Singular and Degenerate Cauchy Problems
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Singular and Degenerate Cauchy Problems
ACADEMIC PRESS RAPID MANUSCRIPT REPRODUCTION
This is Volume 127 in MATHEMATICS IN SCIENCE AND ENGINEERING A Series of Monographs and Textbooks Edited by RICHARD BELLMAN, University of Southern California The complete listing of books in this series is available from the Publisher upon request.
Singular and Degenerate Cauchy Problems R.
w. Ccxrroll
Department of Mathematics University of Illinois at Urbana-Champaign
R. E. Showalter Department of Mathematics University of Texas at Austin
ACADEmIC PRESS
New York
San Francisco
London
1976
A Subsidiary of Harcourt Brace Jovanovich, Publishers
BY ACADEMIC PRESS,INC. ALL RIGHTS RESERVED. NO PART OF THIS PUBLICATION MAY B E REPRODUCED OR TRANSMITTED IN ANY FORM OR BY ANY MEANS, ELECTRONIC OR MECHANICAL, INCLUDING PHOTOCOPY, RECORDING, OR ANY INFORMATION STORAGE AND RETRIEVAL SYSTEM, W I T HOU T PIIRMISSION IN WRITING FROM T HE PUBLISHER.
COPYRIGHT 0 1976.
ACADEMIC PRESS, INC.
111 Fifth Avenue, New York. New York 10003
Unifed Kingdom Edifion published by ACADEMIC PRESS, INC. (LONDON) LTD. 24/28 Oval Road, London N W l
Library of Congress Cataloging in Publication Data Carroll, Robert Wayne, Date Singular and degenerate Cauchy problems. (Mathematics in science and engineering ; ) Bibliography: p. Includes index. 1. Cauchy problem. I. Showalter, Ralph E., joint author. 11. Title. Ill. Series. QA 3 74.C 38 5 15’.353 76-46563 ISBN 0-12-161450-6
PRINTED IN TH E UNITED STATES O F AMERICA
Contents
vi i
PREFACE
1
Singular Partial Differential Equations
of EPD Type 1 .l. 1.2. 1.3. 1.4.
Examples Mean values in R" The Fourier method Connection formulas and properties of solutions for EPD equations in D ' a n d S' 1.5. Spectral techniques and energy methods in Hilbert space 1.6. EPD equations in general spaces 1.7. Transmutation
1 5
a 29 43 66
80
2
Canonical Sequences of Singular Cauchy Problems
a9
2.1. The rank one situation 2.2. Resolvants 2.3. Examples with mzo, = 0 2.4. Expressions for general resolvants 2.5. The Euclidean case plus generalizations
96 101 130 136
V
CONTENTS
3
3.1. 3.2. 3.3. 3.4. 3.5. 3.6. 3.7.
Degenerate Equations with Operator Coefficients
Introduction The Cauchy problem by spectral techniques Strongly regular equations Weakly regular equations Linear degenerate equations Semilinear degenerate equations Doubly nonlinear equations
143 147 166 179 187 206 229
Selected Topics 4.1. 4.2. 4.3. 4.4. 4.5.
Introduction Huygens' principle The generalized radiation problem of Weinstein Improperly posed problems Miscellaneous problems
237 237 245 251 264
REFERENCES
215
INDEX
329
vi
Prefoce
In this monograph we shall deal primarily with the Cauchy problem for singular or degenerate equations of the form (u' = u, = au/at) (1.1)
A(t)u,, + BWu, + C ( t ) u= g
where 4-1 is a function of t , taking values in a separated locally convex space E, while A ( t ) , B ( t ) , and C ( t ) are families of linear or nonlinear differential type operators acting in €, some of which become zero or infinite a t t = 0. Appropriate initial data u ( 0 ) and u,(O) will be prescribed a t t = O , andg is a suitable €-valued function. Similarly some equations of the form
(1.2)
(A(-)u),+ , (B(.)u), + C(.)U = g
will be considered. Problems of the type (1.1) for example will be called singular if at least one of the operator coefficients tends to infinity in some sense as t + 0. Such problems will be called degenerate if some operator coefficient tends to zero as t + 0 in such a way as to change the type of the problem. Similar considerations apply to (1.2). We shall treat only a subclass of the singular and degenerate problems indicated and will concentrate on well-posed problems. If (1.1) or (1.2) can be exhibited in a form where the highest order derivative appears with coefficient one, then the problems treated will be of parabolic or hyperbolic type; if the highest order derivative cannot be so isolated, the equation will be said to be of Sobolev type. We have included a discussion of not necessarily singular or degenerate Sobolev equations for completeness, since it is not available elsewhere. The distinction between singular and degenerate can occasionally be somewhat artificial when some of the operators (or factors thereof) are invertible or when a suitable change of variable can be introduced (see, e.g., Example 1.3); however, we shall not usually resort to such artifices. In practice E frequently will be a space of functions or distributions in aregionclCclR"andO=ztcb 0 where i t i s assumed t h a t t K ( x , t )
0 for t
p
0 w h i l e K(x,O)
= 0; s u i t a b l e i n i t i a l data u(x,O)
>
and
u (x,O) w i l l be p r e s c r i b e d on t h e p a r a b o l i c l i n e t = 0. This ext ample w i l l be t r e a t e d as a degenerate problem i n Chapter 3 b u t i s d i s p l a y e d here because o f t h e connection o f (1.3) EPD equation o f i n d e x 2m
+ 1
t o a (singular)
= 1 / 3 under t h e change o f v a r i a b l e
T = 2/3 t3l2f o r t > 0 which y i e l d s
(1.5)
I
UTT +
3-r
UT
= uxx
S i m i l a r l y i n t h e r e g i o n t < 0 t h e change o f v a r i a b l e ~ = 2 / 3 ( - t )3/2
4
1.
SINGULAR PARTIAL DIFFERENTIAL EQUATIONS OF EPD TYPE
1 i n t o uTT + 3* u
transforms (1.3)
+ uxx
~
= 0 which i s a s i n g u l a r
e l l i p t i c e q u a t i o n o f t h e type s t u d i e d by Weinstein and o t h e r s i n t h e c o n t e x t o f g e n e r a l i z e d a x i a l l y symnetric p o t e n t i a l t h e o r y (GASPT); c f . Remark 1.2 f o r some r e f e r e n c e s t o t h i s ,
The r e f e r -
ences above i n Remark 1.2 a l s o deal w i t h mixed problems where s u i t a b l e data i s p r e s c r i b e d on v a r i o u s types o f curves i n t h e e l l i p t i c ( t < 0) and h y p e r b o l i c ( t > 0) regions; such problems occur f o r example i n t r a n s o n i c gas dynamics.
We w i l l n o t deal w i t h t h e
e l l i p t i c r e g i o n i n general b u t some o t h e r types o f s i n g u l a r hyperb o l i c problems f o r EPD equations w i l l be discussed b r i e f l y i n Chapter 4. 1.2 (1.1)
-
Mean values i n Rn (1.2)
for x
E
tribution T
E
D
= DX I
We r e f e r now t o t h e EPD problem
Rn and 2m Let 0 with
SINGULAR PARTIAL DIFFERENTIAL EQUATIONS OF EPD TYPE
1.
Theorem 2.1 px(t) (i.e.,
sd!-t
Let T
E
D i be a r b i t r a r y .
Then u:(t)
=
T s a t i s f i e s t h e EPD e q u a t i o n (1.1) w i t h 2m + 1 = n - 1 n 1) and t h e i n i t i a l c o n d i t i o n s um(o) = T w i t h m =
*
-
X
P ( o ) = 0. x Proof:
We r e c a l l t h a t (S,
T) +
*
S
T:
E
I
D
x
I -+
D
I
i s s e p a r a t e l y c o n t i n u o u s so e q u a t i o n ( 1 .l)r e s u l t s i m m e d i a t e l y by
-
d i f f e r e n t i a t i n g (2.5) and u s i n g (2.5) px(t)
-+
6 ( D i r a c measure a t x = 0 ) when t
dt d (px(t) *
T) = ; t AAx(t)
t (F) ( A x ( t ) * AT)
-+
0 as t
*
T = (:)[A6
-+
0.
Evidently
(2.6).
*
0 and we n o t e t h a t
-+
Ax(t)
*
TI = QED
For i n f o r m a t i o n a b o u t d i s t r i b u t i o n s and t h e d f f e r e n t ia t ion o f v e c t o r v a l u e d f u n c t i o n s l e t us r e f e r t o C a r r o l l [141, A. Friedman [Z],
v
Garsoux [l],G e l f a n d - S i l o v [5;
6;
71
H o r v l t h [l],
Nachbin [l],Schwartz [l; 21, Treves [l].
A function f:R
D e f i n i t i o n 2.2
E, E a general l o c a l l y
-+
I
I
convex space, has a s c a l a r d e r i v a t i v e f s ( t )E (E )
d
-df t( * ) ,
I
I
I
e > = for a l l e
b r a i c dual o f F).
I
E
E
I
(F
*
topology o f E then f
if
denotes t h e a l g e -
On t h e o t h e r hand i f d f = f I ( t ) I
*
E
E i n the
i s c a l l e d t h e s t r o n g d e r i v a t i v e o f f.
The f o l l o w i n g lemma i s a s p e c i a l case o f r e s u l t s i n C a r r o l l
[2; 5; 141 and Schwartz [2].
7
SINGULAR AND DEGENERATE CAUCHY PROBLEMS
If f:R
L e m a 2.3
+
E
I
I
has a s c a l a r d e r i v a t i v e f S then f i s
strongly differentiable with f Proof: l i m i t s as t
I
I
= fs.
F i x to and s e t g ( t ) = -t
-
.
f ( t ) f ( to>
-
to
I n discussing
to i t s u f f i c e s t o c o n s i d e r sequences tn +. to s i n c e
any to has a countable fundamental system o f neighborhoods. hypothesis g ( t n )
I
+.
By
Using t h e f a c t t h a t E i s a
f s ( t o weakly. )
Monte1 space and a v e r s i o n o f t h e Banach-Steinhaus theorem ( c f . C a r r o l l [14])
i t follows t h a t g(tn)
I
+.
f;(to) s t r o n g l y i n E ' so
I
t h a t f s ( t o )= f ( t o ) . Remark 2.4
QED
A g r e a t deal o f i n f o r m a t i o n i s a v a i l a b l e r e l a t i n g
" s p h e r i c a l " mean values and d i f f e r e n t i a l equations, b o t h i n Rn and on c e r t a i n Riemannian manifolds.
I n a d d i t i o n t o references
i n c l u d e d i n t h e I n t r o d u c t i o n , Example 1.1,
Remark 1.2,
and work
on GASPT, r e l a t i v e t o such mean values, l e t us mention Asgeirsson
[l],C a r r o l l [l],Fusaro [l], Ghermanesco [l;21, G i n t h e r [l], Helgason [l;2; 3; 4; 5; 6; 7; 8; 9; 10; 111, John [l],O l e v s k i j
[l],P o r i t s k y [l], Walter [2], t h e i r bibliographies.
Weinstein [12],
Willmore [l]and
T h i s w i l l be discussed f u r t h e r i n a L i e
group c o n t e x t i n Chapter 2.
For some r e f e r e n c e s t o complex func-
t i o n t h e o r y and v a r i o u s mean values and measures r e l a t e d t o d i f f e r e n t i a l equations see Zalcman [l;21 and t h e b i b l i o g r a p h i e s there. 1.3
The F o u r i e r method.
We w i l l work w i t h a more general
e q u a t i o n than (1.1) which i n c l u d e s (1.1)
8
as a s p e c i a l case.
Thus
1.
SINGULAR PARTIAL DIFFERENTIAL EQUATIONS OF EPD TYPE
where the aa a r e constant (and r e a l ) and the sum i n (3.1) i s over
a f i n i t e set of mu1 t i - i n d i c e s a transform F:T
A
-+
T = FT:S
Schwartz [ l ] ) and f o r $ A
$ =
1
fm
-m
I -+
E
S
I
= (a1
. . ., a n ) .
The Fourier
i s a topological isomorphism ( c f .
S f o r example we use the form F$
$(XI exp - i<x,y>dx where <x,y>
=
A
n
1 xiyi '
=
Hence in par-
t i c u l a r F(Dk$) = ykF$ = y k $. We r e c a l l a l s o t h a t AxT = A 6 w i t h F6 = 1 and will w r i t e A(y) = F(Ax6.) = 1 aaya = 1 aayl8l a n I t w i l l be assumed t h a t A(y) s a t i s f i e s yn
*
T
...
