(1.5) by
viii
R . W . CARROLL
(1.7)
z(t)
=
(1.8)
z(t)
=
],I
)1
where the kernels K
(1.11)
H(t.T)
=
I, and H are given by
-1 2 2tn (t
+
2 -1 T )
(and one checks easily that (1.8) follows from ( 1 . 6 ) and (1.7)).
On the other
hand in terms of operational calculus ,the solutions can be expressed formally as (1.12)
u(t)
=
2
exp(A t)uo;
w(t)
=
cosh(At)uo;
z(t)
=
exp(iAt)u
0
such that the formulas (1.6) - (1.8) are known integral transforms when replaced by a parameter
*A.
A
is
One objective in this monograph will be to report on
some relationships of this sort and show how they arise via transmutation. I n fact we develop in Chapter 2 a formalism based on the transmutation concept which connects eigenfunction transform kernels of ODE and displays the transmutation operator accordingly; numerous examples are given. As a byproduct one obtains a new derivation of inversion formulas such as the Hankel transform. Using the idea of a generalized convolution this also yields elegant expressions for the solutions of certain PDE. An abstract treatment of generalized translation operators is also given and the application of transmutation methods to problems in inverse scattering theory is indicated.
An abstract transform theory is also developed.
In order to make the abstract presentation of differential problems more accessible we have included an appendix wherein certain basic facts from functional analysis, distribution theory, etc. are outlined. Such material is of course standard knowledge for specialists but it should be helpful for graduate students or others
PREFACE
t o have i t r e a d i l y a v a i l a b l e h e r e .
ix
We s u g g e s t t h a t t h e r e a d e r g l a n c e a t t h e
appendix b e f o r e r e a d i n g t h e t e x t s i n c e s p e c i f i c r e f e r e n c e s t o t h e appendix w i l l n o t be made.
Standard n o t a t i o n s w i l l be used throughout and i n t h i s r e g a r d i f a
r e f e r e n c e t o formula chapter.
(x.y)
If a reference
while a reference
(1.y.z)
= Introduction (resp.
labeled consecutively.
a p p e a r s i n a g i v e n c h a p t e r i t means
(x.y.z)
a p p e a r s i t means formula
( r e s p . (A.y.2)) means formula
Appendix 1).
(y.2) (y.2)
(x.y)
of t h a t
of Chapter
x
of t h i s P r e f a c e
Theorems, L e m m a s , Remarks, e t c . w i l l be
We have a l s o found i t u s e f u l t o s t a t e v a r i o u s "formal"
theorems wherein n o t h i n g i s r e a l l y proved o r assumed beyond t h e formalism b u t p r e c i s e v e r s i o n s of t h e r e s u l E a p p e a r l a t e r a s theorems o r examples; such theorems a r e denoted by t h e symbol Theorem x . y ( F ) .
The a u t h o r would l i k e t o e x p r e s s h i s g r a t i t u d e t o Leopoldo Nachbin f o r h i s encouragement t o w r i t e a monograph f o r t h e Notas de Matematica s e r i e s . r e s u l t i n g c o n t r i b u t i o n i s d e d i c a t e d t o my c h i l d r e n David and Malcolm.
This
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TABLE OF CONTENTS
PREFACE
V
CHAPTER 1.
GENERAL TECHNIQUES
1.1. Preliminary results
1
1.2. General systems
12
1.3. Examples
25
1.4. Other techniques
43
CHAPTER 2.
TRANSMUTATION 2.1. Preliminary remarks and examples
69
2.2. Further examples
83
2.3. Separation of variables and transforms
98
2.4. More on kernels and transforms
117
2.5. Generalized translations
131
2.6. Miscellaneous topics
154
APPENDIX 1
181
APPENDIX 2
205
REFERENCES
209
INDEX
2 39
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CHAPTER 1
GENERAL TECHNIQUES
1.1
Preliminary results.
In order
to
deal with ODE having operator coefficients
there are many techniques available when Remark 1.9 for some references).
F
is a Banach or Hilbert space (see
However for various reasons, some of which are
connected with properties of solutions, it is essential to work in "larger" spaces
F, which for convenience will always be assumed to be complete, separated, locally convex topological vector spaces. It will be instructive to begin our discussion with some constructions having immediate application to problems of obvious interest and to proceed from such particulars to more general theorems.
For background
material on functional analysis let u s refer to Bourbaki [l; 2 1 , Carroll [ 4 ] , HorvLth 111, K&the [ l ] , Reed-Simon [l], Schaeffer [l], and Treves [l]. N
11 E C ( W ; F)
(i.e.
x E R, h
is N
h(x) E F
for x
R , and as an F valued function of
E
times continuously differentiable in the topology of
N topological vector space C (R;F)
and
6c
W.
It is known that
N C (IR; F)
=
N
C (W)
CN(R) 8 F
denotes the completion of
5N
As a
E
CN(IR)'
on compact N C (lR)
F where
in the
convergence on products of equicontinuous sets in [ 2 ; 31 and Treves [l]).
F).
is assumed t o have the topology of uniform
convergence of the functions and their derivatives of order subsets of
First let
=
N C ( a ;C)
topology of uniform X
F'
(cf. Schwartz
N
N
f" E F
(finite sum in
K C IR is compact then C (K) = C ( K ; E) has the N N natural Banach topology with C (K; F) = C (K) 6E F a s above. Let now h = 8 :f
Lh:
+
h
in CN(R)
If
gE F
where :h
Then we are able t o define <S,h>
S
E
CN(R)'
since taking
E ' C F'
<
,f'>= L CY a this asserts that V
+
=
CN(lR)
E
and
= lim a
=
V
lim Cf U
C
equicontinuous, one knows that
converges in
=
<Sxf',h>
uniformly for
F, to what we define a s
is a closed convex disced neighborhood
(nbh)
of
0 in
V).
for
Y
=
f' t B'.
But
< h , S > , since if
I: then
V
=
V
00
with
R. W. CARROLL
2
Vo = B'
(Vo
equicontinuous
-
denotes F
F
polarity) and the topology of
F'
coincides with the topology of uniform convergence on equicontinuous subsets of
F'. This leads to N Definition 1.1 If h E C (R;F)
and
S E CN(R)'
then
<S,h>
E F
=
is
well defined, Remark 1.2 where
It is of particular interest to apply this construction to the case
El
S E
=
€'(w)
Set K = {x : 1x1 CN(K)I
5u
5
is of order
€1
<S,h> E F
=
5
<
ci
m}.
Then by Schwartz [l] S E
and let h E CN(K; F).
i
and we can define
supp S C L = i x : 1x1
with
N
as above.
This will be of use when
working with hyperbolic problems (cf. Carroll 15; 6 ; 12; 13; 141 and Carroll-
[a]).
Showalter
tinuous for any h
Proof:
C
N
(L)
E
N
C (L; F)
fixed.
is a Banach space, hence barreled, and bounded sets in
Sn
consequently equicontinuous. Let :
I S E CN(L)'
ChV Q fl
v a
-+
h
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 -
for
n
2
n (u,V).
1
S E CN(Lf'
But for
sufficiently
Consequently
sufficiently large.
<S,h> : CN(L)'
space CN(L)'
[ll)
+
U)
and
C fv E V
Since our construction of : CLY(L; F)
a
by taking
fixed one can make each u n
-
This means that
also equicontinuous. Then if ha
(finite sum in
2
CN(L)'
and set A ' =
0 in CN(L)'
+
B' C F'
with
N
uniformly small for u
large so that n'<
11 SI1 2 11 in
disced nbh V c1
* F is con-
compact the composition S * <S,h> : CN(L)'
Lemma 1.3 For L C €3
QED fixed shows that
h
+
<S,h> =
is continuous one has established now that the bilinear map N
x C (L; F)
is barreled
so
+
F
is separately continuous.
But the Banach
we know for example (cf. Bourbaki [2] and Horvith
3
PRELIMINARY RESULTS
Lemma 1.4 The bilinear map
(S,h)
hn
h
+
in
Sn
<Sn,hn>
then
N
F
-+
is hypocontinuous
N
In particular this means that if hn
N
< S , h > : C (L)' X C (L; F)
C (L; F).
relative to bounded subsets of
N C (L; F)
+
-+
+
in
S
<S,h>
and
CN(L)'
is a sequence with
hn
F. We note in this respect that
in
will be a Cauchy sequence automatically and hence bounded (cf. Bourbaki [l; 21,
Carroll 1 4 1 , HorvLth [l], etc.). d/dx : CN(L)
QE
C d/dx h: Q :f
V
CN-l
(L; F)
F
QE F
CN-'(L)
-f
One remarks also that the continuous map d/dx ZhV 8 :f
defined by
v a
(dldx Q l)Chv @ :f v(y.
=
CN(L; F)
extends by continuity to be a continuous linear map
for N
2
=
-+
1. a s indicated
When working with abstract differential problems in general spaces F
it is extremely useful to utilize the concept of locally equicontinuous group or semigroup as developed by T. Komura [l] and Dembart [l] (cf. a l s o Bablola 111, Komatsu [l], Lions [l], Lb'fstrGm [ 2 ] , Miyadera [l], Oucii [I], Schwartz [ 4 1 , Waelbroeck [l], Yosida [l], etc. for various other general semigroups
-
for the
standard strongly continuous semigroups in Banach spaces see Hille-Phillips 111). We recall that a closed densely defined linear operator A generate a group T(x)
I, and
lim(T(x)
-
E
L(F)
I)f/x = Af
primarily with groups T(x) way except that
T(x)
(continuous) seminorm p
on
5
+
0 for
is only defined for x x
p(T(x)f)
x
as
T(x)T(y)
q(f)
=
T(x+y)
f E D(A).
is said to
for x,y E IR, T(0) =
We will be concerned
and simply mention that semigroups arise in the same
locally equicontinuous if
such that
provided
in F
+
T(x)f
F
E
Co(F)
2 0.
The group
for any
fE F
is said to be
while given any
there exists a (continuous) seminorm q
for
1x1
5 xo
<
m
(any € k i t e x
group is equicontinuous if this last condition hqlds for all called quasi-equicontinuous if
T(x)
p(T(x)f)
5
q(f)exp
cllxl
will do). x
E
for some
IR il
= T(x)Af
Example 1.5 Let subsets of
IR.
when
F
=
F
The
and will be (x E IR).
I n general with any of these groups or semigroups i t is well known that
AT(x)f
on
d/dx T ( x ) f =
f E D(A). Co(lR)
with the topology of uniform convergence on compact
The operator A
=
d/dx generates a locally equicontinuous group
4
R. W. CARROLL
T(x)f(y)
=
f(x+y)
which is not equicontinuous.
We will now sketch formally an important technique due to Hersh [l; 21 for treating differential equations with operator coefficients (precise theorems are spelled out when needed).
The method was first developed by Hersh for Banach spaces F
and later extended to general F
by Carroll [ 5 ; 61 using the construction <S,h>
of Definition 1.1; it has proved very useful in the study of various abstract
differential problems (cf. Bobisud-Hersh [I], Bragg 111, Carroll [ S ; 6 ; 12; 13: 1 4 ;
15; 16; 171, Carroll-Showalter [ E l , Donaldson 11; 2; 3; 4 ; 5; 7; 81, DonaldsonHersh [ 6 ] . Hersh [l; 2; 3 1 ) .
The idea goes as follows, assuming everything makes
sense. In (1.1.1) let A.(t)
=
a.
3
operator in
F
1 ajk (t)Ak
where A
is a densely defined linear
T(x) = exp Ax.
generating a suitable one parameter group
We set
m
so P(t,Dt,A)
=
1 A.(t)D:
O
and consider the abstract Cauchy problem for
t
2- 0,
J
u E Cm(F),
M. where
u
j
E
D(A ')
for suitable M
for distributions Gn(t,x),
where
0
5
6nk is the Kronecker symbol and 0 < k < m-1
+ 111, Gelfand-Silov [l; 2; 31). Gn(t,x)
Now replace A by j' n 5 m - 1 , satisfying
-D
D't,x
able distribution pairing or bracket
h(x) = T(x)u,
Z)'(R)
for large enough
but we will employ this notation anyway
E F
and appropriate h ( * ) Q
be illustrated by examples below). written as
and look
(cf. Carroll [ 4 ] , A. Friedman
since no confusion is likely to arise. Assume now such G n
E
-d/dx
There i s an obvious abuse of notation in writing
for a distribution G E n
corresponding Gn(t,*)
=
E
exist and that a suitcan be defined for t h e of the form
C4(E; F)
(this may involve growth at
m,
etc., and will
Then formally a solution of (1.2) can be
PRELIMINARY RESULTS
5
m- 1
(1.4)
u(t
since we have then formally
m- 1
1 <6nk6(*),
=
T(*)un>
=
uk
n=O for
0
5
k
k
5 m - 1 while, since
k
A T(*)un = DxT(*)un,
the differential equation in
(1.2) takes the form m
(1.6)
P(t,Dt,A)u
=
1 A.(t)D'u
j=o
=
J
m- 1
1
The key fact of course is that <(-Dx)Gn(t,.).
h(.)>
Theorem 1 . 6 ( F ) . and let
u(t),
Example 1.7
F.
>
=
AT(*)un
=
DxT(-)un
while
We summarize this in
Under the types of hypotheses indicated let the given by ( 1 . 4 ) . be well defined.
=
Gn
satisfy (1.3)
u satisfies (1.2) formally.
Then
A s a simple but important application of this technique let us apply A
generates a locally equicontinuous group
T(x)
We are led to consider
Gtt - Gxx
(1.7)
= 0;
G ( O , x ) = 6(~);
Gt(O.x)
The (unique) solution of (1.7) is evidently u
0
for a reasonable distribution pairing.
it to the problem (1.1.4) when in
T(*)u
2
E D(A )
=
0
1
G ( t , x ) = -[6(x+t) 2
we write the solution of ( 1 . 1 . 4 ) as
w(t)
=
+
6(x-t)]
T(*)uo>
and for where
6
R. W. CARROLL
upon r e s t r i c t i n g
t o be suitably f i n i t e (e.g.
t
d e f i n e d r i g o r o u s l y a s i n Remark 1 . 2 .
T < x
+
x
have s u p p o r t at
-
t = 0
and t a k e s a y
L = { x : 1x1
2
TI
and
5
T <
so
with
1x1
2
T
< G ( t , * ) , T(.)uo>
i n (1.1.12).
we o b t a i n
K = { x : 1x1
and l o o k a t
1
C (K)'
in
F.
is 9
write
xol
Dt
lim
-
where
T(x).
Dt < G ( t , . ) ,
a s indicated
Since
AG/At
w
tt
-
A2w = 0
f o r t h e s o l u t i o n of
2 C (F)
in
-
Gtt
with
Gxx = 0 ,
in this
u E D ( A 2) .
for
-
AG = G ( t , * ) -
Gt(to,')
T ( * ) u > =
c o n s t r u c t e d above i s t h e unique s o l u t i o n of (1.1.4)
w(t)
is
< G ( t , * ) , T(*)uo>
f o r e x a m p l e we w r i t e
Gt(to,*), T(-)uo>.
x
Upon e v a l u a t i o n
T(-t)]uo = cosh(At)uo
A < G ( t , * ) , T(*)uo7 = < G ( t , * ) , A T(*)uo>
s o l u t i o n of
G(t,s,x)
5
2
i t f o l l o w s by Lemma 1.3 t h a t
To show t h a t t h e w
+
2
Finally i n order t o obtain
in
w(t) = T[T(t)
Note h e r e t h a t from t h e s p e c i f i c n a t u r e of
c a s e it i s c l e a r t h a t
G(to,*)
1
G
w h e r e w e w i l l assume
t h e number a s s o c i a t e d w i t h t h e l o c a l l y e q u i c o n t i n u o u s g r o u p of
t h e b r a c k e t can b e
m)
Indeed i n t h i s event t h e &-functions i n
x
t = 0
t
w(0) = w t ( 0 ) = 0.
+
0
T(*)u >
suppose Let u s
G ( s , s , x ) = 6 ( x ) , and
1
+
G(t,s,x) = y[6(x + t - s) x+t-s S(C)dC ( i n t e g r a l i n E ' ( l R ) 6(x - t s)]. Similarly l e t H(t.s,x) = x-t+s s e e e . g . B o u r b a k i 13; 4 ; 51 o r C a r r o l l [ 4 ] f o r v e c t o r v a l u e d i n t e g r a t i o n ) b e t h e
Gt(s,s,x) = 0
(0
5
s
5
t)
so t h a t e v i d e n t l y
I
+
s o l u t i o n of
Htt
-
Hxx = 0 ,
I t i s e a s i l y checked t h a t
H ( s , s , x ) = 0 , and Hs = -G
and
G
H ( s , s , x ) = 6(x) t
= -H
tt
= -H xx.
(0
5
s
2
t).
We u s e now a t e c h n i q u e
f o r u n i q u e n e s s i n t h i s s i m p l e example which i s a s u b s t a n t i a l v a r i a t i o n of a p r o c e d u r e o f F a t t o r i n i [ l ] and H i l l e [ Z ] and was d e v e l o p e d by C a r r o l l [ 5 ; 6 ; 1 2 ; 1 3 ; 1 4 1 and C a r r o l l - S h o w a l t e r
[ 8 ] t o a p p l y t o much more g e n e r a l s e c o n d o r d e r s i n g u l a r and
d e g e n e r a t e h y p e r b o l i c p r o b l e m s ( c f . a l s o C a r r o l l [ 1 6 ; 1 7 1 , Donaldson [ 4 ; 5: 7 1 , and l a t e r s e c t i o n s of t h i s c h a p t e r ) . b2(t s) (1.8
where
We c o n s i d e r t h e f u n c t i o n
@ ( t , s )= @ , ( t , s )+
7
PRELIMINARY RESULTS
The brackets make sense as before and for completeness we will check the continuity
Q1
and differentiability of recall that
w
ss
2 A w
-
0 with
=
Q2 as functions of s
and w
E
2
C (F)
that one need only consider sequences sn
Ab2
example
-
T(*)W(S,)>
T(*)w(so)>.
~(T(x)w(s~)) <
m
2
q(w(sn))
for suitable x
G(t,so,*) Aw
=
-+
0
in
@2(t,*)
and that of
we consider first
1'
as
above).
q(w(sn))
But
K
5
q(Aw)
+
0 for
@l(t,*)
-f
T(x)Aw(s
)
will be bounded in
is clearly sufficient to verify that
F)
Thus
AG
=
+
(i.e. T(x)Aw
Then with
since
5
xo
T(x)w(sn)
G(t,sn,') 0.
is
-
Similarly for +
0 in
For differentiability
+
As
=
(a/as)@,(t,s)
T(x)w(sn).
is continuous when
=
sn - so we l o o k at
Gs(t,so,*)
(d/dx)T(x)w(sn)
as well as
Aw(-)
1x1
0. This proves the continuity of
T(*)ws(s)>.
Co(K;
(here
is a Cauchy
which formally can be written
will be bounded in C1(K; F)
)
w(sn)
follows by a similar argument.
F we have
F
on
as above and
xE K
-
T(')w(sn)>
will be bounded.
Now, with a view to applying Lemma 1.4 again, AG/As T(x)w(s
=
hence by Lemma 1.4
+
and consider for
so
+
+
for suitable compact
(a/as)@,(t,s)
sn
T(')w(so)>
< x
when T't
Co(K)';
and hence
to determine continuity, etc. of
T(*)w(s
Here we
~ ~ ( =0 0. ) First, recall
for some (continuous) seminorm q
w(sn) - w(so),p(T(x)Aw)
Co(K; F))
=
5 t).
s
is a (continuous) seminorm on
sequence in F, hence bounded, so bounded in Co(K; F)
-
)>
=
so
w(0)
(cf. Carroll [ 4 ] ) , so let
iR
functions defined on
+
and
5
(0
in
C1(K)'
= AT(x)w(sn)
and =
To see this it w(*)
and
2
A w(*)
are continuous. This "obvious" fact actually needs a little proof which we supply
for completeness. Thus (cf. T. Komura [l]) when (T(x)
-
I)w(sn)
T(n)Aw(sn)dn
= ' 0
and
x
5x
( T ( q ) - I)Aw(sn)
for example one can write ri 2 = T ( 5 ) A w(sn)dE so that
I
0
8
R. W . CARROLL
(1.10)
-
(T(x)
I)w(s ) =
T(E)ALw(so)dSdq. that
Aw(sn)
+
Aw(so)
Thus
xAw(sn)
since
A
and hence
Aw(sn)
Aw/As
-t
ws(so)
Co(K; F)
in
t o t h e desired expression.
+
< H ( t , s , * ) , T(*)wss(s)>.
Next we l o o k a t
2
Dx
( a / a s ) @ , ( t , s ) =
S i n c e now
( t , s ) = < H S ( t , s , - ) , T(*)ws(s)>
Here w e o b s e r v e t h a t f o r s u i t a b l e transport
K
H
<
= -G
G
= -H
,>
supp H ( t , s , . )
xx
we o b t a i n
C
L
C
K
we can
i n a d i s t r i b u t i o n s e n s e a s was done
$ ( t , t ) = w ( t ) = $(t,O) = 0
Therefore
and
+
a s above w i t h
around under t h e b r a c k e t
formally i n (1.6).
and hence t h e b r a c k e t t e n d s
I n view of p r e v i o u s arguments t h e r e i s o b v i o u s l y no
problem i n j u s t i f y i n g t h i s formula.
(1.11)
so
F o r t h e second b r a c k e t i n (1.9)
2 w E C (F)
since
w(sn)
Aw(*) E Co(K; F ) .
is c l o s e d and c o n s e q u e n t l y
T h e r e f o r e Lemma 1.4 a p p l i e s t o t h e f i r s t b r a c k e t . we know
converges w i t h
and
w ( t ) :0.
This
concludes o u r d i s c u s s i o n o f Example 1 . 7 and we s t a t e t h e r e s u l t a s
Theorem 1 . 8
Let
F
be a complete s e p a r a t e d l o c a l l y convex space and l e t
g e n e r a t e a l o c a l l y e q u i c o n t i n u o u s group i n
in
C
2
(F)
i s given by
w(t)
=
F.
A
Then t h e unique s o l u t i o n of ( 1 . 1 . 4 )
cnsh(At)uo.
T h i s theorem i s a s p e c i a l c a s e of more g e n e r a l r e s u l t s t o f o l l o w b u t i t h a s been proved as an i l l u s t r a t i v e example t o i n t r o d u c e v a r i o u s t e c h n i q u e s .
We w i l l g i v e
a n o t h e r such r e s u l t f o r ( 1 . 1 . 3 ) b e f o r e proceeding t o g e n e r a l e q u a t i o n s and systems
9
PRELIMINARY RESULTS
in Section 2 (see Example 1.10). Remark 1.9
Differential equations with operator coefficients and operational
differential equations in various spaces have been studied extensively in the last 25 years or so and a complete bibliography would be virtually impossible to assemble.
In addition to the references already cited in this respect, let u s
mention however a few others which contain valuable background material, provide other points of view, or have in some way influenced our thinking.
For material in
book or lecture note form, information involving both ordinary and partial differential operators in Hilbert space (linear and nonlinear) can be found in Akhiezer-Glazman 111, Berezanskij [l] , Bre>is [l],
Caldera% [l] , Coddington-Levinson
[l], Ekeland-Teman [l], Foias-St. Nagy [2], Garding [l], Garnir [l], Gohgerg-Krein [l], Lax-Phillips [l], Naimark [l], Showalter [l], and Stone [I]; developments of the theory in Banach and other spaces appear in Aubin [l], Barbu [l], Berger [l], Browder [l], Colojoara-Foias [l], Courant-Hilbert [l], Daleckij-Krein [l], DunfordSchwartz [l], Duvaut-Lions [l], Ehrenpreis [l], A. Friedman [2; 31, B. Friedman [I], Goldstein [ 4 ] , Hille [ 3 ] , Hb'rmander [l], Kato [4], Krein [l], Ladas-Lakshmikantham [ 11,
Ladygenskaya-Uraltzeva [ 11, Ladygenskaya-Solonnikov-Uraltzeva
[ 21, Lakshmikan-
them-Leela [l], Lions [4; 5 ; 6; 71, Lions-Magenes [81, Martin [l], Massera-Schaeffer [l], Maurin [l; 21, Palamodov [l], Pascali [l], Schechter [l], Sobolev [l], Strauss [l], Treves [2; 3 ; 4; 51, and Zaidman [l].
Some further papers dealing with differ-
ential problems in general spaces or in distribution spaces are Carroll [l; 2; 31, Dettman [ 2 ; 41, DuChateau [l], Fattorini [2], FoiaS-Gussi-Poenaru [l], MillionEFikov [l], Ovsyannikov [l], Schwartz [ 5 ] , Sebastiao-Silva [l; 2 1 , Steinberg-Treves [l], and Treves [5; 6; 71. We will omit any discussion of hyperfunctions, ultradistributions, microfunctions, etc. beyond mention of Komatsu [2] and Shapira [l].
For
problems in abstract operator equations and abstract boundary value problems see Aubin [2], Browder 12; 5 ; 6 71, Calkin 111, Carroll 17; 27; 28; 301, Cordes [l], DaPrato 11; 2 ; 41, Dezin [l , Milman [l], Vigik [2], and Wyler [l].
For linear
abstract evolution problems and semi-groups see for example Balakrishnan [l; 2 1 , Beals [l; 2 1 , Emami-Rad [I] Feller [l], Goldstein [I; 2; 3 ; 51, Goldstein-Sandefur
R. W . CARROLL
10
[ 7 ; 8 1 , G r i s v a r d [ l ] , Hackman [l], Kato-Tanabe [l], Kato [2; 31, K i s y n s k i [l], Ladyzenskaya [ 3 ; 4 1 , Lyubi? [ l ] , P h i l l i p s [l], P o u l s e n
111,
Raskin-Sobolevskij
S o b o l e v s k i j [ l ] , Tanabe [ l ] , Vis'ik [l] , and Vizik-LadyFenskaya
[l],
Some a b s t r a c t
[3].
Cauchy problems a r e t r e a t e d by s p e c t r a l methods i n C a r r o l l [18; 1 9 ; 20; 2 1 ; 221, Carroll-Neuwirth
[ 2 3 ] , Carroll-Wang [291, Sandefur [l], and Walker
[I]. V a r i a b l e
domain a b s t r a c t e v o l u t i o n problems a r e s t u d i e d i n Baiocchi [l; 2 1 , B e r n a r d i [ l ] , Bernardi-Brezzi
[ Z ] , C a r r o l l [ 3 1 ] , Carroll-Cooper [ 2 4 ] , C a r r o l l - S t a t e
Mazumdar [ 3 2 ] , Cooper [ l ] , Kato [6; 71, and Mazumdar 11; 2; 3; 41.
[25], Carroll-
Some fundamental
p a p e r s on n o n l i n e a r e v o l u t i o n problems, n o n l i n e a r semigroups, v a r i a t i o n a l i n e q u a l i -
[I], B r & i s
t i e s , e t c . a r e Bardos-Brffzis
Crandall-Pazy [ 2 ] , C r a n d a l l - L i g g e t t Lagnese 11; 2 1 , Leray-Lions
131, Browder [ 3 ; 4 1 , C r a n d a l l [I],
[ 3 ] , Dorroh [l], Kato [ 5 1 , Y. Komura [l],
[l], and S e p a l [l].
For Sobolev e q u a t i o n s s e e t h e book
of C a r r o l l - S h o w a l t e r [8] and f o r example Galpern [l], Lagnese [ 3 ; 4 ; 51, Levine
[ 4 ] . Rao [l], Showalter [6; 7 ; 8 ; 9 ; l o ] , Sobolev [ 2 ] , Ting [l]. c a l c u l u s l e t u s mention Berg [ l ] . Boginov [l], Bozinov-Dirnovski
For o p e r a t i o n a l
[2], C h u r c h i l l [I],
Ditkin-Prudnikov [ I ] , Doetsch [l], T. Donaldson [l], E r d e l y i [ 2 ] , B. Friedman [ 2 ] , Groetsch [l], Hirschman-Widder [l], J u r y [l], Krabbe [l], Laptev [l], L i g h t h i l l [l], Maslov [ Z ] , Mikusinski [ l ] , van d e r Pol-Bremmer
[l], Rubel [l], Sneddon [l],
S t r u b l e [l; 2 ; 31, Ungar [l; 2 ; 31, Zemanian [11.
Example 1.10
We w i l l now s o l v e ( 1 . 1 . 3 ) and produce t h e c o n n e c t i o n formula (1.1.6)
r e l a t i n g s o l u t i o n s of (1.1.3)
and ( 1 . 1 . 4 ) .
Such f o r m u l a s have l e d t o t h e concept
of r e l a t e d d i f f e r e n t i a l e q u a t i o n s s t u d i e d mainly i n a c o n c r e t e a n a l y t i c a l s e t t i n g
by Bragg [l; 2 ; 3 ; 4 ; 51, Bragg-Dettman [ 6 ; 7 1 , and Dettman [l; 2 ; 3 ; 4 1 .
Such
problems a r e a l s o connected w i t h t h e i d e a of t r a n s m u t a t i o n o f o p e r a t o r s developed by D e l s a r t e [l], D e l s a r t e - L i o n s [ 2 ;
3 1 , Hutson-Pym [l; 21, L e v i t a n [l; 3 1 , Lions
[ 2 ; 3; 4 1 , MarFenko 11; 31, anctPovzner and f u r t h e r r e f e r e n c e s g i v e n . t e c h n i q u e a s i n Example 1 . 7 .
111;
t h i s w i l l be d i s c u s s e d in d e t a i l l a t e r
F i r s t l e t u s look a t (1.1.3) Thus one c o n s i d e r s
and a p p l y t h e Hersh
11
PRELIMINARY RESULTS
A solution to this is easily seen to be
x
form
1 K(t,x) 7
where
K
is given by
This can be derived for example by taking Fourier transforms in (1.12)
(1.1.9). in
G(t,x) =
to obtain
F @ ( s )=
$(s) =
2
exp(-s t)
6+ +
2-
t(0,s)
s G = 0 with
L
@(x)exp
where we use the Fourier trans-
1
=
( f @ ' ()s ) = - i s $ ( s > .
isxdx s o that
1 2 Fx K(t,x) = exp(-s t) 2
and it is known that
(cf. Titchmarsh [ l ] ) .
generate a quasiequicontinuous or equicontinuous group
Let now A
complete separated space F
(recall
5 q(f)exp
p(T(x)f)
Thus c(t,s) =
T(x)
in a
alxl in this event).
We
consider formally then as a solution of (1.1.3) (1.13)
u(t)
=
T(*)u >
=
O
I
-1
(m K(t,x)T(x)uodx
2 I-,
m
=
m
K(t,x)cosh(Ax)uodx
K(t,x)w(x)dx
= 0
0
where
is the unique solution of (2.1.4) obtained in Theorem 1.8. Because
w(x)
of the nature of
K(t,x)
=
2
(nt)-+exp(-x
/4t)
differentiate under the integral sign for Theorem 1.11
+ m
and the growth estimates
t > 0. This leads u s to state
F be a complete separated locally convex space and let A
Let
generate a quasiequicontinuous group in in C (F)
x
it is easily seen that the integral in (1.13) makes sense and one can
for T(x)u
1
as
F.
Then the unique solution of (1.1.3)
is given by (1.13).
Proof: It remains to prove uniqueness s o we look a t the solution of t
2
s
with G(s,s,x)
Then let
u E C1(F)
=
6(x)
given by
G(t,s,x)
@(t,s)
=
m
G( t ,s , x) T (x) u (s)dx. It is clear that
T[n(t
be any solution of (1.1.3) with
t,s,o (1.14)
=
1
G s = -Gt
so
formally
=
- s)]-'exp(-x
2
Gt
=
Gxx
/4(t - s ) ) .
u ( 0 ) = 0 and consider for
for
12
R . W . CARROLL
since
2 T(.)u(s)> = < G ( t , s , * ) , DxT(*)u(s)>.
make s e n s e f o r q u a s i e q u i c o n t i n u o u s
T(x)
and now from
The c a l c u l a t i o n s e v i d e n t l y $(t,s)
c o n s t a n t we h a v e
=
= 0.
$ ( t , t ) = U ( t ) = $(t,O)
QED
There is no need t o d i s c u s s a fancy b r a c k e t
<, >
h e r e i n t h e manner o f Example
1 . 7 s i n c e w e p u t e v e r y t h i n g i m m e d i a t e l y i n t h e form o f i n t e g r a l s .
This could a l s o
h a v e b e e n d o n e i n Example 1 . 7 o f c o u r s e b e c a u s e o f t h e n a t u r e of t h e d i s t r i b u t i o n s
H.
and
G
However i n g e n e r a l v a r i o u s b r a c k e t s c l e a r l y must be d e f i n e d and we
c o n t i n u e i n t h i s s p i r i t i n C h a p t e r 2.
1.2.
General systems.
F o r n o n h y p e r b o l i c problems t h e d i s t r i b u t i o n s
a r i s e i n ( 1 . 3 ) w i l l n o t h a v e compact s u p p o r t i n pairings in
x
and d i s t r i b u t i o n s i n
F
valued functions
d e l i m i t e d by g r o w t h c o n d i t i o n s a t
x
example t h e c o n s t r u c t i o n i n Example 1 . 1 0 ) .
which
and one i s o b l i g e d t o d e f i n e
x
r e l a t i v e t o s u i t a b l e s p a c e s of
< G n ( t , * ) , T(*)un>
Gn
m
(cf. f o r
I n t h i s s e c t i o n we u t i l i z e p a i r i n g s of
t h e t y p e employed by Donaldson [ 4 ; 5 ; 7 1 when
F
was a Banach s p a c e .
c o n t i n u e t o work i n g e n e r a l c o m p l e t e s e p a r a t e d l o c a l l y convex
We w i l l
F, thus generalzing
D o n a l d s o n ' s c o n s t r u c t i o n s , and w i l l s p e l l o u t t h e d e t a i l s more c o m p l e t e l y a s i n C a r r o l l 116; 171.
F o r h e u r i s t i c p u r p o s e s i t seems more a p p r o p r i a t e t o d e a l w i t h
s i n g l e e q u a t i o n s s e p a r a t e l y and t h e n e x t e n d t h e t r e a t m e n t t o s y s t e m s .
L (F)
b e t h e s p a c e o f c o n t i n u o u s l i n e a r maps
o p e r a t o r topology ( i . e . f E F). a
jk
We c o n s i d e r ( 1 . 2 )
independent of
ajk(-)
Q,
t
-t
in
Q
(with
with
Ls(F)
F
means
T(x)
in
F
Q,f
provided w i t h t h e s t r o n g +
Qf
f o r each fixed
m u E C (F)) and f o r c o n v e n i e n c e t a k e
ajk(t) =
A ( t ) Z 1; e x t e n s i o n s of t h e t h e o r y f o r c o n t i n u o u s
m
are i m m e d i a t e ( c f . Theorems 2 . 6 and 2 . 8 ) .
continuous group
+
Thus l e t
F
Let
A
generate a quasiequi-
f o r example s o t h a t i n p a r t i c u l a r
T(*) E Co(Ls(F)).
13
GENERAL SYSTEMS
b e a n y ( c o n t i n u o u s ) seminorm i n
q(u(t)
-
u(t)
+
u(to)
for
x
near
u ( t o ) ) exp a l x l
values i n
for
u E Co(F)
x
F
DtT(x)u(t)
L D(A )
F
valued function of
+
(xo
for
t
and o b s e r v e t h a t
and h e n c e
0.
-t
0
T h i s shows t h a t
and
whereas f o r
x
finite.
fixed
t
f o r a l l terms i n (1.2)
-
q(u(t)
q
in
*
x
*
0
+
V(t,x)
t o make s e n s e .
u(to)) t
+
5
t
0'
uniformly
Now o p e r a t e on
the
is continuous with
T(x)Dtu(t) =
and we want
D:T(x)u(t)
=
Let
T ( x ) E Co(Ls(F))
Next i t i s e v i d e n t t h a t
T(x)AJu(t)
and l e t
As
F.
u ( t o ) ) e x p ctlxl
(t,x)
-
p(T(x)(u(t)
On t h e o t h e r hand s i n c e
finite).
2
F
(t,x)
T(x)u(to) - T(xo)u(to).
f o r some ( c o n t i n u o u s ) seminorm
(T(x) - T(xo))u(to)
term
i s an
U(t,x)
U ( t , x ) - U( t ,xo) = T ( x ) u ( t ) - T ( x ) u ( t o )
us write p
so
U(t,x) = T(x)u(t)
Set
u(t) E
(1.2) with
T(x)
to
obtain
(2.1)
0
for
5k5
m-1
be working with
k
D U(0,x) = T(x)uk
= 0;
P(Dt,Dx)U(t,x)
t
w h e r e we h a v e w r i t t e n P(D , D ) t x
i n s t e a d of
Note h e r e we w i l l
P(Dt,A) = P ( t , D t , A ) .
as i n t h e H e r s h t e c h n i q u e i n
P(Dt,-Dx)
S e c t i o n 1.1.
Now i f ( 2 . 1 ) c a n b e s o l v e d t h e n
u ( t ) = T(-x)U(t,x)
( 1 . 2 ) s o l e t u s c o n c e n t r a t e f i r s t on ( 2 . 1 ) . regard
U ( t , . ) E Ls(O; F )
tions.
Here
Q,
in
+
Q
Ls(O: Ls(O; F)
w i l l b e d e n o t e d by
For f i x e d
means
L ( @ ; F)
Q,($)
-t
$ ( a ) >
E F
formally a
s o l u t i o n of
i t is u s e f u l h e r e t o
t
0 2 D(IR)
f o r some s u i t a b l e s p a c e
denotes
F)
w i l l be
of
Cm
test func-
with t h e s t r o n g operator topology ( i . e .
Q(@)
in
F
and s i n c e
f o r each
@ E 0). The p a i r i n g is a continuous
U(t,.)
F
03
v a l u e d f u n c t i o n w e can w r i t e integral in
F)
$(*)>
=
I,
U(t,x)@(x)dx
and t h i s c a n a l s o h e w r i t t e n f o r f i x e d
t
as
(Riemann t v p e (rT(x)@(n)dx)u(t) -m
with t h e i n t e g r a l i n n a t u r e of t h e g r o u p ;,(t,s)
and
5
Ls(F). T(x)
(t,T,s)
The s p a c e
O must b e c h o s e n i n a c c o r d a n c e w i t h t h e
and w i t h t h e growth p r o p e r t i e s of c e r t a i n m u l t i p l i e r s d e f i n e d below - examples w i l l be gi ven.
t a k e F o u r i e r t r a n s f o r m s s o l e t u s assume
C
S'
with
F
: O
+
We w i l l w a n t t o
i
i s o m o r p h i s m and i t w i l l b e u s e f u l t o make a b l a n k e t h y p o t h e s i s t h a t c o m p l e t e and h a r r e l e d .
Let
IJS
a topological
0 and
are
r e c a l l n e x t t h e P a r s e v a l f o r m u l a ( c f . S c h w a r t z [l])
R. W. CARROLL
14
=
Q,
If
spaces.
1
s
Q, E
for
= F($)
and
1
$(x)exp i s x dx =
and one h a s
given t h a t
Definition 2 . 1 * *
Q E Ls(O;
6
0
isomorphism
+
FQ
&
where
-1-
F Q, + , F-'$
L s (O ; -
Q
F)
=
8E
=
+
Q, i n
=
I$
and hence
+
0.
Then
d e f i n e d and
Q
-+
6 : Ls(O;
F)
so t h a t f o r f i x e d
Ls(O; F)
follows t h a t continuous.
[l]).
l i n e a r operator
FQ, and
by t h e r u l e
F is a topological
+
F
a Banach s p a c e and w e w i l l
4, 6
+
+
in
8
in F
so t h a t
since
Q E
Zn i s a ( u n i q u e l y d e f i n e d ) c o n t i n u o u s l i n e a r map
t h e a c t i o n of which w e d e n o t e by
F
Ls(i; F)
First l e t
F.
n
: 0
( c f . Gelfand-Silov
6.
show now t h a t i t makes s e n s e f o r g e n e r a l =
Hence v
f o r m a l l y as an F-valued
FQ
and d e f i n e
@ E 0
for
=
= Zn
T h i s d e f i n i t i o n was used by Donaldson 14; 5 ; 7 1 f o r
@,
and we o b s e r v e t h a t
F).
Q E Ls(@; F)
Let
Zn E F
=
=
$ ( - < ) e x p ( - i s < ) d c = 2n(F-l$(-x)) ( s ) .
I t i s i n t h i s way t h a t one d e f i n e s
6
a r e t h e s t a n d a r d Schwartz
-m
(FF$)(x) = 2n$(-x)
F0 =
s'
and
m
-m
on
(S
now one o b t a i n s
m
(F$) ( s ) =
S'
Q E
+
@ E 0
in
F in
+
and t h e map Ls(6;
F)
8
Consequently
is c l e a r l y onto.
one h a s
6, 6
Similarly i f
F)
Ls($;
<e,&. Q
then
+
+
Now l e t
Q,
+
Q
in
F)
in
Q, + Q
6 : Ls(O;
is w e l l
in -+
F; i t F)
Ls($;
L s ( O ; F)
is
s o we
can a s s e r t
P r o p o s i t i o n 2.2
The map
Q
-+
6 : L s ( @ ; F)
+
Ls($; F)
of D e f i n i t i o n 2 . 1 i s a topo-
l o g i c a l isomorphism f o r any complete s e p a r a t e d l o c a l l y convex
We observe i n p a s s i n g t h a t t h e s t a n d a r d formulas s e r v e d s i n c e one h a s f o r example
<(DxQ)*,$> A
=
F.
(DxQ)" = -is6, e t c . , a r e pre-
2n = -2n =
h
2n = <;,(Dx@)^> = -is. Now a p p l y t h i s F o u r i e r t r a n s f o r m to
for
U(t,.>
*
(2.2)
for
f i x e d i n (2.1) s o t h a t w r i t i n g
t
P(Dt,-is)U(t,s)
0
5
k
5
m-1.
= 0;
kA
$(t,s)
for
Fxu(t,*)
Q
-f
;
we have
n
D U(0,s) = T ( * ) u ( s ) t k
In c o n j u n c t i o n w i t h t h i s problem w e c o n s i d e r t h e o r d i n a r y
d i f f e r e n t i a l equations ( c f . (1.3))
15
GENERAL SYSTEMS (2.3)
(t,s) = 0 ;
P(Dt,-is)G
h
DtGn(O,s) kA
=
6 nk
(0
5
n,k < m - 1).
Assume that the equations ( 2 . 3 ) can be solved for
6
are multipliers in
for each
t
and suitably differentiable in
t
6 (t,s)
which
with values
A
L (4)
in
as multipliers (examples will be given later).
Then the solution of
( 2 . 2 ) can be written formally as
and taking inverse Fourier transforms we obtain the formula
(Gn(t,*)
=
F-16n(t.s))
m
mi1J n-0
Gn(t .S)T(x - 5)undS
-m
There is some conscious abuse of notation in writing integrals for distribution action but this should cause no confusion. We will continue in a formal manner to prove the following theorem; a more precise statement and a rigorous proof are contained in Theorem 2 . 6 to follow but this result seems worth stating separately. Theorem 2 . 3 ( F ) .
F
: 4
*
i
Let
0
T(x)
and
T(*)un
E
functions (i.e. the continuously on U(t,x)
t
and
D
C
C
S'
with
be a suitable group generated by a closed densely
Ls(@; F ) .
Suppose
kDtGn(t,s)
are multipliers i n
in L s ( $ ) ) .
0
$ are also assumed to be barreled
in the complete separated locally convex space F
defined operator A
II
E L(F)
test functions where
Cm
a topological isomorphism; @
and complete. Let
D(A )
be a space of
Then
t
+
G(t,s)
(t,s) E Cm(Ls(i)) for 0 5 k
with
un E
as multiplier
5
m
depending
is a solution of ( 2 . 2 ) in L
(8;
F),
satisfies (2.1) in Ls(@; F ) , and a solution of (1.2) is given formally by
the formula m- 1
=
provided
n=O
Gn(t ,S)T(-S)undS -co
T(-x)[Gn(t,*)*T(-)unl
= [Gn(t,*)*T(-)un]x=O
makes suitable sense
R. W. CARROLL
16
(1
Proof:
(A
C a l c u l a t i n g f o r m a l l y one h a s f r o m ( 2 . 6 ) m
(2.7)
P(Dt,A)u(t)
m-1
=
1
i=o
AiDt
i
1
=
A.(t) =
1
a
Ak = P j ( A ) )
jk
=
m
1 < 1 D:Gn(t,
=
n=O i = O
m- 1
s i n c e from ( 2 . 3 ) ( 2 . 3 ) we o b t a i n
P(Dt.Dx)Gn(t,x) k D u ( 0 ) = uk t
0.
=
for
0
Further because
5
k
5 m-
To o b t a i n u n i q u e n e s s t h e o r e m s f o r t h e s o l u t i o n
k DtGn(0,x)
= 6nk6(x)
1.
hy
QED
u(t)
o f ( 2 . 6 ) f o r ( 1 . 2 ) we l o o k a t
u n i q u e n e s s f o r ( 2 . 2 ) and employ a v a r i a t i o n of a t e c h n i q u e d e v e l o p e d i n C a r r o l l [ 1 2 ; 1 3 ; 141 and C a r r o l l - S h o w a l t e r [8] which was i n d i c a t e d h e r e a l r e a d y f o r t h e The u n i q u e n e s s argument i n Donaldson [ 4 ; 5 ; 7 1 f o r
wave e q u a t i o n i n Example 1 . 7 .
Banach s p a c e s i s i n c o m p l e t e ( a l t h o u g h c o r r e c t l y d i s p l a y e d i n Donaldson
[a])
and
t h e p r e s e n t e x p o s i t i o n f o l l o w i n g C a r r o l l [ 1 6 ; 1 7 1 , i n t h e r i g o r o u s form g i v e n b e l o w f o r s y s t e m s , s e r v e s t o g e n e r a l i z e and c o m p l e t e t h i s a r g u m e n t , s i n c e i n t h e p r e s e n t c o n t e x t t h e two a p p r o a c h e s a r e shown t o b e e q u i v a l e n t .
The u n i q u e n e s s q u e s t i o n h a d
b e e n open f o r a number o f y e a r s and e a r l i e r p r o o f s by Hersh [ l ; 2 1 and C a r r o l l [5: 61 r e q u i r e d ( u n n e c e s s a r y ) h y p o t h e s e s on
F'
and
We p r o c e e d f o r m a l l y a t f i r s t
A*.
h
and w i l l make e v e r y t h i n g r i g o r o u s l a t e r .
(0
5
( t , i - , s ) = 0;
P(Dt,-is)E
(2.8)
n
2
m-1;
0
5
T
< t < T < -
a s a r i s i n g from a system
(0
2
a;
Thus Let
Dkb (T,T,s) t n 0
5k5
k,n < m - 1)
=
m-1).
Gn(t,T,s)
satisfy
Ank One c a n t h i n k of t h e s e e q u a t i o n s
GENERAL SYSTEMS
+ (6
where
17
h
nk
)
i s t h e column v e c t o r w i t h e n t r i e s
kA
We are assuming
(DtGn(t,T,s)).
= A (t) Z 1
A
m
m
Ank
+
and
i s t h e column v e c t o r
G
so t h e matrix
P(-is)
has the
form
(2.10)
P(-is)
f o
1
0
...
0
0
0
1
...
0
=
'
. . . 0
-i
-A,
-A,
1 6 ( t , ~ , s ) = (G ( t , T , s ) )
kDtGn(t,7,s)
(0
column index.
with
0
5
h
-*m- 1,
k,n
5
m-1)
with
k
d e n o t i n g t h e row i n d e x and
I
a l s o depends on
t
Such a formula g e n e r a l i z e s e a s i l y t o t h e c a s e
(cf below).
I n p a r t i c u l a r , a p p l i e d t o t h e columns
h
t ( t , T [ , s ) , we w i l l have v a r i o u s formulas f o r t h e terms
of
o f which t h e most r e l e v a n t =re
(2.13)
where t i o n s of
(0
5
-6
D ~ G ~ ( ~ , T=, S ) n- 1( t , T . s )
E 0. '1.
the
and i t i s w e l l known t h a t
-+
Gn(t,r,s)
n
Then one h a s
~(T,T,s) =
lP
...
b e t h e fundamental m a t r i x w i t h e n t r i e s
( c f . C a r r o l l [ 4 ] , Schwartz [ 5 ] ) . when
1
-+
h
Now l e t
0
Let
Suppose now t h a t $(T,s)
T
kDTDtGn(t,T,s)
n < m - 1)
+
P ( - i s ) C m- 1( t , T , s )
+
C
(t,T.s) E
1
c (L~($))
be 9 s o l u t i o n of ( 2 . 2 ) w i t h
t
a s m u l t i p l i e r funcr e p l a c e d by
T
and
18
R. W. CARROLL
for 0 < k
DY(0,s) = 0
Clearly $(t,t)
=
5 m - 1.
Eo(t,t,s)6(t,s)
m-1, E G (t,T,s)D;+%(T,s) n=O n m-1, k, k~lGk-l(t,T,s)DTU(T,s) But
For
fixed consider the expression
t
while
$(t,O)
=
0. However, formally, using ( 2 . 1 3 )
( t ,T ,s ) D
can be written in the form so
F ( T ,s )
+
that (2.14) becomes mrl
(2.15)
$T(t,T)
=
6m-1 (t,T,s)[Dy + 1 Pn(-is)D:]6(T,s) n=O
which vanishes identically. Consequently G(t,s)
=
G(t,O)
=
0 and fi(t,s) E 0
Thus formally we can state (cf. also Theorem 2.8 below) Theorem 2.4(F). E
C1(Ls(o))
Assume the hypotheses of Theorem 2.3(F) and that
as multipliers. Then the u(t)
(1.2) in Cm(F)
with
T
+
En(t,T,s)
of (2.6) is the unique solution of
T(*)un E Ls(@; F).
We consider now the case of general first order systems and will be more rigorous in our treatment. Thus consider
* -t
(2.16)
where
dt
+
u(t)
=
IP(t,A):;
+
G(0) = uo
is a column vector with entries u.(t) J
i s an
mxm
matrix with entries
(0 5 j
5 m-1)
and
IP(t,A)
GENERAL SYSTEMS
k
where
5 Mij 5
and t h e
are c o n t i n u o u s .
cijk(*)
g e n e r a t e a s u i t a b l e group
A
defined
M
f o r t h e v e c t o r w i t h components
T(x)
T(x)u.(t). J -+
a s s u r e d by a p p r o p r i a t e h y p o t h e s e s on T(x)
19
u
Let t h e closed densely -t
as b e f o r e and w r i t e
We a s s u m e
(cf.
E D(A ) , w h i c h c a n b e
ui(t)
(2.22))
U(t,x) = T(x)z(t)
M
on ( 2 . 1 6 ) w i t h
and o p e r a t e
t o obtain d - t -t U(t,x) = T(x)P(t,A)u = P(t,Dx)G(t,x)
(2.18)
;
+
-t
U(0,x) = T ( x ) u o m L (Q; F)
D e n o t e by
t h e s p a c e of m - v e c t o r s
F o u r i e r transform (2.18) i n
x
w i t h components in
Ls(O; F )
and
t o get
A
d - t U(t,s
(2.19)
h
G(0,S)
=
h
for f ( t , * ) E
where
~"'(6
E(t,s)
i s an
multipliers in of
t
F).
In c o n j u n c t i o n w i t h ( 2 . 1 9 ) w e c o n s i d e r
mxm
$ f o r each
matrix.
If the entries
and b e l o n g t o
t
(which w i l l b e a s s u r e d i f
t
-+
C1(Ls($))
;..(t;)
0)
Ls(O; F)
can be w r i t t e n as
,S)U(O,S)
Oh
(2.22)
and p o l y n o m i a l s are
h
E(t
-+ U(0,s) = T ( * ) u (s) E Lz(8; F ) .
g i v e n by
are
-t
=
0
h
where
i(t,s)
m h
t h e n a s o l u t i o n of (2.19) i n
h
+ U(t ,s)
(2.21)
of
as m u l t i p l i e r f u n c t i o n s
E Co(Ls(8))
1 1 1
h
multipliers i n
iij(t,s)
A s o l u t i o n of ( 2 . 1 8 )
-t
U(t,x) = F
-1-t
U ( t , s ) = G(t,*)*u(O,.)
-t
-+
u ( t ) = T(-x)U(t,x)
in
L:(Q;
F)
is then
-+
=
and f o r m a l l y
T(-x)[G(t,*)*T(.)u
'I
i s a s o l u t i o n of (2.16)
L e t u s examine t h e c o n s t r u c t i o n s and see what c a n b e a s s e r t e d p r e c i s e l y .
First
R. 14. CARROLL
20
+ M note that ( 2 . 2 2 ) formally displays u ( f ) E F and shows that ui(t) E D(A ) + whenever the components of u belong to D(AM). Now under our assumptions + i(t,s) E Ls($m) for fixed t and we have defined a map (t(t,s), Q ( s ) ) + h
h
A
: Ls($m)
t(t,s)$(s)
; $ ( : L
X
F)
h
<$(.),iT(t,.);(*)>
$E
for
defining amap from Ls(Qm) (6(t,.)
* $(.I
+
L : ( $ ;
(OT
?? X
F)
*
6(*)) =
since F(E(t,.)
@ transpose).
denotes
L:(O;
to
Lm(O; F) s * &(t,s)d(s).
in Ls(Qm).
But under our definitions it is then clear that
cation (resp.
(resp. Ls(Lz(O;
means
$&$)
+
A
for + h
h
in
F
for
acting by multipli-
in Ls(L:(8;
F))
6E
Lm($; F); but this is equivalent to
+ Hence we can say that Et(t,*) * T(.)uo= + T(*)uo in Lz(O; F). Let us explicitly
8".
@ E
*
*T(O):~ =P(t,A)E(t,-)
P(t,Dx)E(t,*)
Ga-+6
Note here that
F))).
in Lm(i; F)
$($)
h
6,
h
<s,C$),$> + <$(a),$> A
satisfies
satisfies
A
A
+
acting by convolution) satisfies ( 2 . 2 0 ) (resp. ( 2 . 2 3 ) ) in
(6,
F))
Ls(L:(6;
(E(t,x),d(x))
We know & ( t , s )
(8")
C(t,x)
=
This corresponds to
by the action
(2.20) in L
so that
h
by the rule
F)
observe here that one can pass the verification of continuity and differentiability for t(t,s)
in the above spaces to transpose action. Thus if ?(t,s)
is a typical multiplier element and
L~($; F )
since
<;(t,s)G(s),$(s)>
we have
A h
plier implies
AVW +
o
=
in
G(t,*) E C'(L~(~)) Remark 2 . 5
Setting A;
<w,Av@>
so
~ ~ ( F8) ; as
that t
t to
+
G(t,*)
+
or
t
+
<(Aj/At)G,$>
= <;,(&/At)$>
+
€
=
Co(Ls($))
.G(t,*) E h
h
<w,vt $>
G(t,s)
Let u s note here that if
f
E
Lz(O; F)
;6
and
and applying the Parseval formula one obtains h
h
+-+
< IP(-is)T,@>
= 271<
-
CO(L~(L~(&; h
=
as a multi-
h
++ lP(Dx)T,$(-x)>
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
=
);](-x)>
=
A
+ T +
. i *(t,s)
These formulas show for example that using the multiplier "adjoints" G * (t,-is)
= 6
"adjoints"
T
m h in L (Q; F)
(t,-is)
G * (t,x)
=
L * (t,Dx)
=
=
is consistent with using the operator
T
L (t,-D
in LZ(0; F).
)
+ + Now we have shown that Et(t,.) *T(')uo =P(t,A)E(t,*) *T(*)uo in L:(Q; F ) with + + E(O,.) *T(.)uo = uo and it remains to particularize to x = 0 in order to obtain
(2.22).
It is to be noted that in this process the space
Q virtually disappears
from consideration in (2.22) but it really occurs implicitly in two ways in that T(*):(t)
L:(Q;
E
F)
and
E(t,*) E Ls(Qm)
under convolution (satisfying (2.23)).
In general there may be many suitable 0 , or perhaps none, €or which this is the The requirement T(.):(t)
case, and this question will be examined later.
E
is obviously not too much of a restriction in any event. In order to + -f + set x = 0 in U(t,x) = L(t,*) *T(.)u we want U(t,*) to be a function of x, + + say continuous, and to satisfy (2.16) we must have DtU(t,O) = [D,U(~,X)]~,~=
LZ(0; F)
[Et(t,.) E(t,*)
+ M + M m Now for uoED(A 1, T(-)u E C (F ) by assumptions and thus + + Et(t,-) * T(-)uo = IP(t,A)E(t,.) * T(*)uo involves the convolu-
* T(*):olx=O.
* T(*):o
and
tion of elements evaluation
E
L(t,-)
maps
p(h(x))
5
g..(t,*)
at
1J
x
Ci(Fm)
kp exp alxl
in
E(t,*)
5
m)
for
f E F.
Hence
=
0 will be possible for example when, under convolution,
+
Co(Fm)
where
h(.)
for any seminorm
for quasiequicontinuous groups). (N
with terms T(-)f
means
E CZ(F)
p
in
F
h(.)
E
Co(F)
(recall p(T(x)f)
Further requirements oE the type
played off against the order o f the distributions g . . ( t ; )
5 u
and q(f) expalxi
ok
in
D(AN) E(t,.)
1.l
will give other situations where evaluation would be possible (cf. A. Friedman [l], Gelfand-gilov [ 3 ] ) .
Under the present hypotheses one expects a suitable evaluation
to exist for parabolic and hyperbolic systems for example and this will be illustrated later.
For now we will simply assume
u
ok
E
D(AN)
for some N
such
R. W. CARROLL
22
that evaluation E T(*):o].
+
E[E(t,*) *T(-)uo] = DtE[E(t,.)*
Then we can state
Theorem 2.6
F
x = 0 is possible and
at
: @
8
-t
Let
be a space of
@
C"
test functions where
a topological isomorphism; 0
and complete in general.
Let T(x)
and
F with the components uok
Assume E(t,*)
evaluation E -t
at
x
= 0
maps
in the complete locally convex space
C:(Fm)
N
for suitable N
)
c1 ( L ~ ( $ ~ ) )as +
a
l*multiplierl'satisfy-
-t
*T(*)uo]
=
DtE[C(t,*) * T ( * ) c o ] .
Then u(t)
given by (2.22) is a solution of (2.16).
Remark 2.7
It was mentioned earlier that if the entries iij(t,s)
are continuous multiplier functions of then t
* iij(t,s)
The integrands values in
8
T
E
-t
since
C (Ls(8))
t
and polynomials are multipliers in
,.
h
* iLk(~,s) E Co(Ls(8))
as
and
EC1(Ls(&)) jk
multipliers and
p
T
with
(T,-is) =
in.
Hence by standard facts about
vector valued integration (cf. Bourbaki [ 3 ; 4 ; 51)
8
6
as a multiplier. To see this write from (2.20)
Z C ~ ~ ~ ( T ) ( -is ~ ~a )polynomial ~ (cf. (2.17)).
in
of t(t,s)
~~~(',-is)g~~(~,s)~(s)are continuous functions of T
and
under convolution and that
Co(Fm)
satisfies E[Et(t,.)
1
S' with
are also assumed to be barreled
belonging to D(A
E ~~ ~~ ( F) 0. ; Suppose t * i(t,s) E ~ ( * ) u
ing (2.20).
C @ C
be a quasiequicontinuous group generated by
a closed densely defined linear operator A
* of u
8
D
(A;.
Jk
/At)$
+
D
2
t jk
(t,s)$
as a multiplier.
Now for uniqueness we will give two proofs and show them t o be equivalent in the present context. The first is based on the technique of Carroll [12; 1 3 ; 141 whereas the second is the author's version of the approach in Donaldson [ 4 ; 5; 71 modeled on Carroll [16; 171.
First we consider the ordinary differential equations
( c f . (2.20))
and as with (2.12) one knows that
GENERAL SYSTEMS
Theorem 2 . 8
23
Assume the hypotheses of Theorem 2 . 6 and suppose further that as a multiplier while polynomials are multipliers in
i(t,T,s) E Co(Ls(fm)) -+
solutions satisfying T(*)t(t)
E L : ( @ ;
that
T
h
-+
be any solution of (2.19)
will show that
im
E
-+
E 0. First consider
U(t,s)
5
0
fixed and
T
5
t
so
that
+ with U(0,s)
in L z f f ; F)
n
E
* &(t,T,S)
1 C (Ls(Cm)).
A
Let now U ( t , s )
h
f.
F).
Proof: Following Remark 2 . 7 we know from ( 2 . 2 9 )
t
*
of ( 2 . 1 6 ) given by ( 2 . 2 2 ) is unique in the class of
Then the solution u(t)
for
T
$(t,T,s)
0 and we
-+ =
m h
= i(t,T,s)lj(T,s)
n
+
$(t,t,s)
=
n
-+
U(t,s)
E Ls(@; F)
-+
while
+(t,O,s)
= 0.
Let
n
and look at
T
+
q ( t , ~ )= <;(t,T,s),
$(s)>
E
To check continuity and
F.
differentiability it suffices to consider sequential AT
= T
-
T
(cf. Carroll [ 4 1 )
+
F
is separately
and in an obvious notation we write
We recall that the bilinear map continuous and since
f
(6,i)
+
:
Ls($; F)
X
f
is barreled it follows that this map is hypocontinuous (cf Horvith [l]).
relative to bounded sets in LS'\$; F)
*
Now the sequence
h
h
U(T + A T , s ) n T -+ GT(t,T,s)$(s)
is Cauchy in
im
in
Lm($; F).: and hence bounded while
ZT(t,T,s);(s)>
+
fo the first bracket in ( 2 . 3 0 ) converges to n
n
h
<;(T,S),
(AiT/AT)J
in F.
I
Similarly the sequence &/AT
is Cauchy in
Lm(f; F) and hence bounded so the second bracket in ( 2 . 3 0 ) converges to s, + in F. Hence we obtain ,
h
+T(t,T)
(2.31)
This shows that $JT(t,T)
$(t,t)
=
9
h
9
$(s)>
is continuously differentiable and from ( 2 . 1 9 ) and ( 2 . 2 9 ) yields
$
h
-b
d ( s ) > +
0. Hence the continuous function -+
=
=
*
i(s)> = $(t,O)
=
T
+
$J(t,T) is constant so that
0 for any fixed
t.
Since
$E@
is
n
+
arbitrary this implies that U(t,s) E 0 i n LZ(6; F ) . This entails uniqueness + + for ( 2 . 1 6 ) since if u ( t ) were any solution of (2.16) with u ( 0 ) = 0 and
24
R. W.
+ T(*)u(t) = $(t,.)
CARROLL h
E Lm(O; F)
+
then t h e corresponding
s a t i s f i e s (2.19)
U(t,s)
in
Lz(6; F ) .
+ U(0,s) = 0
h
Second p r o o f :
h
be any s o l u t i o n of ( 2 . 1 9 ) as above w i t h
$(t,s)
Let
+ h
and s e t
(0
5
5
t
h
T $ ( T , t , s ) = C ('r,t,-is)$O(s)
E
bm
s e e Remark 2 . 5 f o r t h e n o t a t i o n
T;
+
b0
for
6
8"
a r b i t r a r y and
fixed
T
-T
(GT(T,t,-is)= C ( T , t , s ) ) .
Then from
( 2 . 2 9 ) we have
so that in
@,$(T,t,s)
satisfies
h
Consider now
- t
At = t
For
A
-+
e ( t , ~ )= < $ ( t , s ) , @ ( ~ , t , s )E> F
which makes s e n s e f o r
fixed.
t
sequential
Using a g a i n t h e h y p o c o n t i n u i t y r e l a t i v e t o bounded s e t s i n
L r ( b ; F)
of t h e map h
(8,;)
: Ls(b; F) x
+
A
h
h
+
F
h
,$> = -
6+
PT(t,-is)$>
6
for A
+
= 0
Thus
A
= < i ? ( ~ , s$,(s)> ),
e(T,T)
"
h
any
h
+
(from ( 2 . 3 3 ) and ( 2 . 1 9 ) ) . h
= ~(o,T)=
o
h
-+
@o 6 $m
and s i n c e
=
h
i s c o n t i n u o u s and c o n s t a n t so
e(t,T)
et
b a r r e l e d one s e e s t h a t
+
is a r b i t r a r y it follows t h a t
0
U(T,s)
in
Lm(8; F )
for
QED
T.
Remark 2 . 9
The e q u i v a l e n c e of t h e s e two p r o o f s in t h e p r e s e n t c o n t e x t i s s e e n + + -+ T + from t h e f a c t t h a t $ ( t , T ) = < P ( t , T , - i s ) U ( T , s ) , @ ( s ) > = < U ( r , s ) , P ( t , T , - i s ) @ ( s ) > = A
+
8(T,t) =
+
i(t,T,s)>
where
$(s)
A
i s taken for
h
-+
io(s).
A
The t e c h n i q u e of *
t h e f i r s t proof however, as used i n C a r r o l l [12; 1 3 ; 141 w i t h s i n g u l a r o r d e g e n e r a t e problems, i n v o l v e s no r e c o u r s e t o F-valued d i s t r i b u t i o n s and could b e so a p p l i e d
.
h e r e d i r e c t l y t o (2.16).
Remark 2 . 1 0
Thus t h e e q u i v a l e n c e i s a consequence of c o n t e x t .
We n o t e t h a t t h e s t a t e m e n t of Theorem 2 . 4 ( F ) i s somewhat s t r o n g e r than
t h e theorem which would r e s u l t a s a c o r o l l a r y of Theorem 2 . 8 .
A s t r i c t corollary
25
EXAMPLES
would require
7
k * Dten(t,7,s)
need this statement for k 2.4(F).
=
E
1 C (Ls(6))
0< k,n < m - 1 whereas we only
for
0 as indicated in the formal demonstration of Theorem
This can be made rigorous following the formulation of Theorems 2 . 6 and
2.8 but we will omit an explicit statement of this fact.
1.3.
Examples. We will give first some results €or singular and degenerate problems
using the pairing of Definition 1.1 (cf. Carroll [ 1 2 ; 1 3 ; 141 and Carroll-Showalter [ S l ) and then turn t o some examples of general parabolic and hyperbolic systems
following Donaldson [ 4 ; 5; 7 1 .