.
Condition 3.1 IYI >
A(y)
=
1 aaya
i s real w i t h A(y) 2 a > 0 f o r
R o w
In p a r t i c u l a r Condition 3.1 holds when Ax = -A = -1 i 2D 2 -X k 2 1 0; so t h a t F(-A6) = 1 yk = lyI2. I t can a l s o hold f o r c e r t a i n h y p o e l l i p t i c operators A since then IA(y)l
-+
a s IyI
(cf.
-+
Hb'rmander [ l ] ) o r even nonhypoelliptic operators (e.g., A(y) = 2 2 a + y2 + + y n ) . One m i g h t a l s o envision a p a r a l l e l theory
...
when Re A(y) 2 a > 0 f o r IyI > Ro b u t we will not i n v e s t i g a t e t h i s here.
Let us f i r s t consider the problem f o r
on 0 5 t 5 b
0.
SO
that
^cm
Let
i s a fundamental m a t r i x f o r t h e f i r s t o r d e r system
corresponding t o (3.4)
+
(3.14)
ip =
w r i t t e n i n t h e form A
0; M(y,t) =
As i n Schwartz [3] i t i s e a s i l y shown t h a t
from which f o l l o w s i n p a r t i c u l a r
Given t h a t
^Rm
and
irnhave
s u i t a b l e p r o p e r t i e s ( s t a t e d and v e r i -
f i e d below) i t f o l l o w s t h a t i f
"m with w (T) = mulas (3.18)
A
T and ;"(T) t
-
(3.20)
i s a s o l u t i o n o f (3.4)
= 0 then u s i n g (3.16)
below make sense.
14
-
(3.17)
i n S' the f o r -
SINGULAR PARTIAL DIFFERENTIAL EQUATIONS OF EPD TYPE
1.
- ^m R (y,t,-c)?
t im(t)
-
= i m ( t ) "m R (y,t,T)?
From (3.20) i t then f o l l o w s f o r m a l l y t h a t any s o l u t i o n (3.4) i n S
I
"m with w (T) =
A
T
and
^m
w ~ ( T )=
0
(T
2 of
> 0) must be o f t h e
form i m ( t ) = i m ( y y t y T ) ? . We w i l l now proceed t o j u s t i f y t h i s and t o a t t e n d t o t h e case
T
= 0 as w e l l .
W e remark a l s o t h a t i f one
considers
w i t h i n i t i a l conditions given a t t =
A
T
as a b o v e ( f ( - ) c C o ( S I ) ) then
t h e procedure o f (3.20) f o r m a l l y y i e l d s t h e unique s o l u t i o n as (3.22)
$(t)
= i m ( y y t y T ) ?+ r ? ( y , t , < ) ? ( f ) d < T
We w i l l develop a procedure now which can a l s o be used i n o t h e r s i t u a t i o n s (see S e c t i o n 1.5).
It i s h e l p f u l t o have an ex-
p l i c i t example o f t h e technique however and hence we i n t r o d u c e i t here; i n p a r t i c u l a r t h i s w i l l make t h e considerable d e t a i l s more e a s i l y v i s i b l e and s h o r t e n t h e development l a t e r . 1/2 l e t P ( t ) = exp ( -
r y t
Thus f o r m 2
-
d5) = (t/b)2m+1 so t h a t P ( T ) / P ( t ) =
(-c/t)2m+1 5 1 f o r 0 S T 5 t 5 b and P ( t )
-+
0 as t
-+
0.
L e t us
c o n s i d e r equation (3.4) f o r km(y,t,T) w i t h i n i t i a l data a t
15
SINGULAR AND DEGENERATE CAUCHY PROBLEMS
t = T > 0 as i n (3.13) and then, m u l t i p l y i n g by P ( t ) , we o b t a i n
(3.23)
& (Pi!)
+ PA^Rm = 0
This leads t o t h e i n t e g r a l equation (T 5 5 5 u 5 t), e q u i v a l e n t t o (3.4), w i t h i n i t i a l c o n d i t i o n s a t t = T as i n d i c a t e d ,
and e v i d e n t l y T = 0 i s p e r m i t t e d i n (3.24).
A s o l u t i o n t o (3.24)
i s given f o r m a l l y by the s e r i e s (3.25)
03
“m R (y,t,-c)
=
1
k=O
(-l)kJk
1
( c f . H i l l e [ l ] ) where Jk denotes t h e kth i t e r a t e o f the o p e r a t i o n J and Jo i s t h e i d e n t i t y . (3.26) we have R
Writing
imYp = 1 =
- ^RmYP-’
f
k=O
= (-1)’Jp-1
imYo = 1 and (-l)kJk and i f y
1 E
K C Cny w i t h K
compact, then I A ( y ) I C 5 cK and i n p a r t i c u l a r
5 cKF(t,-c) 5 cKF(t,o) = c K F ( t ) where F ( t , T) i s the double i n t e g r a l i n (3.27) and F(tl) 5 F ( t ) 5 F(b) f o r 0 S t, 5 t i b ( n o t e t h a t F ( t ) = have, using (3.27)
,
16
t2
Continuing we
SINGULAR PARTIAL DIFFERENTIAL EQUATIONS OF EPDTYPE
1.
-
so t h a t
1:
h
Rmy'
-
-1
F(a)
0
,..
2 2 cKF ( t ) / 2 ! s i n c e i f F ( t ) =
Ic 5
($)2m+1dSdo t h e n
F
= F F' w i t h F(o) = F ( o ) = 0.
By
i t e r a t i o n one o b t a i n s
Hence t h e s e r i e s (3.25) [O 5
T
5 t S bl
x
converges a b s o l u t e l y and u n i f o r m l y on
K and s i n c e t h e terms Jp
1 are continuous h
and a n a l y t i c i n y t h e same i s t r u e o f Rm(y,t,-r). It ^m i s c l e a r t h a t R g i v e n by (3.25) s a t i s f i e s (3.24) and we can
i n (y,t,T)
state Lemma 3.5
"m w i t h R (Y,T,T)
The s e r i e s (3.25)
"m
= 1 and Rt(y.~,-r)
The maps (y,t,-c)
+
"m R (y,t,T)
r e p r e s e n t s a s o l u t i o n o f (3.24) = 0 (0
and (y,t,-c)
5
T
-t
5 t 5 b and m > -1/2). "m Rt(y,t,T)
are contin-
"m "m uous n u m e r i c a l f u n c t i o n s w h i l e R ( * , t , ~ ) and Rt(*,t,.r)
are anal-
ytic. Proof: for
ty
E v e r y t h i n g has been p r o v e d f o r
im and t h e
f o l l o w i m m e d i a t e l y upon d i f f e r e n t i a t i n g (3.24)
We examine now ( c f .
(3.24))
17
statements i n t.
QED
SINGULAR AND DEGENERATE CAUCHY PROBLEMS
For
> 0 t h e r e i s o b v i o u s l y no problem i n p r o v i n g c o n t i n u i t y and
T
a n a l y t i c t y as i n Lemma 3.5 so we consider
T
-+
0.
Let
(3.31) Clearly
i s n o t continuous as 2m+l "m t h e r e i s no hope t h a t (y,t,i-) * e t ( y . t , . r ) t,T)
-f
H(t,.r)
i:(y,t,o)
from (3.30)
i s continuous on [o,b]
( y o y o ) equal t o - A ( y 2m o )+. 1m 2m+l "m
7 Rt (y,t,-c)
(3.31), x
(y,t)
0, (3.17)
"m "m One has R = A(y)S (i.e., (3.16)) T "m "m holds w h i l e S (Y,T,T) = 0 w i t h St(y,-c,-c)
= 1; finally
P(y,t,o)
=
Proof:
o
f o r m > -1/2. One notes f i r s t t h a t t h e s e r i e s o b t a i n e d by term"m i n T i s t h e same as A(y)S
wise d i f f e r e n t i a t i o n o f (3.25) where m:
i s give n by (3.36).
Since Jk
(y,t,T)
U = 0 a t t = T for k
2
1 w i t h U(T,T) = 0 i t f o l l o w s t h a t &?(y,-c,~) = 0. On t h e o t h e r "m hand S (y,t,o) = 0 f o r m > -1/2 s i n c e U(t,o) = 0 w h i l e (3.17) holds f o r T > 0 s i n d e
im clearly
m "m ( s i n c e S (y,-c,-c) = 0) ST = UT UT(t,-c)
2m+l
= ,-U(t,-c)
-
-
s a t i s f i e s (3.35)
d J d-c
?=
U
T
1 and hence f o r T > 0
20
-
and hence J
irn But T*
1. SINGULAR PARTIAL DIFFERENTIAL EQUATIONSOF EPD TYPE
( fr om (3.35) and t h e d e f i n i t i o n o f J ) , from which f o l l o w s "m St(y,~,~) = 1 f o r
$.
erties o f
> 0 and t h e c o n t i n u i t y and a n a l y t i c i t y prop-
T
To check (3.4)
one can use (3.17)
t o obtain f o r
T > O
which leads t o (3.4)
upon d i f f e r e n t i a t i o n i n t.
"m
Now ST i s determined by (3.17) and as "m
l i m ST may n o t e x i s t .
T
-+
QED 0 a priori
However l e t us l o o k f i r s t a t m
(3.40)
+s^m(y,t,r)
1 1 where 7 U ( ~ , T ) = $1 t 5 b.
=
-
1 (-l)kJk
k=I)
(c)2m3 2 T
Hence, as i n (3.26)
-
U(.,T)
*.
1
f o r m > 0 and 0 5
(3.29),
There i s l i t t l e hope t h a t (y,t,r)
5 5 5
t h e s e r i e s i n (3.40)
verges u n i f o r m l y and a b s o l u t e l y f o r 0 5 Cn.
T
T
+
5 t 5 b and y
-r1 "mS (y,t,T)
E
con-
K C
w i l l be
I 5 t 5 b ] s i n c e e.g., U(t,-r) i s n o t so T 1 continuous ( n o r i s J -r U(-,T), etc.). We can however l e t T
continuous on [0 S
0 i n (3.40)
T
f o r m > 0 t o obtain f o r t > 0
21
-f
SINGULAR AND DEGENERATE CAUCHY PROBLEMS
Consequently from (3.17)
for
Lemma 3.9 "m ST(y,t,o)
while y
+
For m > 0 and m = -1/2 th e 1 "m = -R (y,t,o) i s continuous f o r 2m "m ST(y,t,o) i s a n a l y t i c . F u rth e r ,
l y continuous f o r m
IBq(y,- t , -c)dr
Jci,
sion
2
% -1/2 w i t h S (y,t,B)
f u n c t i o n (y,t)
+
0 5 t 5 b and y "m S (y,t,?)
-
"m
K
E
i s absolute-
S (y,t,a)
=
and f o r 0 5 T 5 t S b and m 2 -1/2 t h e expres-
"m
Ts,(y,t,T)
i s continuous i n (y9t,-c) and a n a l y t i c i n y.
Proof: "m ST(y,t,o)
The c o n t i n u i t y and a n a l y t i c i t y i n d i c a t e d f o r "m and TS,(Y,t,T) f o l l o w immediately from (3.42) and
^m
(3.17). For 0 < ci, 5 B 5 t 5 b t h e formula S (y,t,B) S (y,t,a) = jb: (y ,t,r)d r "m i s obvious and by c o n t i n u i t y one can take l i m i t s as a
+
0.
I n t h e cases -1/2 < m
w i l l have an i n t e g r a b l e s i n g u l a r i t y a t f o r -1/2 < m < 0 and
-
T
5 0 we n o t e t h a t
1 U(t,-c) = 0 since T
QED
log-c f o r m = 0. A
I n o r d e r t o o b t a i n some bounds f o r Sm when
YE
m u l t i p l y (3.17) ( w i t h T re p l a c e d b y . 5 ) by im(y,t,E) d "m 2 -1s I = 2s"m"mS5 one o b ta i n s upon i n t e g r a t i o n d5
22
-
Rn we and s i n c e
1. SINGULAR PARTIAL DIFFERENTIAL EQUATIONS OF EPD TYPE
By Gronwall's lemma ( c f . Section 5 and Carroll [14]) there reA
s u l t s ISm(y,t,-c)12 5 c5 f o r 0 5
5 t 0 and one
0 i n t h e r e s u l t i n g formulas ( c f . Lemnas 3.5,
1 "m
3.9 and n o t e t h a t terms - S (y,t,c)@(y,€,,T)
5
h
s i n c e @ s a t i s f i e s (3.21)
with
a r e continuous
l A
To work
@(Y,~,T) continuous).