The singular and degenerate problems considered do
n o t fall into the framework of Section 2 since they are not of the form (1.2)
(either A.(t)
is singular at
3
Example 3 . 1 .
t
=
0 or
Am(t)
vanishes at
Euler-Poisson-Darboux (EPD) equations.
t = 0).
As the basic equation we
consider
+ 2m+l ut t
2
-
(3.1)
u
where A
generates a locally equicontinuous group
A
2
=
A
in
tt
Rn
=
A u; u(0)
= u
0’
u (0)
= 0
t
T(x)
in
F.
The case where
has been of considerable interest and an extensive survey of such
equations appears i n Carroll-Showalter [ 8 ] (cf. also Carroll [ 3 3 ; 341 and CarrollSilver [9; 10; 111 for group theoretic aspects). > m-
-4
We will treat here only the case
and denote the corresponding solutions u
easily proved for Re m > -%
m u ; parallel results are
by
while for other values of
bit (cf. Carroll-Showalter [ 8 ] ) .
m
the theory changes a
In the study of such equations with
AL
=
A
various growth and convexity properties of the solutions were shown to be of interest and to handle this it was absolutely necessary containing objects of unrestricted growth (e.g.
F
=
E
=
to
work in spaces F
Cm(IRn))
.
Thus our
program of phrasing the theory of operator differential equations in general locally convex spaces F
has a very realistic motivation.
Banach or Hilbert spaces
will simply not suffice to naturally enrompass certain interesting types of solutions (using weighted Banach spaces suggests itself but this distorts the operator
R. W. CARROLL
26
realization A2
-
(cf. (1.2)
of
A).
Thus following the procedure of Section 1 we consider
(1.3)) 2m+l m +R t t
(3.2)
t:R
= Rm
Here G
has been replaced by
Rm(O,x) = 6(x);
*
xx'
Showalter [81 where R"
R
iyt+-
2mf1
Gm
where
=
t
0.
was called a resolvant. One can work with
ii"t +
FRm . The
=
in conformity with the notation of Carroll-
and Fourier transform (3.2) in x (3.3)
m Rt(O,x)
s2fi" =
Rm(t,-)
€
S'
to obtain
-m 0; R (0,s) = 1; $ y ( O , s ) = 0
solution of (3.3) is
Now a version of the Sonine integral formula can be employed here in writing for m >
-4
(cf. Carroll-Showalter [8]) where
1 [6(x 2
+
t)
+
6(x
-
= Cos(st)
@(t,s)
with
R-'(t,x)
=
ux(t)
=
Consequently
t)].
E'
where the integral is taken in
=
Ei. We set cm
= r(m
+ l)/r(+)r(m+ 4)
so
that (3.6) can be written (3.7)
m R (t,x) = c m
1
1 (1
-
-
c2)m-46(x
Et)dc
-1
and express a solution of (3.1)
as
1 (3.8)
um(t) =
(bracket of Definition 1.1).
T(*)uo>
=
(1 - E2)m-4<&(x
c
For m =
m
- [t),
T(x)uo>dE
-1
-4
(3.1) is a wave type equation and we
EXAMPLES
27
have already expressed the unique solution as 1.8).
u-&(t)
=
cosh At uo
(cf. Theorem
It is known that
which correspond to recursion relations
-ARmfl(t,x) 2 (mtl)
(3.11)
Rm(t,x) t
=
(3.12)
R:(t,x)
= -
in
€:(A
=
2m
2 Dx).
Similarly u
m
m-1 [R (t,x) - Rm(t,x)]
t
Consequently
2m
m-1
= - [u
t
t
- um ]
and combining this with (3.13) one obtains (3.1).
The calculations can all be justified as in Example 1.7 and we can state Theorem 3 . 2
A
Let
Then for m >
-3
generate
a locally
equicontinuous group in
(m >
-4)
resolvants Rm(t,T,x)
and
2mtl m m ytt + t yt =
6(x),
=
Sm(t,T,X)
P.ym
=
5b
For
satisfying in
0 < T
2
t
and constructs
E'
m yxx
m R~(T,T,X) = 0 ,
Carroll-Showalter [8] f o r details).
< t
E D(A2).
is answered by the following
argument (see Carroll [ 1 2 ] , Carroll-Showalter [a]).
wit11 R"'(T,T,~)
u
a solution of (3.1) is given by (3.8).
The uniqueness question for this problem
(3.14)
F with
the distributions Rm(t,'r,.)
sm(T,-r,x)
When and
T
=
0,
Sm(t,T;)
=
0 , and
St~ ( I , T , ~ ) = ~ c x ) (cf.
Sm(t,O,x) E 0 , while for
0
5
T
have supports contained In a
R . W. CARROLL
28
fixed compact set K < 2. of order -
=
{x : 1x1
5
xo}
and, with their x
derivatives in
Furthermore one has
(3.15)
m RT(t,T,x)
(3.16)
S (t,T,X) =
= -AS
m T
m
(t,T,X) 2m+l + __
m -R (t,T,X)
m
s (tJ,X)
and the constructions which follow all make sense as in Example 1 . 7 . be any solution of (3.1) with u ( 0 )
u E C2(F)
(bracket of Definition 1.1).
= u (0) =
t
Setting $(t,T) = R(t,T)
T
@(t,T)
-f
Theorem 3 . 3
Then, referring to Example 1 . 7 to justify the calcu-
(a(0)
S(t,T)
we have $(t,O)
is continuous with
$T = 0).
= 0 =
u(t)
since
u(t)
of ( 3 . 8 )
is the unique
We consider problems of the form
= 0)
s(t)ut
=
(3.16))
2 C (F).
Degenerate Cauchy problems.
+
$(t,t)
-
This proves
Under the hypotheses of Theorem 3 . 2 the
solution of ( 3 . 1 ) in Example 3 . 4
+
Thus let
0 and set
lations (cf. also Carroll-Showalter [ a ] ) , we can write (cf. (3.15)
(i.e.
D', are
+
Ar(t)u
(3.21)
utt
(3.22)
u ( 0 ) = ut(0) = 0
-
2
A a(t)n
+ b(t)u
= f
EXAMPLES
where A
29
is the generator of a locally equicontinuous group
T(x)
in F.
The
discussion here follows Carroll 113; 141 to which we refer for bibliography.
Such
problems are an extension of the classical Cauchy problem for the Tricomi equation in the hyperbolic region and thus are of interest in transsonic gas dynamics for example. Various abstract versions have been treated by Carroll [26], Carroll-Wang [29], Krasnov [I], Lacomblez [l], Walker [I], and Wang [l; 21 while some classical results can be found in Berezin 111, Bers [l], and Protter [l] (cf. also Barantzev
[I], Bitsadze 11, Brgzis [2], Carroll-Showalter [81, Friedman-Schuss [4], Schuss Following Section 1
[l], Showalter [2; 3 ; 4; 51, Smirnov [l; 21, Weinstein [l; 21.
-
(cf. (1.2) (3.23)
( .3))
Gtt
+
S(t)Gt
for G(t) E S i . with
a(0)
(3.24)
=
we replace A
-
r(t)Gx
-
by
-d/dx
+
a(t)Gxx
It is assumed that
in (3.21) and consider first
r, a, and
s,
0. Taking Fourier transforms x
ett + s(t)et
+
iyr(t)e
+
= 0
b(t)G
a(t)y2e
+
-f
b
are continuous while
a(t) > 0
(3.23) gives
y
b(t)e
=
0 A
and it will be convenient to eliminate the b(t)
term as follows. Let
G(t)
=
rt g(t)
exp joy(c)dc
(3.25)
y'
+
sy
where y
+ y2 + b
satisfies the Riccati equation
= 0;
y(0)
= 0.
In order to produce a suitable y we note that if one sets y
= ci'/ci
then
ci
satisfies (3.26)
a"
+
s(t)a'
+
b(t)ci
=
0
and we choose a
to be the unique solution o f (3.26) satisfying a ( 0 )
a ' ( 0 ) = 0. Then
y ( 0 ) = 0 and the continuous function y
< t < T < t some interval 0 -
where
then restrict our attention to
[O,T] since f o r
t
2T
a(t).
We can
the problem is not
degenerate and can be handled by the methods of Section 2. consider
1 with
will remain finite on
is the first zero of t
=
Thus on [O,T] we
30
R. W. CARROLL
i'' +
(3.27)
+
(s
2y)g'
+
(ay
2
+ iry)g^ =
0
which w e w r i t e as a system
?
We look f o r s o l u t i o n s
0<
where and
T
5t5
z^(t,T,y)
2
and
of ( 3 . 2 8 ) s a t i s f y i n g
By w e l l known theorems t h e r e e x i s t s o l u t i o n s
T.
of (3.27) s a t i s f y i n g t h e p r e s c r i b e d i n t i a l c o n d i t i o n s a t
and t h e s e f u n c t i o n s are c o n t i n u o u s i n 0
5
T
5t5
T
2T =
(3.30)
and
y E
(ay2
+
11 5 c
in particular
Lemma 3.5 0
5
T
5
t
ij
-Z
+
(s
+
+
-
T)
11 11
where
(c
Q ( T , T , ~ )= I
when
Q = (q. .)). 1J
?(t,T,y)
and
while for
Hence one can s t a t e
?(t,T,y)
e n t i r e a n a l y t i c f u n c t i o n s of
y
(t,T)
a r e continuous i n
fixed,
A
A
yY, yZ, Y t ,
of e x p o n e n t i a l t y p e
A
$
=
Cp
+
iJ, one h a s
+ ay 20 + yr@ = 0. and add t o o b t a i n
Cp"
+
(2y
+
s)$'
and
(t,T,y)
it
for
are
5 "c.
For v a r i o u s r e a s o n s we need t o e s t a b l i s h growth e s t i m a t e s on writing f i r s t
then
d e n o t e s t h e m a t r i x norm ( s o t h a t
h
y E
for
31 i f one w r i t e s t h e s o l u -
+
g ( t , - r , y ) = Q ( t , T , y ) g ( T , ~ , y ) where
1 5 11 Q((
and
y
T
f
2y)(T)f
By a n argument now i n Gelfand-Silov
The f u n c t i o n s
5T
fT =
iyr)(r)?;
exp c l y l ( t 14
and a n a l y t i c i n
t
I t i s e a s i l y shown a l s o t h a t
(c.
t i o n of (3.28) i n t h e form Q(t,T,y)
(t,T,y)
Y
(cf. Carroll [ 4 ] ) .
11
?(t,T,y)
+ ay 2 4 -
h
Y , 2, e t c . and
yrJ, = 0
with
M u l t i p l y t h e f i r s t ( r e s p . second) e q u a t i o n by
J,"
Cp'
t
EXAMPLES
31
This type of inequality can be handled by the use of Gronwall type lemmas (cf Carroll [ 4 ] ) . Thus set P on
[O,T].
ay21?12
We add
+
dc
r )y
Q
and
=
1
-
+
2(2Y
so that
s)
to the right side of (3.32) and set
r
=
191 5
[?,I2
+
that
so
t
+
T< 1
(3.33)
(a'
=
1
t
+
P1?I2dc T
dc.
T
By Gronwall's lemma (3.34)
< E(t,T)
where
E(t,S)
+ j:PIE\
'E(t,C.)dS
exp c(t - 5).
=
litl 2
Forgetting the
term in
and following a Gronwall type procedure we obtain for P
A
P
where
Lemma 3.6
2
+
r )/a.
Given
aE C
(a'
=
satisfying ?(T,T,Y)
=
2
for the moment
0
This yields 1
,
$> 0, and
b,r,s E C o ,
?
the solution of (3.27)
qt(~,~,y)= 1 it follows that for
0 with
y
real and
O < T z t< T (3.36)
I2 5
a(T)y21?(t,T,y)
i
E(t,T)exp
t
(r2/a)dS.
T
Let now
F(t,T)
=
exp(-[
t
2 (r /a)d<)
with
F(T)
=
F(T,T)
that
so
F(T)
5
F(t,T).
JT
Then since E(t,T) < exp ? T
Note that T > 0
F(T)
both
=
k
we have from (3.36)
may tend to zero as
F(T)
and
a(T)
T +
0 while
are positive.
a(T)
+
0
by assumption but for
Similarly one obtains from the above
inequalities
(3.38) where
I?,(t,T,y)
<
- , . Q(T)
=
k max a(t)
IYtt(t,T,y) I
I2a(T)F(T) on
<
[O,T].
Setting
G(T)
=
(a(T)F(T))'
from the differential equation. Now writing
one can bound c(t,T,y)
=
32
R. W .
i t follows t h a t
G(T)?(t,T,y)
and i s bounded u n i f o r m l y f o r that
?(t,T,*)
say,
IG(t,T,y)
lC(t,T,y)l
5
r e a l and
5T5
0
5
is a l s o of e x p o n e n t i a l t y p e
I
is bounded by c o n t i n u i t y i n
W(t,T,*)
0 <
i n a f i x e d compact s e t f o r
T
2
t
T.
2
ET
A s i m p l e argument shows
T.
(t,T,y)
y
r e a l with
(yI
2
R
w h i l e from t h e a b o v e
From t h e Paley-Wiener-Schwartz
F-%(t,T,y)
=
5
t
ZT. A l s o f o r
l y l > Ro.
i s bounded f o r
k'/lyl
5
i s e n t i r e of e x p o n e n t i a l t y p e
yij(t,T,y)
+
y
theorem i t f o l l o w s t h e n t h a t
Wtt
y
CARROLL
E
Ei
with
supp W
contained
Similar conclusions apply t o
w h i c h w i l l i n d e e d r e p r e s e n t t h e d e r i v a t i v e s of
W
in
€4
and
Wt
( c f Carroll 1 4 1 ) .
There r e s u l t s Lemma 3.7
W, Wt,
Under t h e h y p o t h e s e s i n d i c a t e d
s u p p o r t s c o n t a i n e d i n a f i x e d compact set f o r W(t,T,'),
W t ( t , T , * ) , and
W
tt
(t,T,.)
0
Co(F)
with
a bracket
z(t) = s(t) t 2y(t).
by
s(t)
2
f ( t ) E D(A )
while
5
T
5t2
Let
Ah(*)
and
b(t)
T.
Moreover
A h(.)
<W(t,S.*), T(*)h(c)> a s i n Section
t h e map
(S,0)
bounded s e t s i n
S E
E'
of o r d e r
<
< S , 0 > : E'x CL(F)
-f
C2(F).
2
+
Consider f o r
with F
belong t o
u'(t) =
T(
-
-f
€4
Co(F).
for
h(.) E
We d e f i n e
1 ( c f . Lemmas 1 . 3 and 1 . 4 , D e f i n i t i o n
supp S C
2
Here
compact a n d f o r
W(~,T,.)
0 E C2(F)
w i l l a g a i n be hypocontinuous r e l a t i v e t o T
> 0
Then t h e f o l l o w i n g c a l c u l a t i o n s :an b e j u s t i f i e d a s i n S e c t i o n 1.
(3.40)
( t , ~ )
a n d assume
1.1, and Remark 1 . 2 ) and t h e a r g u m e n t s p r o c e e d as i n Example 1 . 7 . corresponds t o
and h a v e
t e r m i n view o f ( 3 . 2 5 ) and
h(t) = f(t)/o(t)
2
€4
belong t o
a r e continuous with values i n
Going b a c k t o ( 3 . 2 1 ) - ( 3 . 2 2 ) now we o m i t t h e replace
Wtt
and
) h ( E ) >dS
EXAMPLES
1
t
Similarly A2u(t)
=
2
There is no trouble in passing to the limit as
f.
=
transform back to the original equation. Theorem 3.8 s
and
Let
+
Eu'
+ Aru
-
belong to
f
with
Co
= (a'
(aF)'
with
F(T)
=
exp(-(
+
2
T
* 0 and using
a(0) = 0 and
a E C 1; let
2 r )/a > 0; and choose T
(r /a)dc)
y
to
Hence
t > 0 with
act) > 0 for
t =
and we obtain u"
T(*)h(c)>dS
T
2 A au
6
33
and assume h
=
b, r,
as indicated.
f / i j E Co(F)
with
Let
Ah(*)
JT
and
F.
A2h(.)
E Co(F)
where
A
generates a locally equicontinuous group
Then, after modification by a factor expIty(6)dS,
u(t)
T(x)
in
given by (3.39) with
0
is a solution of (3.21) - (3.22) on [ O , T ] .
T = 0
A
To treat uniqueness we need some estimates f o r with
" Y
Set P
=
and arguing as before
we obtain
(a'
+
r2)y2
and
adding the term (3.45)
Z(t,r,y)
< a(T)y2
=
dS
+
1
1 - 2s as before with
x"
=
" 2 IZt1
+
ay21?I2
to the right side of (3.44) we obtain
t
T
There results
Q
P1.il2dt
+
?j
t^
T
dC.
so that
R. W. CARROLL
34
Lemma 3.9
z^
Given the hypotheses of Lemma 3.6 and
1t (T,T,Y) =
satisfying ~"(T,T,Y) = 1 with
the solution of (3.27)
0 it follows ;hat for y
real and
O c T c t c T t
Iz"(t,T,y) 1'
(3.48)
5
E(t,T)eXpj
(r2/a)dS 'I
which can be written as
F(T)[.?(t,T,y)l':
E(t,T).
iT and iT
Next one can establish estimates on
?
already obtained for and
i.
and
Reasoning as before we also conclude that
are entire functions in y
?T
from (3.30) using the estimates
of exponential type
5
A
yT
ZT and one can easily
prove then Under the hypotheses indicated Fs (T)Z = F4(~)F-lz", q(-r)ZT
Lemma 3.10
Ei
~(T)Z), and G ( T ) Y ~ belong to 0
for (t,?)
5
-f
5
T
F4Z
t
5
or
T.
The
T
with supports contained in a fixed compact set
derivatives can be taken in
GZ, aZT, and
QYT
(and
Ei
T > 0 and
for
Ei.
are continuous with values in
Now for uniqueness we extend the technique of Section 1 and define
is any solution of the modified equation (3.21) (i.e.
where u by
E(t)
s(t)
=
+
2y(t)
and
b(t)
=
0) with
f = 0.
s(t)
is replaced
By our lemma above (3.49)
and (3.50) make sense as do the calculations which follow (cf. Example 1.7). (3.51)
R
5
=
5'
Letting now $(t,c) (3.53)
$5
=
Tu>
+
=
Tu> =
=
R(t,c)
T(u"
+
+ S(t,E)
+ gu' +
-
r(E)ATu>
2
-
< Y , a(5)A
we obtain from (3.51) 2
rAu - aA u> = 0.
2
Tu>
-
(3.52)
Thus
EXAMPLES
Theorem 3.11 Let
F-'(T)U(?)
-f
u
0 and
a, b, r, s hold.
35
be any solution of the modified equation ( 3 . 2 1 ) such that ;-'(T)U'(T)
Then u
-+
0 as
T +
0 while the other hypotheses on
is unique.
6 1. 0
Remark 3.12 The condition
a' > -r2
or
allows a(t)
not to be monotone
(which had been assumed in some earlier work on such equations). t
about this and about the behavior of
2
Some remarks
as
F ( T ) = exp(-/ (r /a)dE)
T
-t
0 are
JT
given in Carroll-Wang [29]. In general the requirements of Theorem 3.11 regarding the growth of as
u'(T)
(3.21). T
+
T
-f
0 are somewhat strong although possible for suitable f
It is therefore of interest to consider the case when
and
U(T)
F(T)
+
in
0 as
and the relation of this to results of Krasnov [l] and Protter [l] has
0
been discussed briefly in Carroll-Wang [29]. u ( 0 ) = 0 and
Theorem 3.11 are only that
examine the feasibility of this let with
f
=
0, u(0)
=
0, and
u
In this event the requirements of
a-'(T)u'(T)
+
0 as
T
0. To
+
satisfy the modified equation (3.21)
u ' ( 0 ) = 0. Multiply this equation by the term
t
s(5)dE)
exp(1
and integrate to obtain
0
IM
on
[O,T], for any continuous seminorm
p
on
F
have
Now (Au
p(Au)
and
and 2
A u
2 p(A u)
belong to
will be bounded for a solution u Co(F))
and
E
2 C (F)
of (3.21)
we
36
R. W. CARROLL
whereas
fiad5
((jtadc)4)2.
=
Hence for
0
and
a-'(t)u'(t)
+
0 if
t a(c)dS)'
! ) t ( ' a
+
0. This condition is examined in
0
Carroll-Wang [ 2 9 ] and Wang [l; 21 and since oscillations in a(t)
are permitted
h
> 0 it is not automatically satisfied. by the stipulation P -
near
t
0 it is obviously valid.
F
Theorem 3.13
+
a
is monotone
Consequently F(T) > 0
Assume the hypotheses of Theorem 3.11 with
and suppose a-'(t)(jta(E)d()'
If
0 as
t
-f
0.
Then if u
on
[O,T].
on
[O,T]
satisfies (3.21) -
0
( 3 . 2 2 ) with
f
=
0 it follows that
u E 0
We go next to some examples illustrating Section 2 and recall briefly some con-
"
structions of Gelfand-Silov [l; 2; 31 (cf also A. Friedman [l]). Example 3 . 1 4 (3.59)
5
IxkDP@(x)I > A
for
The space S B y B consists of ,A
functions @
satisfying
-k-p ka PO c(A,i,@)A B k p
> B.
and
C"
For
6,p =
I,$,%,
...
the topology is defined in
terms of norms
G;Bm With this topology If a,B > 0 For
p > 1
with
a
FSa
c1
+
= S i
S
= S i
c1
= US
",A,,,
while
S:
where
Am,Bm
-t
-.
is nontrivial.
S1-'/' and i t consists of functions UP as entire functions satisfying
P
(c
is a Monte1 space and
B> 1 then FS B = -SB c 1 c 1
the space Zp
extendable to
Since
a,A
we have
is defined as
FZp P
=
zpP
= Zq
q
where
1 P
-
+ -41 = 1 .
Multipliers
37
EXAMPLES f : Sa6
-+
Sa6
are determined by functions f
is continuous and if
q > p.
for any
5
If(z)I
@
such that the map
c exp(alzlP)
then
f
+
f@ : S t
is a multiplier
+
Sa6
Zq + Zq 4
9
Zp. We mention
Evidently polynomials are multipliers in any
also that the spaces Zp are sufficiently rich in the sense P with iIg@ldx < m , for all @ E Zp implies g E 0. P The space 0
Remark 3.15
FU
= 2
k
W
K
is denoted by
in Gelfand-Silov [l; 2 ; 31 and
is the space of entire functions of exponential type satisfying
1s $(u
+
i.r)I
5
s
for
c exp alTl
=
u
+
i?.
Now to work with equations of the form ( 2 . 1 6 ) let of
and let
t
for 1
max Re Xk(s)
Q
so
that
hk(s)
max clq j
so that
and for
where
jk
follows.
5
1J
p
5m
I 5 (1
Ip..(-is)l
5
k
2
1 1 1
5
CClqjkl
2
11 011
.
P(A)
= 0.
be independent
Set
A(s)
=
the operator norm of a matrix
and we let
for large
=
The entries in the matrix P(-is)
= Zcijk(-is)k
clslp
-XI)
det(lP(-is)
and denote by
pij(-is)
05 t < T <
po
5 2
will have the form k
be the roots of
P(t,A)
p
Then
Is/.
be the highest index
IllP(-is)II(
cllslp
+
c2
m
is called the reduced order of the system and is determined as
If one writes det(lP(-is) - h I )
=
(-lImXm
+
p 1(s)Xrn-’
+ ... + p,(s)
8
with degree p (s) m
=
p
m
then po
is defined
as
max pk/k
Definition 3.16 The system is called hyperbolic if for
A(s)
5
in which case
11
(3.63)
+ lull
p(m-1)
clsl
+
.
11 5
c(1
+
1 s1
p(m-l)ebt
Is[)
The system is called parabolic if
which case ( 3 . 6 2 ) holds with
11
s = u
+
iT
with
11
exp t P(-io)
5 c2
cl; A(u)
exp t P(-is)
15 k 5 m.
for
exp t P ( - i o )
11 5
c(1
+
A(u)
5
h -cIuJ
+
c1
lul)P(m-l)exp(-atlul
in
h
).
11
< -
38
R. W. CARROLL
In the case of a hyperbolic system the elements of i(t,s) h
will be multipliers in @ = 2 the elements of
E(t,x)
< t < T < set for 0 -
@ = K =
so
D
=
exp t IP(-is)
can be taken in Section 2.
Actually
will be distributions with supports in a fixed compact + + and U(t,x) = E(t,*)*T(-)u in Section 2 is defined for
m
(t,*) are of bounded order ij -t M and T(*)C0 will belong to C (F) when the elements of uo belong to D(AM). + will be the evaluation E[E(t,.) *T(S):~] Hence for suitable assumptions on u locally equicontinuous groups T(x).
The entries g
I
appropriate in Theorem 2 . 6 (see A. Friedman [l] or Gelfand-Silov [ 3 ] for further details in this direction). > 1
For parabolic systems one knowns p
0< h
and
5
po
even while for so called Petrovskij parabolic systems h
are integers with
=
p
8
= 2’’
5
1.
= p.
If
h
p1 > po
s o @ = Zql with P1 41 is indicated. Now there is another index associated with parabolic
then the elements of 1 +
i(t,s)
will be multipliers in
1 = 1 p1 1 systems called the genus, !J, where
1 - (po
-
h)
systems this can be defined directly from A(s)
1 ~ 51
the domain
k(l
+
(01)’
one has
h(s)
5
2
!J
as the largest p -cIuI
and various useful properties of the e ements of
h
If p > 0 for example and
+ cl.
E(t,x)
h
this.
For constant coefficient
p = po/po-!J with
such that in
It can be shown that
can be established from Poh > 3‘ where 0 =
e
b^Tpo/p then (3.65)
for
I\ E(t,x) 11 5
t E [O,T].
c exp
Thus the entries in E(t,x)
are functions with exponential can be taken in the classical
growth as indicated and convolution E(t,x) *T(*)C0
A
sense for continuous function elements in for suitable bo.
T(*)Zo
of growth bounded by
exp bolxlP
Similarly there will be estimates A
(3.66)
with
11
D:D:E(t,x)
c and sj
p
(1 5
cqi exp(-~lxlP)
depending on
t
(becoming
m
as
t
-f
0).
Thus €or t > 0
EXAMPLES
* T(O):~
one can differentiate E(t,*)
T(*)Z0
39
in the classical sense for continuous
of suitable growth so that
E[E(t,*) *T(-)Zo] + To insure continuity of E(t,-)*T(*)uo
Theorem 2 . 6 .
of differentiability on
Remark 3.17
with
as
t
+
0 suitable hypotheses
have to be imposed (not s o for Petrovskij
T(*):,
parabolic systems however). c exp(-a]xI$)
will be appropriate in
Analogous remarks hold for
= h/h-v
v2
0
where
,x) 11 2
11 E(t
and estimates of the form ( 3 . 6 6 ) are valid.
Using the notions just discussed of hyperbolic, parabolic, etc. and
the formalism of Section 1.1 (cf Theorem 1 . 6 ( F ) ) Hersh [l] gives versions of the following theorems where
F
is a Banach space.
-
equation to a system transpires as in (2.8)
Here the passage from a single
(2.10) and the results will also
follow from the technique of Section 1 . 2 as indicated in the discussion just concluded following Remark 3.15. P(Dt, -Dx)
or
Recall also that the Hersh technique involves
in (2.8); one can retain the notation i(t,T,s)
P(Dt, is)
however and simply use P(is),
P.(is), J
etc. instead of B(-is),
the discussion concerning the roots Xk(s).
etc.
P.(-is),
etc. in
J
An equation is said to be hyperbolic,
parabolic, etc. if the corresponding system has these properties.
-
constructed then as in ( 2 . 3 ) (with P(Dt, is)
cf. ( 1 . 3 ) )
The
Gn
are
and the solution is
represented in the form ( 1 . 4 ) . Theorem 3.18
Let
F be a Banach space and P(Dt. -D
a strongly continuous group and blem ( 1 . 2 ) has a solution for Theorem 3.19
One notes here that
p^
> 1
M exp wlxl
t 6
E
Dm(A)
(-m,
be hyperbolic with
&(t,x)
k
given by ( 1 . 4 ) .
m)
(cf. ( 3 . 6 5 ) ) .
11 unll
P(Dt, -Dx)
t < T
decays like
exp(-klxlY)
On the other hand it is well known that
(cf Hille-Phillips [l])
so
A(0)
<
c
for
11
t < T.
where
T(x)unll:
everything fits together in (1.4).
We mention also that a system of the form (2.20) where called Petrovskij correct if
be
Then ( 1 . 4 ) represents a solution of (1.2) for for fixed
T(.)
nD (A) for example. Then the pro-
=
Under the other hypotheses of Theorem 3.18 let
parabolic of positive genus.
y =
u
)
u
i(t,s) = exp t lP(-is)
real in which case one has
is
R. W. CARROLL
40
1)
5
t(t,o)ll
u
C(1
+
IUl)h
where
such that in the domain
p(m-1).
The genus is defined as the largest
I T [ 5 k(l + 1 ~ 1 ) '
for example one has then for
t
and this leads to estimates on the G (t,.)
5
h
C
one has
A(s)
5
c.
For
!.I > 0
T
11 G(t,x)(l
permitting compositions in (1.4) of
with suitably restricted T(*)un
(cf. A. Friedman [l], Gelfand-
V
Silov [31).
It is pointed out in Hersh [ l ] (cf. also Bragg [I]) that the techniques
Remark 3.20
of Section 1.1 can be extended to situations involving several commuting operators Ai
generating groups T (x) = exp A.x. i
(The case of several commuting operators
has also been treated in Hilbert space by spectral methods in Carroll [ 4 ; 20; 2 1 ; 2 2 1 and we refer here to section 1 . 4 . )
A
and
(3.68)
(0
5
i
B
Thus let u s take two commuting operators
to illustrate the matter and set
Bk
P(Dt, A, B) = Icijk D:A'
Lm).
As in (1.3) let the distributions Gn
Then for suitable u
(0 < n< m - 1)
satisfy
the function
will formally satisfy
The bracket in (3.70) is a suitable pairing as before between Gn(t,*,*) which will be
and vector functions TA(x)TB(y)un appropriate selection of the u
.
as
E
",y
smooth as necessary upon
The proof follows the lines of (1.6) exactly.
A slightly different approach to the same kind of problems is indicated in Bragg
EXAMPLES
41
[l] where the formula
is exploited. Let for example P(D) order form
=
P(D1,
...,D
be an elliptic operator of
)
2k with real constant coefficients where the highest order term has the (-l)k-'Po(D)
degree
2k.
with
a positive definite homogeneous polynomial of
One considers the Cauchy problem
and following Rosenblum (3.74)
Po(<)
v(5,t)
[i]a
solution
K(S-U,
=
to
(3.73) is given by
t)f(U)dU
L n where
K
is a Green's function for which growth estimates are obtained.
Symbol-
ically (3.74) says
Let then
r$
(3.76)
u(t)
nD(Aik)
6
with
T.(x)
=
exp A.x
=
K(-u,t)
1
as above and consider
.
n =
exp[tP(A)]r$
Ti(Ui)r$dU.
1
L.n
This is a formal generalization of (3.72) with replaced by
Ti(Ci)
f(a)
=
nTi(ai)@
(exp aiCi
is
in (3.72)) and will agree with the corresponding result
obtained via the Hersh technique. Remark 3.21 A variation on the Hersh technique is presented in Donaldson [31. One considers a Banach space group.
A solution Y(t)
(3.77)
F
in which
A
generates a strongly continuous
of the equation
fj(tA)AjD:-jY j=O
is required. Set m consider
=
max m. J
(0 5 j
5
n)
and
y
=
m
+
n.
Let
I$ E D(Ay)
and
42
R. W. CARROLL
(3.78) over
a
Y(t)
= j:F(x)T(xt)Qdx
suitable contour from a
to
B
in t
such that
B (3.79)
{AnP(P(x),
= 0
T(xt)$ll
a where
and
P
Q(D)
is a suitable bilinear concomitant defined below (cf. Ince [l]). is the formal adjoint of
(3.81)
so
F is a solution of
d
P(u,v)
=
vQ(D)u
-
Q*(D)
=
C Ca xn-jDP
P
and
P
Thus
is defined by
uQ*(D)v
that in the present situation one can write explicitly
n
"'j p-1
1 1 1
(-l)kap{DP-k-l[T(xt)m]}D
k n-j [x F(x)]
j = O p=O k=O
Theorem 3 . 2 2 Proof:
Under the hypotheses above Y(t)
We insert Y(t)
For finite intervals
is a formal solution of ( 3 . 7 7 ) .
given by ( 3 . 7 8 ) into ( 3 . 7 7 ) to obtain
(a,B)
of
IR
there is no problem in justifying the
43
OTHER TECHNIQUES
calculation; for infinite intervals one needs some growth condition on general contours in
(c
one requires that
T(x)
F(x);
for
have a suitable analytic exten-
sion.
QED
Various applications of this formalism are given in Donaldson [3] but they are covered here by other techniques. 1 . 4 Other techniques.
I n this section we will sketch some results involving
differential equations of the form (4.1)
Ln(Dt)v(t)
=
0
We will not deal with the case n
=
1 as such since the equation v
t
+ Av
=
0 is
essentially the classical linear semigroup problem which is covered exhaustively for example in Carroll [4], Hille-Phillips [l], Yosida [l], etc.
Such results as
are needed for dealing with higher order equations will simply be cited. seem worthwhile to include here some results for the case stant Ak
and variable A (t) k
n = 2
It does
both for con-
since a lot of this has not yet appeared in book
form and we will draw upon Baiocchi [ 3 ] , Bruk [ l ] , Carroll-State [25], DaPratoGiusti [ S ] ,
Gearhart [l], Goldstein [2; 3 1 , Kato [ 6 ; 7 1 , Kleiman [l], Krein [l],
Levine [l; 2; 3 1 , Levitan [ 4 ] , Lions [4; 91, Mazumdar [4], Raskin-Sobolevskij [ l ] , Sobolevskij [2; 3 1 , Torelli [l], Treves
[a;
9 1 , Vigik [l], Vigik-Ladyyenskaya [ 3 1 ,
Yakubov [l; 2; 3 ; 4; 6 ; 7 1 , Yosida [2], etc. (cf. also Lax-Phillips [l; 2 1 ) . n > 2
For
there are methods due to Abdulkerimov [l], Bairamogly-Khalilova [l],
Carroll [la; 19; 20; 2 1 ; 221, Chazarain [ l ] , Djedour [ l ] , Dubinskij [l; 21, Gasymov [l] , Gorba$uk-Ko&bei
131, Mazya-Plamenevskij [l], Raskin "21, Yakubov
[ 5 ; 91, etc. which we will discuss and indicate results.
Finally we mention some
interesting new results on operator differential equations with noncommuting operator coefficients due to Hersh-Steinberg [ 7 ] (cf also Grabovskaya-Krein [ l ] and Krein-Fikhvatov [2]).
In fact we will begin the section with this latter
material since it seems to be of more current interest.
R. W. CARROLL
44
The Hersh-Steinberg [ 7 ] theory goes as follows.
N (4.3)
Lu
k t
1 Pi(A)DtN-i u;
=
D u(0)
i=o where
P.(A)
=
05 k
Consider for
5
N- 1
Uk
is a polynomial sum of ordered products of generally noncommuting
One densely defined linear operators A in a Banach space F (1 5 j 5 n). j are a basis for a finite dimensional Lie algebra of operators A assumes the A j over IR which is "integrable". (Standard elementary facts about Lie theory are assumed here and we refer to Helgason [ Z ] and Varadarajan [l] for details.) Explicitly
(4.4)
[Ai,Ajl
" k 1 cij\
=
k= 1
and to explain the word integrable we recall that if sentation of a Lie group then
operators on g'(0)
=
d.rr
induces a map
TI
F. Here if
aE
on
G
:IR
g
G
g :
IR
-f
G
on G.
and an operator L that
in IRn
into densely defined linear
TI
a
= g'(0)
E
9
G
of
on
g(0) = e
and
into a nbh of
The right translation Rh
so
then
that the diffeomorphism
a
such that
one defines for
A
h
j
=
E G
is defined by
is said to be right invariant if
e e
F
9.
is seen to be right invariant. Now if
local coordinate system at 0
S
is a smooth curve with
form a basis of
as above and
for smooth functions f
so
G
and a representation j
f (gh-')
is a repre-
5 . The algebra A is said to be integrable if
for all j where the a
Now with
L(F)
then
there is a Lie group
Lf(g)
-f
+
(i.,e. a continuous homomorphism into L(F))
taking the Lie algebra
is defined on a dense set of
d?T(a.) I
F
: G
TI
s =
-1
Rt,
LRhf(g)
.
\f(g) =
( s ~ , .. ,sn)
@(s)
is a
takes a nbh of
has an expression as a first order differ-
ential operator in these coordinates. Take the basis
a
of
in the form
=
45
OTHER TECHNIQUES
ak
=
ISzo
a/as,$(s)
and one has then
Indeed in a nbh of
e
one can write
f
=
f
@
0
tion of euclidean coordinates and one writes To evaluate
notation.
P,i(s)
so that
of
@-I.
(P,i(s))
m
g.
clearly n(g)y
y
One knows that
A
where
E
F
defined by
L
Cm(G)
(note that
=
'f, Lh>
a*
=
-a + c
extend to be operators on Definition 4.1
If
Q(T)S
=
for where
Cm(n)
is a func-
in standard
$;I'
denotes the
?(@-'(@(r)$(s)h)) ith coordinate
is
.rr(g)y
dll
A.A. j i
G
F and
so that
n(g)y
E
F
valued
D(A.)
while
J
a . r i ( g ) = A.n(g) J
3
DR
a i2 j
as an
so
of right invariant in this correspondence).
D(G),
E'(G),
E(G),
denote the right invariant Haar measure
L
: D(G)
f ,h E D ( G ) c
=
Cm
-f
D(G)
where
is a constant.
a (formal) adjoint is
< f , h> = IfKdu.
Evidently
a
One shows
and
5"
E'(G).
T E €'(G)
operator with domain
(4.8)
Let
is a linear operator
then easily that
a^i/as,
0 we consider
is dense in
Cm'(n)
D ' ( G ) , etc are all well defined.
Then if
where
=
such that
One has a standard notion of distribution on
G.
@-I
for
is isomorphic to the Lie algebra
differential operators on
on
r
0
A simple calculation then yields
6 D(aj).
n(g)Cm(n)
7
=
is clearly invertible.
C (IT)be the set of
function of
I$-'
af/ask
at
Irzo
a/ark@il(@(r)$(s)h)
=
The matrix
Let next
that on
a/arkf(@(r)@(s)h)
0
C
the quantization
F
Q(T)
is defined as a linear
by the rule
IT(g)n(g)Fdu. J
c This is an abuse of notation of course and means valued
E' - E
bracket relative to dli. w
easily shown that for
To
seP
this consider
€ C
(n)
ir(g)t>
for a vectnr
This conrept is very useful since it is
R. W. CARROLL
46
The property (4.9) allows one to transform (4.1) into a partial differential equation on
from (4.3) replace A
Now in Lu P(T,U)
=
G.
Let P
LPi(U)~N-i.
terms of order N)
N
j
(T,U)
by
uj
and
Dt by
T
be the principal part of
to obtain a polynomial (i.e. the
P(T,U)
and define (4.3) to be of hyperbolic type if for all real
0 the equation P ( T , U ) N further
U
=
0 has
N distinct real roots T.(U).
Define
J
I
Theorem 4.2
If (4.3) is hyperbolic then
L
is hyperbolic as a partial differen-
tial operator in local coordinates s * @ ( s ) . The proof is straightforward and we omit details. Next recall that for T,S
E'(G)
E
one defines (with abuse of notation)
Then by patching together local solutions and using standard facts one can prove
If (4.3) is of hyperbolic type the Cauchy problem
Theorem 4.3 (4.13) has is
a
Lf.(t,g) 3
=
k 0; D f (0,g) t j
unique solution with
ern
f.(t;) J
in
=
E
(0 5 j, k
5
N-1)
Sjk6(g) E'(G).
ib
If
E Cm(G)
t.
Finally using (4.9) it is easily seen that for
5
E Cm(Trr)
then
I
fj(t,g)Q(g)dp(g)
OTHER TECHNIQUES
47
and t h e n e x t theorem f o l l o w s immediately from t h e p r e c e e d i n g r e s u l t s .
Let ( 4 . 3 ) be of h y p e r b o l i c t y p e w i t h t h e
Theorem 4 . 4
a l g e b r a of an i n t e g r a b l e a l g e b r a
A.
Then f o r
u
k
Pi(A)
i n t h e enveloping t h e problem ( 4 . 3 ) h a s
E Cm(7i)
a s o l u t i o n g i v e n by N-1
14-15>
1Q ( f j ( t , g ) ) u j .
u(t) =
j=O
I t i s a s s e r t e d i n Hersh-Steinberg
Remark 4.5
171 t h a t i f t h e a l g e b r a g e n e r a t e d by
A t i s a l s o i n t e g r a b l e t h e n t h e s o l u t i o n (4.15) of (4.3) i n unique. I t seems J p r o b a b l e t h a t a u n i q u e n e s s theorem based on t h e t e c h n i q u e s of S e c t i o n 2 could a l s o the
be d e v i s e d and we w i l l r e t u r n t o t h i s l a t e r .
The r e s t r i c t i o n t o h y p e r b o l i c equa-
t i o n s above i s o n l y t o i n s u r e t h e e x i s t e n c e of s u i t a b l e
f
j
such t h a t a l l b r a c k e t s
and i n t e g r a l s make s e n s e ; o t h e r s i t u a t i o n s can e a s i l y b e e n v i s i o n e d .
Example 4 . 6
A number of i n t e r e s t i n g examples and a p p l i c a t i o n s a r e g i v e n i n Hersh-
Steinberg [7].
The Weyl-Heisenberg a l g e b r a h a s a s a b a s i s t h e skew a d j o i n t opera-
tors
(4.16)
A
=
j
a/ax.; J
B~ = i x k ;
iI
each of which g e n e r a t e s a u n i t a r y group i n
(4.17)
where
tA. Jf(x) = f(x
e
ek
+
te.); J
e
t iI
o f t h e form
L2(Rn)
f(x) = e
it
itx
tB
f(x);
d e n o t e s an orthonormal b a s i s i n IRn.
e
kf(x) = e
k f (x)
Any l i n e a r p a r t i a l d i f f e r e n t i a l
e q u a t i o n L i t h polynomial c o e f f i c i e n t s can be w r i t t e n a s an o p e r a t o r e q u a t i o n of However t h i s i s n o t always t h e b e s t a l g e b r a t o
t h e form (4.3) u s i n g t h i s a l g e b r a . take.
(4.18)
It i s noted f o r example t h a t t h e e q u a t i o n u
tt
=
-U
2 xxxx + x uxx + 2 x u
4
1 4
- x u + - u
i s n o t h y p e r b o l i c i n t h e u s u a l s e n s e and would n o t l e a d t o a h y p e r b o l i c o p e r a t o r e q u a t i o n i n terms of t h e Weyl-Heisenberg a l g e b r a .
(4.19)
A
2
=
ix ;
B =
xnx +
1 -;
2
c
= i D
2 Y
However i f one t a k e s
R. W . CARROLL
48
t h e n ( 4 . 1 8 ) becomes sees that
[A,B]
A
tt
+ B2 +
= (A2
-2A,
.
which w i l l b e h y p e r b o l i c . [B,C] = -2C
while
Here one a r e formally
A,B,C
F u r t h e r t h e y e x t e n d t o u n i q u e skew a d j o i n t o p e r a t o r s on
g e n e r a t i n g u n i t a r y groups. and by Simon [l] A
= fA,B,C}
2 C )u
[A,C] = 4B, and
Cz(lR)
skew a d j o i n t on LL(IR)
=
u
is a t w o f o l d c o v e r i n g of
The H e r m i t e f u n c t i o n s a r e a n a l y t i c v e c t o r s f o r
is i n t e g r a b l e .
.
SL(2,lR)
I n f a c t t h e a s s o c i a t e d group
G
A n o t h e r a s p e c t o f c h o o s i n g a "good" o p e r a t o r
v e r s i o n of ( 4 . 1 8 ) i s t h a t one o b t a i n s n a t u r a l l y an e n e r g y
y i e l d i n g good a p r i o r i e s t i m a t e s .
with e l l i p t i c
L
A f u r t h e r a p p l i c a t i o n t o e q u a t i o n s of t h e form
is a l s o given.
We g o now t o some s e c o n d o r d e r e q u a t i o n s and w i l l s k e t c h v a r i o u s m e t h o d s .
f i r s t t h e e q u a t i o n ( 1 . 1 4 ) i n a Banach s p a c e w'(0) = ul. -f
(4.21)
Assume
w1 = w
Setting
wt = A:;
[ 1; 1;
+ w(0)
w2 = w '
and
=
A
=
F
with i n i t i a l d a t a
A
p(A) = t h e r e s o l v a n t s e t of
and
w(0) = u o
one h a s
[0
1
A
).
0
generates a s t r o n g l y continuous group
A
Consider
A
one e x p e c t s
T(*)
in
0 E
and a s s u m i n g
F
t o generate a strongly continuous
g r o u p g i v e n by
(4.22)
[
exp t A = cosh A t
I 0 I
]
-k
A t A.
A-'sinh
D(A)
I n f a c t i t is proved i n G o l d s t e i n 131 t h a t i f
t h e n t h i s i s indeed t r u e i n t h e s p a c e (but not i n F
X
F).
F x F
since
For v a r i a b l e
1
A- s i n h A t A
A(t)
i f € o r any
Isl,It/ < T
(1
'r
> 0
B ( t ) - B(s)II
x
F
where
First for
t h e r e is a c o n s t a n t
5
mlt-sl.
D(R)
D(A2)
X
D(A)
h a s t h e g r a p h norm
d o e s n o t d e f i n e a bounded o p e r a t o r i n
w i t h c o n s t a n t domain
t h e o r e m i s proved i n C o l d s t e i n [ 2 ] .
Lip(E,F)
D(A)
is t a k e n t o b e
Also i f
D(A(t))
R(t) m
E =
D
=
L(E,F)
m(T)
I)(A) C D(B)
t h e following
B(-) E
one says
such t h a t f o r with
A-1
E L(F)
49
OTHER TECHNIQUES
then
BA-l E L ( F )
Theorem 4 . 7
from
0
Let
2
(B(t)x,x)
by t h e c l o s e d g r a p h t h e o r e m .
F = H
be a H i l b e r t space with
c ( t ) I I x1I2
for
x E D(B(t))
o n bounded i n t e r v a l s .
(4.23)
+
w"
6
with
w
Here
D($(t))
E
t i o n term
w(0)
B(t)w = 0;
and
Assume
t
+
(= D ( A ( t ) ) = D
self adjoint satisfying
c(t) > 0
where
B(t)B-l(O)
i s bounde d away
E Lip(H).
2 w E C (H).
6
is automatically constant with
P(.)
with
The n t h e p r o b l e m
w ' ( 0 ) = w1
has a unique s o l u t i o n
w1 E D
P ( t ) E L(D,F)
wo;
=
6
=
B(t)
E Lip(D,F)
s k e t c h some o f t h e s t e p s of t h e p r o o f .
c a n b e added t o
and a p e r t u r b a B(t).
We w i l l
A p r e l i m i n a r y lemma b a s e d o n K a t o [ 2 ; 81
( c f . a l s o M i z o h a t a [l]) s t a t e s
Lemma 4 . 8
Let
F
b e a B anach s p a c e w i t h
t i n u o u s semigroup i n L(F)
with
Q(t)C(t)Q
Q(-) E Lip(F). tive
i
L(F)
-1
(t)
i ( - )E
d o ma i n i n d e p e n d e n t of
t.
an d o n e s u p p o s e s
(4.24)
dU
dt
: B+
Lip(F),
F
- C(t)U;
P r o o f of Theorem 4 . 7 :
R-l(t)
-
E(O))-'
w(t) E F = D x H
R(*)
:' R I
E L ( F ) , a nd
-+
L(F)
L(F)
Q-'(t)
€
and
with strong deriva-
E ( t ) = R(t)C(t)R-'(t)
has
E L(F)
From t h e s e h y p o t h e s e s i t f o l l o w s t h a t t h e r e i s
satisfying
U(0)
=
f
E
D(C(0)).
( 4 . 2 3 ) is equivalent to
-f
wh e r e
be such t h a t
Then
D(*) E Lip(F). +
Q ( t ) E L(F)
g e n e r a t i n g a c o n t r a c t i o n semigroup i n
D ( t ) = (I - E ( t ) ) ( I
u
Let
Suppose t h e r e i s a f u n c t i o n
such t h a t
a unique
t > 0.
for
t h e g e n e r a t o r of a s t r o n g l y c on-
C(t)
with
D(A(t)) =
8x0.
Let
Ft = F
w i t h t h e norm
50
R. W. CARROLL
and t h e n f o r f i x e d
11
w i t h norm
A(t)
t,
g e n e r a t e s a u n i t a r y group i n
I( A ( t ) f l (
fll =
(D
has the
argu me n t t h a t t h e r e i s a n o p e r a t o r Qo(t) : Dt
+
Set
2 = Qo(t)
To(t)
norm).
Q ( t ) E L(D)
is u n i t a r y , and
D = D
Do
considered i n
: D
Qo(t)
Do
with
To(t)
T > 0
Let
6
w i t h doma in
D
=
E L(D ),
T o ( t ) = B-'(O)B(t). (this takes a
T ( * ) E Lip(D ) [O,T]
CI
5
To(t) < c-lI
A standard formula ( c f . C a r r o l l [ 4 1 , Yosida
[I]) a l l o w s o n e t o w r i t e now f o r
1
Q,'(t)
so t h a t
b e f i x e d and t h e n o n
Do.
s e l f a d j o i n t on
Dt
is p o s i t i v e s e l f a d j o i n t .
D
+
Let
One shows by a r o u t i n e
such t h a t
Th e h y p o t h e s e s o f Theorem 4 . 7 w i l l i m p l y now t h a t
little calculation).
L(Ft).
f E D = D
m
(4.26)
Qo(t)f =
s4(sI
+
To(t))-'To(t)fds.
0
U s i n g t h i s f o r m u l a o n e c a n show t h a t p o s s i b l e t o a p p l y Lemma 4 . 8 .
Then
and
Q(t)
unitary.
Q-'(t)
He n c e
A(t)
First with
belong t o
Q(t)A(t)Q-'(t)
Q ( t ) e x p s A( t) Q - ' (t )
Q o ( * ) E Lip(D)
L(F),
F = D
Q(*)
H
X
( c f . Kato [ 4 ] ) . ( D = Do)
E L i p ( F ) , and
g e n e r a t e s a u n i t a r y group i n
(s E W ) .
Next s e t
R ( t ) = I.
g e n e r a t e s a u n i f o r m l y bounded g r o u p i n
L(F)
One knows with
I t i s now
define
: Ft
Q(t)
F
+
is
L ( F ) , namely
1 E p(A(t))
D(A(t))
=
5
x
since
D.
Hence
i n Lemma 4 . 8 o n e c a n w r i t e
a n d t o show
D(*)
E
Lip(F)
i t s u f f i c e s t o show
f o l l o w s w i t h a small c a l c u l a t i o n .
A(.)A-'(O)
E Lip(F)
which
Hence Lemma 4 . 8 applies a n d Theorem 4.7
is
QED
p rov e d .
We c o n s i d e r n e x t some v a r i a b l e domain p r o b l e m s f o l l o w i n g C a r r o l l - S t a t e
[25].
The
b a s i c problem w i l l be phrased a s
(4.28)
w"
+
B(t)w
=
f
i n a separable Hilbert space
H
and w e w i l l all-ow t h e doma ins
D(B(t))
to really
51
OTHER TECHNIQUES
vary i n time.
T h i s i s a c c o m p l i s h e d by u s i n g a t e c h n i q u e o f C a r r o l l [ 4 ; 3 6 ; 3 7 ; 381
t o d e s c r i b e t h e v a r i a t i o n o f s u b s p a c e s i n terms o f o p e r a t o r s ( c f . a l s o C a r r o l l Cooper [ 2 4 1 , Cooper [l; 2 1 , Mazumdar [l; 2 ; 3 ; 4 1 ) .
Other a b s t r a c t techniques
f o r v a r i a b l e domain p r o b l e m s a p p e a r i n B a i o c c h i [ l ; 2 1 , B e r n a r d i [ l ] ,B e r n a r d i B r e z z i [ 2 ] , G e a r h a r t [ l ] , K a t o [ 6 ; 7 1 , L i o n s [ 4 ; 5 ; 10; 11; 1 2 1 w h i l e some r e c e n t
[3],
work w i t h a p p l i c a t i o n s t o s c a t t e r i n g t h e o r y c a n h e f o u n d i n Bandos-Cooper Cooper [ 3 ; 4 ; 5 ; 6 1 , Cooper-Medeiros
[ 7 ] , Cooper-Strauss
[ 8 ; 9 1 , Inoue [ l ] ,
Strauss [ 3 ] .
We w i l l d e a l w i t h a weak v e r s i o n of ( 4 . 2 8 ) f i r s t .
H
f a m i l y of H i l b e r t s p a c e s d e n s e i n t h e a n t i d u a l s o one can w r i t e b e d e t e r m i n e d by
D(S(t))
=
V(t) C H C V ' ( t ) .
(x,y) = ((R(t)x,y))t
details).
one t o one o n t o
The o p e r a t o r s
H
for H
with
L 2 ( V ( t ) ) = { u E L (H) product
on
Let
x E H and
R(t)
and
: H
Let
V'(t)
V(t)
be
(into)
Then
y E U(t). R -4( t )
S(t) =
+
be a
R-l(t)
maps i t s domain (see Carroll [ 4 ] f o r
( ( x , ~ ) =) ~(Sx,Sy)
were c a l l e d " s t a n d a r d " o p e r a t o r s a t o n e t i m e b u t
S(t)
p e r h a p s a b e t t e r name would b e c a n o n i c a l
2
V(t) C H
with continuous i n j e c t i o n s .
is a p o s i t i v e s e l f a d j o i n t o p e r a t o r i n V(t)
Thus l e t
[O,T],
( ( U , V ) )= ~ (T(Su,Sv)dt.
V(t)
u ( t ) 6 V(t)
related operators.
Let
2
Su E 1, (HI)
a.e.,
W =
with s c a l a r
One makes h y p o t h e s e s w h i c h i n s u r e t h a t
J
11
S-l(t)II
5
c
with
t o be measurable f o r
;-'(*)h
measurable i n
k E H).
Then
H
W C L (H).
c o e r c i v e , s e l f a d j o i n t s e s q u i l l i n e a r form on
cII
xllt
11
yllt
and
b(t,x,x)
2
for
2
2
all x l l t .
Let
V(t)
x
h E H
(i.e.
b(t,.,.) V(t)
(S-'(-)h,k)
be a c o n t i n u o u s ,
with
Ib(t,x,y)I
2
Such a form i n d u c e s a t o p o l o g y on
V(t)
e q u i v a l e n t t o i t s o r i g i n a l t o p o l o g y a n d we i n t r o d u c e a new H i l b e r t s t r u c t u r e on V(t)
with s c a l a r product
B(t) : V ( t )
+
antiduality).
V'(t)
b(t,x,y).
Clearly
b(t,x,y)
i s l i n e a r and c o n t i n u o u s ( < , > t
= t
denotes
T h i s d e t e r m i n e s a n unbounded o p e r a t o r i n
where
V(t) - V ' ( t )
H , d e n o t e d a g a i n by
B ( t ) , w i t h domain
and f o r
x E D(B(t))
one h a s
b(t,x,y) = (B(t)x,y).
Since
b(t,x,y)
= h(t,y,x)
is
52
R. W.
we s e e t h a t
B(t)
Next we l e t
O(t) : V ' ( t )
is a s e l f adjoint operator i n
<x,y> = ((€J(t)x,y)), S
-2
(t)
-+
B(t)
V(t)
so t h a t
when r e s t r i c t e d t o
work w i t h t h i s
CARROLL
be t h e i s o m e t r i c isomorphism determined by
B ( t ) = O-'(t)
H.
H , c l o s e d , and d e n s e l y d e f i n e d .
Thus
in (4.28).
2
i s s e l f a d j o i n t and we w i l l
B(t) = S ( t )
S u i t a b l e p e r t u r b a t i o n s of
be p e r m i s s a b l e ( c f . C a r r o l l - S t a t e
O(t) =
and one shows e a s i l y t h a t
B(t)
w i l l of c o u r s e
[25]).
Now a weak v e r s i o n of ( 4 . 2 8 ) i s t o f i n d
w E W , with
w'
E
and
L2(H)
w(0) = 0 ,
so that
(4.29)
for a l l
v
E
W
with
v'
E
LL(H)
and
v(T) = 0.
I n g e n e r a l a term (wl,v(0))
w i l l be added t o t h e r i g h t s i d e o f ( 4 . 2 9 ) b u t we assume
(wl
city
for
C1 (S
-1
-
~ ' ( 0 ) ) . We assume t h a t
h , k E H)
S-l(.)
i s weakly
here f o r simpli-
w1 = 0 C1
(i.e.
i n which c a s e t h e r e a r e s e l f a d j o i n t o p e r a t o r s
11
( t ) h , k ) ' = (8-'(t)h,k),
S-l(t)
11 5
Lipschitz continuous ( c f . Carroll [ 4 ] ) .
11
cl,
i-'(t)
11 5 c 2 ,
and
(S
-1
(*)h,k)
8-'(t)
with
S-l(-)
is
Now i n making ( 4 . 2 8 ) o r ( 4 . 2 9 ) i n t o a
f i r s t o r d e r system an a d d i t i o n a l term a p p e a r s i n o r d e r t o t a k e i n t o account t h e v a r i a t i o n o f domain.
when
0
( ),
w;
Thus s e t
makes s e n s e .
w
Let
w = S -'wl
=
2 ), [z:]€H=L(H 2 )xL(H
and c o n s i d e r t h e problem of f i n d i n g
++
-(w,v')
(4.30)
for a l l
-t
v
E
W
+ );,;(A
with
+
+
v'
E
H
and o b s e r v e t h a t i n
-L
w E H
D'(H)
w = W x W ,
+ f =
such t h a t
+ + ++ (Aw,sv) = ( f , v )
and
+
v(T) = 0.
Here a
A
term is added on f o r
convenience i n t h e argument; i t does n o t a f f e c t e x i s t e n c e o r u n i q u e n e s s .
To s e e
t h a t ( 4 . 3 0 ) i s a c o r r e c t t r a n s p o s i t i o n of ( 4 . 2 9 ) w e w r i t e o u t t h e e q u a t i o n s a s
is
OTHER TECHNIQUES
(4.32)
53
(wl,Sv2)dt
Th u s t a k i n g setting
v 1 = S-'@
X = 0
X = 0
define
(0,T)
on
w e have
v '1 =
and
i S-'@'
( 4 . 3 1 ) says
w2 = (S-'w1)'
wh ic h means
@ E D(H)
for
=
w
-1
D'(H).
w2
so that
w1
S
=
in
+ w
Hence g i v e n a s o l u t i o n =
w'
and
(wl,Sv2)
of ( 4 . 3 0 ) w i t h
= (Sw , Sv2) = b ( t , w , v
).
2
Then ( 4 . 3 2 ) s a y s
(4.34)
+
-IT(w',v;)dt '0
wh ic h i s ( 4 . 2 9 ) w i t h
d e f i n e d as L'
D(L)
with
L'u
=
=
v2 = v .
As i n C a r r o l l - C o o p e r
Lu = u '
/Tb(t,w.v2)dt ' 0
W'
[24] = {u E
-u'
2
= L (V'(t))
=
O-lW
2
W ; u ' E L (H): u ( 0 )
with
D(L')
=
proved t h a t
Ls = L = ( L ' ) *
Co op e r [ 2 4 ] , n a me l y i f
=
S-2(.)
L
.
i s mo n o to n e i f f o r
(rake here
F
Similarly
2
L (H);
S-l(.)
is w e a k l y
C1
then
D(Q)
D(Q) = F
W'
Both
L
C'
by is
a nd
i t is
T h i s is a v e r y Q : F
We r e c a l l n e x t t h a t a map
-f
u,v E D(Q),
dense we s a y
Q
When
Q
is l i n e a r
i s ma xima l monotone i f i t i s n o t t h e A (nonlinear)
Q : F
+
i s bounded i f i t t a k e s bounded s e t s i n t o bounded s e t s a n d hemi-
c o n t i n u o u s i f i t i s c o n t i n u o u s from l i n e s i n is c o e r c i v e i f
W'
-t
b e i n g weakly
Ls = Lw.
p r o p e r r e s t r i c t i o n of a n o t h e r l i n e a r monot one o p e r a t o r . with
u ( T ) = 01.
t o b e a r e f l e x i v e Banach s p a c e f o r e x a m p l e ) .
and monotone w i t h
: W
L'
-f
A c t u a l l y a s t r o n g e r r e s u l t i s proved i n C a r r o l l -
u s e f u l c r i t e r i o n and t h e p r o o f i s n o n t r i v i a l . F'
E
a r e d e n s e l y d e f i n e d a n d a s a c o n s e q u e n c e of
-
0).
=
{u E W; u '
L : W
and one d e f i n e s
F
t o t h e weak t o p o l o g y of
F';
Q
F'
54
where
R. IJ. CARROLL
$(x)
-f
x
-+
($(x)
m
can b e n e g a t i v e f o r
[ 2 4 ] , by a r e s u l t of B r g z i s [ Z ]
Carroll-Cooper Ls = Lw.
as
m
x
small).
Now, f o l l o w i n g
i s maximal monotone s i n c e
L
The f o l l o w i n g s p e c i a l c a s e of a r e s u l t of Browder [8] i s then needed
f o r o u r development.
Lemma 4 . 9
Assume
i s a r e f l e x i v e Banach s p a c e .
F
d e n s e l y d e f i n e d l i n e a r maximal monotone map and t i n u o u s bounded c o e r c i v e map.
L
Then
+Q
Let
Q : F
maps
D(L)
L : F +
F'
C
F
+
F'
be a closed
a monotone hemicononto
F'.
Consider now t h e r e g u l a r i z e d s t r o n g p a r a b o l i c problem a s s o c i a t e d w i t h ( 4 . 3 0 ) ( c f Lions [ 5 ] f o r p a r a b o l i c r e g u l a r i z a t i o n )
Problem 4 . 1 0
(4.35)
E '
w
Find
+
LzE +
such t h a t
6 W
P;"
+
+
EKZ'
= f
Let u s check t h a t a s o l u t i o n o f ( 4 . 3 5 ) s a t i s f i e s ( 4 . 3 0 ) w i t h a s u i t a b l e E term + + added, L e t v E [U w i t h :' 6 H and v(T) = 0. Then t D(L') i n an obvious (0 - Id'
n o t a t i o n and t a k i n g
brackets i n (4.35) with
+ v
we o b t a i n
Indeed t h e i n d i v i d u a l e q u a t i o n s i n ( 4 . 3 5 ) a r e
(4.38)
L
(4.39)
L
E
WE
+
hw
WE
+
xw
s l
s 2
1 E
2
+
0
and
s
v E V(t),
x
F
s w1
+ swE1 +
and one n o t e s t h a t , f o r V'(t)
-1 -1.-1
Ee
E
H
-1
- sw; +
EO
-1
w
E
1
=
0
E
w2 = f
and
v E V(t),
<x,v> = (x,v)
< x , v > = O OX,^)) = (SHx,Sv).
while, for
x E
Hence i n a p p r o p r i a t e s p a c e s
OTHER TECHNIQUES E ' + +E = (Aw , S+v ) ,
e t c . so t h a t
GE
t o find solutions
'
E '
55
E'
= (Sw
o f (4.35) w i t h
+ , Sv) ,
I[
E4 Ew '
and
etc.
[Iw
The p r o g r a m now i s E4ISw +E
=
Iff
bounded.
Then by weak c o m p a c t n e s s w e c a n t a k e weak l i m i t s i n ( 4 . 3 7 ) which l e a d s t o a s o l u t i o n of ( 4 . 3 0) . Q = XI
for
+P+
EK
x E V(t).
F = IU
T h i s i s a c c o m p l i s h e d by t a k i n g
i n Lemma 4 . 9 w i t h
and making t h e h y p o t h e s e s
GE
X > B one o b t a i n s
Then t a k i n g
s o l v i n g (4.35)
lGElff
r o u t i n e c a l c u l a t i o n t o v e r i f y c o e r c i v i t y ) and t h e e s t i m a t e s E
4ISw E'
IH 5
follow e a s i l y .
Theorem 4 . 1 1
Let
V(t)
V(t)
X
Assume t h e c a n o n i c a l
weakly
with (4.40).
for
and p u t on V(t)
h E H
V(t)
Examples a r e p r o v i d e d t o show t h a t
V(t)
V(s) C V ( t )
2
,
( 4 . 4 0 ) h o l d s , and
a bounded o p e r a t o r i n
H.
exists a solution
w
(Sw)' E W ' , u = eXtw
w' E W , and
then
u'(0) = w'(0)
w E
u"
+
+
2
S u
Xw(0).
S-'(-)
S
-1.-1 S
of
w"
w" E =
+
H
eXt[w"
2Xw'
E
with
f ' E ff
+ X 2w + s2w
=
f
and
0
=
2
s.
using f u r t h e r
w(0) = w ' ( 0 ) = 0.
+
+ X 2w +
2
S w]
.-2
with
S
-1
S , extends t o be
S
f(0) = 0
satisfying
while 2Xw'
t
I n p a r t i c u l a r h e assumes
S, o r equivalently
f E H
Then i f
for
need n o t be c o n s t a n t .
Mazumdar [ 4 ] w o r k s w i t h ( 4 . 3 5 ) d i r e c t l y f o r
C
satisfy
!251 t h a t ( 4 . 4 0 ) i s e q u i v a l e n t t o
h y p o t h e s e s and some r e s u l t s o f Mazumdar [ l ; 2 ; 31. i s weakly
S(t)
Then t h e r e e x i s t s a s o l u t i o n of ( 4 . 3 0 ) .
and t h i s l e a d s t o t h e r e q u i r e m e n t t h a t
Remark 4 . 1 3
t h e corresponding Hilbert
related operators
I t i s shown i n C a r r o l l - S t a t e
Remark 4.12
and
be a family of continuous s e l f a d j o i n t c o e r c i v e
structure. C'
< c -
Consequently one h a s
b(t,.,.)
s e s q u i l i n e a r f o r m s on
( a f t e r some
there
Sw E W,
Note h e r e t h a t i f
u ( 0 ) = w(0)
and
Examples show t h a t t h i s i s a l s o a t r u e v a r i a b l e domain
situation.