5
3.6,
A
4
i n E we r e c a l l t h a t (S,T) + ST : x E E i s separately Y Y Y continuous w i t h E a Frechet space and hence t h i s map i s conY tinuous (see Bourbaki [2] and f o r v e c t o r valued i n t e g r a t i o n see Bourbaki [3; 41, C a r r o l l [14]). (3.18)
-
(3.20)
Lemma 3.13
Hence a l l of t h e o p e r a t i o n s i n
make sense i n E Y'
and (3.22) For 0 5
T
+
TS;(*,t,r)
C 1 (OM) w h i l e , f o r 0
0 where B i s the type o f 2 1/2 lYlC = Yk) IC (1 lYk1E)1/2
any
5 6 exp(B+E)lzlC f o r
E
I Bt 5
5
ZlYklC =
Since
o f z.
llyllc
One
has then
Bb = B
The conclusion o f t h e Lemma now f o l l o w s from t h e Paley-WienerSchwartz theorem Theorem 4.2
see Schwartz [l3) I f Ax = - A x ,
then wm(t) = Rm(-,t)
*
2
QED
-1/2
E
E I
DI.
-
(3.3)
E
in
D
I
D’
I
D on (t,T).
By Lemma 4.1 supp Rm(-,t) i s contained i n a f i x e d
compact s e t f o r 0 2 t 5 b so t h a t Rm(.,t)
T
0 5 t 5 b, and T
T i s a s o l u t i o n o f (3.2)
depending continuously i n Proof:
m
.
*
T makes sense f o r
Furthermore t h e s e t {Rm(*,t); 0 i t 6 b} i s bounded i n
(cf. Ehrenpreis [3],
[l]t h a t t h e map (S,T)
Schwartz [ l ] )
*
and one knows by Schwartz
D’ i s hypocontinuous.
T : El
x
DI
To check t h e d i f f e r e n t i a b i l i t y o f t
-+
Rm(.,t)
-+
S
30
-f
i n E l we r e c a l l
SINGULAR PARTIAL DIFFERENTIAL EQUATIONS OF EPD TYPE
1.
t h a t on bounded s e t s i n E
I
t h e topology c o i n c i d e s w i t h t h a t i n -
I
duced by Oc ( c f . Schwartz [l], p. 274, where t h e F o u r i e r t r a n s form i s shown t o e s t a b l i s h a t o p o l o g i c a l isomorphism between E
I
M and n o t e t h a t on s e t s B C Exp OM o f bounded exponen-
and Exp 0
t i a l t y p e t h e topology i s t h a t induced by OM).
The r e s u l t f o l -
QED
lows immediately.
For uniqueness i n Theorem 4.2 we can go t o a d i r e c t express i o n f o r "R"(y,t,r)
d e r i v e d i n C a r r o l l [2;
51 and then f o l l o w
Thus one can w r i t e f o r m
t h e procedure i n v o l v i n g (3.20).
2 -1/2
n o t an i n t e g e r
where t = tm,z1 =
TM, and
y,
=
r(m+l)r(-m)/Z,
while f o r
m 2 -1/2 an i n t e g e r we have
where Nm denotes t h e Neumann f u n c t i o n ( c f . Courant-Hi1 b e r t [Z], Watson [l]).Some a n a l y s i s o f t h e Bessel and Neumann f u n c t i o n s , which we o m i t here (see C a r r o l l [2; L e m a 4.3
Wnen A(y) = I y I
2
31
51 f o r d e t a i l s ) , leads t o
^m t h e r e s o l v a n t s R (y,t,T)
are
SINGULAR AND DEGENERATE CAUCHY PROBLEMS
e n t i r e f u n c t i o n s of bounded e x p o n e n t i a l t y p e f o r 0 5 T 5 t 5 b w i t h {Rm(*,t,=)}
a bounded s e t i n E l .
i s bounded f o r 0 5
6 5
Lemmas 3.5,
we can say t h a t given A(y) = IyIL,
C E l i s bounded f o r 0 5 6 5 t 5 b and c e r t a i n l y 3.6,
3.7,
3.8,
3.9,
3.10,
3.11,
and 3.12 remain v a l i d .
Lemma 3.13 holds w i t h OM r e p l a c e d b y Exp OM = FE
-
we t a k e i n v e r s e F o u r i e r transforms i n (3.18) i n g c o n v o l u t i o n formulas w i l l be v a l i d f o r T Theorem 4.4
E
E
S i m i l a r l y t h e problem (3.52)
D
I
, given
I
c OM
(3.20) D
.
I
*
The s o l u t i o n wm(t) = R m ( - , t )
(3.3) when Ax = -Ax and T
and W:(T)
El
t 5 b.
Using f o r example (3.39) {S"(*,t,S)l
S i m i l a r l y {R:(*,t,S)}C
T
the resultHence we have E
by Theorem 4.2,
w i t h f ( * )E Co(D1),
= 0 i s u n i f o r m l y w e l l posed i n D
I
and if
w"(T)
D I o f (3.2)i s unique. = T
E
D',
w i t h s o l u t i o n given
b y (3.53). Going back now t o t h e r e s o l v a n t e x p l o i t t h e Sonine i n t e g r a
In R (y,t)
i n OM o r Exp OM we
Watson formula ( c f . Rosenbloom [l],
[l]) t o o b t a i n f o r m > p Z -1/2
We n o t e i n passing t h a t f o r m-p i n t e g r a l (4.6) consequence o f t h e r e c u r s i o n r e l a t i o n (3.12)
i s also a d i r e c t (cf. C a r r o l l - S i l v e r
[15; 16; 171, S i l v e r [l])and t h i s w i l l be u t i l i z e d i n Chapter 2 where an e x p l i c i t d e r i v a t i o n i s given; analogous formulas i n 32
1.
SINGULAR PARTIAL DIFFERENTIAL EQUATIONS OF EPD TYPE
o t h e r group s i t u a t i o n s w i l l a l s o be derived.
We r e c a l l n e x t
the formula ( c f . Watson [l])
2 0 an i n t e g e r .
for p
provided m f -1, -2,
T h i s leads t o t h e formula
. . ., -p.
We observe t h a t (4.8)
g i v e s re-
s o l v a n t s f o r va ues o f m < -1/2 and i t should be noted here t h a t
zrnJ-,(z
jm(z) =
-2m+l s a t i s f i e s jm + (T)jm.+
I
jm= 0 so t h a t ( c f .
A
R-m(y,t) = 2-mr(l-m)zmJ-m(z), f o r z = t m and any
(3.6))
nonintegral m
2
A
0, s a t i s f i e s t h e equation R i F + h
( -2m+l 7)Rtm A
A
A(y)"Rm = 0 w i t h i n i t i a l data R-m(y,o)
= 1 and R-m(y,o)
t
=
+ 0.
Now
from (4.8) w i t h n o n i n t e g r a l m < -1/2 we can choose an i n t e g e r p
so t h a t m + p 2 -1/2 and express I Rn i n terms o f d e r i v a t i v e s o f A
RmP,
whose p r o p e r t i e s a r e known from S e c t i o n 1.3.
I n particular
holds i n E and a l s o i n OM o r Exp OM(cf. Lemma 3.13 and t h e Y remarks before Theorem 4.4) so we can s t a t e , r e f e r r i n g t o (3.11), (4.8)
Theorem 4.5 o f t h e form (3.6) "m o f the R f o r m
For n o n i n t e g r a l m < -1/2 r e s o l v a n t s Am R (y,t) obey (4.8)
2 -1/2.
and thus i n h e r i t t h e p r o p e r t i e s
Also (4.6)
holds i n OM o r Exp Or,,.
Taking i n v e r s e F o u r i e r transforms i n (4.6) composing t h e r e s u l t w i t h T
E
S
I
or T
obtain
33
I
E.
and (4.8)
and
D (when Ax = -Ax) we
SINGULAR AND DEGENERATE CAUCHY PROBLEMS
(3.3)
in S
I
2
For m > p
Theorem 4.6
-1/2 t h e (.unique) s o l u t i o n o f (3.2)-
I
(or D ) f o r index m i s g i v e n by
-
where wp s a t i s f i e s (3.2)
(3.3)
i n S I ( o r 0') f o r index p.
-
S i m i l a r l y i f wm+P i s t h e (unique) s o l u t i o n o f (3.2) S
I
in
D ) f o r index m + p 2 -1/2 ( p an i n t e g e r ) then
(or
-
i s a s o l u t i o n o f (3.2)
m
(3.3)
I
+
-1,
. . ., -p.
Remark 4.7
(3.3)
in S
I
(or
I
D ) f o r index m provided
Formulas o f t h e form (4.9)
discovered by Weinstein ( l o c . c i t . ) , u s i n g d i f f e r e n t methods.
I n p a r t i c u l a r Weinstein observed t h a t
w i t h i n d e x m.
(3.54)
, leads
(4.9)
involves p = n
t o (4.10)
of descent (see e.g.,
-+
w"(t)
=
Weinstein's v e r s i o n o f
1 and i s o b t a i n e d by a g e n e r a l i z e d method
Weinstein [3]).
We n o t e a l s o here t h a t
t h e r e i s no uniqueness theorem f o r s o l u t i o n s o f (3.2) when m < -1/2 s i n c e ifm = - s (s>1/2) then t s a t i s f i e s (3.2)
-f
- (3.3)
um(t) = t
2s s w (t)
f o r i n d e x m w i t h um(o) = uF(o) = 0; here ws i s
t h e s o l u t i o n o f (3.2) Remark 4.8
t-2"CI-m(t)
This fact, plus a version o f
f o r example.
-
(4.10) were f i r s t
i n a classical setting,
i f w - ~s a t i s f i e s (3.2) w i t h i n d e x -m then t s a t i s f i e s (3.2)
-
-
(3.3)
f o r index s.
The e x c e p t i o n a l cases m = -1, -2,
34
. . . have
1.
SINGULAR PARTIAL DIFFERENTIAL EQUATIONS OF EPD TYPE
been t r e a t e d by Blum [l;23, Diaz-Weinberger [Z],
[lo],
Diaz-Martin
B. Friedman [l], M a r t i n [l],and Weinstein (.loc. c i t . )
we r e f e r t o these papers f o r d e t a i l s .
and
Here f o r m = -p w i t h p 2
1 an i n t e g e r we may t a k e
h
( c f . B. Friedman [ l ] ) . (3.5)
T h i s r e s o l v a n t R-p s a t i s f i e s (3.4)
f o r i n d e x m = -p w i t h T = 1 and one may w r i t e m
(4.12)
zpNp(z) =
1
k=O
ckzZk + zZp l o g z
^R'p
Consequently t h e 2pth d e r i v a t i v e o f mic s i n g u l a r i t y a t t = 0. (4.13)
-
h
w-P(t)
= R-P(-,t)
m.."
1
k=O
ckz
2k
i n t w i l l have a l o g a r i t h -
However, w r i t i n g f o r m a l l y
*
m
T
=
1
k=O
cktZk(AxS
T! we see t h a t t h e l o g a r i t h m i c term varnishes i f A
*
T)
= 0, i n which
event w - ~w i l l depend smoothly on t. We n o t e t h a t i f one takes p = 1 i n (4.8),
Remark 4.9
m r e p l a c e d by m
-
with
1, t h e r e s u l t i s e x a c t l y (3.12).
We w i l l now discuss some growth and c o n v e x i t y theorems f o r s o l u t i o n s o f (3.2)
-
(3.3)
when Ax = -Ax.
f i r s t discovered by Weinstein [9;
Such r e s u l t s were
10; 151 i n a c l a s s i c a l s e t t i n g
and subsequently g e n e r a l i z e d and improved i n c e r t a i n d i r e c t i o n s by C a r r o l l [l;2; 51 i n a d i s t r i b u t i o n framework; we w i l l f o l l o w
35
SINGULAR AND DEGENERATE CAUCHY PROBLEMS
(Other k i n d s o f growth and c o n v e x i t y
t h e l a t t e r approach.
theorems f o r s i n g u l a r Cauchy problems were developed by C a r r o l l [18; 19; 251 and w i l l be t r e a t e d l a t e r . )
R e f e r r i n g t o (4.6),
t h e b a s i c f a c t one r e q u i r e s f o r these theorems i s t h a t Rp(*,n)
2 0 f o r some p
2 -1/2 (i.e.,
Rp(-,n)
measure), i n which case a l l Rm(-,t)
should be a p o s i t i v e
2 0 f o r m 2 p.