Reamrk 4 . 1 4
I n K a t o [ 6 ; 71 e v o l u t i o n e q u a t i o n s of t h e t y p e
w'
+
A(t)w = f
are
56
R . W. CARROLL
s t u d i e d where t h e
-A(t)
generate
n e c e s s a r i l y a n a l y t i c semigroups.
s e m i g r o u p s i n a Banach s p a c e
Co
F
but not
Such e q u a t i o n s a r e t h e n s a i d t o b e o f h y p e r b o l i c
t y p e and h a v e some a p p l i c a t i o n s t o symmeyric h y p e r b o l i c s y s t e m s of d i f f e r e n t i a l e q u a t i o n s ( c f . C r a n d a l l [ 4 ] , F r i e d r i c h s [ 1 ] , P h i l l i p s [l; 2 1 ) .
The domain
i s a l l o w e d t o v a r y b u t i t is assumed t h e r e is a d e n s e l i n e a r s u b s p a c e
D(A(t))
D C D(A(t))
on which
a c t s "smoothly".
A(t)
The R u s s i a n s c h o o l h a s t r e a t e d s e c o n d o r d e r p r o b l e m s of " p a r a b o l i c " t y p e e x t e n s i v e l y and e x t e n d e d t h e t e c h n i q u e s t o h i g h e r o r d e r e q u a t i o n s . K r e i n [ l ] we l e t a Banach s p a c e
(4.42)
Since
E
D(A(t)) = D
and h e n c e
(4.43)
u(0)
t
with
6 L(E)
A-l(t) E L(E).
i t follows t h a t
-
If
A(0)u'
u ( 0 ) F- D
from
i s continuous.
with
A ( O ) u ( * ) E C'
= w(t)
A(O)u(t)
v' = A(t)v
u(o)]
+
(A(O)u(t))' = A(O)u'(t)).
A ( * ) y E C1(E)
B(t)A-'(O)w;
for
L e t now
so t h a t
w' = A(O)v
S o l u t i o n s of ( 4 . 4 2 ) c o r r e s p o n d t o s o l u t i o n s of assumes
1'
is closed
u(t) E D
and
A(O)A-'(t)
u'(0) = u
uo;
=
i n d e p e n d e n t of
c o n t i n u o u s and A(0)
be closed densely defined l i n e a r operators i n
B(t)
and c o n s i d e r
A(O)[u(t)
u' = v
and
u" = A ( t ) u ' i- B ( t ) u ;
Suppose
A(.)u'
A(t)
Thus g o i n g t o
y E D
with
A t)
).
x'
=
+
c(t)x.
The f i r s t t e c h n i q u e
s u i t a b l y a p p r o x i m a b l e by Y o s i d a
approximations
A ( t ) = -nA(t)RA(t)(n)
(recall
R (1) = ( A A
-
hI)-')
=
while
A(t)(I -
R(t)A
1 -1 ; A)
-1
(0) E L(E)
with
B(-)A-'(O)
strongly
57
OTHER TECHNIQUES
differentiable.
Then
i s thought of a s
Q(t)
A(t) t 8 ( t )
where
A(t)
(resp.
The p r o b l e m i s v i e w e d as a
8 ( t ) ) h a s z e r o s i n t h e s e c o n d ( r e s p . f i r s t ) column.
+ p e r t u r b a t i o n of
= A(t)y
and a u n i q u e s o l u t i o n
using r e s u l t s f o r f i r s t order equations. y E D
11
while
RA(t)(A)II
5
l/l+h
If
A(.)y
h > 0
for
uo,ul E D ,
s obtained f o r E
1 C (E)
1 y E C (E)
for
for
( i n s t e a d of t h e approxi mat i on
p r o p e r t y ) and t h e o t h e r h y p o t h e s e s are u n c h a n g e d t h e n t h e s a m e c o n c l u s i o n f o l l o w s The t h r u s t of t h e m a t t e r i n
by i n v o k i n g o t h e r r e s u l t s on f i r s t o r d e r e q u a t i o n s .
t e r m s o f o p e r a t o r s t r e n g t h is d i s p l a y e d f o r c o n s t a n t o p e r a t o r s i n t h e f o l l o w i n g C
t h e o r e m w h e r e we s a y t h a t a n o p e r a t o r
11
and ql/
AXII
Cx)Iz
cII
Ax11
x E D(A).
for
is subordinate t o
if
A
I f i n addition f o r small @,
f o r a c o ntin u o u s convex f u n c t i o n a l
then
B
D(C) 3 D(A)
11
ri
+
C x I I I @,(x)
is s a i d t o be completely
subordinate.
Theorem 4.15
Let
and
A.
completely subordinate t o
A.
A
=
-BA-l
g e n e r a t e a n a l y t i c semigroups with D(B) n D(A)
uo
Then f o r
*O
u1 E D(A)
and
there
e x i s t s a u n i q u e s o l u t i o n of ( 4 . 4 2 ) .
One a l s o s p e a k s of a weak s o l u t i o n o f ( 4 . 4 2 ) i n r e q u i r i n g and
B ( * ) u E Co
o n l y on
(O,T]
i n s t e a d of
4.15 such s o l u t i o n s e x i s t uniquely f o r b e a n a l y t i c i n some s e c t o r c o n t a i n i n g
Theorem 4 . 1 6 and s a t i s f y Re h > w
and
A(t)
and
Let
(0,m).
Ao(t) = B(t)A
IIRA(t)(A)ll(M/IA-wl
.
E D(A)
A'(t)A-'(t),
and
-1
u1 E D(Ap)
for
p >
CY
2
,
A(-)u' E Co,
I n t h e s i u t a t i o n of Theorem and
u1 € E
and
u(t)
will
S i m i l a r l y one h a s
(t)
h a v e d o m a i n s i n d e p e n d e n t of
)IRno(t)(X)IICMo/lA-w
Ao(t)Aol(0) a n d A o ( t ) A - a ( 0 )
h e bounded and s a t i s f y a H 6 l d e r c o n d i t i o n .
[O,l]
and
Let
u
[O,T].
u E C
Then f o r
uo
I
for
f o r some f
Re (Y
t ).
> w
E
D(A) n D(B)
t h e r e e x i s t s a u n i q u e weak s o l u t i o n o f ( 4 . 4 2 ) .
I t i s i n t h i s s p i r i t t h a t Gne f i n d s c e r t a i n r e s u l t s f o r h i g h e r o r d e r e q u a t i o n s .
F o r e x a m p l e f o l l o w i n g Yakubov [ 5 ] w e c o n s i d e r f o r c l o s e d unbounded d e n s e l y d e f i n ed
%
58
R. W . CARROLL k D u ( 0 ) = uk
with
(0
t
[O,T]
( 4.45) on
on
[O,T]
k
5
n-1)
u E Cnhl(E)
for
[ O,T] ,
while
=
1.
u E Cn(E)
A1(-)Dt
( O ,T] , and A,(.)D:-k~
on
n- 1 u E Co(E)
and i s
(O,T]
on
The f o l l o w i n g t h e o r e m s are p r o v e d by w r i t i n g (4.45)
[O,T].
1 satisfy
S o l u t i o n s of t y p e
i s a Banach s p a c e h e r e ) w h e r e a s s o l u t i o n s
(E
on
1. 2
n
A.
wh er e
u E Cn(E)
with
of t y p e 2 s a t i s f y Co(E)
5
L
E
1
on
as a f i r s t o r d e r s y s t e m
and i n v o k i n g r e s u l t s on f i r s t o r d e r e q u a t i o n s .
Theorem 4.17
(6 > 0).
Let
S uppos e Assume
[O,T].
D(Al(t))
D
=
A l ( * ) A i l ( 0 ) E C 2 (Ls(E))
uk E D
with
11 RAl(t)(A) 11 5
dense with
f = f(t)
and
%(*)A;l(O)
a function i n
C1(E).
l / ( X + 6)
1 > -6
for
E C1(Ls(E))
on
Then ( 4 .4 5 ) h a s a
unqiue type 1 solution.
Let
Theorem 4 . 1 8 Re
> 0.
(k = 2 ,
Let
...,n)
k < n- 2
with
D( Al( t) )
A1(*)A-'(O)
E C1(Ls(E))
u
n-1
E D(A')
with
f o r some
c1
+ A ( t) Dnt - lu
( 4.47)
=
F(t,u
with s u i t a b l e d a t a
[Al(-)A-'(O)]' [ O ,T] . and l e t
Let
c/(l+
and
for
A,(t)A-'(O)
uk E D
f = f(t)
1x1)
for
0 <
satisfy a
,... ,Dn- 1u ) L e t u s m e n t i o n i n p a s s i n g t h e p a p e r of
[ 11 wh er e t h e a b s t r a c t Love e q u a t i o n
+ A(t)[u + Vu"]
u"
RAl(t) ( X ) I I (
are o b t a i n e d i n t h e framework
f
b u t we w i l l n o t b e more s p e c i f i c h e r e . Kharitonenko-Yur&k
> 0
I'
Then ( 4 . 4 5 ) h a s a u n i q u e t y p e 2 s o l u t i o n .
Theorems f o r more g e n e r a l
DYu
dense with
s a t i s f y i n g a HElder c o n d i t i o n on
HElder c o n d i t i o n .
( 4.46)
= D = D(A)
u(0)
=
and
f
u'(0)
is t r e a t e d i n a H i l b e r t space.
W e go n e x t
t o a t e c h n i q u e of R a s k i n [ 2 ] ( c f . a l s o D u b i n s k i j [l; 21, Gasymov [ l ] , G r i s v a r d
[Z])
f o r e q u a t i o n s o f t h e form
where t h e conditions
Ak
a r e c l o s e d d e n s e l y d e f i n e d o p e r a t o r s i n a Banach s p a c e
k Dtv(0 ) = vk
(0
5k 5
n-1)
a r e required f o r suitable
E.
v,.;.
Initial Assume
OTHER TECHNIQUES
E
L(E)
and set
-Bk
=
2
n
(1 5 k < n)
-%A&
59
with resolvant %(A)
= (A1+Bk)-l
satisfying
where
n- 1 < 6
i%
=
as before).
(A-XI)-'
(note we are taking
Let
Ln(A)
=
ZXn-k4,
RA(A)
(XI -A)-1
=
here instead of
and the main tool in this work is
the following lemma. Lemma 4.19
1'
11 x I I
BkX!I
for x ED(Bk)
commutes with > u1 Re X -
Proof:
for k
Sk
and
11
R < Bo
c > 0 and
For some
Lil(A)ll
=
=
6 - (n - 1)
n D(Bk+l)
let
11
...,n -
k = 1,
and
2,...,n. Then Ln(X)
Bk+l~I(5
1.
Assume
-1 has an inverse Ln ( A )
B1
=
A1
for
5C~~XI-~.
One writes first
Since
BkBk-l
B1
where
il > i2 >
=
..- >
4,
it follows that
i . Hence writing m
we can express (4.51) in the form
One can establish the estimate and hence
11
K(X)ll(
1
11
K(A)
11 5
c2
I I
for sufficiently large
guarantees the existence of
-1 Ln ( A )
13-Bo for
IAl
Re X
(since
2
o1
B < Bo).
(see below) This
from (4.53) which we can write in the form
R. W. CARROLL
60
with a norm estimate as indicated. To obtain the estimate on
1x1
that for
11 K ( X ) I I
observe
large
i > 1 it follows that (note 1
Hence for any
Rn(X) for Similarly Bi Bi R1(X) 1 2 Rn(X) Rn(X) = R3(X) B2B1R1(X)
il > 2
0 . .
..*
-
XR2(X
D < 1)
can be estimated in terms of * a *
R,(X)
-
hB2R1(X)
Rn(X)
and using (4.56) it follows that
II K ( X ) II
-
QED
sni
n-i-1 Let now v
E
i
D(As),
=
.
0,. . ,n- 1, and write
Then there results Theorem 4.20 Under the hypotheses of Lemma 4.19 the function (4.59)
is a
v(t) Cm
=
I
o+im exp XtL~l(X)$o(X)dX Zni o-im
solution of (4.48) for
t
> 0.
n
Theorem 4.21
Let
"
(7
while
D(Bs))
I,!J~(X). for
Then t > 0
v(t)
with
v
6
for
D(As)
i 1 vk+2,...,~ n-1
i
= 0,.
=
v
j
with
n- 1 v ~ + ( ~n D ( A s ) )
u
1 satisfy the condition used in constructing
given by (4.59) is k + l D:v(O)
..,k
for
0< j< k
times continuously differentiable
+
1.
61
OTHER TECHNIQUES
Somewhat more a b s t r a c t l y and g e n e r a l l y D u b i n s k i j [ 2 ] c o n s i d e r s t h e m a t t e r a s f o l l o w s ( c f a l s o A t k i n s o n [ I ; 2 1 , Browne [ I ] , C a r r o l l [ 4 1 , C h a z a r a i n [l], C o r d e s
[ 2 ] , DaPrato-Grisvard
[ 2 ] , I c h i n o s e [l; 2 ; 3 ; 4 1 , KsllstrGm-Sleeman [I]).
[:I,
[ 3 ] , D u b i n s k i j [ l ] ,B . F r i e d m a n
Gasymov 111, G r i s v a r d
[ l ] ,Mohammed [ l ] , Roach-Sleeman
Let s
(4.60)
.
S
1 A.DJ*
P(Dt) =
1 AjX j .
P(h) =
j =O
j = J~ t '
The i d e a i s t o c o n s i d e r f i r s t a g e n e r a l o p e r a t o r e q u a t i o n
s (4.61)
.
1 A.BJu
P(B)u =
j=o
= h
J
A. J
with closed densely defined operators that the spectra
o(B)
and
(B-A)x=O
f o r some
B
i n a Banach s p a c e
r
a r e d i s j o i n t with
i f 3 x E D(P) = nD(A.) J
s e p a r a t i n g them. 'n = o ( P ) if
o(P)
and
E.
Suppose
a s u i t a b l e contour i n
(c
P(A)x = 0; A E o(B)
such t h a t
x E D(B) - o n l y p o i n t s p e c t r a a r e b e i n g c o n s i d e r e d .
Then f o r m a l l y a s o l u t i o n t o ( 4 . 6 1 ) w i l l b e
where
k
i s a n i n t e g e r c h a r a c t e r i z i n g t h e g r o w t h of
P-l(A)
at
m.
This is an
e x t e n s i o n o f m e t h o d s of G r i s v a r d [ 2 ] ( c f C a r r o l l [ 4 ] and s e e a l s o C h a z a r a i n [ l ] , DaPrato-Grisvard
Ichinose [ l ; 2 ; 3; 4 1 ) .
[3],
curve i n t h e halfplane to
lAl;
r.
Then I :
let
S+
( r e s p . S-)
Assume
O(P) C S - ;
111:
Re A > 0
D(A.B) J
going out t o
El
-f
c(l
and
E
D(P)
be
+
: E
F
1-1
ixl)k-j
Theorem 4.22
-f
-t
with a r c length proportional
m
=
D(BA.) J
with
E2
satisfying
E2
where
)I ( B
F = E
ans a n a l y t i c with E2
1
X
$
1
n E2; further for
E
and
: F
F
Let t h e above h y p o th e se s h o ld .
-f
+
El.
El
and
C E
(B - XI)-'
S+
- AI)-lll> c / ( l
A.P-l(A) J -f
E
Let
11:
A . B = BA: J J'
Assume t h e r e a r e Banach s p a c e s
i n b o t h norms
b e a smooth
b e t h e r e g i o n l y i n g t o t h e r i g h t ( r e s p . l e f t ) of
tinuous i n j e c t i o n s s u c h t h a t f o r a n y (B - XI)-'
r
More p r e c i s e l y l e t
: El
+
]XI)
1
S_
and
11
c
U(B)
E2 C E +
S+
and
w i t h conis 1-1 w i t h
D(B)
i n b o t h norms let
P-l(A)
: E2
-f
AjP-'(A))IZ
Then i t i s I m m e d i a t e t o v e r i f y
'Then f o r a n y
h E D(Bk+')
with
62
R . W . CARROLL
Bk+'h
E F
k+l.
by
C l I AjBju1IE2
F u r t h e r one h a s
then (4.62) gives t h e s o l u t i o n with
Wk+l
S e t now
of ( 4 . 6 1 ) g i v e n by ( 4 . 6 2 ) w i t h
u
there exists a solution
k
cII
B
k+l
Bkh E F
If in addition
hllF.
replaced
unchanged.
u E D ( B k+l P ( B ) ) ; Bk+'P(B)u
= (u;
k
E F}
ik+L = ( u ;
and
A B'uED(B k+l ) ;
1
Bk+lA.Bju E F I . J Theorem 4 . 2 3
u
The s o l u t i o n
is dense i n
un E Wk+l
Then it f o l l o w s t h a t
of
P(B)u = h
i n t h e sense t h a t f o r any
Wk+l
such t h a t
u
-
Wk+l
t h e r e is a sequence
u E Wk+l
B ~ + ' P ( B )*~ B~+'P(B)~ ~
and
u
+
as a b o v e is u n i q u e p r o v i d e d
E.
in
T h e s e t h e o r e m s c a n b e r e f i n e d somewhat b y making f u r t h e r a b s t r a c t h y p o t h e s e s b u t we o m i t t h i s .
I n t h e c a s e when
E = H
with s p e c t r a l measures
G C H
with a s c a l a r product
(h
C
p);
11
11
EX
As = I
, )F
(
h
and f o r
AjPp-l(X)II
5
+
C(1
is a s e l f a d j o i n t o p e r a t o r i n a H i l b e r t s p a c e one c a n r e f o r m u l a t e t h e m a t t e r a s f o l l o w s .
h l l a = j( d E Ah , dEhh)F <
I holds with
1-1 w i t h
B
E
F
one d e f i n e s
=
(h E H;
(EX - E ) h E G
1J
where t h e i n t e g r a l i s o v e r
a]
O(B)
IXl)k-j
assume t h e map i n the
F
+
H
P-l(X)
norm.
If
U(B).
: F
+
Assume
D(P)
is
Then from s t a n d a r d
s p e c t r a l c o n s i d e r a t i o n s there r e s u l t s
Theorem 4 . 2 4 u E D(P(B))
(4.63)
u =
Under t h e a b o v e h y p o t h e s e s i f of
P(B)u = h
Wk
For u n i q u e n e s s l e t
=
k
'
u
+
h
a s above a r e u n i q u e .
let
X
B h
F
then a solution
i s d e t e r m i n e d by t h e f o r m u l a
Zll A j B J ~ l I H (
B A.B'un J
+
c(ll hlIF+
k
11
{ u E D ( P ( B ) ) ; B P ( B ) u E F} u E Wk
Assume f o r a n y
BkA.B3u E F ) . J and
k
with
P-'(X)dEhh JO(B)
and t h e r e i s a n e s t i m a t e
u
h E F
k
'
B A.BJu J
in
k
B h(lF).
Gk
and
t h e r e is a sequence H
( j = 0 ,...,s ) .
=
unf
{u
f
wk
D(P(B)):
such t h a t
Then s o l u t i o n s o f
P(B)u =
To a p p l y t h e s e c o n s i d e r a t i o n s t o d i f f e r e n t i a l e q u a t i o n s
b e a H i l b e r t o r Banach s p a c e a n d d e f i n e
63
OTHER TECHNIQUES
H (s,P)(X) q
Let
-1
<
for y
H (s,P,y)(X) 4
u(t)exP(-Yt)
{u : I
=
+
D(P) = nD(A.): I
Y
A
+
=
y
11
is 1-1 with
norm denotes Y
+
h with
=
A
is called y-regular if o(P)
and there is a Banach space Y
D(P)
an interval;
u
q,Y
Consider now (4.60),and equations P(Dt)u
Re
R
such that k . Further let H(q,k,I,y)(X) = {u; CII DJ(u(t)exp(-yt))1Iq 0 = L (X) = Lq(X) with weight function exp(-yt)).
(note H ( q , O , I , y ) ( X )
P(Dt)
C
a real number be the space of
H (s,P)(X). q
Definition 4.25
I
AjP-'(A)II
5
X
C
I.
lies outside the line
such that for Re X
+
c(y)(l
=
=
y, P - l ( X )
..., s )
where the
(j = 0,
IXl)k-j
:
X.
It is straightforward then to verify the following theorem. Theorem 4.26 _____
Let
P(Dt)
=
t
h
and
in L 2 (IR, X).
he the spectral measures for -iDt the equation P(D ) u
Y
be y-regular with
X
Hilbert spaces.
Then for any
has a solution u E H2(s,P,y)(X)
Let
EX
h E H(Z,k,I,y)(Y)
given by
.m
If
= {u E
A u
D(P);
3
is dense in
Yl
E
______ Definition 4.27 The operator P(Dt) that
O(P)
C
{A;
Re
<
is 1-1 and analytic for c(a)(l
+
IXl)k-j
(0
t
Definition428
P(D ) t
is called parabolic if there is a
Re X > yo s)
while if
with norm Y
Re X
+
X.
2
11 AjP-'(A)
a > Yo,
P(Dt)
:
Y
yo +
such
D(P)
[I 5
is called backward
is parabolic.
is called hyperbolic if it is simultaneously parabolic and
backward parabolic. Thus there exists yo < Re strip yo -
then this solution is unique.
y 1 and on some Banach space Y CX, P-'(X)
I j5
parabolic if P(-D )
D(P)
X < yl while for Re
of Definition 4.27 holds.
5
y1
X < a < y
such that and
Re A
O(P)
1a
lies in the > y1
the estimate
64
R . W.
y
o r closed) s t r i p <
+
c(a ,al)(l
2
5
(0
y1
w i t h norm
s)
5
yo < a.
while f o r
5
j
O(P)
Y
+
X.
l i e s o u t s i d e some (ope n
5
Re h
11 AjP-'(A) 11
a l < Y1
is c a l l e d quasi-
P(Dt)
l i e s o u t s i d e of a f i n i t e number of (open o r c l o s e d ) s t r i p s
G(P)
2 Y2p+l
1
Re
5
< Re A o -
IAOk-j
hyperbolic i f
YZp
is called quasielliptic i f
P(D ) t
Definition 4.29
CARROLL
( p = O,..,,N;
1) w h i l e i n e a c h o f t h e s e " r e s o l v a n t "
N
s t r i p s t h e e s t i m a t e above holds.
T h e r e a r e a number of t h e o r e m s f o r t h e s e t y p e s of o p e r a t o r s t h e p r o o f s of w h i c h a r e a l l s h o r t and s t r a i g h t f o r w a r d .
Theorem 4 . 3 0
Let
h ( t ) E H(q,k
u ( t ) E H (s,P,y)(X)
t
a n d one may a ssume
Proof: u(0)
=
Let
0).
E
=
B
and
with
D u(0) = 0 t
T(D t )
a nd
for
0
5
k
5
be Parahas a
s - 1
i s t h e H e a v y s i d e f u n c t i o n (* d e n o t e s c o n v o l u t i o n i n
> 0
w i t h o u t l o s s of g e n e r a l i t y ) .
yo -
L (R+,X) 4.Y
with
Bu
=
Dtu
a nd
S i m i l a r l y f o r t h e backward p a r a b o l i c
y < a < -y
1
P (Dt )
a solution
D(B) = {u; u ( t ) E H(q,l,T,y)(X):
(E = E
Then Theorem 4 . 2 2 c a n b e a p p l i e d
y < -yo; t h e n f o r
Y > Yo
for
k
given by t h e formula
9
yo < a < y
whe r e
1, I , y)(Y)
P(D ) u = h t
b o l i c ; then t h e equation solution
+
First
let
a nd
E =L (R+,Y)). 2 4rY
h(t) E H(q,k+l,I,y)(Y)
u ( t ) E Hq(s,P,y)(X)
of
a nd
P(Dt)u = h
i s g i v e n by
Theorem 4.31 generality. P(Dt)u
=
h
Let Let with
t h e n a s o l u t i o n of
P(D t )
be h y p e r b o l i c and t a k e
h ( t ) E H(q,k+l,T,y)(Y) Dku(0) t
=
0
P(Dt)u = h
for
0
5
(same
k h)
with
5
y,
5
0
5 y1
w i t h n o loss o f
y > yl. Then a s o l u t i o n of
s - 1 i s g i v e n by ( 4 . 6 5 ) .
If
Y
<
Yo
w i t h no i n i t i a l c o n d i t i o n s i s g i v e n by
(4.66).
R e s u l t s f o r q u a s i e l l i p t i c and q u a s i h y p e r b o l i c
P(D t )
a r e a l s o given i n Dubinskij
65
OTHER TECHNIQUES
[ 2 ] where c o r r e c t problems a r e i n d i c a t e d . s u c h as
(A)
L
i n Lemma 4 . 1 9 o r
P(A)
For t h e s t u d y of o p e r a t o r " p e n c i l s "
i n t h e D u b i n s k i j work and some r e l a t e d
t o p i c s we m e n t i o n f o r e x a m p l e Abdukadyrov [l], A l l a k h v e r d i e v - G a s a n o v
[ll,
Corbaruk-GorbaFuk [ 2 ] , Gorbaruk-Vainerman
[ 4 ] , Markus-Mereutsa
M a t s a e v [l], Vornpaeva-Maslov 111, Yakubov-Balaev
Remark 4 . 3 2
v
[I; 2 1 , Corbacuk
[ 4 1 , Gohberg-Krein
[ 2 ; 31, Gasymov [ I ; 2 ; 3 1 , Gasymov-Maksudov
[l], Burk
[ l ] , Virozub-
[S].
The s p e c t r a l t h e o r y of C a r r o l l [ 1 8 ; 1 9 ; 20; 2 1 : 221 h a s a l r e a d y b e e n
s k e t c h e d i n book form i n C a r r o l l [ 4 ] s o we w i l l n o t d e v e l o p i t i n d e t a i l h e r e . However a few comments a r e a p p r o p r i a t e h e r e f o r c o m p l e t e n e s s and some m o t i v a t i o n was p r o v i d e d i n t h e i n t r o d u c t i o n .
4\
i D k = (-A) Ca
a
ja
(t)Dx
\
the
\
+
(-A)'
and
A
problem w i t h
a
hk
and
hi;'
(1
5
L(H).
E
5
ak,
N)
one can w r i t e
i s p o s i t i v e and s e l f a d j o i n t w h i l e
H
Thus l e t
t.
be a s e p a r a b l e H i l b e r t space
a family of p o s i t i v e s e l f a d j o i n t o p e r a t o r s with
(1 5 j < M)
b. E L(H) J and
ak
2
L (Rn)
F o r s i m p l i c i t y we t r e a t t h e c o r r e s p o n d i n g a b s t r a c t
i n d e p e n d e n t of
ja
Let
and w i t h t h e by t h e
a l l commute.
k
A = 1
where
=
a r e c o n t i n u o u s so t h a t i n p a r t i c u l a r
where t h e R ie sz o p e r a t o r s
= Zajka(t)RnAk
F
Recall t h a t i n
/ik.
A
Let
b . , and t h e i d e n t i t y
b e t h e c o m m u t a t i v e Bdnach a l g e b r a g e n e r a t e d We w i l l u s e a few f a c t s from Banach a l g e b r a
e.
be t h e j o i n t spectrum of t h e g e n e r a t o r s OA
(il($),
i
5
N
+ M)
of t h e
on
u n d e r t h e map
ak
y
OA.
EM).
in
EM
Clearly
ak
(resp.
b.)
( a A ( b ) = prM(OA) w h e r e zk =
a,(@) 2 0
in
OA
hi:
thus
=
15k
zk, and
(2' =
U A C (c
N+M
uA i s t h e image of
G ( $ ) ; then
v($)
< N -
OA
=
i s compact and
(resp.
z.
1'
N + l
5
aA(b) denotes t h e j o i n t spectrum
prPI d e n o t e s t h e p r o j e r t i o n of
a n d we s e t
Xk = z
the equation
is t h e c h a r a c t e r i s t i c polynomial
Let
d e f i n e d by
-f
@(x)
The s p e c t r a l v a r i a b l e s
correspond to
bi
and
= OA
..., g N ( @ ) , cl($), ...,c M ( @ ) ) w h e r e
homeomorphic t o
--
b e o p e r a t o r s commuting among t h e m s e l v e s
t h e o r y h e r e and r e f e r t o C a r r o l l [ 4 ] f o r t h e n e c e s s a r y d e t a i l s .
the c a r r i e r space
ak
( z ~ + ~ , . ,. z.
N+M
))
-1
.
EN+M
Associated t o
66
R. W. CARROLL
(4.68
(cf.
1 . 2 . 1 0 ) and the associated framework).
Here one thinks of ( 4 . 6 7 ) as a
c'
first order system as in Section 1 . 2 of the form
a matrix like (1.2.10) with bottom row of the form
generally let u s take a compact set
roots of
-
and write $(A,z')
Q(C,X,z')
+
it is I
on which
IcN+M
=
correct if for
2'
zk
Cs(h,z'),
max Re 5
b, while $ ( A , z ' ) < c -
P
is and
is real and
s = 1,.
. . ,m
z
> 0;
k-
be the
(X.2').
hyperbolic if for z ' E
$(A,z')
5
A > c real. The equation k- k h -alXI + b for real A > c * k - k'
E prMI, $ ( A , z ' )
2
c
parabolic if for z ' E prM 2 ,
is
0 where
...,Pm-,(A,b))
(Po(A,b),
The equation ( 4 . 6 7 ) will be called
$(x,z') 5 alAl
prM=,
zC
but this is not assumed. Let
Definition 4 . 3 3
=
Now one can develop the theory here in terms of aA but more
1 replaced by -1.
usually aA C
+P(A,b)u
for
for real
Xk > c k' +P(A,b)c
Now the spectral form of the Green's operator for the system
is 5 ( t , ~ , A , z ' ) = exp[-(t - T ) P ( X , Z ' ) ] and one wants to show that domain of a suitable functional calculus relative to
5
=
0
is in the
I with suitable
(t,T)
dependence. Then a matrix G(~,T) with entries in L(H) will arise yielding a + -+ This involves considerable solution of ( 4 . 6 7 ) in the form u ( t ) = G(t,T)u(T). analysis which we omit here (cf Carroll [ 4 ; 18; 19; 2 0 ; 2 1 ; 2 2 1 ) .
A typical
theorem is cited for completeness however. Thus we will say that 2 i s
5 $
convex if for any Ip(z)I p
< 1
for
one has
11
z E
I. We say I is
p(a,b)llc
a
Theorem 4.34 Let
there is a polynomial p
he a
p
spectral for A
p
c suplp(z,z')l
over
convex p
such that
=.
L
p(<)
=
1 and
if for any polynomial
Then
spectral set for A
with ( 4 . 6 7 ) a
parabolic equation. Then there exists a weak Green's operator 9 ( t , T ) (t,T) +G(t,T)
E
Co(Lw(H
m
)),
G(t,T) E C1(Ls(Hm)),
while 9
Remark 4 . 3 5
P(X) = A
( A -k B)u = h
Taking
in the form
G(t,sfi(s,~) =G(t,-r), t
+P(A,b)G
+
A in
=
p
G(T,T)
=
I, and
with t
0.
( 4 . 6 2 ) one obtains the solution of
+
z
67
OTHER TECHNIQUES
k
for
=
0
r).
( w i t h s u i t a b l e t r a v e r s e of a s u i t a b l e
T h i s i s t h e f o r m u l a of
G r i s v a r d [ 2 ] and h a s b e e n s y s t e m a t i c a l l y e x p l o i t e d i n D a P r a t o - G r i s v a r d
[3].
They
d i s t i n u g i s h two main t y p e s o f p r o b l e m s a s f o l l o w s .
D e f i n i t i o n 4.36
The o p e r a t o r
are invertible for
X > 0,
for
parabolic i f
ZA
= {A:
X > 0
with
...,
with
m = l,2, (A - A )
larg X I
5
TI
+
L = A
M
- OA1
CB
and
A
independent of
(B - A )
and
is hyperbolic i f
B
(A
and
-
{A:
5
larg X I
OBI
n -
(B - A)
and
The o p e r a t o r i s
m.
CA
are invertible in sectors =
A)
CB w h e r e
and
RA
with
+ BB
< n,
while i n these respective s e c t o r s
Thus f o r
L
above w i t h
hyperbolic
OA,OB < n/2
- A)
(A
and
11
and
(B - h )
( A - X)-'l(
5
are invertible in sectors as with
M/Re X
11
Thus t h e h y p e r b o l i c c a s e is a l i m i t of p a r a b o l i c c a s e s where limit.
T i m e d e p e n d e n t e v o l u t i o n e q u a t i o n s of t h e f o r m
-u'
+
(B -
OA
=
n
A(t)u(t)
=
f(t)
+
B
and r e s u l t s a n a l o g o u s t o t h o s e o f K a t o [ 2 ; 5 : 7 1 , Kato-Tanabe
[ l ] a r e obtained.
a r e s t u d i e d a l s o i n t h e C o n t e x t of noncommuting
For noncommutative
A
and
B
can a l s o be u t i l i z e d under s u i t a b l e hypotheses, e f f e c t i v e l y w i t h a s o l u t i o n of with
B
r e p l a c e d by
B - 5
I n t h e commutative c a s e
+
OB
u"
A ( t ) u ( t ) - hu = f ( t )
X)-'II 5 M/Re A.
S
r,
o n e a s s u m e s f o r example t h a t
Lu - i,u = h
(L
(c
- C)-'
(B - A ) - '
and A
and
[ l ] , and Tanabe
a formula s i m i l a r t o (4.69) One c a n work i n e i t h e r c a s e most =
A
+
R)
(for technical reasons).
=
in the
e x p r e s s e d by ( 4 . 6 9 )
Thus set f o r
5 > 0
whereas i n t h e noncommutative c a s e i f D(A)c D ( A )
for
A t p(B)
with
68
f o r suitable
R. W. CARROLL
4
will be small f o r
(0 = arg A ,
5
8'
large and
=
arg z )
(f - 5 ) -1
ity a s in Kato 16; 71 are also utilized.
(E - 5)s
then = S
5
(1
+ R$-l
=
5
.
1
+
R
where
5
R
5
Hypotheses of s t a b i l -
CHAPTER 2
TRANSMUTATION
2.1.
Preliminary remarks and examples.
The idea of transmutation of differential
operators will be useful in dealing with related abstract differential equations and we will discuss this from various points of view following at first Delsarte [I], Delsarte-Lions 1 2 ; 31, Levitan [l; 3; 51, and Lions [ Z : 3: 41 (cf. also Andros'c'uk [l], Braaksma [l], Braaksma-deSnoo 121, Carroll 1351, Carroll-Showalter [81, Chebli [l], Gasymov-Magerramov 151, Helgason [l], Hersh [3], Hutson-Pym [ I ; 21, LeBlanc [l], L&fstrgm-Peetre [l], Marrenko [l; 2; 3 1 , Povzner [l],
Thyssen [ I ; 21). polynomials
Let
P(D)
or distributions.
into
we have
u(x) in
onto
-1
u = B
E
F
Q(D)
Q(D)Bu
X
from
v
and P
Q
=
Y
0 so
v
=
then from
F and
P(D)u(x)
=
then setting v(y)
solutions of
F Q(D)v(y) ODE
=
=
Q(D)B
Bu
Y
and
of functions
B, usually an integral operator, transmutes =
BP(D).
Then for example if
satisfies Q(D)v
Q(D)v
=
0 we get
=
(Bu)(y)
g(A)v(y).
0. If
P(D)B-'v
0. More generally if
g(A)u(x) =
and consider (linear) differential
need not have constant coefficients nor be of the
if formally
satisfies P(D)u
so that in
a/ay
acting in suitable spaces X
Q(D) and
or
We s a y that an operator
same order. P(D)
D = a/ax
u
=
B
P(D)u
=
0
is an isomorphism
B-'Q(D)v
=
0 so
is a vector function with
for a suitable operator function
g(A)
acting
we have
It is in this spirit that one can connect
with operator coefficients and one s p e a k s then of related
differential equations. Let u s cite a few results on transmutation first to gain some perspective.
X
=
Y
be the space
H
of analytic functions of one complex variable
the topology of uniform convergence on compact subsets of
69
(c.
Let
z
P(D)
Let with
=
70
R. W. CARROLL
m
1 aj(z)D'
where
j =O
a.(.)
D = d/dz, and
H,
I
am(z)
=
1. Then Delsarte-Lions
[ Z ] prove
Theorem 1.1 BP(D).
There exists a continuous isomorphism B : H
Explicitly
B
H
-+
such that DmB
=
has the form p+km (p+km) !
'
m A s a corollary it follows immediately that if
Q(D)
1 b.(z)Dj
=
j=O
b (z) m
and
=
1 then there exists an isomorphism
: H
with
bj(-)
E
Q(D)B
=
H
J
H
-+
such that
The formal verification of (1.1) is straightforward and one need only con-
BP(D).
firm the convergence in order to prove Theorem 1.1.
Such results are however
special for analytic situations and the general question of just what be transmuted into what
Q(D)
is unsolved;
P(D)
can
it cannot always be done (cf.
Levitan [l]). Remark 1.2
Delsarte [l] was also concerned with generalizing the idea of trans-
lation and this is developed further in Levitan [ l ; 31. order differential operator for example of the form
A@
with
@(O,A)
= 1
and
@'(O,A)
= 0.
Lf
Thus let =
Write formally
@
f"
-
pf. k
=
m C
L be a second Let
L@
A @k with
=
L@, =
0
$k-l
and
so that
L@o = 0
L
Then the generalized translation operator associated with
for
f E Cm.
f(x)
while
Since bk(0)
@i(O)
0
=
=
0 for
for k
2
translation follows upon taking xn/n! that
s o that
LxT:f
(x)
DQn =
=
L TYf (x) Y X
k
1 with
@o(0)= 1
0 implies D TYf(x)ly=O Y X L = D, @(x,A) = exp Ax
= =
is defined by
we have
Tof(x)
=
0. The analogy with n n
I A x /n!, and
"k k and TYf(x) = ED f(x)y / k ! = f(x + y). 0 (x) while since LXT', = Z,@,(y)L,k+lf
an(x)
We note also
=
PRELIMINARY REMARKS AND EXAMPLES
71
For further comments in this direction see Remark 3.18. Now following Lions [2] we consider the Cauchy problem, for
f.
for suitable
priate space and define B P(Dx)@(x,y)
so
-
of order
m
(1.4) has a unique solution in some appro-
by the rule
(Bf)(y)
=
@(O,y).
-
-
Setting +(x,y)
=
we have
Consequently, given (1.3)
-
Assume that (1.3)
Q(D)
B
defined as above and the uniqueness of solutions of
(1.4) and (1.5) - (1.6), it follows that
that
B
transmutes P(D)
into Q(D).
We will utilize precise versions of
this later and for now simply state (cf also Remark 2.18) Theorem 1.3(F). and define
B
Example 1.4
Assume (1.3) - (1.4) and (1.5) as indicated.
where
=
(1.6) have unique solutions
BP(D).
Let u s apply this technique to (1.1.3)-(1.1.4)
in Example 1.1.10. find
Then Q(D)B
-
Thus let
P(D)
=
D2
and
Q(D) = D
with
toproduce (1.1.6) as g(A)
= A
2
.
To
we look at (cf. (1.3) - (1.4))
B
f
is a function in say
Remark 2.18).
s'
which we take (or extend) to be even ( c f .
A s with (1.1.12) one can take Fourier transforms in
x
to obtain
R. W. CARROLL
72
$t + s
2-
Q,
2
0, $ ( s , O ) = ? ( s ) , from which &(s,y) = F(s)exp(-s amd recalling that f~1 K(y,x) = exp(-s 2 y) we have =
(1.9)
Q,(x.y)
=
y)
where
&
FQ,
=
m
1
- C)d5
K(y,S)f(x -m
rm
a formula which makes sense even if
is of say exponential growth as 5
f(5)
Now apply this to the vector valued functions P(D)w
Thus
=.
2
A w
Q(D)Bw
= BP(D)w
BA2 w
=
f
(1.10) (cf. a l s o (1.1.13)).
5
-f
= A
Bw;
-
(1.10) corresponds t o the condition
u = Bw
we obtain ( 1 . 1 . 6 ) from the form of
w(S)
One should emphasize that
and be integrable against
m
2
in (1.8)
0. Consequently setting
=
t
of (1.1.3) - ( 1 . 1 . 4 ) .
w
and
K(y,c)
so we must take
must make sense a s
T(S)
to be a quasi-
equicontinuous group for example (local equicontinuity is not enough). Remark 1.5
With Example 1 . 4 in mind let us look at some other aspects of the
transmutation question.
Suppose we assume
DB
=
BDL
B
where
is an integral
operator of the form
I
where
is
or
(-m,m)
suitable behavior at
I
For
(0,m).
= (0,~)
we obtain formally, assuming
a),
1
m
(1.11)
(BD2f)(y)
=
b(y,x)f"(x)dx
=
0
+ bx(y,O)f(0)
j:bxx(y,x)f(x)dx
-
b(y,O)f'(O)
m
whereas
(I)Rf)(y)
=
[
b (y,x)f(x)dx.
Hence i E
'0
and K(y,x)
f'(O) =
=
(ny)
m.
and
the use of even functions w (0)
u
+
0 we will have
-L2 2 cxp(-x /4y)
DD
=
BD2
b
Y
acting on s u c l l
=
b
f.
satisfies these requirements.
xx
with
bx(y.O)
Clearly For
I
b(y,x)
= (-m,m)
=
0
=
thc,
AND EXAMPLES
PRELIMINARY REMARKS
s i t u a t i o n i s similar e x c e p t t h e c o n d i t i o n
I
intervals
2 A w
w (0) = 0 , and w e want t o f i n d
with
by a f o r m u l a
u
u(y) = j.b(y,x)w(x)dx.
satisfying
Then
u
=
on
(1.1.3)
A2u
b
if
Y
0 s i n c e from (1.11)
hx(y,O) = 0
f
s a t i s f i e s (1.1.4),
w
On t h e o t h e r h a n d s u p p o s e
m
w
d o e s not a r i s e ; t h e trjo
f'(0) = 0
a r e e q u i v a l e n t i f we c o n s i d e r e v e n f u n c t i o n s
f ' ( 0 ) = 0.
that
73
=
Y
(-m,a)
so
namely
wxx =
connected t o bxx
with
m
A2u(y) =
(1.12)
(y,x)w(x)dx
The i n i t i a l c o n d i t i o n a s is the case with
F
t > 0
i s a c h i e v e d when
b(0,x) = 6(x)
(one s i d e d )
b(y,x) = K(y,x).
Let us consider (1.l.S)
Example 1 . 6 in
u ( 0 ) = w(0)
u (y Y
=
when
A
g e n e r a t e s a n equicontinuous group
and f o l l o w t h e p r o c e d u r e o f S e c t i o n 1.1.
Thus f o r
--LD
< x <
and
a'
we a r e l e d i n t u i t i v e l y t o t h e h a l f p l a n e D i r i c h l e t p r o b l e m
It i s e a s i l y checked t h a t
r a t h e r t h a n a n i m p r o p e r l y p o s e d Cauchy p r o b l e m . G(t.x) =
1
-
2
H(t,x) = t n - l ( t 2
+
x2)-'
i s a s o l u t i o n o f ( 1 . 1 3 ) and we w r i t e m
(1.14)
1
z ( t ) = <- H ( t , . ) , 2
T(')u~>
= -
r"
=
I
I_.,
H(t,x)T(x)uodx
I-
H ( t , x ) c o s h Axu
'0 to o b t a i n (1.1.8)
formally.
Since
p(T(x)uo)
5
q(uo)
for seminorms
i t follows t h a t t h e b r a c k e t , or i n t e g r a l , i n (1.14) makes s e n s e f o r (since
IH(t,x)l
d o m i n a t e d hy
5
2t
;-(x
c(t)/xn
2 (n
+ 2
2 -1
t )
).
Similarly
2)
as
x
+ m
Ht,
Hx,
and
Htt
=
p
and
t > 0
-H x x
w i l l be
and o n e c a n d i f f e r e n t i a t e u n d e r t h e
i n t e g r a l s i g n i n ( 1 . 1 4 ) and i n t e g r a t e by p a r t s .
Thus
q
74
CARROLL
R. W.
(0,m)
( w h e n ' w o r k i n g on
note that
Hx(t,O) = 0
Thus w e c a n
wx(0) = 0 ) .
and
s t a t e ( l e a v i n g open t h e q u e s t i o n o f u n i q u e n e s s )
Theorem 1 . 7 while z
w
Let
g e n e r a t e an e q u i c o n t i n u o u s group i n
A
is t h e unique s o l u t i o n of ( 1 . 1 . 4 )
g i v e n by (1.14)
uo 6 D(A2 )
with
from Theorem 1.1.8.
Then t h e f u n c t i o n
(or e q u i v a l e n t l y (1.1.8)) i s a s o l u t i o n of (1.1.5).
L e t u s l o o k a t Example 1 . 6 from t h e p o i n t o f v i e w of t r a n s m u t a t i o n .
Remark 1.8
F o l l o w i n g Formal Theorem 1 . 3 we c o n s i d e r f o r
$xx
(1.15)
F
+
@lYY
and
Q(D) = -D 2
@(X,O) = f ( x ) .
= 0;
$ (x,O) = 0
Imposing a c o n d i t i o n
P(D) = D2
Y
would l e a d t o a n i m p r o p e r l y p o s e d Cauchy p r o b l e m
s o w e want t o f i n d some o t h e r way t o p r o d u c e u n i q u e s o l u t i o n s o f ( 1 . 1 5 ) . f
t o b e a n e v e n f u n c t i o n on
(1.15) t o o b t a i n as
y
+
$yy
+
2s @ = 0
i n s u c h a way t h a t
$(s,y) = P(s)exp(-sy).
(x2
-
(-m,m)
y2)-l
But
( c f . Remark 2.18) and F o u r i e r t r a n s f o r m
with $(s,y)
$(s,O) = ? ( s ) . +
F ?1 H ( y , x )
0 =
$
exp(-sy)
where
1 7 H(y,x)
$(x,y)
+
i s g i v e n by =
y~
-1
and h e n c e
z = Bw
a s i n (1.14).
The s p e c i f i c a t i o n o f Cauchy d a t a i n ( 1 . 4 ) i s o n e way o f a s s u r i n g a
unique solution B.
I f one r e q u i r e s
a s w e l l then a unique
A p p l y i n g t h i s t o v e c t o r f u n c t i o n s a s b e f o r e we o b t a i n
Remark 1 . 9
Extend
@
of ( 1 . 3 ) - ( 1 . 4 ) s o a s t o d e t e r m i n e as w e l l d e f i n e d o p e r a t o r
O b v i o u s l y a n y s p e c i f i c a t i o n of d a t a a t
uniquely determine
@
y = 0
p l u s growth c o n d i t i o n s which
w i l l s e r v e t o d e f i n e a t r a n s m u t a t i o n o p e r a t o r ( c f Remark
2.18).
Example 1.10 z
of ( 1 . 1 . 3 )
L e t u s e x a m i n e now t h e f o r m u l a (1.1.7) c o n n e c t i n g s o l u t i o n s
and ( 1 . 1 . 5 ) .
Setting
P(D) = D
and
Q(D) = -DL
f o l l o w i n g Theorem 1 . 3 ( F ) t o c o n s t r u c t a t r a n s m u t a t i o n o p e r a t o r
u
and
i s n a t u r a l and
B
one c o n s i d e r s
0
75
PRELIMINARY REMARKS AND EXAMPLES
w i t h some f u r t h e r s t i p u l a t i o n n e c e s s a r y i n o r d e r t o i n s u r e a unique s o l u t i o n . want t o produce a s o l u t i o n i n v o l v i n g as a k e r n e l which i t s e l f s a t i s f i e s
Ix = I
(1.1.8) i n
x
has t h e form (is)'
=
lfT
A(s) = 0
w e have G(s,y)
(1 + i ) s 4
=
$
I ( y , x ) where
Fc
A
=
,
G(s,y)
F
and
Thus
exp(-(is)'y)
I(y,Ix[)
I"
for
x
5
=
0
B
an i n t e g r a l o p e r a t o r w i t h k e r n e l
+ m
b(y,x).
z(y) = =
while
b(y,O)
=
0
u
= A
2
u
r O
0
I ( y , 1 x l ) e x p i s x d x = FI-.
Remark 1.11 I t i s i n s t r u c t i v e t o c o n s t r u c t
should have w e assume
FcI
and
z = Bu
x
as
y
+
then
m
d e n o t e t h e F o u r i e r c o s i n e and s i n e t r a n s f o r m s re-
Applying t h i s to v e c t o r f u n c t i o n s w e o b t a i n
as
0
exp(-s 4y / f i ) S i n s'y/fi
exp(-s 4y / f i ) C o s s4 y / f i = F,I(y,x),
I(y,x)exp(-isx)dx =
b(y,x)
+
Hence we t a k e
t o be t h e d i s t r i b u t i o n
z and assume
Now F o u r i e r t r a n s f o r m i n g
A
and i f w e s t i p u l a t e s a y t h a t
s p e c t i v e l y ( c f . E r d e l y i [l]).
I-
2
- is@= 0 w i t h @ ( s , O ) = f(s) t h e s o l u t i o n of which @YY A ( s ) e x p ( i s ) 4y + B ( s ) e x p ( - ( i s ) 4 y ) . C o n s i d e r i n g s real
is indicated.
But i t i s known t h a t
I(y,x) = 1 ~ r - ~ y x - ~ / ~ e x p2 (/4x) -y
( c l a r i f i e d below).
YY
A
W e
iFsI for
and i f w e t a k e x > 0
Consequently
which i s (1.1.7)
i n Example 1.10 by assuming i t is
W e can i n f a c t go d i r e c t l y t o
b(y,x)u(x)dx. and t h e n i f
it follows t h a t
FcI-
then
u
and
To s e e what p r o p e r t i e s b(y,x)
h a s s u i t a b l e growth
76
R. W . CARROLL
Thus
and we a r e l e d t o l o o k f o r a d i s t r i b u t i o n o r f u n c t i o n
(1.22)
b
YY
-
b
=
x
0;
b(0,x) = 6 ( x ) ;
Weknow a s o l u t i o n o f ( 1 . 2 2 ) i s Laplace transform (1.22)
6
(1.23)
where
YY
(x
b(0,s)
= sb;
b ( y , s ) = Lb(y,x) =
,
+
=
b(y,O)
=
b(y,x) = I(y,x) s)
b(y,x)
such t h a t
0 b u t t o d e r i v e t h i s we c o u l d
t o obtain
1
b(y,x)exp(-sx)dx
( s e e D o e t s c h [l], S c h w a r t z [l],
Sneddon [l], Widder [l], Zemanian [ l ] f o r L a p l a c e t r a n s f o r m s ) . (1.23) is
$ ( y , s ) = exp(-ys')
r e a l (the function
i f we r e q u i r e say
exp(ys4)
-
f o r Laplace transforms
+
as
0
But
LI(y,x)
and
u(t)
Let
s
for
(cf.
b(y,x) = I ( y , x ) .
g e n e r a t e an e q u ic o n tin uous group i n
F
a s follows.
with
u
b e t h e u n i q u e s o l u t i o n of (1.1.3) g i v e n by Theorem 1.1.11.
g i v e n by ( 1 . 1 . 7 )
Proof:
A
+
e x p ( - y s 4)
=
For c o m p l e t e n e s s we w i l l f o r m u l a t e a t h e o r e m r e l a t i v e t o ( 1 . 1 . 7 )
Theorem 1 . 1 2
y
c a n b e e x c l u d e d a l s o by g e n e r a l g r o w t h r e q u i r e m e n t s
c f . S c h w a r t z [l]).
E r d e l y i [l]) which y i e l d s
b(y,s)
The s o l u t i o n of
E D(A2)
Then
z
is a s o l u t i o n of ( 1 . 1 . 5 ) .
One n e e d s o n l y t o c h e c k t h a t t h e v a r i o u s i n t e g r a t i o n s , e t c . , make s e n s e
1 / 2 n4
We n o t e f i r s t t h a t w i t h
c
while
i s a seminorm i n
I
seminorm
= Ix.
YY
q
and
If
p
=
F
then
p(w(t))
5
q(uo)
f o r some
PRELIMINARY REMARKS AND E W L E S
where since
i,
h ( x ) = (TX)-'
u
=
2
A u,
(cf. Erdelyi
ill)
exp(-t
p(ux(x))
2
5
and
Similarly
2
h(x)q(A u o ) .
( c f . Theorem 1.1.11).
Consequently, from ( 1 . 1 . 7 )
s o t h a t ( 1 . 1 . 7 ) makes s e n s e .
w e l l d e f i n e d and by ( 1 . 2 5 ) t > O
by ( 1 . 1 . 6 )
/4x)dt
77
Ix(y,x)u(x)dx
Similarly
Similarly, t > 0
for
I(y,x)ux(x)dx
is
j'b
can be checked by n o t i n g t h a t f o r
n > 2
zt
and
ztt
can be o b t a i n e d by d i f f e r e n t i a t i n g under t h e i n t e g r a l
QED
sign in (1.1.7). Remark 1.13
The t h r e e formulas ( 1 . 1 . 6 )
-
( 1 . 1 . 8 ) , w i t h ( 1 . 1 . 9 ) - (1.1.111,
examined t o g e t h e r i n a c o n c r e t e a n a l y t i c a l s e t t i n g by Dettman [ 3 ] .
They a r e a l s o
[ 6 ; 7 ; 81 and Dettman
d i s c u s s e d s e p a r a t e l y i n v a r i o u s c o n t e x t s i n Brags-Dettman [l].
are
C l a s s i c a l L a p l a c e t r a n s f o r m and complex v a r i a b l e methods a r e used.
The
matter i s t a k e n up a g a i n i n Dettman [ 2 ] ( c f . a l s o Dettman [ 4 ] ) i n g e n e r a l s p a c e s u s i n g t h e H i l b e r t t r a n s f o r m and we w i l l s k e t c h some of t h i s .
For t h e c l a s s i c a l
H i l b e r t t r a n s f o r m see B u t z e r - T r e b e l s [ l ] , C o t l a r [ l ] , Horvgth [ 2 ] , O k i k i o l u [l], Orton [l; 2 1 , S t e i n [ l ] , and Stein-Weiss [ 2 ] .
The i d e a i s t o r e p l a c e t h e c l a s s i c a l
Hilbert transform
(1.29)
tf(x - t)dt
Hf = l i m -
by a formula ( c f a l s o Westphal [l])
(1.30)
where
Hf
T(t)
=
1 lim y+o
w
J-,
tT(t)fdt t
+y
i s an e o u i c o n t i n u o u s group of o p e r a t o r s i n
F
is t a k e n t o be a complete, b a r r e l e d , s e p a r a t e d LCS o v e r IR. be proved a s i n t h e c l a s s i c a l c a s e where
2
H f
=
-f
g e n e r a t e d by Then
F
A.
-2 H f = -f
( s u i t a b l e h y p o t h e s e s on
will
f
7%
R. W. CARROLL
are i n d i c a t e d below).
-
blems (1.1.3)
s o t h a t t h e pro-
A
(1.1.5) are uniformly w e l l posed i n t h e sense t h a t f o r D
dense subspace
One wants h e r e enough h y p o t h e s e s on
t h e r e e x i s t s a unique s t r o n g s o l u t i o n f o r
i n i t i a l v a l u e s i n a s t r o n g s e n s e , and such t h a t i f
u
CI
0
-+
t
in
i n g s o l u t i o n s converge uniformly t o z e r o on compact s u b s e t s of
3
u
in a
0 , assuming t h e t h e correspond-
D
+ lR .
Also t h e
problem
should b e u n i f o r m l y w e l l posed f o r t h i s t h e o r y . r e s o l v a n t s e t of setting
S(t)
=
A
and
A
g e n e r a t e s an e q u i c o n t i n u o u s group
1
cosh A t = $ T ( t )
a l s o equicontinuous (note
+ T(-t)]
v ( t ) = lJ(t)v
0
I n particular i f
i t follows t h a t
for (1.31)).
T(t)
is in the in
F
then
U(t) =
Under t h e s e assumptions
a l l of t h e problems a r e uniformly w e l l posed ( c f B a l a k r i s h n a n [ 2 ] , Bragg-Dettman [6; 7 1 , Dettman [l; 2 1 , F a t t o r i n i [ l ] , G o l d s t e i n [ 6 ] and o u r p r e v i o u s d i s c u s s i o n
Now d e f i n e
of t h e s e problems).
Then
@(y)f
is defined f o r
and
f E F
Y(y)f
e x i s t s as a Cauchy p r i n c i p a l
value i n t h e sense t h a t
(1.33)
I” %
Y(y)f = l i m N m
dt
-N t + y
1 l i m 7T
=
N*
IN 0
t“T(t) -T(-t)lf 2 2 t +y
-
T(-s)]ds = 2
dt
where
a(t) =
:1
[T(s)
is equicontinuous.
S(n)dn
and
Mote h e r e t h a t R(A)
J:
s i n h Asds
a t)
i s dense.
=
2[T(t/2) - T(-t/2)]U(t/2)
On t h e o t h e r hand
where
79
PRELIMINARY REMARKS AND EXAMPLES
0
The f i r s t t e r m on t h e r i g h t s i d e i n (1.34) t e n d s t o
is equicontinuous. Theorem 1 . 1 4
+
= since
and
For
f E D(A)
y > 0
and
for
0
case since
15 p <
m.
(Of)" = -A Of
and
f = u ).
f E D(A).
is defined f o r a l l
6
an i n v e r s i o n formula f o r
2 f E D(A )
Now f o r
(Yf)" = -A I f ; i n p a r t i c u l a r
is e a s i l y seen t o be w e l l defined f o r if
A.
2
( 1 . 1 . 5 ) and y i e l d s t h e formula ( 1 . 1 . 8 ) ( f o r
operator then
and
A = -d/dx
However t h e c l a s s i c a l t h e o r y i s n o t a s p e c i a l
i s n o t i n t h e r e s o l v a n t s e t of 2
evidently
one h a s
( Y ( y ) f ) ' = -AO(y)f.
The c l a s s i c a l Cauchy Reimann e q u a t i o n s are e x h i b i t e d i n t h e case F = Lp(-=,-m)
a(t)
By a s t r a i g h t f o r w a r d c a l c u l a t i o n one now o b t a i n s
[Cauchy-Riemann e q u a t i o n s ) .
( O ( y ) f ) ' = A.Y(y)f
N
as
let
f E F
g = Hf
z = Of
fi
and assume
satisfies
The H i l b e r t t r a n s f o r m
Further i f
and
one h a s
A
is a continuous
i s continuous. Hg
exists.
To p r o v i d e
Then f o r
f' E
F' < f ' , T(-x)Y(y)g>
as
y
-+
-+
I f one shows t h a t
0.
w i l l follow.
-2 < f ' , T(-x)H f > -2 < f ' , T(-x)H f > = - < f ' , T ( - x ) f >
-2
then
H f = -f
To do t h i s c o n s i d e r
I
m
$(x,y) = y
(1.36)
TI
--m
m
so t h a t
< f ' y T2( - t ) 2f > d t ; (x-t) + y (x-t) dt 2 2
$ ( x , y ) = < f ' , T(-x)O(y)f>
and
$(x,y) =
< f t r
T(-x)v(y)f>.
Y
As
+
5
$(x,y)
-+
< f ' , Tf-x)f>
while
$(x,y)
+
< f ' , T(-x)Hf>;
bounded harmonic f u n c t i o n s i n t h e h a l f p l a n e
y > 0.
further L e t now
@
and
$
are
0
80
R. W. CARROLL
which i s a bounded a n a l y t i c f u n c t i o n f o r
y > 0
and h a s t h e r e f o r e a Poisson
integral representation
Now l e t
R*(z) f
@*(x,y) = < f ' , T ( - x ) @ ( y ) g > and
=
+
@*(x,y)
i$*(x,y)
g = fif.
r e p l a c e d by
so t h a t
$*(x,y) = < f ' , T ( - x ) Y ( y ) g >
h a s a P o i s s o n i n t e g r a l r e p r e s e n t a t i o n l i k e (1.38) w i t h
Then
R e R*(z) = ImR(z)
so that
ImR*(z) = -Re n ( z )
to
w i t h i n a c o n s t a n t which must b e z e r o s i n c e
Hence
$*(x,y) = -@(x,y)
(1.40)
lim $*(x,y)
=
P O
and
consequently
= - 1 i m @ ( x , y ) = - < f ' , T ( - x ) f > . Y+0
T h i s p r o v e s t h e i n v e r s i o n formula
Theorem 1 . 1 5
Remark 1 . 1 6
If
if
-2 -f = H f .
has a Hilbert transform then
These i d e a s a r e extended i n Dettman [ 4 ] t o g e n e r a l i z e d H i l b e r t
t r a n s f o r m s r e l a t e d t o t h o s e of Heywood
[I], Kober [ l ] , and O k i k i o l u [I]. He
c o n s i d e r s t h e problems
(1.41)
u"
2v +u' t
(1.42)
w"
+
t
2
= A u;
2 w ' = -A w;
u(0) = f ;
w(0) = f
u'(0) = 0
(u
< 1/2)
a g a i n i n some complete b a r r e l e d s e p a r a t e d LCS e q u i c o n t i n u o u s group
T(t)
with
0
3.14.
where
i n t h e r e s o l v a n t s e t of
d i s c u s s e d (1.41) i n Example 1 . 3 . 1 (where m = -1/2)
o v e r IR
F
2u
=
2m
+
1
with
A.
g e n e r a t e s an
A
We have a l r e a d y
R e m > -1/2
or
and i t w i l l come up a g a i n i n Example 2 . 6 , Example 3 . 1 1 , and Example
I t p r o v i d e s a n i c e example of a v a r i a b l e c o e f f i c i e n t o p e r a t o r
s u s c e p t i b l e t o v a r i o u s t e c h n i q u e s of i n t e r e s t .
The problem ( 1 . 4 1 )
D
2
2 m + lD + __ t
i s a Cauchy
81
PRELIMINARY REMARKS AND EXAMPLES
p r o b l e m f o r t h e EPD e q u a t i o n w h e r e a s ( 1 . 4 2 ) i s a n a b s t r a c t D i r i c h l e t t y p e p r o b l e m f o r t h e e q u a t i o n of g e n e r a l i z e d a x i a l l y symmetric p o t e n t i a l t h e o r y ( c f . Wei nst ei n The s o l u t i o n t o ( 1 . 4 1 ) i s g i v e n by (1.3.8) when
[3]).
A
generates a l o c a l l y
e q u i c o n t i n u o u s g r o u p and t h i s f o r m u l a a l s o h o l d s of c o u r s e f o r
A
i n which c a s e t h e problem ( 1 . 4 1 ) w i l l b e u n i f o r m a l l y w e l l posed. (1.1.3)
equicontinuous The p r o b l e m
f o r t h e a b s t r a c t h e a t e q u a t i o n serves a s a l i n k b e t w e e n ( 1 . 4 1 ) and ( 1 . 4 2 )
( c f . Theorem 1.1.11 f o r t h e s o l u t i o n o f ( 1 . 1 . 3 ) g i v e n by ( 1 . 1 . 1 3 )
which a l s o
s p e c i a l i z e s t o t h e case o f a n e q u i c o n t i n u o u s g r o u p ) ; t h e p r o b l e m (1.1.3) w i l l b e a l s o u n i f o r m l y w e l l p o s e d as i n Remark 1.13.
Now l e t t i n g
( 1 . 1 . 3 ) one c a n w r i t e ( c f . B r a g g [ 3 ] , Bragg-Dettman
v
be t h e s o l u t i o n of
[ 6 ; 7 ; 81, D e t t m a n [ l ] )
and composing t h e s e f o r m u l a s o n e o b t a i n s
where
a(.,.)
is the beta function.
I n p a r t i c u l a r when
V = 0
u ( 0 ) = c o s h Auf
and ( 1 . 4 4 ) c a n be w r i t t e n t l - 211
(1.45)
When
w(t) =
R(%-
F = Lp(-m,m),
u,%)
Im J-m
1< p <
m,
T(U)fdU
2
2 1-1J
and
T(o)
( t +o )
i s t h e t r a n s l a t i o n g r o u p t h i s becomes
which i s t h e " P o i s s o n f o r m u l a " f o r t h e s o l u t i o n o f t h e g e n e r a l i z e d a x i a l l y symmetric p o t e n t i a l problem i n t h e h a l f p l a n e f(x)
(cf Weinstein 131).
The f u n c t i o n
t > 0
I$(x,t)
i n t e r m s o f boundary v a l u e s
i s t h e r e a l p a r t o f a pseudo-
a n a l y t i c f u n c t i o n i n t h e upper h a l f plane with conjugate f u n c t i o n
82
For
R. W. CARROLL
-4 < u
<
4
one h a s
which, e x c e p t f o r a c o n s t a n t , is t h e g e n e r a l i z e d H i l b e r t t r a n s f o r m of Heywood [ l ] , Kober [l], and O k i k l o l u [ Z ] .
u
-112
(1.49)
(1.50)
With t h i s background Dettman [ 4 ] sets f o r
112
Q (t)f
u
=
m
t l - 2lJ @(+-lJ&
-m
T((J)fdU ( t2+(J2 ) 1-lJ
YP(t)f =
and d e f i n e s a g e n e r a l i z e d H i l b e r t t r a n s f o r m as t h e s t r o n g l i m i t (when i t e x i s t s )
(1.51)
H f = lim \Y ( t ) f . t + O lJ
Using t e c h n i q u e s s i m i l a r t o t h o s e of Remark 1.13 i t is proved t h a t Theorem 1 . 1 7
If
f E D(A)
t > 0
then f o r
(Y,,(t)f)'
=
-t2'AQ
lJ
(t)f
and
AYlJ ( t ) f = t 2 1 J ( Q p ( t ) f ) ' .
Theorem 1.18
Theorem 1
If
. If
Remark 1.20
f E D(A)
H f
lJ
then
H f
e x i s t s and
lJ
h a s a g e n e r a l i z e d H i l b e r t t r a n s f o r m then
and
L1
L2
(H f ) = - f .
P
of second o r d e r a r e s a i d t o b e
connected o r r e l a t e d i f t h e r e a r e smooth f u n c t i o n s
,...,x,),
-lJ
Pagkovskij [l; 21 examines an i d e a of r e l a t e d e q u a t i o n s a s f o l l o w s .