For general
Ax n o t t o o much i s known i n t h i s d i r e c t i o n b u t when Ax f o l l o w s from S e c t i o n 2 t h a t Rn/'-'(=,n)
=
ux(n) 2 0.
Another
case o f i n t e r e s t i n v o l v e s t h e metaharmonic o p e r a t o r Ax = -A m n where R ( - , n ) L 0 f o r m > 7 r = (1~:)"~
-
Indeed f o r m >
1.
X
-
2 a
1 and
5 t one has (see C a r r o l l [5] and c f . O s s i c i n i [ l ] )
- r ( m + l ) ( t 2 -r2 ) m-n/2 ITn/2
while
n 7 -
it
= -Ax
p(*,t)= 0
for
Irl
t2m
m
1
k=O 2
2k [a2$-r2) Ik2 0 k! r(m-$+k+l)
2 t; i n p a r t i c u l a r then R m ( * , t )
E
El.
Now, general theorems o f growth and c o n v e x i t y i n v o l v i n g ask sumptions o f t h e form (-Ax) T
;1
0 were developed i n C a r r o l l [2;
51 u s i n g t h e n o t i o n s o f value and s e c t i o n o f a d i s t r i b u t i o n i n troduced by L o j a s i e w i c z [l].T h i s can be s i m p l i f i e d somewhat i n d e a l i n g w i t h Ax = -A
X
s i n c e AT
2
0 i m p l i e s t h a t T i s an almost
subharmonic f u n c t i o n . (see Schwartz [l]) and hence T a u t o m a t i c a l l y has a "value" almost everywhere (a.e.).
We r e c a l l b r i e f l y some
n o t a t i o n s and r e s u t s o f Lojas ewicz [l].
36
1.
SINGULAR PARTIAL DIFFERENTIAL EQUATIONSOF EPD TYPE
A d i s t r i b u t i o n T i s s a i d t o have t h e value
D e f i n i t i o n 4.10 c at x
i s nondecreasing f o r C,t' A t 2 0. Consequently f o r t > 0 f i x e d <wm $ (C) 2 C,t' A + The r e s u l t f o r m = and l e t t i n g A + 0 we o b t a i n wm 2 Tx
uous f u n c t i o n t
-
+
<wm(t),$A> = <wm
X0,t
1 i s now immediate from (2.5)
0
.
and t h e remarks above.
L e t us now combine (4.8) w i t h Lemma 3.3 when A,
= -Ax
QED
in
order t o o b t a i n a formula f o r Rm(-,t) (m 2 -1/2) i n t e r n s of
38
SINGULAR PARTIAL DIFFERENTIAL EQUATIONS OF EPD TYPE
1.
Rm+P(*,t) and Rm+p+l(-,t),
where t h e i n t e g e r p i s chosen so t h a t
2 - 1. Some s i m p l e b u t t e d i o u s c a l c u l a t i o n s y i e l d (see C a r r o l l [2; 51 f o r d e t a i l s )
m + p 2
-
(3.2)
(3.3)
-
F o r -1/2 5 m < n/2
Lemma 4.13
T
(i.e.,
= 6
i n (3.3))
1 t h e r e s o l v a n t R"(*,t)
for
given by F-l applied t o
(4.8) can be w r i t t e n Rm(-,t) = (
(4.17)
+ (
q-1
q- 1
1
k=O
1
k=O
ak(Fl)t2k(A6)k)
Bk(n)t2k(A6)k)
*
*
Rm+P(-,t)
Rm+P+l(*,t)
where p i s an i n t e g e r chosen so t h a t m + p
[q]([u]
n 2 7 - 1,
q = 1 +
denotes t h e l a r g e s t i n t e g e r i n u ) , ak(m) 2 0,
Bk(m) 2 0, and
~1
0
(m) = 1.
n F o r - 1 / 2 5 m < 7 - 1 l e t t h e i n t e g e r p be n chosen so t h a t m + p 2 7 - 1. L e t q = 1 + and assume Theorem 4.14
T
E
D ' s a t i s f i e s (A)
-
s o l u t i o n o f (3.2) section t
+
T2
0 f o r 1 F k 5 q.
(3.3)
f o r Ax = - A ( t 2 0 ) t h e n a.e.
i n x the
rn
0
for t
->
0.
xO
Proof:
We c o n v o l u t e (4.17) w i t h
f e r e n t i a t e , u s i n g (3.11). p+p+2
T t o g e t wm(t) and d i f -
m+p+l (*,t), Since Rm+P(*,t), R
and
( - , t ) a r e p o s i t i v e measures we have a g a i n t h a t w T ( t )
Hence, as i n Theorem 4.12,
i n Dx f o r t 2 0. a.e.
I f wm(t) denotes t h e
wx ( t ) i s a nondecreasing f u n c t i o n o f t w i t h
wm ( t ) 2 T xO
k
[q]
m
for w
x,t
E
D
I
x,t
and t
+
w
X0't
39
= w
0
we may f i x x = xo
m ( t ) w i l l be a xo
->
SINGULAR AND DEGENERATE CAUCHY PROBEMS
nondecreasing f u n c t i o n o f bounded v a r i a t i o n (t
*
wm(t) = Rm+P(.,t)
+ l ( * , t ) where l ( . , t )
T
(we r e c a l l here t h a t ao(m) = 1). Rm+'(=,t)
*
w!+P(t)
L Tx
0
_> 0 i n D i f o r t L 0
From Theorem 4.12 we know t h a t
T = wm+P(t) s a t i s f i e s wm+P(t) 2 T
AT 2 0 (which holds since q 0
Further
0).
xO
L 2).
Hence, a.e.
i n x when
a.e. xO
i n x, w
m xO
.
(t) L QED
C o r o l l a r y 4.15
For -1/2 5 m
-
Given w"(t)
xO
0.
xO
Proof:
We note f i r s t t h a t p L 1 so t h a t q
-
1 =
Then f r o m (4.17) one w r i t e s again wm(t) = Rm+P(-,t)
*
I
where l ( * , t ) 2 0 i n Dx f o r t 2 0 under our hypotheses.
L e t us w r i t e now Lm =
a /at
+2n a/at; t +
1.
T + l(*,t)
As i n a.e.
the p r o o f of Theorem 4.14 i t f o l l o w s t h a t wm ( t ) L T xO
[q] L i n x.
xO
then as noted
by Weinstein, f o r m f 0, L i = (2m)2t-2(2m+1) a2/aa(t-2m)y w h i l e
2 a 2/a 2 ( l o g t ) .
f o r m = 0, L: = t-
A
X
D
I
= -A we have Lkwm =
, for
m > n/2
-
R e f e r r i n g now t o (3.2) w i t h
awm and using (4.15), composed w i t h T
1 i t follows t h a t
40
E
SINGULAR PARTIAL DIFFERENTIAL EQUATIONS OF EPD TYPE
1.
since Lt = C2LCt.
Thus i f AT
t i v e measure f o r t t 0. m
0. Hence by Schwartz [l] xo ,t QED t -+ w l ( t ) i s a convex f u n c t i o n of t. 0
41
SINGULAR AND DEGENERATE CAUCHY PROBLEMS
Remark 4.17
O f p a r t i c u l a r i n t e r e s t i n t h e EPD frameword o f
course i s t h e wave e q u a t i o n when m = -1/2 and we see t h a t i n t h e hypotheses o f C o r o l l a r y 4.15 -1/2 + p
n
2 2
-
1 means p 2
-.n-12
Note here t h a t i n dimension n = 1, -1/2 5 m < -1/2 i s impossible so t h a t we are i n t h e s i t u a t i o n o f Theorem 4.12.
[q]so t h a t ,
n-1 p = 2;
1 =
g i v e n n even, p = n/2, while, f o r n odd ( n t 3),
thus f o r n even q
[!$!These I.
-
Now q
-
1 =
[-1n4+2
-
w h i l e f o r n odd q
1 =
r e s u l t s on t h e number o f p o s i t i v e Laplacians o f T
s u f f i c i e n t f o r a minimum ( o r maximum) p r i n c i p l e improve W e i n s t e i n ' s estimates and a r e comparable w i t h r e s u l t s o f D. Sather [l;2 ; 31. n even and
N
=
Sather i n [3] f o r example sets N =
n-3 for 2
n odd w i t h a =
fornodd; w h i l e a = [%'I n even w h i l e b =
1 [nI 4
=
f o r n odd.
2 T ( x ) f o r t 2 0.
for
[?If o r
N+l s i m i l a r l y he s e t s b = [T]
f o r 1 5 k 2 a and Akw;'/2(o) w-l/'(x,t)
[N+2 I 2
9
=
n even
[,In
for
Then he r e q u i r e s t h a t A kT t 0
2 0 for 0 5 k
5 b i n order t h a t
Thus our r e s u l t s seem a t l e a s t
e q u i v a l e n t and c e r t a i n l y more concise than those o f Sather.
For
f u r t h e r i n f o r m a t i o n on maximum and minimum p r i n c i p l e s f o r Cauchy t y p e problems we r e f e r t o Protter-Weinberger [l]and t h e b i b l i o graphy there; c f . a l s o Agmon-Nirenberg-Protter P r o t t e r [7],
[l],Bers [l],
Weinberger [l]. For growth and c o n v e x i t y p r o p e r t i e s
o f s o l u t i o n s o f p a r a b o l i c equations w i t h subharmonic i n i t i a l data see Pucci-Weinstein [l]. Remark 4.18
The e x i s t e n c e o f p o s i t i v e elementary s o l u t i o n s
( o r r e s o l v a n t s ) f o r second o r d e r h y p e r b o l i c equations i s i n 42
1.
SINGULAR PARTIAL DIFFERENTIAL EQUATIONS OF EPD TYPE
general somewhat n o n t r i v i a l and we mention i n t h i s regard D u f f
[l;21, i n a d d i t i o n t o C a r r o l l [2],
where t h e problem i s d i s -
cussed. Spectral techniques and energy methods i n H i l b e r t
1.5
F r s t we w 11 present some t y p i c a l theorems o f Lions f o r
space.
EPD t y p e equations, u s i n g "energy" methods, which were d i s cussed i n C a r r o l l [8].
Lions proved these r e s u l t s i n 1958 i n
l e c t u r e s a t t h e U n i v e r s i t y o f Maryland (unpublished).
There-
a f t e r f o l l o w some theorems i n H i l b e r t space based on a s p e c t r a l technique developed by C a r r o l l [4; 6; 8; 10; 111 (suggested by Lions as a v a r i a t i o n on C a r r o l l ' s F o u r i e r technique).
The two
approaches (energy and s p e c t r a l ) a r e n o t d i r e c t l y connected and l e a d t o weak and s t r o n g s o l u t i o n s r e s p e c t i v e l y . Thus, r e g a r d i n g energy methods ( c f . C a r r o l l [14] and Lions [5] f o r a more complete e x p o s i t i o n ) , l e t (u,v)
V
x
V
+.
B
a(t,u,v)
-+
:
be a f a m i l y o f continuous s e s q u i l i n e a r forms, V c
H,
where V and H a r e H i l b e r t spaces w i t h V dense i n H having a f i n e r topology. a(t,u,v)
D(A(t)) H).
One can d e f i n e a l i n e a r o p e r a t o r A ( t ) such t h a t
= (A(t)u,v),,
if V
+.
Assume t
a(t,u,v)
+.
a(t,u,v)
V (i.e.,
for u
E
D ( A ( t ) ) C V and
:V
-+
B i s continuous i n t h e topology o f
E.
1 C [o,b]
w i t h a(t,u,v)
V E
= a(t,v,u)
u
and
consider Problem 5.1
t
+.
A(t)u(t)
Find u ( - )
E
Co(H) on [o,b],
C2(H) on [o,b], and
43
u(t)
E
E
D(A(t)),
SINGULAR AND DEGENERATE CAUCHY PROBLEMS
where k t
-+
E
a,
q(-)
0, and f
E
D(A(t)) f o r t
Co(o,b]
E
i s r e a l valued w i t h q ( t )
Co(H) on [o,b].