Two p a r t i a l d i f f e r e n t i a l o p e r a t o r s
(xl
H
such t h a t f o r any
u , v E C'
ai(x)
and
bi(x).
x =
one h a s
n
(1.53)
!L2vL1u = LluL2v
+ ~Dit$i(x,u,v,uX,vX) 1
(1.54)
!Llu = Eb D.u i l
+
n
L1
!L2v = l iD iv
0
Boundary v a l u e problems f o r s a i d t o be r e l a t e d i f
la
b u;
L1
and
and L2
L2
+
a 0v .
i n a region
G
w i t h boundary
a r e r e l a t e d and whenever
u
and
v
aG
are
satisfy
83
FURTHER EXAMPLES
a p p r o p r i a t e homogeneous boundary c o n d i t i o n s n = 2
Taking
ja,Lm,(.,u,v,u
I,,
f o r convenience t h i s means
b e t h e formal a d j o i n t i t i s e a s y . t o s e e t h a t
L?L2 = IIfL2.
if
L v = 0 2
Thus i f
then
w
- 4 2 dy
aldx
and
L1
II2v
=
x
, v ) c o s nxido = 0. x
Lz
Then l e t t i n g
= 0.
are r e l a t e d i f and o n l y
L2
A number
L*w = 0.
satisfies
1
of s t r a i g h t f o r w a r d c a l c u l a t i o n a l r e s u l t s are proved.
2.2.
FURTHER EXAMPLES.
The p r e c e e d i n g s e c t i o n h a s g i v e n a n i d e a of t h e t y p e s
of r e s u l t s one can o b t a i n about t r a n s m u t a t i o n and r e l a t e d e q u a t i o n s and we w i l l F i r s t l e t u s t a b u l a t e what
compile some f u r t h e r i n f o r m a t i o n h e r e i n t h i s r e g a r d .
h a s been l e a r n e d about some s i m p l e o p e r a t o r s and e n l a r g e t h e list of examples. From Example 1.4 and Remark 1 . 5
From Example 1 . 6 and Remark 1 . 8
2 2 -D B = BD ;
(2.2)
(Bf)(y) =
The f i r s t e x p r e s s i o n f o r
a
satisfying
g(A)B if get
-
w
D
2
Bf
g(A)a
CY =
= A
in
2
zt t
and
wt t
c
ly.
The second e x p r e s s i o n s f o r
apply t o z = Bw
CY
(2.1) and (2.2) a p p l y t o any v e c t o r f u n c t i o n =
or
6
=
B.?
defined f o r
as b e f o r e f o r
Ba
then s a t i s f i e s
2
w
and
w, u , z
1.10 and Remark 1.11 one h a s
=
B6
B
6
=
5
or
Similarly taking
r e s p e c t i v e l y then
5 2 0
DR = g(A)B
a(<) must be d e f i n e d f o r
= A
= -A2z.
H(y,S)f(-S)dc;
-m
and
and
= Bz
m
B(0) = a ( 0 ) ; however
with
i s any s o l u t i o n o f
Gt
1
$
=
B6
g ( A ) = -A2
2-A v
and
<
-m
ctt
2 -D B =
or
5 <
a.
Thus
r e s p e c t i v e l y we if = A
ztt 2
6
= -A2?
respective-
i n (2.1) and (2.2) are r e f i n e d v e r s i o n s which
satisfying
at ( 0 )
=
0
a s i n the Introduction.
and we g e t
u = Bw
or
F i n a l l y from Example
R . W. CARROLL
84
The f o r m u l a s from s p e c i a l f u n c t i o n t h e o r y c o r r e s p o n d i n g t o ( 2 . 1 ) (setting
g(A) = A L
(2.4)
e -A
2
-
A
with
-
(2.3)
are
iX)
m
J 0exp(-T 2 / 4 t ) c 0 s
= (Tt)-’
ATdT;
The f i r s t two f o r m u l a s a r e w e l l known F o u r i e r c o s i n e t r a n s f o r m s a n d t h e l a s t f o r m u l a c a n b e o b t a i n e d f r o m a t a b l e of L a p l a c e t r a n s f o r m s ~ ( s / C ) ~ K + [ ~ 4( ]S ~ and )
K4(z) = ( ~ / 2 z ) + e x p ( - z )
-
(L(x-~” exp(-c/x))
=
c f . Magnus - 0 b e r h e t t i n g e r - S o n i
[11)* Example 2 . 1
QX = $yy
Thus one c o n s i d e r s t h i s becomes
fyy
?(s)exp(-(-is)
4y )
+
F I(y,x)
+
+
I (y,x) = 0
since
with
(-is)’
$(s,y)
+
0
x < 0.
Hence i f
2 Du = A u satisfies
then D
2
2
Bu =
= -A
2
= ?(s).
Q(D)
We t a k e as a s o l u t i o n
=
D2
.
;(s,y)
y
-+ m
where
( c f . Example 1 . 1 0 ) .
I+(y,x) = I(y,x)
for
e x p ( - ( i s ) 4y )
But x
2
0
and
which y i e l d s
I(y,S)f(-S)dS.
w
2.
satisfies
D2G = A
However t h e v a l u e s
2
6
while i f
~ ( - 5 ) or
2
Dv = -A v
v(-c)
then
which a r i s e
from ( 2 . 5 ) may i n t r o d u c e c o m p l i c a t i o n s .
A f e w o t h e r s i m p l e e x a m p l e s are w o r t h m e n t i o n i n g h e r e .
Q(D) = -D 6(x
-
y)
so o n e c o n s i d e r s
* f(x)
$x = -by
with
F i r s t let
$(x,O) = f ( x ) .
so t h a t m
6(s - y ) f ( - L ) d E
$(O,y) = ( B f ) ( y ) = -m
=
(1 - i ) s 4 and t h i s w i l l b e u n i q u e u n d e r
$(x,y) = I (y,.) * f ( * )
1,
and
= D
and a f t e r F o u r i e r t r a n s f o r m a t i o n
+
Thus
( B f ) ( y ) = @(O,Y) =
fi as
+
for
$(x,O) = f ( x ) ;(s,O )
=
i F s I ( y , x ) = FI ( y , x )
(2.5)
? = Bv
with
is; = 0
the stipulation that =
P(D)
The k e r n e l I w i l l a r i s e a g a i n i f we t a k e
=
f(-y).
P(D)
Clearly
= D
and
$(x,y) =
FURTSER EXMlPLES
Da
Applied t o vector functions if B(y)
=
a(-y).
Example 2 . 2
where
$
6 =Ba satisfies Df? = -A2f? and
then
A more interesting example is Let
P(D)
F'$.
=
A 2a
=
85
Thus
m
=
D3
and
$(s,y)
e-isxei s 3yds
(2.7)
so we consider
Q(D) = D
=
F(s)exp
is3y
=
K(3y.x)
=
and one knows that 4
(3y)-'j3Ain(x(3y)
3
)
-co
where Ain(z) = Ai(-z)
denotes the Airy
function
(cf. Roetman [l], Magnus-Oberhettinger-Soni *f(*)
[l]).
Consequently @(x,y)
K(3y,.)
and
By results of Roetman [ l ] this will make sense '->rcontinuous f with c exp kl El 3/2
-' as
5
+
m
and
@(x,y)
will satisfy ( 2 . 6 ) .
can state
.
g(A)
be a well defined operator function in F with
D(g(A)).
u(t)
satisfies D 3u
Assume p(u(t))
= g(A)u
for
-m
I f(c) I 2
In particular we
Theorem 2.3 Let
while
=
< t <
u
6
m
with
u(0) = u
F.
Then
v(t)
0
3/2 -F.
5
kp exp cp)tl
for any seminorm p
in
=
m
KOt,S)u(-E)dS
=
(Bu)(t)
satisfies Dv
=
g(A)v
with
v(0)
=
u
-m
The formulas of Example 2 . 2 are perhaps better exploited in solving the problem Du
=
-A3u
(t > 0)
Thus we assume A Gt = Gxxx
with
with
u(0)
=
uo E D(A3)
generates a group
T(.)
by the technique o f Section 1.1. and replace A
By (2.6) - (2.7)
G(0,x) = 6(x).
C(t,x)
solution of the abstract Cauchy problem is m
(2.9)
u(t)
=
= O
j
-m
K(3t,c)T(c)uodS
by
-D
= K(3t,x)
to obtain and a
R. W. CARROLL
86
which makes s e n s e s i n c e
p(T(C)uo)
- 6)
T(E)uo(x) = uo(x
t h e group
u(t) = u(t,x)
solves
5
I f we let
q(uo)e"lCI.
A = -Dx
i n a s u i t a b l e s p a c e of f u n c t i o n s
ut = uxxx
with
u ( 0 , x ) = u (x)
generate then
F
a s i n Roetman [ l ] .
Summarizing
Theorem 2.4 u
3
6 D(A ) .
Remark 2.5
A
Let
Then
g e n e r a t e a q u a s i e q u i c o n t i n u o u s group u(t)
d e f i n e d by ( 2 . 9 )
T(*)
ut = -A 3u
satisfies
in
with
F
with
.
u(0) = u
E v i d e n t l y t h e t r a n s m u t a t i o n method of r e l a t i n g s o l u t i o n s of a b s t r a c t
d i f f e r e n t i a l problems i n v o l v i n g a n o p e r a t o r e r a t i n g a group.
i s q u i t e independent of
A
A
gen-
Groups a r o s e i n Example 1.4. Example 1 . 6 , and Theorem 1 . 1 2
because of t h e f a c t t h a t Ij(t,x)w(x)dx
w ( t ) = cosh A t u
wtt
a r e known s o l u t i o n s of
and t h e c o r r e s p o n d i n g =
A 2w
and
= A2u
u
u(t) =
respectively
t
when
g e n e r a t e s a group.
A
Example 2.6
An important example of t r a n s m u t a t i o n was f u r n i s h e d by Lions [2: 3: 41
in treating
D2
and
L = D m
i n Section 1 . 3 we take Q(D) = Lm(D) = Lm.
m >
2
+
-4
together with related generalizations.. A s
D
h e r e f o r convenience and c o n s i d e r
Thus we l o o k f o r
@
R e f e r r i n g t o Example 1 . 3 . 1 w e s e e t h a t
and
satisfying
OY(X,0) = 0.
@(X,O) = f ( x ) ;
Qy:
P(D) = D L
@ ( x , y ) = Rm(y,x)
* f(x)
so that (cf.
(1.3.6) )
where
c
m
=
I'(m
+
w ( t ) = cosh Atuo
l)/r(l/2)r(m then
+
1/2).
um = B w
m
formula ( 1 . 3 . 8 ) from (2.11) ( i . e . a c t u a l l y c o n s t r u c t s an isomorphism for
L
m
If
2 2 D w = A w
satisfies u m ( t ) = 2c Bm
with
w (0) = 0 t
so t h a t
2 L um = A um and we r e c o v e r t h e
*J'
(1
-
m O with inverse
2 m-+i
E, )
Bm-1
cosh ActuodC).
Lions
i n s u i t a b l e spaces
and v a r i o u s g e n e r a l i z a t i o n s b u t we w i l l n o t s p e l l t h i s o u t h e r e ( c f .
FURTHER EXAMPLES
87
Examples 3.11 and 3.14 f o r o u r v e r s i o n of t h i s ) .
Remark 2 . 7
Various t r a n s m u t a t i o n o p e r a t o r s can be e n v i s i o n e d f o r h i g h e r o r d e r
s i t u a t i o n s such a s
P(D) = (-l)n+1D2n
Q(D) = D
with
s o t h a t one c o n s i d e r s
A f t e r F o u r i e r t r a n s m u t a t i o n t h i s becomes
@y + s
(2.13)
with s o l u t i o n
2 n ~
(0 = 0 :
G(s,O) = ? ( s )
$(s,y) = ?(s)exp(-s
2n
s p a c e of e n t i r e f u n c t i o n s s a t i s f y i n g clexp(-alxlP) [ 2 . 31.
for
I t is known t h a t
$ ( s , y ) = exp(-s with
real so
x
2n
2n
y) E ZZn
q = 2n/2n-1 = 1
@(x,y) = $(x,y) * f ( x )
+
Z;
‘-
FZ - Z P
=
P
A s i n S e c t i o n 1 . 3 we d e n o t e by
y).
-
5
(g(z)I
’-”’
slip 2
9
- S
c exp b l z l P
with
P
the
5
i n t h e n o t a t i o n of Gelfand-Silov
1-UP
where
1 P
-
+ -q1 =
corresponds t o a transmutation kernel 1/2n-1.
If(x)I
Zp
Thus f o r s u i t a b l e
1 and t h u s @ ( x , y ) E 2;
f , which can b e s p e c i f i e d ,
w i l l yield a transmutation operator
(Bf)(y) = @(O,y).
T h i s w i l l b e p e r f e c t l y r i g o r o u s b u t a n e x p l i c i t d e s c r i p t i o n of
$(x,y)
g e n e r a l l y n o t be a v a i l a b l e from a t a b l e of F o u r i e r t r a n s f o r m s .
Thus s i n c e h i g h e r
will
o r d e r d i f f e r e n t i a l problems do n o t u s u a l l y a r i s e i n p r a c t i c e anyway we w i l l n o t concern o u r s e l v e s w i t h t h e q u e s t i o n of t r a n s m u t i n g g e n e r a l
P(D)
(which is n o t always p o s s i b l e i n any e v e n t a s a n example w i t h Q(D) = D’
-
q(x),
q
not a n a l y t i c , i l l u s t r a t e s
-
into
Q(D)
P(D) = D3
c f . Levitan [ l ] ) .
and
Rather l e t
u s c o n c e n t r a t e on second o r d e r d i f f e r e n t i a l polynomials where e x p l i c i t f o r m u l a s can be o b t a i n e d .
Some g e n e r a l r e s u l t s f o r t h i s s i t u a t i o n a r e developed i n
L e v i t a n [ l ] and L i o n s [ 2 ;
3 ; 41 but s i n c e w e a r e concerned p r i m a r i l y w i t h opera-
t i o n a l c a l c u l u s and r e l a t e d e q u a t i o n s w e w i l l proceed r a t h e r d i f f e r e n t l y v i a some examples of i n t e r e s t i n a p p l i c a t i o n s and d e v e l o p g e n e r a l theorems i n t h a t context.
Example 2 . 8
A n example of r e a l t e d non-homogeneous e q u a t i o n s connected by a
g e n e r a l i z e d S t i e l t j e s t r a n s f o r m is provided i n Bragg 141 ( c f . a l s o Dettman [ l ]
88
R. W. CARROLL
and see Remark 2 . 1 0 for further information on non-homogeneous related equations). The discussion in Bragg [ 4 ] takes place in a Banach space F an equibounded Co
group
T(t);
Consider the problems
(2.14)
utt
-
a 2 t A u = f:
u(0) = u'(0) = 0
(2.15)
v
+
t 0A 2v = g;
v(0)
where
and
f
[ (a+2)T/21 2/o+2
UTT
(2'17)
' T T
where etc.
+
2
0, 0
2
0)
0
= [(a
+ 2 ) ~ / 2 2/a+2 ]
in ( 2 . 1 4 ) and
One
t =
in ( 2 . 1 5 ) to obtain
&UT -
2
A U = f*
2 VT + A V = g * .
+
U(7) =
=
(a
are
are suitable continuous functions with values in D(A2).
g
makes a change of variables t
(2.16)
generates
suitable extensions to more general F
clearly possible but we omit this.
tt
A
where
U([T] ( a +
2)T 2/0.+2 ) ,
f*(T)
=
(2/(af2)T)2"/a+2f([~]2/a+2),
The equation
(2.18
W
2 -AIJ=h
is taken as an intermediate link and in keeping with a general formalism indicated below (see Bragg [ 3 ] and Remark 2 . 1 0 ) one takes E,
in ( 2 . 1 8 ) to obtain
where
a
=
a+2
provided that
.
One can then write
<
= (1/4).r2
in (2.16)
and
T =
89
FURTHER EXAMPLES
Here
Lil{
ting
T =
where
means t h e i n v e r s e L a p l a c e t r a n s f o r m w i t h
<-'
b =
i n ( 2 . 1 7 ) and
T = 1/45
-+
5.
S i m i l a r l y set-
i n (2.18) one o b t a i n s
B / B + 2 , provided
Eliminating
W
between (2.20) and (2.22) t h e r e r e s u l t s
11
B+2)+ K(a,O) = C ( a , B ) i ( a + 2a) -( o
where
s
T h e c o r r e s p o n d i n g r e l a t i o n between
with
and
f
g
is
Now t o produce a s a t i s f a c t o r y theorem from t h e s e c a l c u l a t i o n s Bragg [ 4 ] f i r s t constructs explicitly a solution with
f ( t ) E D(A2).
u(t)
One w r i t e s f o r
t o ( 2 . 1 4 ) when
f E Co(F)
on
[O,m)
I$ E D(A2)
a+2
~
where
m(o,t) = 2t
where
K-1 ( a )
i1(0)
=
'$, and
=
<'/(a+2)
2 B(-, a+2
a+l a+2
--).
ai(0) = 0
and
Then while
U1(t)$
=
i,(t)
U2(t)@= i 2 ( t )
s a t i s f i e s (2.14) with s a t i s f i e s (2.14) w i t h
f
=
f = 0,
0,
R. W. CARROLL
90
&(o)
= 0,
Now let G
Jill
4.
A solution to (2.14) is then given by
be the space of
dt <
f(t)II
-
$;(O)
and
Theorem 2.9
on
f E Co(F)
with
[0,m)
2 f(t) E D(A )
and
There results
m.
Let
f
and a >
G
E
f by (2.26)
defined in terms of
8.
Then a solution v
of (2.15) with
is given by putting the u(t)
g
of ( 2 . 2 9 ) i n
(2.24). Remark 2 . 1 0
A class of related differential equations connected to generalized
hypergeometric functions are studied in Bragg [ 3 ] and this of course will yield corresponding transmutation formulas (which we mercifully will not spell o u t ) . Let 4 (2.30)
for
6
Ol(q,8,t,Dt) =
(a,, ...,Bq)
tDt
=
1
-
(tDt + B
j
1)
and P
(2.31)
02(p,cx,t,Dt)
t7T (tDt + a i ) .
=
1 Let
P
= P(x,D)
k1
orders are
= Ca (x)Dx
with
x
and
!L2
Q
=
Q(x,D)
o
t
e same form; suppose
:ir
t
respectively. Equations of the type
are considered and here it i s seen that
P and
Cj
could be suitable more general
abstract operators in a space F with no essential change in technique. Laplace transforms are used and some typical theorems are Theorem 2.11
2
with
p
max(p
+ R1,
Let
q, ai
q
+
u(x,t)
and
be a solution of
8 . positive, J
k 2 ) , and
u
u
E
Cr
in
(x.t)
for
and all derivatives of order
5
t > 0
where
r bounded.
r =
Let
FURTHER EXAMPLES
c(
P
91
> 0; then t h e f u n c t i o n
s a t i s f i e s (2.32). then
v(x,t)
+
If
u(x,t)
as
$(x)
t
+
+
Let
ai
-P
0
I$
with
bounded and c o n t i n u o u s
S i m i l a r l y one h a s
6.
and
t
0.
The proof is s t r a i g h t f o r w a r d .
Theorem 2.12
@ ( x ) as
be positive with
J
Bq
> 1.
Let
IJ(x,t)
be a
s o l u t i o n of
t > 0
for
If
V
exists,
V
then
s
and set
V E C r , and
Theorem 2.13
Let
p
a negative integer.
for
t > 0.
s a t i s f i e s (2.32) f o r Re s > k > 0
for
with a l l derivatives o r order
V
t > 0
such t h a t
then a s
5
q
Let
+ u
t
+
>
1, E
U(x.t)
0
U(x,l/s)
CY
P
V
s a t i s f i e s (2.32) f o r
i s bounded and a n a l y t i c i n
equal t o zero or
satisfy
$(XI
t > 0
Rl,...,Bq-l
> 0 , and no
as
0
$
with
t
+
1
- 0)
a Then
a r e bounded
l i m V ( x , t ) = $ ( x ) = U(x,O).
cr-l
-P
If
< r -
continuous.
Define
V
-a -1 U(x,ta)do.
and
Numerous a p p l i c a t i o n s of such formulas a r e g i v e n i n Bragg [ 3 ] and t h i s formalism
i s extended t o nonhomegeneous problems i n Bragg [ 2 ] .
We c i t e a few t y p i c a l
r e s u l t s r e l a t i v e t o Theorems 2 . 1 1 - 2 . 1 3 where by (2.32) = f [SO, - P02]u = f .
f o r example we mean
R. W. CARROLL
92
Theorem 2.14 by ( 2 . 3 4 ) .
b e a bounded s o l u t i o n of (2.33) = f
U
Let
V
Then
s a t i s f i e s (2.32) = g
and l e t
V
b e given
provided
a -1 (2.39)
g(x,t) =
Let
Theorem 2.15 e x i s t s as
t
U
0.
-+
r-'(a
P
re-ao
)
f(x,to)da
'0
be a bounded s o l u t i o n of (2.35) = f
Let
be given by ( 2 . 3 6 ) .
V
V
The
such t h a t
l i m U(x,t)
s a t i s f i e s (2.32) = g
provided
1-E
2-8 (2.40)
Theorem 2.16
U
Let
6 - a > 1.
and
q
where
qr(6q)Li1{s
g(x,t) = t
Gq =
P
(Bq
Example 2.17
s a t i s f y (2.37) = f
Then
P
- a
P
qf(x,;)ls+t.
-
and d e f i n e
s a t i s f i e s (2.32) = g
V
l)r(Rq)/r(ap)r(Oq
The S c h r o d i n g e r e q u a t i o n
-
ut
V
by (2.38) w i t h
a
P
> 0
provided
ap).
iu
=
V
viewed i n t h e Gelfand-Silov
xx
framework ( c f t h e d i s c u s s i o n around D e f i n i t i o n 1 . 3 . 1 6 and Theorem 1 . 3 . 1 9 ) l e a d s x(s) = -is2
to
and
Petrovskij correct. t
2
-is $
=
A(s)
=
T = 0
is zero f o r
Under t h e F o u r i e r t r a n s f o r m
which l e a d s t o
( E ( 0 , s ) = 1).
20-r
Since
A(s)
2
2 ( t , s ) = exp(-is t )
F
so t h a t the equation is
f o r a fundamental s o l u t i o n
1 ~ 51 k
i s bounded i n t h e domain
16(q)(t,~)5 l cq(l
+
we have i n f a c t
u =
(1
+
lOl)-'
the
genus
p = -1.
Q > 0
t h e one d i m e n s i o n a l Cauchy problem is w e l l posed i n t h e c l a s s of f u n c t i o n s
u(x)
Further
101)'
and one can s a y t h a t f o r any
h a v i n g , t o g e t h e r w i t h t h e i r d e r i v a t i v e s of o r d e r
of o r d e r
2
Q
-
2
(while
u(x,t)
5
9,
+
2 , polynomial growth
h a s polynomial growth of o r d e r
5 a).
Regard-
i n g t h e a b s t r a c t Schrodinger e q u a t i o n we c o n s i d e r
(2.42)
2 ut = i A u;
u ( 0 ) = uo
and t h e Hersh t e c h n i q u e l e a d s t o shown ( c f . T r e v e s [ 3 ] ) t h a t
Gr = i G
xx
with
G(0.x) = 6 ( x ) .
It i s easily
93
FURTHER EXAMPLES
Consequently
u(t)
=
T(x)uo>
is given by
T(x) uodx
(2.44)
u(t)
where A
generates a suitable group
=
T(*)
in a space F.
IJe note that if ;(t)
is the solution of the abstract heat problem (1.1.3) then ;(it)
u(t)
=
and
.
formally ( 2 . 4 4 ) can be written (2.45)
u(t)
K(it,x)cosh
=
Axuodx.
We note that there are some errors in Polyanskij-Fedoryuk [l] where the formula (n/p)’-
exp(-clp’)
=
1,
2
T-’exp(-
!& - pT)dT
is used (cf. Bragg-Dettman [ 6 ; 7; 8 1 ) .
Their formulation is however easily modified to give a connection formula between the u
of ( 2 . 4 2 ) and the
z
of (1.1.5) using spectral hypotheses on
A
2
.
Remark 2.18 The construction of transmutation operators by the method of Theorem 1.3(F) usually involves the extension of f(x)).
f
to
in some manner (eg
(-m,r))
f(-x)=
The resulting transmutation operator will depend on the extension and
the following discussion and remarks should clarify the matter.
First, there are
a number of results available concerning the solution o f half ulane or quarter \r
plane differential problems (see eg. Dikopolov [ 1 1 . Dikopolov-Silov [ 2 ; 31, Hersh V
[4; 5; 61, Palamodov [ 2 ] , Pavlov [l], Sarason [I: 21, Silov [l], ZarnitskayaSelezneva-Eidelman [l] and cf. a l s o Fattorini [3], Fattorini-Radnitz [4], Gasymov [3]) and we will give some sample theorems to indicate natural frameworks. Let u s say that for some space
Q
is P-admissible if
P(Dx)$
=
O(Dy)@,
with
Y , has a unique solution in some space @
@(x,O)
where
@
=
f(x) E Y
will usually
involve some conditions Di$(x,O) = 0 and/or growth conditions (see Section 2.3 Y also where this is subsumed under condition C). For such a problem to make sense one will usually have t o extend the space y - 0.
f(x),
given for x
2
0, to
(--,O)
0 deals with functions o r distributions in the half space
Then
Bf(y)
=
$(O,y)
determines the transmutation operator B
and -m
< x < m,
such that
94
R. W. CARROLL
QB = BP.
Note t h a t i f
Ek E L(Y,Yk)
{ O , m - l } , is g i v e n , where
c
P
+
Q
same e q u a t i o n f o r
6
$(O,y) = g(y) E A
h a s a unique s o l u t i o n i n s a y -m
i n d i c a t e d and one d e f i n e s for
$
< y <
m,
3
require global solutions i n
t = y
E =
(cf (1.6)
P(Dx)$ = Q(D ) $ Y
5
where
PR = KQ.
g(y)
with
@
is l i k e
but
y < 0
to
i s now
The h a l f s p a c e problems v
and t h e s t a n d a r d Gelfand-Silov t h e o r y t < T <
f o r example i s n o t immediately a p p l i c a b l e (cf S e c t i o n 1 . 3 where a r i s e s i n fixing spaces).
then c o n d i t i o n s
S i m i l a r l y working w i t h t h e
x > 0: an e x t e n s i o n of
R g ( x ) = @(x,O) w i t h
Yk
k E
BEf(y) = $(O,y),
i s Q-admissible i f
P
Ekf(x),
=
P(Dx)Ek = EkP(Dx)
and s e e L e v i t a n [l; 31 f o r simple v e r s i o n s of t h i s ) .
d e f i n e d over a r e g i o n
Y and
Y
for suitable
a s b e o f r e provided
one can s a y
Dk $(x,O)
s a y and d a t a
@, and t h e o p e r a t o r
would change of c o u r s e , as would
( E k ) , would transmute
m
i s of o r d e r
Q
m
often
Some m o d i f i c a t i o n s of t h i s t h e o r y a r e p o s s i b l e and w e v
c i t e f i r s t a few t y p i c a l r e s u l t s f o l l o w i n g Dikopolov-Silov
[2].
Consider f o r
ex ample
m-1 (2.46)
DTu =
1 Pk ( i D x ) D kt u
k=0 where t h e
Pk
a r e polynomials and we work i n t h e r e g i o n
--m
< x <
with
m
t > 0.
I n o r d e r t o r e l a t e t h i s t o o u r format l e t Pk be c o n s t a n t f o r k > 0 w i t h m-1 Po(iDx) = P(Dx). S e t t i n g Q(Dt) = DY - X P Dk w e have Q(Dt)u = P(Dx)u. Let m- 1 1 k t k C PkX = P o ( a ) , o r d e r e d by t h e r u l e A . (a) be t h e r o o t s of Am J 1
-
(2.47)
R e X,(o) <
... -< Re
r e a l , and l e t
for
U
G1 2
... 2 Gm.
Let
G
H
d e r i v a t i v e s (computed i n
1
Xm(u)
be t h e s e t of such
be t h e space of
P') so that
m u l t i p l i e d by a r b i t r a r y polynomials.
where
R e ),.(a) J
f u n c t i o n s on R
L2 H =
0
FH
c o n s i s t s of
5
0
so t h a t
together with a l l L2
f u n c t i o n s on R
These a r e c o u n t a b l y normed s p a c e s i n t h e
Y
s e n s e of Gelfand-Silov [I: 2 : 31 and one w r i t e s f o r example i s t h e H i l b e r t space w i t h s c a l a r product
H = UHp
where
Hp
95
FURTHER EXAMPLES
h
A sequence
fn
Similarly
fn
+
0
+
Fn E L 2 , and
in
0
Fn
H
H
in 0
+
i f f o r some
p
?n
€
and
Hp
(1
an(/p+ 0
i f f o r some d i f f e r e n t i a l polynomial
in
L
2
.
P(D),
A t y p i c a l theorem r e l a t i v e t o (2.46)
n
as
+ m.
f n = P(D)Fn, is
v . ( u ) be g i v e n on G where v is t h e p r o d u c t o f J j j 2 a polynomial times a f u n c t i o n i n L ( G . ) (one writes v . E H(G.)). The e q u a t i o n J J J ( 2 . 4 6 ) h a s a s o l u t i o n u ( x , t ) w i t h [D:-'u(x,O)]~ = vk(a) on Gk (1 5 k 5 m). Theorem 2.19
For
0
t
Let functions
u(.,t) E
H
and, as
no f a s t e r t h a n some power of and depends c o n t i n u o u s l y i n
Example 2.20 gives
=
H
i o with
+
G2 =
X2
- 0'
=
@.Thus
0
v (a)
1
X,(o)
= -1~11
(0
5
k l m-1)
grows i n
H
i s u n i q u e i n t h e c l a s s of s u c h f u n c t i o n s
so t h a t
X2(o) = -io.
and
u
on t h e d a t a
Cauchy problem f o r t h e wave e q u a t i o n . where
k Dtu(-,t)
-,
This
t.
2 2 Dtu = D u
Consider
X,(u)
t
Thus
v
k
E H(Gk).
Po(iDx) = -(iDx)2 GI = G 2 =
X,(o)
can be p r e s c i r b e d b u t n o t
=
= 0
2
2
2
D u = -D u = (iD ) u t
1u1, one h a s
v2(U)
+ u2
IR and we have t h e f a m i l i a r
On t h e o t h e r hand f o r with
X2
and
G
1
=IR
with
and w e have a w e l l posed
D i r i c h l e t problem f o r t h e Laplace e q u a t i o n .
I f more r a p i d growth is allowed
Remark 2.21 placed by
GC = {Re X.(o) < c ) ri 3 -
specified for
5
Is1
a r e nonempty f o r some works l o c a l l y i n
t
p o n e n t i a l growth and
(O(tPexp c t ) )
the
and i n t h e L a p l a c e example above
G
j
can be re-
v2(s)
can be
On t h e o t h e r hand f o r any e q u a t i o n (2.46) t h e s e t s
c. c
and t h e r e w i l l b e n o n t r i v i a l c o r r e c t problems.
GC
j
I f one
t h e Cauchy problem i t s e l f w i l l be c o r r e c t f o r s u i t a b l e ex-
0 < t < T
( c f . t h e end of S e c t i o n 1 . 3 ) .
Various r e f i n e m e n t s of such r e s u l t s a r e p o s s i b l e b u t we w i l l n o t e n t e r i n t o a d i s c u s s i o n of t h i s . Y
Remark 2 . 2 2
L e t u s mention now € o r completeness some c r i t e r i a of S i l o v [l] f o r
q u a r t e r p l a n e problems and s p e c i a l i z e h i s r e s u l t s t o some second o r d e r e q u a t i o n s for simplicity.
Consider
96
R. W . CARROLL
u(0,t) = w (t);
(2.49)
Dxu(O,t) = w ( t ) 1
where t h e number of c o n d i t i o n s imposed a t
=
0
may v a r y h e r e a s i n d i c a t e d
F i r s t o r d e r terms i n (2.48) a r e o m i t t e d h e r e f o r convenience.
below.
Xj(s)
a t the roots
A
(2.51)
2
=
( j = 0,l)
branch p o i n t s .
of
a
Thus i f
s =
?(-a/y)'
=
where t h e
X.(s)
have a l g e b r a i c
J
s = ? i ( a / y ) !5
y have t h e same s i g n we have
and
whereas i f t h e y have o p p o s i t e s i g n s
5B
One l o o k s
+ YS 2
a
and t h e r e are c r i t i c a l p o i n t s
Ims
t
s = ?(-a/y)'
is r e a l .
In e i t h e r case
la/yI4 f o r b o t h r o o t s and one works i n t h e h a l f p l a n e
t h e r o o t s a r e s i n g l e valued and d i s t i n c t .
A number
r
Ims >
(r = 0,1,2)
8
where
i s then
determined s p e c i f y i n g how many c o n d i t i o n s (2.50) a r e allowed, such t h a t , f o r
Ims >
6,
R e X.(s) 3
l e a s t f o r some permitted.
(2.52)
s
2
0
0 < j < r-1
for
with
>
Ilns
One w r i t e s (2.48 2 t
D u
= C~U
-
B.
r
For
and f o r =
2
j > r
Re A.(s) > 0 J
at
both c o n d i t i o n s i n (2.50) a r e
i n t h e form
2
y[D u - w 6 - w 6 1 1 1 0
and t h e q u a n t i t y
w i l l be of importance i n t h e t h e o r y .
for
u > r
(r = 0,l).