03
as
Note here t h a t i f u(o) = T
E
then v i t ) = u ( t ) - T s a t i s f i e s (5.1)
[o,b]
E
-+
w i t h f replaced b y f ( * ) - A ( - ) T (assuming t h e l a t t e r i s continuous on [o,b]
- which
w i t h values i n H
holds i f A ( t ) = A).
T h i s can be transformed i n t o a weak problem (which i s more
d = t r a c t a b l e ) as f o l l o w s (= Problem 5.2 2 L (H) on [o,b],
Find u
).
L2(V) on [o,b],
+kq(u',h)"
0
for a l l h
E
u ( o ) = 0, u l f i
E
such t h a t g i v e n f/GE: L2(H)
jbla(t,u,h)
(5.2)
E
'
L2(V) w i t h hJii
Theorem 5.3
Let t
-+
E
-
I
1
( u ,h )H
L2(H), hl/J;i
a(t,u,v)
E
E
-
(f,h)Hldt = 0
L2(H), and h(b) = 0.
C 1 [o,b],
a(t,u,v)
= a(t,v,u)
Co(o,bl, q > 0, q ( t ) -+ as t + 0, Re k > 0, and a(t,u,u) 2 2 c l l l u l l v f o r a > 0. Then t h e r e e x i s t s a s o l u t i o n o f Problem 5.2. q
E
03
Proof:
Set h = e-Yt+l
I
f o r y r e a l ( t o be determined) and
b
(5.3)
Ey(u,+)
Let
=
0
1
{a(t,u,e-Yt+l)
+
k q ( t ) ( u l ,e-Yt+l)H
-
( u ,(e Y t + l ) l ) H l d t
F be t h e space o f f u n c t i o n s u
E
L2(V), u l G
2 u(o) = 0, w h i l e G i s t h e space o f 4 E L ( V ) , I I 1 2 2 9 4 E L (H), 9 / 4 E L (H), $(o) = 0, and
44
+' E + (b)
E
L2 (H),
2 L (V),
I
= 0.
Clearly
SINGULAR PARTIAL DIFFERENTIAL EQUATIONS OF EPD TYPE
1.
G CF and we p u t on G t h e induced topology o f F d e f i n e d by
,f)(
IIull;
+ q l l u ' l ! i ) d t = IIuIIF 2 so t h a t G i s a p r e h i l b e r t space.
0
Then i f u
E
F s a t i s f i e s E (u,$) =
Y
/
b
( f , e Yt$')Hdt f o r a l l $
G i t f o l l o w s t h a t u i s a s o l u t i o n o f Problem 5 . 2 .
i s a s e s q u i l i n e a r form on F x G w i t h u tinuous f o r 4
G fixed.
E
-
+ 2a(t,$,e-Yt$') (5.4)
b
(5.5)
yt$)
t
4
con-
= at(t,$,e-Yt$)
+Y
at(t,$,$)e-Ytdt
I
b
[-e-yt
0
=
y
i ( $ ,(e-Yt$')l)Hdt
0
=
: F.+
E (u,$)
= a(b,$(b),$(b))e-yb
0
b
-2Re
Now Ey(u,$)
so t h a t ,
ya(t,$,e-Yt$)
I
-
-f
Further, a(t,$,e
2Re I:a(t,$,e-Yt$')dt
E
0
&11$11: b
11$'(0)11
Hence, f o r y > c/a,
+ 2ye-ytII$'lli]dt
+
IEy($,$)I
0
e-ytII
2 811$11;
$
'
2
I] H
dt
'
( n o t e t h e second term
i n 2Re E ( $ & ) ) and by t h e Lions p r o j e c t i o n theorem ( c f .
Y
C a r r o l l [14], u
E
Lions [5],
Mazumdar [l;2 ; 3; 41) t h e r e e x i s t s
F s a t i s f y i n g E (u,$) =
Y
f(f,e-yt$')Hdt
(f,e-Yt$')Hdt,
-f
i s a continuous s e m i l i n e a r form on G.
0
Remark 5.4
since $
QED
For an i n t e r e s t i n g g e n e r a l i z a t i o n of t h e Lions
45
SINGULAR AND DEGENERATE CAUCHY PROBLEMS
p r o j e c t i o n theorem and a p p l i c a t i o n s see Mazumdar [l;2; 3; 41. We not e t h a t i n Problem 5.2 q c o u l d become i n f i n i t e a t o t h e r p o i n t s on [o,b]
p ro v i d e d t h e "weight" f u n c t i o n s h a r e s u i t a b l y t h e problem can be solved d i r e c t l y as
chosen; i f q > 0 b u t q i n Lions [5].
1 Theorem 5.5 L e t t + a(t,u,v) E C [o,b], a(t,u,v) = 1 a(t,v,u), q E C (o,b],q > 0, q ( t ) + - as t + 0, Re k > 0, and 2 a(t,u,u) 2 a l l u l l v f o r a > 0. Then a s o l u t i o n o f Problem 5.2 i s unique. Proof:
L e t h be g i v e n by h ( t ) = 0 f o r t
= u on [o,s);
E +
0, l i m a(E,h(E),h(E))
Taking r e a l p a r t s i n (5.2),
+ I
where a(t,h,h)
I
s with h
I
-
kqh
then some c a l c u l a t i o n shows t h a t h i s admissable
i n Problem 5.2 and, as
= 2Re a(t,h,h
(2aq Re k
0
and s i n c e q ( t ) t h a t 2aq Re k u = 0 i n [o,so].
-+
-
(4
as t
-
z
2 0.
= 0 I
) + at(t,h,h).
Consequently
2 c)llhllVdt 5 0
+
0 one may choose so small enough so
which w i l l i m p l y t h a t h = 0 and
c > 0 i n [o,so]
I n t h i s step o n l y q
extend t h e r e s u l t u = 0 t o [so,b] methods ( c f . Li o n s [5])
= 8
w i t h f = 0, one o b t a i n s w i t h t h i s h,
j o at(t,h.h)dt
S
(5.7)
2
where q
E
Co(o.so]
i s needed; t o
one can r e s o r t t o standard E
46
C'[s,,bl
i s used.
QED
1.
SINGULAR PARTIAL DIFFERENTIAL EQUATIONS OF EPD TYPE
Remark 5.6
There a r e a l s o versions of Theorems 5.3 and 5.5 2
2
due t o Lions when a ( t , v , v ) + Al1vllH 2 cill-vllv (cf. Carroll [81) b u t we will omit the d e t a i l s here.
We note t h a t no r e s t r i c t i o n s
have been placed on the growth of q ( t ) as t * 0; however the solution given by Theorem 5.3 i s a weak solution and not necess a r i l y a solution of (5.1).
We consider now the operator
for 0 5 as
T
-+
T
5 t 5 b
0, Yt(A,T,T)
Proof: t h a t Ut =
and represents a continuous
= 0, Y(A,t,o)
and f o r T > 0
= 0,
and, f o r
= 1.
T h i s goes as i n t h e p r o o f o f Lemma 3.8
P(T)/P(t),
UT
rt
= -1
( n o t e here
+ (a+B)U, and from (5.301,
56
cf.
1.
SINGULAR PARTIAL DIFFERENTIAL EQUATIONS OF EPD TYPE
Now YT i s d e f i n e d b y (5.30) t o i n f i n i t y as T
and does n o t n e c e s s a r i l y tend
o r a,(-c)Y(A,t,T),
a(T)Y(A,t,T),
We need o n l y examine
0 ( c f . (3.42)).
-+
as
T
As i n (3.40)
0.
-+
m1
al(T)Y(h,t,T)
=
I
where B 2 ( o , ~ ) =
1 (-l)kJk
C X ~ ( T ) U ( * , T so ) we l o o k f i r s t a t
k=O
aB2( 0 ,
and $1 = a;/a:
a0
small arguments (e.g.,
Lemma 5.11).
al(q)dn)
f i r s t t h e term A(t;-c) =
can compare here w i t h S e c t i o n 3 where a ( t ) = = (T/t)2m
-+
Thus assume A(t,-c)
0 as -+
o
t 5 s ) ; here B 2 ( o , ~ ) = exp
i s Cm i n ( a , ~ w ) i t h I B 2 ( o , ~ ) I 5 c14 (cf.
A(t,.r)
(-i
= $ for sufficiently
w
T
-+
0 as T
as
+
aZ(rl)do) We examine 0 and one
2m 7 and +
f o r m > 0 (Bz(t,r) -+
T
T
1 i n Section 3).
0 i n which event, from (5.31)
I
t h e r e r e s u l t s ( s i n c e B2/Bz = -az(.)) (5.32)
a,(.)
t
Bz(o,~)C1 +
az(4
+
(+:a1
$(o)le
(
0
do
~ +
1
T
as
T
-+
0, independently o f t.
But a,(a)/a,(o)
+
@ ( o ) (and B2(a,.r) 1 1) we have
(5.33)
l i m al(T)U(t,T) T-+O
=
0 as u
-+
0 and
+ $(a)
t a k i n g t small enough so t h a t 1
-+
1
rn 57
SINGULAR AND DEGENERATECAUCHY PROBLEMS
provided $(o) $ -1.
This checks w i t h Section 1 . 3 ( f o r m > 0) where 1/(1+$(0)) = o r 2m + 1 U ( t , T ) -t 2m T h u s when 2m T A ( t , - c ) * 0 a s T * 0 w i t h $ ( o ) $ -1 we have by (5.30) as T * 0
-
-. +
YT(X,t,o) = - ( $ ( o ) / ( l + $ ( o ) ) Z ( X , t , o ) ( c f . (3.42) and the d e f i n i t i o n o f Z i n Lemma 5.8). In general we cannot expect of course t h a t
A(t,T)
-t
0 as
* 0 o r t h a t $(o) $ -1 ( c f . Section 1.3). However, as w i t h ?(y,t,.r), ye can show t h a t YT(A,t,r) E L 1 in T ( w i t h Y ( A , t , . ) T
absolutely continuous) and we follow again Carroll [8]. m
consider al(T)Y(X,t,-r)
=
T
5
f o r some upper l i m i t
1
1(-l)kal(~)Jk
k=O
U(-,T)
Thus
and define
2
..,
(5.34)
where J n
Kn =
t
0
U(*,T)
0 by d e f i n i t i o n i f
a l ( ~ J)n =
U(0,T)d-c ..,
(Jn
T >
=
U)(t,-c)
(note here t h a t ( J n
U)(t,.c)
t). W e s e t again 1B2(u,~)1 5 c14 and
lal(~)l5 k l R e a l ( T ) so t h a t
..,
( f o r Fubini-Tonelli theorems see McShane [ l ] ) .
Similarly, s e t -
t i n g [ X I C 5 R and ( c 2 + Rc12) = c , we have ( c f . (5.13) f o r J )
58
=
1.
SINGULAR PARTIAL DIFFERENTIAL EQUATIONS OF EPD TYPE
. "
5 c,4k,cbF(t) 5 c14klcbF(b) Upon i t e r a t i o n we obtain ( c f . Section 3)
and t h u s
m
1 (-1)"'
m
1 (-l)n
jt
a l ( ~ ) ( J n U ) ( i , T ) d T converges n=O n=O 0 uniformly and absolutely. An elementary argument ( c f . Carroll =
[8]) then shows t h a t al(*)Y(A,t,*)E L1 w i t h
I
. "
(5.38)
t
m
..,
al(T)Y(A,tyT)dT
0
1 (-l)"Kn
=
n=O
. "
Lemna 5.14 -Z(A,t,T)
I f [ a l l 5 klReal on [o,s] 1 + (a+@)(T)Y(A,t,T) E L for 0 5
i s absolutely continuous w i t h Y ( h , t , - r )
+ = a'/,*
E
Co[o,s] and +(o)
4
=
then
I
T 5 T
T
-+
YT(X,t,.r) =
t dS
which coincides w i t h the formula e s t a b l i s h e d by Donaldson [l] using a d i f f e r e n t technique. wrn(t) = 2cm
/
2 m - -1 (1-S ) 2 cosh(A =
To v e r i f y t h a t (6.4) represents a s o l u t i o n o f (6.1)
one can o f course work d i r e c t l y w i t h the v e c t o r i n t e g r a l s (6.5)
or (6.6) b u t we f o l l o w Hersh [l]and C a r r o l l [18; 19; 24; 251 i n t r e a t i n g (6.4) as a c e r t a i n d i s t r i b u t i o n p a i r i n g . R:(*,t),
= AxRm(*,t) a r e a l l o f order less than o r
and R:,(-,t)
equal t o two i n E
Thus Rm(*,t),
I
w i t h supports contained i n a f i x e d compact
68
StNGULAR PARTtAL DIFFERENTIAL EQUATIONS OF EPD TYPE
1.
set
K
xo +
=
Bl
Ix;
1x1 5
xol
for 0 5 t 2 b
0.