T h i s i n t e g r a l i s defined f o r
provided f o r example t h a t t h e One l o o k s f o r s o l u t i o n s
Indeed s e t
u
wi(t)
Ims >
B
and
Re Xv(s) > 0
have a t most polynomial growth as
i n a space
/ft
t
+
m.
d e l i m i t e d by c o n d i t i o n s of t h e form
rCO
exp(-ZBx) l I q u ( x , t ) I2dx < JO
The space
13
H+ =
FH:
m
where
Iqu
d e n o t e s a s u i t a b l e " p r i m i t i v e " of
on t h e o t h e r hand c o n s i s t s of f u n c t i o n s
r$(s), analytic
u.
97
FURTHER EXAMPLES
Ims > f3,
for
B
Ims >
(s = 0
Theorem 2.23 Gv(s)
+
2
A
wo,
q , uniformly i n
T =
-
s
(2.50) p r o v i d e d t h e Ims > f3
t o the half plane
a s func-
B
Let
+
f o r some
H+-
2
R e i(O
m
There e x i s t s a unique s o l u t i o n of (2.48)
Example 2.24 obtains
<
iT).
can be c o n t i n u e d a n a l y t i c a l l y i n
tions in
and
2
I @ ( s ) I /lsI2‘do
such t h a t
=
iT)
y = -1
and
s o t h a t (2.48) i s t h e wave e q u a t i o n .
-s , X
= is,
h
< 0
while
R e A ( s ) = Re[-i(5
= -T
wl,
0
CY =
u
and
1
f3 = 0 , and f a r
-is,
=
1
+
Ims > 0
Re Xo(s)
= 7 > 0.
iT)]
One =
r = 1
Hence
can be p r e s c r i b e d which i s a s t a n d a r d mixed problem f o r
t h e wave e q u a t i o n .
Example 2.25
h
2
Re
= s
2
, X
X 1( s )
Let
CY
= s,
h
=
1
and
f3
-s,
y
=
=
0,
1 which makes ( 2 . 4 8 ) a Laplace e q u a t i o n where and f o r
Ims > 0
( n e i t h e r of which i s always n e g a t i v e ) .
= 0
defined f o r
0
=
Res > 0
Res < 0
and
Re h ( s ) = Thus
r = 0
-0
while
and t h e
GV(s),
respectively, are
Y
An a n a l y s i s of t h e Cauchy Riemann e q u a t i o n s i n t h e p r e s e n t c o n t e x t ( c f S i l o v [l]) G (s)
a l l o w s one t o conclude t h a t t h e functions W+(x,t)
W+(t) = wl(t) for
x > 0
and
w
a r b i t r a r i l y when both
Remark 2.26
+_
iwi(t) t > 0.
and
w i l l be s u i t a b l y e x t e n d a b l e p r o v i d e d t h e
can b e extended t o b e a n a l y t i c € u n c t i o n s However n e i t h e r
w1
II
nor
u1
can be p r e s c r i b e d
are t o be s p e c i f i e d .
Consider a g a i n t h e wave e q u a t i o n
@xx = Q t t
with
O(x,O)
=
f(x)
Ii
and
Qt(x,O)
=
0
x > 0.
for
,
;
’
Extend
Qt
by
0
and
f(x)
by
T(x)
for
x
c’
0
R . W.
= f(0)
and
CARROLL
a t least
(?(XI = f ( x )
7 [f(t) +
Q(0,t) =
1 = 2 f(t)
?(-t)
I f w e c o n s i d e r t h e q u a r t e r p l a n e problem of Example 2.24 w i t h
h(t).
i n t e r c h a n g e d i t is s e e n t h a t
h(t)
could b e p r e s c r i b e d a l o n g w i t h ~
@,(x,O) = 0.
and
x
for
1 -
-
2h(x)
f(x)
Hence t h e r e is an e x t e n s i o n
such t h a t
B f ( t ) = h(t) g
o p e r a t o r determined by t h i s e x t e n s i o n .
f
B
where
of
f , namely
0).
Then
+
g(t)l =
and
x
t
@(x,O) = f ( x )
-
f(-x) = g(x) =
denotes the transmutation
g
W e note a l s o t h a t the extension
g = f
= Bf is i n t h i s c a s e a g e n e r a l i z e d t r a n s l a t i o n g o p e r a t o r of t h e t y p e s t u d i e d by L e v i t a n [l; 31 ( c f S e c t i o n 2 . 3 ) . Formally l e t us
corresponds t o
h = f
f E Y
consider then
o=
where
here.
f(0) = 0
if
B (f) =
g
Y
+
Bg
such
1
7f. is
and
with
B
I$E 0
Note f i r s t t h a t g
B
g
+
Ag : Y
+
h =
Thus
1
7
f
while
B-f(f) = 0,
$(O,t) E 0
-
h = 0
g = -f
and
B f ( f ) = f , and
is t h e zero operator then
Bg(g) = 0 = g
and
Further
T h e r e f o r e t h e o p e r a t o r d e f i n e d by and
g
-
g = 0
appear admissible.
In particular i f
1-1.
a s i n Remark 2.18 and l o o k a t
L(Y)
is
A (f) = B (f) g g
1-1 s i n c e
A
g
=
-
Bo(g)
implies
0
is linear, A (g) = g
g
A
g
- -21 g
E
L(Y'), 1 2
= - g = 0.
B ( f - g ) = Bof f o l l o w s A ( f - g) = Bo(f - 8 ) . From g g 1 1 1 A (f) = - [f g ] - - g = - f of c o u r s e we s e e t h a t A = is t r i v i a l b u t i n g 2 2 2 g 2 p a r t i c u l a r A E L(Y) ( c o n t i n u i t y ) . Thus g B : Y + A(") where A ( " ) denotes g g c o n t i n u o u s a f f i n e o p e r a t o r s Y + Y and B ( f ) = f i m p l i e s f = g. I t could b e g of i n t e r e s t t o examine s u c h mappings i n more g e n e r a l i t y f o r o t h e r e q u a t i o n s (cf Note a l s o t h a t from
+
+
Section 2.5).
2.3.
S e p a r a t i o n of v a r i a b l e s and t r a n s f o r m s .
Let u s f i r s t make some o b s e r v a t i o n s
about t r a n s m u t a t i o n and s e p a r a t i o n of v a r i a b l e s which l e a d t o a s y s t e m a t i c formal-
i s m f o r h a n d l i n g many problems of i n t e r e s t i n a u n i f i e d manner.
The p r e s e n t a t i o n
99
SEPARATION OF VARIABLES AND TRANSFORMS
w i l l b e e s s e n t i a l l y f o r m a l a t f i r s t b u t w i l l b e i l l u s t r a t e d by numerous e x a m p l e s t o j u s t i f y i t and a more r i g o r o u s t r e a t m e n t w i l l f o l l o w .
A preliminary version
3; 41).
a p p e a r s i n C a r r o l l [35] ( c f . a l s o L e v i t a n [l; 51 and L i o n s [ 2 ; s u p p o s e we c a n t r a n s m u t e
P(D)
with appropriate conditions solution
C
into
Q(D)
(0,m)).
w i l l denote
i Y
for
The c o n d i t i o n s
i E {l,m-
f E Y
i n g on f u n c t i o n s
B
by
(Bf)(y) = @(O,y).
f o r s u i t a b l e b r a c k e t s on
11
C
(a,-) (<,
or
(0,m)
W e assume >+
( c f Remark 2.18) w i l l i n v o l v e some combin-
Q or suitable derivatives plus stipulations that
a t i o n o f g r o w t h r e q u i r e m e n t s on
D @(x,O) = 0
v i a a s o l u t i o n of
( u n s p e c i f i e d a t t h e moment) t o g u a r a n t e e a u n i q u e
ip(x,y), which d e t e r m i n e s t h e n
(Bf)(y) =
Thus
where
m =
order
Q(D).
Suppose Y
f o r some r e a s o n a b l y l a r g e s p a c e
QB = BP
act-
and c o n s i d e r t h e
eigenvalue equations
If
h ( * , A ) E Y’
then
an eigenvalue f o r h(x,A)B(O,X)
QBh = BPh =
Q(D).
-A 2 Bh
Conversely c o n s i d e r
( c f . L e v i t a n [l]).
If N(B)
O(0,A) = 0
(Bh)(y,A) = < b ( y , x ) , h(x,X)>
is
@ ( x , y ) = w(x,y,X) = @ ( x , O ) =
c
Given c o n d i t i o n s
u s u a l l y b e p r o v i d e d by c o n d i t i o n s on h(O,A)e(y,X).
so t h a t
8, w e o b t a i n
for this
@,
which c o u l d
w(O,y,X) = $ ( O , y ) = ( B h ) ( y , h ) =
Consequently
or
(null-space)
m
we g e t nothing whereas
w h i c h is
fO}
if
B
h(0.X)
is 1-1.
=
0
corresponds t o
I n p r a c t i c e we w i l l u s u a l l y
work w i t h
h ( 0 , X ) = e(0,X) = 1 and summarize t h i s i n
Lemma 3 . 1
Under t h e c o n d i t i o n s i n d i c a t e d t h e e i g e n f u n c t i o n s o f
c h a r a c t e r i z e d t o b e o f t h e form
h E
R(y,h) = < b ( y , x ) . h ( x , h ) >
Q(D)
are
( f r o m now on
h(0,h) =
R. W. CARROLL
100
O(0,A)
=
1 is assumed unless otherwise stated).
Now for P(D) = Cp.(x)DJ
the formal adjoint P*(D)
J
Z(-l)’D’(pj$) (3.5)
is defined by
P*(D)$
=
and we let P*(D)R(x,A)
= -A
2
Q(x,A).
Consider then the formal expression
where P*(Dx)B
Theorem 3 . 2
QB
=
Q(D ) B Y
and define Bf(y) =
The operator Bf(y)
=
f(x)>
f(x)>.
Since
transmutes P
into Q ; i.e.
= BP.
Given
B
defined via (3.1) it will be of interest to know when
proceed as follows. Let
(3.7)
P(Dx)U(x,5)
satisfying U(x,O)
=
=
U(x,c)
and we
be the solution of
P(Dg)U(x,5)
f(x);
(or growth conditions).
B = B
conditions D>(x,O)
The function U(x,[)
= 0
may be imposed for uniqueness
will be a generalized displacement
or translation operator in the sense of Levitan (1; 31 which could be written
5 U(x,F,) = Txf(x)
(cf. Remark 1 . 2 ) and in general we will have U(O,C) = f(E,).
Explicit constructions via Riemann functions are often possible and examples will be given below.
Then we form the generalized convolution
and observe that formally
101
SEPARATION OF VARIABLES AND TRANSFORMS
Suppose now t h a t
i s chosen s o t h a t
R(x,A)
shown t o be n a t u r a l from t h e examples). f(x)
s o , by d e f i n i t i o n of
provided conditions f E Y'
Y'
with
Theorem 3 . 3
Then
@(x,O) =
a r e s a t i s f i e d (which we assume).
r e a s o n a b l y l a r g e w e can conclude t h a t
B
<6(<), U ( x , < ) >
=
( t h i s is U(x.0) =
B,
Under t h e c o n d i t i o n s i n d i c a t e d i f
the kernelof
Remark 3 . 4
C
B(0,x) = = 6 ( x )
is p r e c i s e l y
Since t h i s holds f o r
b(y,x) = B(y,x).
Hence
B(0,x) = < n ( x , A ) , 1>= S(x)
B(y,x) =
and
B
=
then
B.
G e n e r a l i z e d c o n v o l u t i o n s such as (3.8) s h o u l d be u s e f u l i n s o l v i n g
a p p l i e d problems and w e w i l l comment f u r t h e r on t h i s below ( c f . Theorem 3.7 f o r T h i s shows t h e importance of s o l v i n g e q u a t i o n s of t h e t y p e ( 3 . 7 ) a s i d e
example).
from t h e t r e a t m e n t of Hutson-Pym [ l ; 2 1 , Leblanc [ l ] , Povzner [ l ] , e t c . ( c f . Section 2.5).
Let u s d e f i n e now t h e t r a n s f o r m s ( s e e Appendix 2 f o r g e n e r a l t r a n s f o r m s ) .
(3.11)
O_f(X) = = f ( A ) .
We suppose t h a t
from ( 3 . 4 ) PO(y,*)
b ( y , x ) = O(y,x)
G(y,A) = < f i ( y , x ) , h(x,X)> (y
Q
=
PO(y,*)
i s e s s e n t i a l l y a parameter h e r e ) .
v a r i e t y of c h o i c e s of that
so t h a t taking
Q
be chosen s o t h a t
h(0,X) = e(0,A)
=
1 w e have
w h i l e by d e f i n i t i o n s
B(y,x) =
Now t h e r e w i l l u s u a l l y be a
f o r which t h e above p r o c e d u r e s a r e v a l i d and we a s k
9
i n (3.11) i s an isomorphism
H
-+
for suitably
102
R. W . CARROLL
large spaces
H
fi.
and
?
Then f o r
F),
E
PIP?=
f(y)> =
?.
Next w e
can w r i t e
and f o r =
g.
g = B*f = P Q f =IP? EIPfi
Hence
9 +PR.
: IPG
-+
2
we have IPPg =
and s i n c e PIP; =
; there
f ( y ) > = <8(y,x) f ( x ) >
r e s u l t s a l s o IF = P-'
:
We s t a t e t h i s a s
Theorem 3.5
Remark 3.6 that
P =IP
-1
If
B(y,x) = b ( y , x )
It f o l l o w s a l s o t h a t
(B*)-l = Q-lP : IPG
+
t h e n under t h e c o n d i t i o n s i n d i c a t e d IP = P-'. q-'F'B*f
=
f
from (3.12) and Theorem 3.5 s o
H.
We show next t h e r e l a t i o n of (3.8) t o a g e n e r a l i z a t i o n of a p r o c e d u r e of L e v i t a n [ l ] f o r solving (3.1).
Now f o r
$(x,X) =
F i r s t l e t u s w r i t e (3.8) as
U(x,E)>
w e have
P(D5)U(x,S)> =
solutions with
5
P(Dx)$ =
2
U(x,E)> = -A $.
P(Dx)U(x,S)> =
Ph =
-A 2h
has unique
h(0,X) = 1, perhaps s u b j e c t t o some growth c o n d i t i o n s , then
$(x,X) = $(O,X)h(x,X)
with
q(0,X) =
U(O,S)>
=
f(S)> = F(h).
Using t h i s w e can w r i t e from (3.13)
We can a r r i v e a t t h i s formula in a n o t h e r way by g e n e r a l i z i n g a p r o c e d u r e of L e v i t a n
[l].
Thus a s a s o l u t i o n t o (3.1) t r y
-1 Given IP = P
a s i n Theorem 3 . 5 , from t h e r e l a t i o n
103
SEPARATION OF VARIABLES AND TRANSFORMS
it is suggested t h a t
and t h i s w i l l b e c o n f i r m e d i n numerous e x a m p l e s b el o w .
Using t h i s we g e t formally
from (3.15)
Q(Dy)c = < Q ( D y ) @ ( x , y ) , Q ( X , U ) > = < P ( D x ) @ ( x , y ) , n ( x , U ) > = < @ ( x , Y ) ,
Then
If
P*( D x)R (x, p)> = - p 2 c .
2
Q0 = -u 0
p e r h a p s s u b j e c t t o some g r o wth c o n d i t i o n s . i t f o l l o w s t h a t whe re
F(p) = c(0.p)
< f ( x ) . Q(x.u)>
=
( 3 . 1 5 ) we o b t a i n ( 3 . 1 4 ) a g a i n . Theorem 3.7 O(x.y)
Let
and
from ( 3 . 1 7 ) .
in
and s u p p o s e
B(0.x)
c a n b e w r i t t e n i n t h e form ( 3 . 1 4 ) .
If
=
6(x) h ( x ,A )
so t h a t is unique
S i m i l a r l y i f (3.16) h o l d s
i s u n i q u e a s i n d i c a t e d t h e n ( 3 .1 4 ) r e p r e s e n t s t h e s o l u t i o n o f ( 3 . 1 ) .
e(y.h)
Remark 3 . 8
c(y,u)
Thus
F(h) =
@
c(y,p) = F(p)B(y,p)
Putting t h i s
g i v e n by (3.8) r e p r e s e n t s t h e s o l u t i o n o f ( 3 . 1 ) .
a s indicated the
e(0.p) = 1,
has unique s o l u t i o n s with
The q u e s t i o n o f i n v e r t i n g a t r a n s f o r m s u c h as
f ( x ) = H(x,X).
F(X)>
P
i n (3.9) b y a f o r m u l a
is s t u d i e d i n Mer cer [ l ; 2 ; 31 ( c f . a l s o T i t c h m a r s h [l]).
The g e n e r a l e i g e n f u n c t i o n e x p a n s i o n t h e o r y d u e t o Weyl, K o d a i r a , T i t c h m a r s h , e t . a l . w i l l be e x p l o i t e d l a t e r ( c f . T i t c h m a r s h
"21).
B e f o r e d e v e l o p i n g more g e n e r a l r e s u l t s o r r e f i n i n g t h e ab o v e l e t u s i n d i c a t e how t h e s e f o r m a l t h e o r e m s work i n a p p l i c a t i o n s : t h i s w i l l p r o v i d e some u s e f u l g u i d e lines. Example 3 . 9 277
exp(-iXx).
Take f i r s t Let
Q(D)
P(D) = D
2
= P*(D)
with
h(x,X) = exp(ihx)
and
n(x,h)
b e a r b i t r a r y s u c h t h a t t h e r e q u i r e m e n t s i n d i c a t e d ab o v e
a r e s a t i s f i e d and we w i l l s a y i n g e n e r a l t h a t s u c h
Q
a r e P-admissible.
Take
=
R. W. CARROLL
104
e(0,X)
=
B(y,x) =
1 and from (3.6) w e have
1 5
exp(-iXx)>
while i t is W
Recall here that
B(0,x) = 1 <exp(-iXx), 1> = 6 ( x ) .
known t h a t
2T
j-_exp (-iXx) dX
= l i m ( S i n Nx/~rx) = 6 ( x ) . Hence Theorem 3.3 a p p l i e s and from (3.4) w e have NO(y,X) = < B ( y , x ) , exp(iXx)>. The formulas (3.9) - (3.10) w i t h Theorem 3 . 5 t h e n y i e l d t h e F o u r i e r i n v e r s i o n formulas a s a by p r o d u c t .
T h i s cannot r e a l l y be con-
s i d e r e d a s a new d e r i v a t i o n however s i n c e a t r a d i t i o n a l d e m o n s t r a t i o n u s e s t h e formula
6(x
- 5)
l i m ( S i n N(x - E)/TT(XN” B(0,x) = 6 ( x ) .
equivalent t o
=
5))
( c f . Sansone [ l ] ) and t h i s i s
Let us define
Remark 3.10
( t h i s t r a n s f o r m w a s used i n Theorem 3 . 7 ) .
Then under t h e c o n d i t i o n s of Lemma 3 . 1
and Theorem 3 . 3 we can w r i t e
and t h i s i s a kind of P a r s e v a l formula. and a f = F - l f
1
where
-1 F[f(-x)] = F f
‘FQ,
FI)>
=
Let
F denotes the Fourier transform.
by p u t t i n g
P(D) = D2
and
@(y,X) = i m ( y , X )
f(-x)
i n p l a c e of
and
(3.20)
O(y,X) = (B h)(y,X) = Fxbm(y,x) = R
F(X) =
SO
*m
C can b e t a k e n as D @ ( O , h ) Y
’$(O,x) = 6 ( x )
Pf = F f , Z f = F - l f , f(A)
with
=
f
and
and we t a k e F
even
f(x)
i n (3.19). Let
h(x,X) =
by ( 3 . 4 )
m
as i n Example 2 . 6 and Thus
Evidently $f
a s i n Example 2 . 6 .
Q(D) = L,(D)
exp(iXx)
and c o n d i t i o n
Pf = Ff
s o we can i d e n t i f y w i t h t h e P a r s e v a l formula w r i t t e n a s
2n
Example 3.11
Indeed f o r Example 3.9 we have
=
(Y,X) 0
which h o l d s ; h e r e
by Example 3.9 when
LmBm = BmD
2
1 Q(x,X) = 2ll e x p ( - i h x ) .
B(y,x) = < n ( x , x ) , B(Y,X)>+
SO
t h a t if
105
SEPARATION OF VARIABLES AND TRANSFORMS
where
H
d e n o t e s t h e Hankel t r a n s f o r m .
m
or its dual Schwartz
HA
The Hankel t r a n s f o r m works on f u n c t i o n s
( s e e Zemanian [l] and c f . a l s o Dube-Pandey (1; 2 1 , Lee [ l ] , A .
[I], T r i o n e [l]).
Hence f o r s u i t a b l e
f
such t h a t
h-m-'IF(A)
can compute t h e Hankel t r a n s f o r m i n (3.21) w i t h i n v e r s i o n formula
Let u s w r i t e ( r e c a l l
h
g
Rm
on
i s d e f i n e d by ( 3 . 2 2 ) ; s e t t i n g
F(A) = Sm(RmF)( A )
and
F(A) = p f ( X ) = F-lf
?FSm.
0)
km = 1 / Z Z m r 2 ( m + 1 )
we have a formula
( c f . a l s o Levitan-Sargsyan f = FF =
1
7 FS% m f
so t h a t
[2]).
8m
Since
= B-'
m
=
There i s a t e c h n i c a l p o i n t involved h e r e i n t h e i n t e r p l a y between b r a c k e t s
( 0 , ~ ) and
work w i t h even
(-m,m)
which r e q u i r e s
f
F
and
Example 3 . 1 4 ) . D
2
Bm
=
Now t h e o p e r a t o r
EmLm
Smg(A)
even i n
and i n t r o d u c e d a f a c t o r of
t i v e l y we could work s o l e l y on
that
we
0
for suitable
1
2
Hi
F(X) = @ f ) ( A ) )
Then an i n v e r s e S m f o r l e t for
(A
E
( 0 , ~ ) with
Rm
X; 112
w e have chosen t o
in (3.21).
h(x,X) = 'd(x,A) = Cos Ax
Alterna(cf. also
should be a t r a n s m u t a t i o n o p e r a t o r such
and w e can check t h i s d i r e c t l y ( c f Lions [2: 3 ; 41 and Example
106
R. W. CARROLL
3.14 below).
Thus l e t
g(y)
b e given f o r
y
2
0
(g'(0) = 0
is n o t n e c e s s a r y )
and from t h e d e f i n i t i o n s 2(Bmg)(x) = (FS"g)(x)
Li$
=
2 D $
-
km<exp iXx, <(Xy)
2 2 DmBmg = -A Bmg.
we o b t a i n immediately
where
=
(2m
+
On t h e o t h e r hand
Here one h a s
l)D(y-'$).
2nd-1-m R (y,A), g(y)>+>
LG[(Xy)
2m+l~m R (y,X)]
(hy)2m+1Lmim(y,h) and a s i m p l e c a l c u l a t i o n shows t h a t t h e terms a t from i n t e g r a t i o n by p a r t s w i l l v a n i s h ( r e c a l l
2m+l > 0
or
Re(2m
=
y = 0
+
arising
1) > 0 ) .
Hence we can s t a t e ( c f . a l s o Example 3 . 1 4 )
Theorem 3.12
$FSm
23
Bm =
satisfies
D
2
Bm
= RmF-'
LmBm = BmD2
while
Bm
Bil =
=
The P a r s e v a l r e l a t i o n (3.19) w r i t t e n i n t h e form
= BmLm.
2a =
satisfies
F f ( * ) > e s t a b l i s h e s t h e c o n n e c t i o n between (3.21)
and (2.11) and t h e l a t t e r can b e r e w r i t t e n i n terms of
The formula (3.26) i s l i s t e d f o r comparison purposes l a t e r ( c f ( 3 . 6 1 ) ) .
For
p e r s p e c t i v e we w i l l o r g a n i z e h e r e a t a b l e of t h e v a r i o u s t r a n s f o r m s coming i n t o play.
Thus
Remark 3.13
We t a k e
h
so t h a t i f t h e r o l e s of new
h.
Take
R
with
and
Q
were r e v e r s e d t h e n
a s i n (3.5) s o t h a t
Theorem 3.5 holds. (3.27)
P
as i n ( 3 . 2 ) - (3.3) w i t h
0
and
Q*(D)fi =
Then t a k e
8
B(0,x)
=
6(x),
0
h(0.X) = B ( O , A )
B(y,x) = b ( y , x ) , and
so t h a t
I n t h e c a s e of Example 3.11 w e have
km(Xy)2mc1~m(y,A). R e l a t i v e t o
P
w e have t r a n s f o r m s
1
would c o r r e s p o n d t o a
-A28
= 6 ( x ) .
=
8(y,X) =
107
SEPARATION OF VARIABLES AND TRANSFORMS
while the corresponding
t r a n s f o r m s are
Q
I n Example 3.11 t h e t r a n s f o r m s i n (3.29) appear with
Bm
involve
< ,>+
and some f a c t o r s of
1/2
i n c o n n e c t i o n w i t h o u r u s i n g even f u n c t i o n s b u t t h i s s h o u l d
c a u s e no c o n f u s i o n l a t e r .
Now i f e v e r y t h i n g f i t s t o g e t h e r w e s h o u l d have
whereas i n Example 3 . 1 1 s i n c e t h e k e r n e l s are a l l f u n c t i o n s of
XA
or
yA
(i.e.
F o u r i e r k e r n e l s ) we have i n a d d i t i o n
21 Bm
Since
IP
=
Q.9by
F while
(3.23)
(resp
= Q-'IP-'
(Bi)-'
P&(resp. and
=
4
=
=
Sm s o
BG = I P Q
by ( 3 . 1 2 ) ) we have
QP as i n Remark 3 . 6 ) . Bil
=
51 FA
2Bi1
=a-'Q-'=
I n Example 3 . 1 1 $ =
as indicated.
F-I
W e refer also t o Sections
2.5 and 2 . 6 f o r examples of i n t e r p l a y between t r a n s f o r m s ( a l s o Appendix 2 ) .
Example 3.14
L e t now
w e l l d e f i n e d by (3.1)
im(x,A)
(3.32)
and, s e t t i n g
P(D)
=
(such
+
with
Q(D)
w i l l be c a l l e d
Q
e(0.A)
O(y,A) = 2mI'(m
Lm(D)
=
1,
Q(D) = D2
with
t r e a t e d a s even i n
A)
P
admissible).
8(y,A) = < b ( y , x ) ,
Am R (x,A)>+.
Let
B
h(x,X) =
Thus
1)X-m-4Mm[x-m-4b(y,x)l
and t h i s is c o n s i s t e n t w i t h t h e formula f o r when
unspecified but such t h a t
O(y,X) = Cos Ay. with kernel
Ern i n Example 3.11 which a r i s e s
Indeed from
B,(y,x)
w e have
Bm
=
+Sm
(with
Smg(A)
is
ioa
R. W. CARROLL
!;(XX)~'COS
= km~m!,
Xy Jm(Xx)f(x)dxdh
m
xm3""f (x) lHm[i&Cos
where
(3.34)
E
m
= 2mr(m
+
1) and
€,Bm(Y,X)
k
m
=
E
hyldx
-2
.
m
xm3* Mm[X%os
=
Hence we have f o r m a l l y
Xyl
which i s e q u i v a l e n t t o ( 3 . 3 2 ) f o r
B(y,A) = Cos Xy
d e s c r i p t i o n of such d i s t r i b u t i o n s
B,(y,x)
i n t h e c a s e of g e n e r a l
(3.35)
so t h a t
P
admissible
Q
and
E~B(O,X= ) x&lHm[X&]
we take
R(x,h) = km (Ax)
6(x)
and
6(x2)
so t h a t the kernel
with
cf.
Bm(O,x) = 6 ( x ) .
Bm(y,x)
of t h e t y p e reD
2
Bm
= BmL,
-4
Bm(y,x)
=
-
and b e g i n s by s e t t i n g
-1 < R e m <
B,(Y,X)
must b e used
He works d i r e c t l y w i t h t h e t r a n s m u t a t i o n problem
m
R (x,X)
m 6(x) must make s e n s e ( c f . T r i o n e [l] f o r a
For t h i s i t c l e a r l y s u f f i c e s t o show
quired i n ( 3 . 3 4 ) .
(3.38)
2m+lAm
= E
Thus l e t u s summarize L i o n s ' c o n s t r u c t i o n of k e r n e l s
and t h e n f o r
Now a p r e c i s e
i s worth d i s p l a y i n g , e s p e c i a l l y s i n c e
similar formula where a r e l a t i o n between
f o r more g e n e r a l
Bm.
e(y,A)>+ = kmcmxm4* Hm [ x ~ % ( ~ , x ) I
~ ( y , ~ = )
a l s o Bochner [ 2 ] ) .
b =
2 bmy(y
(note t h a t
Bmf(0) = f ( 0 )
-1 < R e m <
-4
h a s t h e form
or
2 -m-?, x )+
m (0,x)
x
=
~
6(x)).
f o r comparison w i t h ( 3 . 3 4 )
~
l
One h a s a formula t h e n w i t h
109
SEPARATION OF VARIABLES AND TRANSFORMS
(cf. Erdelyi et. al. [l]). B,(y,x)
in m
Starting here one wants to analytically continue
to establish formulas for all Re m >
-4.
To do this we write
first
and define recursively
Then for -1 < Re m < n - 4
the continuation of
B,
is given by y) dt
where
c^
m
=
+
(-l)nbm/(2m
1)(2m
-
1)
...
(2m
-
(2n - 3)
.
We note that for
n,l
... 3 - 1 =
since 2"r(m
+
(2n-1)(2n-3) 211-3 3/2
)/r(m - T ) ,
= ( - l ) & ' ~and ,
r(-m
...
2nr(~+k)/r(%, (2m+1) (2m-(2n-3)) = 211-1 2n-3 + ---)r(m 2 - 2) = (-~)~-~+'n, r ( m + 3 / 2 ) r ( - m - + )
the integral in (3.42) equals
T(m
Let u s give a different expression €or the kernel
+
l)r(-m
Bz(y,x)
+ ~211-1 ) / 2 r ( n + 4). of
8"m
as
follows.
It is easy to see that
where
cnl
=
(2n - 1) (2n - 3 )
written for m < n - 4
as
.. .
3
B
-1
=
2?(n
+ 4)/I-(%)
so
that Rnm
can be
R. W. CARROLL
110
(3.44)
B;f(y)
=
cm
n+l
-m+-
- 3) f(k-l)(ty)dt
k=1
- n+l 1 cnky-2n+lJ~x2dk(y2 - x
= c m
(2n
-m+-
(211-3)
1
f(k-l)(x)dx
k=1
Thus we have (3.45)
f(x)>
Summarizing we have Bm
=
2 and D 8
defined for all m > -$
m
Bt(y,x)
is a distribution given by ( 3 . 4 1 ) or ( 3 . 4 5 ) with
-1 < m <
-4 we have by
B;(O,x)
and
1 by Bmf(y)
example express Bmf(y) 1
gm(x,A)
=
=
1
f(x)>
1. 1.
6(x).
For
-m
x J (Ax) m
=
Amg (x,A)
m
Hence for
s o that ( 3 . 4 6 ) holds with
8 2 makes sense for -1 < m < $ and hence
But
( 3 . 4 6 ) ; note here that
(3.47)
=
The kernel
B m (f) is equal to
any of its determinations via formulas ( 3 . 4 1 ) or ( 3 . 4 5 ) for n
Bm replacing Em.
BmLm.
(3.39)
f(x) = X-~J~(AX) is an analytic function of x
Now
=
so
does
where
2-
n
so that ( 3 . 4 6 ) can be written
If
E
one knows m
= Cm[O,m)
+
Em is holomorphic with values in L(E)
Lions [2]) while for Re z > 0 both ( 3 . 4 8 ) holds for
when
P,(y,x)
-1
C
Re m <
4
T(z)
l/I'(z)
and
(cf.
are holomorphic. Hence
and continuing we obtain ( 3 . 3 4 ) for Re m >
d
is identified with a suitable determination B;(y,x).
-4
SEPARATION OF VARIABLES AND TRANSFORMS
We remark here also that the function U(x,c) L,(D,)U(x,S)
= Lm(Dg)U(x,S)
U (x,O)
5
=
where w2
=
indicated in (3.7) satisfying
U(x,O) = f(x)
(and U ( 0 , S ) = f(0,
Ux(0,5)
=
0) can be written explicitly in the form
x2
=
+ 52 +
(cf. Copson-Erdelyi 111)
2xct
Using a P admissible Q
Remark 3.15 B(0,x)
with
111
such as Q(D)
=
D2 and knowing that
the inversion theory for P
+ = 6(x)
in Theorem 3.5
is
equivalent to the Hankel inversion formulas and seems to represent a new derivation. However a certain amount of information about Hankel inversion is implicit in the fact that
B(0,x)
=
6(x).
We will want to deal with formulas of the form (3.16) for example
Remark 3.16
and record here the following observations. Thus take h(x,X) 2m+lAm R (x,X) R(x,X) = km(Xx)
=
im(x,X)
and
to obtain by the Hankel formulas
'0
Similarly by symmetry (3.51)
h(x,X)>+
=
6
We note here also that formally the truth of (3.50) implies
.
=
I>+
= 6(x)
1 but the converse does not seem to follow immediatly by analytical
I
Example 3 . 1 7
The following differenti