E
m
and we l e t
A
K
1x1 5
= {x;
I
i s o f o r d e r l e s s than o’r equal Z A 1 t o two w i t h supp S C K we can t h i n k o f S E C (K) ( c f . Schwartz E
ill). Recall f u r t h e r t h a t on *K,C 2 ( E ) = C Z gA) E (see e.g., E
111) where t h e
E
Treves
2 topology on C @ E i s t h e topology o f u n i f o r m
convergence on p r o d u c t s o f equicontinuous s e t s i n ( C 2 ) ’ x E
I
.
It
i s easy t o show (see C a r r o l l [18] f o r d e t a i l s ) a
Lemma 6.1 tinuous f o r
The composition S + <S,p-: C 2 ( i ) ’
$E
C2(E) f i x e d .
(defined by (A x I )
1 $i
-+
E i s con-
The map A @ 1 : C2(E) = C O G E E
8ei =
1
8’ei)
i s continuous.
L e t us i n d i c a t e f i r s t t h e formal c a l c u l a t i o n s necessary t o show t h a t (6.4) i s a s o l u t i o n o f (6.1), t h e same t i m e some r e c u r s i o n r e l a t i o n s .
while establishing a t Their v a l i d i t y i s
e s s e n t i a l l y obvious upon s u i t a b l e i n t e r p r e t a t i o n ( c f . below). Thus, from (6.4) and (3.11) transformed,
w F ( t ) = =
1 (1-5 2 ) m+- 2
T(Ct)A*edE
S i m i l a r l y from (6.4) and (3.12) transformed
69
SINGULAR AND DEGENERATE CAUCHY PROBLEMS
-
]
=
2mt
-
[wm-l ( t )
Then d i f f e r e n t i a t i n g (6.7) and using (6.7)
-
wm( t ) l (6.8) we
The c a l c u l a t i o n s under t h e b r a c k e t can be
o b t a i n (6.1).
j u s t i f i e d using Lemna 6.1 and i n o r d e r t o t r a n s p o r t A around i n (6.7) one can t h i n k of as a d i s t r i b u t i o n p a i r i n g over the h
i n t e r i o r o f K (cf. Carroll [l8]). 2 and E as above ( e E D(A ) ) Theorem 6.2
The e q u i v a l e n t formulas (6.4) - (6.6) g i v e a
s o l u t i o n t o (6.1) f o r m 2 d i c a t e d by (6.7)
This proves, given A, T ( t ) ,
-
-
1/2 and t h e irecursion r e l a t i o n s i n -
(6.8) a r e v a l i d .
The q u e s t i o n o f uniqueness f o r (6.1) had seemed t o be r a t h e r more complicated than t h a t o f e x i s t e n c e b u t a new theorem o f C a r r o l l [26] (Theorem 6.5) gives a s a t i s f a c t o r y r e sponse.
There a r e several types o f theorems a v a i l a b l e ( c f .
C a r r o l l [18; 24; 25; 261, Carroll-Donaldson [201, Donaldson 111, Donaldson-Goldstein [2], Hersh [ l ] ) .
The f i r s t one we discuss
( f o r completeness) i s based on a v a r i a t i o n o f t h e c l a s s i c a l " a d j o i n t " method and simp1 i f i e s somewhat t h e p r e s e n t a t i o n i n C a r r o l l [18]; i t i s n o t as s t r o n g however, as Theorem 6.5. D e f i n i t i o n 6.3 adapted i f E
I
The space E (as above) w i l l be c a l l e d A-
*
i s complete and D(A ) i s dense.
When E i s A-adapted (A being t h e generator o f a l o c a l l y
70
SINGULAR PARTIAL DIFFERKNTIAL EQUATIONSOF EPD TYPE
1.
equicontinuous group i n E) i t f o l l o w s ( c f . Komura [ll)t h a t
*
generates a l o c a l l y equicontinuous group T ( t ) i n E
I
.
A*
The r e -
quirement o f completeness i s a l u x u r y which we p e r m i t ourselves for simplicity.
Indeed, f o r t h e d i s c u s s i o n o f l o c a l l y equi-
continuous semigroups s e q u e n t i a l completeness i s s u f f i c i e n t ( i n view o f t h e Riemann t y p e i n t e g r a l s employed f o r example).
Thus,
i n p a r t i c u l a r , i f E i s r e f l e x i v e ( n o t n e c e s s a r i l y complete) hence s e q u e n t i a l l y t h e n i t i s quasicomplete ( c f . S c h a e f f e r [l]), complete, as i s t h e r e f l e x i v e space E
I
*
, and
A
*
w i l l generate a
*
I
l o c a l l y equicontinuous group T ( t ) i n E w i t h D(A ) dense ( c f . Komura [ll).We remark a l s o t h a t i f E i s b o r n o l o g i c a l then E
I
i s i n f a c t complete ( c f . Schaeffer [l]). We use completeness
2 2 ^ b a s i c a l l y o n l y i n a s s e r t i n g t h a t C ( E ) = C BEE b u t a v e r s i o n o f t h i s i s p r o b a b l y t r u e f o r E s e q u e n t i a l l y complete; t h e i n crease i n d e t a i l and e x p l a n a t i o n throughout our d i s c u s s i o n does n o t seem t o j u s t i f y t h e refinement however. We c o n s i d e r now t h e r e s o l v a n t s Rm(-,t,.r) S e c t i o n 3 when A(y) = y S e c t i o n 4). Sm(*,t,T)
2
w i t h Ax = -A
and 2
X
= - 3 /ax
2
Sm(*,t,T)
of
(cf. also
We r e c a l l ( c f . Lemma 4.3) t h a t R m ( - , t , r )
and
belong t o E i , w i t h s u i t a b l e o r d e r s ( e x e r c i s e ) , and
Lemna 3.13 h o l d s w i t h OM replaced by Exp OM = F E
I
.
Thus l e t
w m ( t ) s a t i s f y (6.1) w i t h wm(0) = wF(0) = 0 and consider ( f o r an A-adapted E) (6.9)
um(t, -p (p
for f
E
2
For
1 an i n t e g e r ) t h e " c o n t i n u a t i o n " o f I' i s given by
Cp(E) o r f
CP-'(E) w i t h f ( ' ) ( = )
E
E
L 1 (E) " s u i t a b l y " .
= IatBa except p o s s i b l y when B i s a negative One knows t h a t IaIB
i n t e g e r B = - p a where IqPf= f(')a f('-')(o)
i n which case f ( o ) =
i s r e q u i r e d i n order t h a t
= 0
...=
i n general,
=
i s d e f i n e d t o be t h e i d e n t i t y . E v i d e n t l y ( d / d t ) ( I " f ) = and Io I"-1 f f o r any a. We record now some e a s i l y e s t a b l i s h e d f a c t s . Lemna 6.6
If f
E
Cn+'(E) w i t h -n < Re a 5 -n
+ 1 and
a
n o n i n t e g r a l then (6.22)
(I"f)'(t)
= (Ia-'f)(t)
I f f ( o ) = 0 o r a = -n w i t h f (6.23)
(Iaf)l =
-*-+
Cn+'(E)
(Iafl)(t) then
= Ia-lf
Iafl
(when Re a > 0 (6.22) w i t h -n < Re a 5 -n
E
=
-
(6.23) h o l d f o r f
+ 1 or
a = -n ( o r f
E
E
1 C (E)).
If f
E
Cn(E)
Co(E) when Re a > 0)
then (6.24)
t(I"f)(t)
Proof:
= (I"(sf))(t) + a(Iat'f)(t)
Routine computation y i e l d s (6.22)
(6.24) f o l l o w s by an i n d u c t i o n argument.
75
-
(6.23) and
QED
SINGULAR AND DEGENERATE CAUCHY PROBLEMS
Now, r e f e r r i n g t o t h e Sonine i n t e g r a l formula (4.6) ( c f . a l s o (6.3) and (6.6)), (6.25)
Rm(*,-r1/')
we can w r i t e , a f t e r a change o f v a r i a b l e s ,
I
m- 1
f
=
(T-S)
's
_1_ _1_ 'R
2(-,s1/2)ds
0
as before.
where cm =* . (6.26)
p(t) =
w i t h wm(t1/')
m m
L e t then
1/2
= -1/2 we
(6.7),
have under t h e hypotheses o f Theorem 6.2 Theorem 6.9
Let
E be a space o f f u n c t i o n s o r equivalence
classes o f f u n c t i o n s and assume e 0 ( m >_ -1/2).
Proof: (6.36)
2 D(A2) w i t h cosh (At)A e >_
Then w m ( * ) i s a nondecreasing f u n c t i o n o f t. From (6.6)
m
E
wt(t)
=
-
(6.7) we have f o r m
tA’ 2(m+17 w“+’(t)
1
m-+ 1
m+l 78
t
-1/2
2 cosh(A6t)A e dS 2 0
QED
SINGULAR PARTIAL DIFFERENTIAL EQUATIONSOF EPD TYPE
1.
Next we r e c a l l t h a t ( f r o m S e c t i o n 4) L i =
2m
a/at
+
t
a 2/ a t 2
= (2m) t -2(2m+1) a2/a2(t-2m) f o r m
t
0 w h i l e Lot =
t-2 a2/a2(10g t ) . The d i f f e r e n t i a l equation i n (6.1) can be
w r i t t e n as L k P = A2wm so we have from (6.6) 1 1 ni-(6.37)
Lmw t m ( t ) = 2cm
1
(1-5
w i t h m > -1/2 r e a l
cosh(A5t)A2e d5
0
Theorem 6.10
Under t h e hypotheses o f Theorem 6.9,
a convex f u n c t i o n o f t-2m for m Proof:
+ 0 and o f l o g t f o r m
~"(0)i s = 0.
The r e s u l t f o r m = -1/2 i s obvious s i n c e w - ' l 2 ( t )
cosh(At)e and t h e r e s t f o l l o w s from (6.37).
=
QED
L e t E = Co(IR) w i t h t h e topology o f u n i f o r m
Example 6.11
The o p e r a t o r A = d/dx generates a
convergence on compact sets.
l o c a l l y equicontinuous group ( T ( t ) f ) ( x ) = f ( x + t ) which i s n o t
I f f L 0 then e v i d e n t l y T ( t ) f 2 0 f o r t
equicontinuous.
t o f u l f i l l t h e hypotheses o f Theorems 6.9 f i n d e = f such t h a t A 2 f = f
I I
we need o n l y
and we do indeed want t o
+
E t o p e r m i t such growth.
L e t E = C"( R3)
Example 6.12
R so
2 0; such f u n c t i o n s are abundant,
b u t o f course t h e y increase as t work i n " l a r g e " enough spaces
- 6.10
E
x
C"( R3) where C"( R3) has
the Schwartz topology and r e c a l l t h e mean value operators p,(t) and A x ( t ) d e f i n e d i n S e c t i o n 2 which we extend as even f u n c t i o n s f o r t negative. (6.38)
A
=
Define 0 (A
1
) ; A 2 =
79
A (0
0 A1
SINGULAR AND DEGENERATE CAUCHY PROBLEMS
where A i s the three-dimensional Laplacian.
Then A generates a
l o c a l l y equicontinuous group T ( t ) i n E a c t i n g by convolution (i.e., T ( t ) ?
=
T(t)
* 1) determined
by
One notes again t h a t T ( t ) i s not equicontinuous.
From (6.39) i t
follows t h a t cash A t
(6.40)
so t h a t cosh A t
=
72
u,(t) + gt 2 A 2A x ( t ) 0 provided
?2
0 and A2? 2 0.
Given
7
=
fl ( f ) t h i s means t h a t f i 2 0 w i t h A f i 2 0. T h u s i n order t o apply 2 Theorems 6.9 - 6.10 we want ? such t h a t A 2' f 2 0 and A4? 2 0 (i.e., 2 A f i 2 0 and A f i 2 0) w h i c h i s possible e.g., when f i ( x ) = i exp l y j x j . Again the necessity of using "large" spaces E i s apparent.
W e note a l s o from (6.39) t h a t T ( t ) and T ( - t ) behave
q u i t e d i f f e r e n t l y r e l a t i v e t o the preservation of p o s i t i v i t y . Analogues of these growth and convexity theorems f o r other "canonical" singular Cauchy problems appear i n Carroll [18; 19; 251 ( c f . a l s o Chapter 2 ) . 1.7
Transmutation.
The idea of transmutation of operators
goes back t o Delsarte and Lions ( c f . Delsarte [ l ] , DelsarteLions [2; 31, Lions [ l ; 2 ; 3; 4;
51) with subsequent contributions 80
1.
SINGULAR PARTIAL DIFFERENTIAL EQUATIONS OF EPD TYPE
by Carroll-Donaldson [ZO],
Hersh 131, Thyssen [l;23, e t c .
The
s u b j e c t has many y e t unexplored r a m i f i c a t i o n s and i s connected w i t h t h e q u e s t i o n o f r e l a t e d d i f f e r e n t i a l equations ( c f . BraggDettman, l o c . c i t . ,
Carroll-Donaldsoc [20],
Hersh [3]); we expect
t o examine t h i s more e x t e n s i v e l y i n C a r r o l l [24].
This section
t h e r e f o r e w i 11 be p a r t i a1 l y h e u r i s t i c and t h e word " s u i tab1 e"
w i l l be used o c c a s i o n a l l y when convenient w i t h p r e c i s e domains of d e f i n i t i o n u n s p e c i f i e d a t times. One f o r u m l a t i o n goes as f o l l o w s .
L e t Dx = a/ax and Dt =
a / a t and c o n s i d e r polynomial d i f f e r e n t i a l operators P Q = Q(D)
(D = D, o r Dt).
= P(D) and
One says t h a t an o p e r a t o r B transmutes
P i n t o Q i f ( f o r m a l l y ) QB = BP.
B w i l l u s u a l l y be an i n t e g r a l
o p e r a t o r w i t h a f u n c t i o n o r d i s t r i b u t i o n k e r n e l and i n f a c t one o f t e n assumes t h i s a p r i o r i , a l t h o u g h i t i s perhaps t o o r e s t r i c t i v e ( c f . Remark 7.3).
One p i c k s a space o f f u n c t i o n s f ( t h e
choice i s v e r y i m p o r t a n t ) and considers t h e problem
(7.2)
@(x,o) = f ( x ) ;
where 1 5 k 5 m
-
k Dt@(x,o) = 0
1 w i t h m = o r d e r Q.
p u t @ ( o , t ) = ( B f ) ( t ) and w r i t i n g $ ( x , t ) sul t s
81
x,t
2
0
I n o r d e r t o d e f i n e B we = P(Dx)@(x,t) t h e r e r e -
SINGULAR AND DEGENERATE CAUCHY PROBLEMS
One may w i s h t o extend f and P (D )f a p p r o p r i a t e l y f o r x < 0 when d e s i r a b l e o r necessary. (7.5)
There r e s u l t s f o r m a l l y
uJ(o,t)
and e v i d e n t l y i t have const ant c o e f f i c i e n t s .
-
t h a t (7.1)
(7.2)
I t i s however necessary t o suppose
have a unique s o l u t i o n i n o r d e r t o w e l l d e f i n e
-
B; t h e same uniqueness c r i t e r i o n a p p l i e s t o (7.3)
(7.4)
but
t h i s w i l l o f t e n be t r i v i a l i f f i s chosen i n t h e r i g h t space. Theorem 7.1 (7.4))
Suppose t h e problem (7.1)
has a unique s o l u t i o n
+
-
(7.2)
(resp. (7.3)
(resp. $) w i t h f i n a s u i t a b l e
space so t h a t (7.5) makes sense and d e f i n e g e n e r i c a l l y ( B f ) ( t ) = +(o,t).
Then B transmutes P(D) i n t o Q(D). 2 L e t P(D) = D
L e t us i n d i c a t e some simple a p p l i c a t i o n s .
-
and Q(D) = D so t h a t (7.1)
(7.2)
become
We prolong f as an even f u n c t i o n and take p a r t i a l F o u r i e r t r a n s A 2^ s ( F f = f ) t o o b t a i n +t = - s @ w i t h i ( s , o ) = f ( s ) . A
forms x
+
A
The s o l u t i o n i s + (s ,t)
2 f ( s ) exp ( - s t ) and i f A
=
A
82
A
-
1.
SINGULAR PARTIAL DIFFERENTIAL EQUATIONS OF EPD TYPE
(7.7)
2
i t i s known t h a t FxR(x,t) Hence ( R ( * , t )
*
= exp ( - s t ) ( c f . Titchmarsh [Z]).
f ( - ) ) ( x ) = @(x,t) and r e c a l l i n g t h a t f i s even
there r e s u l t s
(7.8)
@ ( o y t )= ( B f ) ( t ) =
( c f . Carroll-Donaldson [ZO]).
(7.5) w i l l h o l d provided f
I
c
f(S)K(S,t)dS
An easy c a l c u l a t i o n shows t h a t
(0) =
0 and o f course f, f
I
, and
f
I 1
must have s u i t a b l e growth a t i n f i n i t y .
Now t h e r e i s no d i f f i c u l t y i n extending t h e transmutation i d e a t o v e c t o r f u n c t i o n s f w i t h values i n a complete separated l o c a l l y convex space E.
For example suppose t h a t A i s a s u i t a b l e
(closed, densely d e f i n e d ) o p e r a t o r i n E w i t h g(A) a reasonable o p e r a t o r f u n c t i o n and l e t P(D)w = g(A)w where w takes values i n E ( a c t u a l l y i n D(g(A)) s i n c e A may n o t be everywhere defined).
I f B transmutes P i n t o Q then f o r m a l l y Q(D)Bw = BP(D)w = Bg(A)w = g(A)Bw p r o v i d e d t h a t w s a t i s f i e s t h e c o n d i t i o n s necessary f o r t h e t r a n s m u t a t i o n and t h a t Bw
E
D(g(A)).
I n t h i s event u = Bw w i l l
s a t i s f y Q(D)u = g(A)u and t h e r e a r i s e s t h e general q u e s t i o n o f when t h i s s i t u a t i o n can p r e v a i l .
I f B i s an i n t e g r a l o p e r a t o r o f
Riemann type w i t h a f u n c t i o n k e r n e l then g(A) can be passed under t h e i n t e g r a l s i g n and t h i s p a r t o f t h e q u e s t i o n i s t r i v i a l up t o t h e p o i n t where i n i t i a l o r boundary values a r i s e .
As an ex-
ample l e t us consider t h e case ( c f . Carroll-Donaldson [ZO])
83
SINGULAR AND DEGENERATE CAUCHY PROBLEMS
(7.9)
2 2 P(D)w = D w = A w;
(7.10)
2 Q(D)u = Du = A u;
w(o) = uo;
wt(o) = 0
u ( 0 ) = uo
IfA2 = B t h e s e a r e o f c o u r s e wave and h e a t e q u a t i o n s and t h e
c o n n e c t i n g f o r m u l a i s w e l l known, b u t we w i l l d e r i v e i t v i a t r a n s mutation.
F i r s t f o r m a l l y we use (7.8)
obtain u = Bw =
(7.11)
t r a n s m u t i n g D2 i n t o D t o
c
w( 0 w h i l e f o r t = 0 one must
require u
0
E
D ( A ~ ) . Consequently
Theorem 7.2 g i v e n by ( 7 . U )
wt,
and wtt
Given w a s o l u t i o n o f (7.9) s a t i s f i e s (7.10),
p r o v i d e d uo
i t follows t h a t u E
2 D(A ) w h i l e w,
grow s u i t a b l y a t i n f i n i t y .
Such r e s u l t s a r e n o t new o f course, a t l e a s t i n more c a s s i c a l forms, and we r e f e r t o Bragg-Dettman, Donaldson [ZO], Remark 7.3
Hersh [3],
loc. cit.,
Lions, l o c . c i t . ,
One can r e l a x (7.2)
Carrol
-
and r e f e r e n c e s t h e r e .
and s t u d y n o t w e l l posed
Cauchy problems ( o r o t h e r problems) w h i c h l e a d t o a unique s o l u t i o n @ ( c f . C a r r o l l - D o n a l d s o n [ZO]);
f o r example growth c o n d i -
t i o n s on @ and f c o u l d be imposed (see h e r e C a r r o l l [24]).
Lions
i n h i s e x t e n s i v e i n v e s t i g a t i o n s , chooses spaces o f f u n c t i o n s f 84
1.
SINGULAR PARTIAL DIFFERENTIAL EQUATIONS OF EPD TYPE
where the transmutation i s an isomorphism in the sense t h a t
there i s an inverse transmutation B - ' sending Q ( D ) i n t o P ( D ) ( i . e . PB-
era
=
B-lQ) b u t t h i s seems t o be an unnecessary luxury in gen-
, although obviously of g r e a t i n t e r e s t ( c f . Section 4 . 3 ) .
The choice of spaces f o r f i s i n any event of paramount importance f o r the transmutation method t o work.
I f B i s an i n t e g r a l opera-
t o r w i t h a function o r d i s t r i b u t i o n kernel one can begin with this a s a s t i p u l a t i o n and then discover the d i f f e r e n t i a l problem w h i c h the kernel must s a t i s f y ( n o t necessarily a simple Cauchy
problem), and we r e f e r t o Carroll [24; 251, Carroll-Donaldson [20],
Hersh [3], Lions, loc. c i t . , e t c . f o r f u r t h e r information
( c f . a l s o (7.25)).
We w i l l conclude this section w i t h a version of Lions trans2 2 mutation method sending P(D) = D i n t o Lm = D + t D (cf. a l s o Section 4.3). T h u s we look f o r a function $ ( x , t ) s a t i s f y ing
(7.12)
@xx = 4 t t
+
?m + 1 t t ;
and again we extend f t o be even w i t h f
I
(0)=
0.
This i s of
course t h e one dimensional EPD equation whose solution is given by ( 6 . 3 ) a s ( c f . Theorem 4.2)
(7.14)
$(o,t) = (Bf)(t)
=
2cmt-2m
1
t
0
85
( t -T
1 ) m-Ff(-r)d.r
SINGULAR AND DEGENERATE CAUCHY PROBEMS
which agrees w i t h Lions [l;2; 3; 51. (7.15)
1
t u = Bw = 2c t-2m ( t m
Consequently
1 ) m-Pw(-c)d-c
-T
0
i s t h e v e c t o r v e r s i o n o f (7.14) and we have Theorem 7.4 g i ven by (7.15) Proof:
I f w s a t i s f i e s (7.9) w i t h uo
E
D(A2) then u
s a t i s f i e s Lmu = A 2u w i t h u ( o ) = uo and u t ( o ) = 0. I
E v i d e n t l y (Bw)(o) = w(o) and (Bw) ( 0 ) = 0 ( c f . L i o n s
[l]where an e x p l i c i t v e r i f i c a t i o n o f (7.5)
i s also given
-
the
I
c o n d i t i o n w (0) corresponding t o f ( 0 ) = 0 i s c r u c i a l here as i n t Again A2 can be passed under t h e i n t e g r a l (7.8) and (7.11)). s i g n i n (7.15) as b e fo re i n (7.11). Remark 7.5
QED
Changing v a r i a b l e s i n (7.15) we o b t a i n (6.6)
when W (T ) i s w r i t t e n as W(T ) = cosh
Awe.
A c t u a l l y i n t h i s case t h e r e i s a l s o an i n v e r s e t r a n s m u t a t i o n o p erat or B - l b u t we w i l l n o t go i n t o d e t a i l s here (see Lions, l o c . cit.. f o r an exhaustive study o f t h e m a t t e r ) . I n Lions [3] 2 i t i s a l s o shown how t o transmute P(D) = D i n t o Q(D) = Mm = 2 D + D + p ( t ) D + q ( t ) and v i c e versa, and we w i l l sketch t some o f t h e c a l c u l a t i o n s here (p and q a r e assumed t o be conti n uous) ( c f . a l s o Hersh [S]).
Thus l e t $(x,t)
be t h e s o l u t i o n
of
(7.16) (7.17)
$(x,o)
= f(x);
$ t ( ~ , ~ =) 0
86
(x,t 2 0 )
1.
SINGULAR PARTIAL DIFFERENTIAL EQUATIONS OF EPD TYPE I
w'here f i s extended as an even f u n c t i o n w i t h f ( 0 ) = 0.
Then
again t he t rans mu ta ti o n o p e ra to r B i s determined by ( B f ) ( t ) = and s a t i s f i e s f o r m a l l y t h e tra n s m u t a t i o n r e l a t i o n MmB =
r$(o,t)
BD2, w h i l e t o c o n s t r u c t B L i o n s uses t h e f o l l o w i n g ingenious
"trick."
He sets
(7.18)
B(cos x s ) ( t ) = 0 (s ,t)
and from t h e t r a n s mu ta ti o n r e l a t i o n p l u s t h e f a c t t h a t ( B f ) ( o ) = f ( o ) we o b t a i n (7.19) w i t h e(s,o)
2 MmO(s,t) + s 0 ( s , t ) = 1 , w h i l e et(s,o)
s o l u t i o n s e(s, t )
= 0 = 0 a u t o m a t i c a l l y (.for unique
see Section 5).
Assuming now (we o m i t v e r i f i -
c a t i o n ) t h a t B i s d e fi n e d by a ( d i s t r i b u t i o n ) k e r n e l b ( t , x ) w i t h support i n t h e r e g i o n
1x1
5 t, which i s l e g i t i m a t e here b u t i s
i n p a r t i n c i d e n t a l t o t h e tra n s mu ta ti o n concept, one obtains, s i n c e f ( x ) = cos xs i s even, and b ( t , * ) w i l l be even ( v e r i f i c a t i o n f o l l o w s from (7.25) f o r example) (7.20)
0(s,t)
= 2
t
b(t,x)
cos xs dx = 2
b(t,x)
cos xs dx
0
from which f o l l o w s , by F o u r i e r i n v e r s i o n i n a d i s t r i b u t i o n sense, (7.21)
b(t,x)
Theorem 7.6
=
1
O(s,t)
cos xs ds
0
The tra n s mu ta ti o n o f D
2
i n t o Mm i s determined
by an operat or B w i t h k e rn e l d e fi n e d by (7.21),
87
where €Ii s known.
SINGULAR AND DEGENERATE CAUCHY PROBLEMS
F u r t h e r p r o p e r t i e s a r e developed by Lions [3;
51 and a gen-
e r a l t ransmut ati o n t h e o r y w i l l be developed i n C a r r o l l [24]. Hersh's technique ( c f . Hersh 131) i s based on s p e c i a l f u n c t i o n s and k e r n e l s and leads f o r m a l l y t o t h e same r e s u l t as t h a t o f Theorem 7.6. w(t,A)
L e t us sketch t h i s f o r completeness.
Thus l e t
be t h e s o l u t i o n o f 2 Mmw + A w = 0;
(7.22)
w(o,A)
= 1;
wt(oyA) = 0
w h i l e u s a t i s f i e s ( c f . (7.10)) 2 2 D u + A u = 0;
(7.23) THen u(t,A)
=
u(0,A)
= 1;
ut(o,A)
=
0
cos A t and Hersh proposes a formula m
(7.24)
w(t,A)
b(t,x)u(x,A)dx
=
b(t,x)
=
cos Ax dx
m ,
Since (7.22) hol d s we have, p ro v i d e d b ( t , - ) w i t h reasonable smoothness (7.25)
,
Mmb(t,x) = bxx(t,x)
Th i s shows symmetry o f b (t,x ) (7.26)
has compact support
w(t,A)
=
2
[
b (t,x )
i n x and (7.24) becomes ( c f . (7.20)) cos Ax dx
Again (7.25) can be solved, as can (7.19),
by t h e methods o f
L
D e f i n i n g now B by ( B f ) ( t ) = b(t,x) f ( x ) dx we 2 have f o r m a l l y from (7.25) MmB = BD The i d e a then i s t o r e p l a c e Section 1.5.
.
A 2 by s u i t a b l e o p e ra to r f u n c t i o n s -g(A).
88
Chapter 2 Canonical Sequences of Singular Cauchy Problems 2.1
The rank one situation.
In t h i s chapter we will
develop the group theoretic version of the EPD equations studied i n Chapter 1.
This leads t o many new classes of equations and
results parallel t o those of Chapter 1 as well as t o some i n teresting new situations.
Moreover i t exhibits the main results
of Chapter 1 i n t h e i r natural group theoretic context.
The
ideas of spherical symmetry, radial mean val'ues and Laplacians, etc. inherent i n EPD theory have natural counterparts i n terms of geodesic coordinates and one can obtain recursion relations,
Sonine formulas, etc. group theoretically.
The results are
based on Carroll [21; 221 in the semisimple case and were anticipated i n p a r t by e a r l i e r work of Carroll [18; 191, CarrollSilver [l5; 16; 17) and Silver [ l ] f o r some semisimple and Euclidean cases.
The group theory i s "routine" a t the present
time and r e l i e s heavily on Helgason's work (cf. Helgason [ l ; 2 ; 3; 4 ; 5; 6; 7; 8; 9; 101) b u t one must of course refer t o basic
material of Harish-Chandra (cf. Warner [ l ;
21
for a sumnary) as
well as lecture notes by Varadarajan, Ranga Rao, e t c . ) ; other specific references t o Bargmann [ l ] , Bargmann-Wigner [2], Bhanu-Munti [l], Carroll [27; 281, Coifman-Weiss [l], EhrenpreisMautner [ l ] , Flensted-Jensen [1], Furstenberg [ l ] , Gangolli [ l ] , Gelbart [ l ] , Gelfand e t a l , [ l ; 2 ; 3; 41, Godement [ l ] , Hermann 89
SINGULAR AND DEGENERATE CAUCHY PROBLEMS
[ l ] , Jacquet-Langlands [l], Jehle-Parke [l], Kamber-Tondeur [l], Karpelevic" [ l ] , Knapp-Stein [l], Kostant [l], Kunze-Stein [ l ; 23, Lyubarskij [ l ] , Maurin [l], McKerrell [l], Miller [ l ; 23, Naimark [ l ] , Pukansky [ l ] , RGhl [l], Sally [ l ; 23, S i m s [ l ] , Smoke [ l ] , Stein
113,
Takahashi [l], Talman [ l ] , Tinkham [ l ] ,
Vilenkin [ l ] , Wallach [ l ] , Wigner [ l ] , etc., as well as the fundamental work of E. Cartan and H. Weyl, are t o be taken f o r granted, even i f n o t mentioned explicitly.
The basic Lie theory
i s developed somewhat concisely i n t h i s section; for a rather more leisurely treatment we refer t o Bourbaki [5], Carroll [23], Hausner-Schwartz [l], Helgason [ l ; 2; 31, Hochschild [ l ] , Jacobson [1], Loos [ l ] , Serre [ l ;
21,
V
Tondeur [ l ] , Zelobenko
Ell, etc. W e will s t a r t o u t w i t h the f u l l machinery f o r the rank one semisimple case, following Carroll [21; 221, and l a t e r will give extremely detailed examples f o r special cases.
T h i s avoids
some repetition and presents a "clean" theory immediately; the reader unfamiliar w i t h Lie theory m i g h t look a t the examples f i r s t where many d e t a i l s and definitions are covered.
We delib-
erately omit the treatment of invariant differential operators acting in sheaves o r i n sections of vector bundles even t h o u g h t h i s i s one of the more important subjects i n modern work (some references are mentioned above).
The preliminary material will
be expository and specific theorems will n o t be proved here. The Eucl idean group cases have been covered in Carroll -Si 1 ver [15; 16; 171 and especially Silver [ l ] s o that we will only give
90
2.
CANONICAL SEQUENCES OF SINGULAR CAUCHY PROBLEMS
a few remarks l a t e r about t h i s a t t h e end o f t h e chapter; t h e b a s i c r e s u l t s a r e i n any event i n c l u d e d i n Chapter 1.
Thus
l e t G be a r e a l connected noncompact semisimple L i e group w i t h f i n i t e c e n t e r and K a maximal compact subgroup so t h a t V = G/K -
.
-
d
L e t g = k + p be a
i s a symmetric space o f noncompact type.
Cartan decomposition, a c p a maximal a b e l i a n subspace, and we
-
-
w i l l suppose u n t i l f u r t h e r n o t i c e t h a t dim a = rank V = 1. One s e t s A = exp a, K = exp k, and
IsA for
N
-
= exp n where n =
A > 0 where t h e gA a r e t h e standard r o o t spaces corres-
ponding t o p o s i t i v e r o o t s a and p o s s i b l y 2a i n t h e rank one case.
1
One s e t s p = - c m A f o r 2 x element Ho
E
X > 0 where mA
= dim gA and we p i c k an
a such t h a t a(Ho) = 1. Thus p =
1 (H ma
+ m2a)a and
we can i d e n t i f y a Weyl chamber as a connected component a + C I
a c a w i t h (o,m) to
u
E
*
a
i n w r i t i n g a ( t H o ) = t where
by u ( t H o ) = U t .
u
E
R corresponds
The Iwasawa decomposition o f G i s G =
KAN which we w r i t e i n t h e form g = k ( g ) exp H(g) n ( g ) where t h e n o t a t i o n at = exp t H o i s used.
I
L e t M (resp. M ) denote t h e
c e n t r a l i z e r (resp. n o r m a l i z e r ) o f A i n K so t h a t t h e Weyl group I
i s W = M /M and t h e maximal boundary o f V i s B = K/M ( t h u s M = {k
E
H
K; AdkH = H f o r
E
a) and M
I
=
{k
E
K; Adka c a )
-
see
t h e examples f o r s p e c i f i c d e t a i l s ) . There are n a t u r a l p o l a r c o o r d i n a t e s i n a dense submanifold o f V a r i s i n g from t h e decomposition G = (kM,a)
-+
G+K (A+
=
exp a+) p r o v i d e d by t h e diffeomorphism
kaK : B x A+-+ V (one c o u l d a l s o work w i t h t h e decom-
p o s i t i o n G = KAK).
Thus t h e p o l a r coordinates o f ~ ( g )=
91
SINGULAR AND DEGENERATE CAUCHY PROBLEMS
ak2)
V a r e (klM,a)
E
g i v e n v = gK -1
E
where
:G
IT
V and b = kM
+ .
V i s t h e n a t u r a l map.
B one w r i t e s A(v,b)
E
k ) and t h e F o u r i e r t r a n s f o r m o f f
=
2 L ( V ) i s d e f i n e d by
E
Y
F f = f where ( u s i n g Warner's n o t a t i o n ) Y
f(u,b)
(1.1) for
u
E
a
*
and b
= E
r 1, f(v) B.
exp (iu+p)A(v,b)dv
--
A l l measures a r e s u i t a b l y normalized i n
t h i s treatment. T h i s s e t s up an i s o m e t r i c isomorphism f f 2 * between L ( V ) and t h e space L2(a: x B) ( c f . here below f o r a+) w i t h i n v e r s i o n formula
Here t h e 1 / 2 comes from t h e o r d e r o f t h e Weyl group (which i s two) and c ( u ) i s t h e standard Harish-Chandra f u n c t i o n .
For a
general expression o f C(M) we can w r i t e ( c f . a l s o (3.14)) c ( u ) = I(ip)/I(p)
where ( c f . G i n d i k i n - K a r p e l e v i c [l],Helgason
[3; 81, Warner [l;21)
Here B(x,y)
= I'(x)r(y)/I'(x+y)
i s t h e Beta f u n c t i o n and one
d e f i n e s (v,a) i n t h i s c o n t e x t i n terms o f t h e K i l l i n g form B(-,*)
as B(H ,Ha)
A(h) = B ( j , H )
V
where f o r X
for H
E
l i n e a r maps o f a i n t o
E
*
aC, HA
E
*
aC i s determined by
a ( n o t e here t h a t aC i s t h e space of R w h i l e aC i s t h e c o m p l e x i f i c a t i o n o f a
which i s f o r m a l l y t h e s e t o f a l l sums A + i u f o r
92
A,u
E
a).
Then
2.
CANONICAL SEQUENCES OF SINGULAR CAUCHY PROBLEMS
*
a+ i s t h e preimage o f a+ under t h e map A chamber i n a later.
* . *
HA and i s a Weyl
S p e c i f i c examples o f c ( p ) w i l l be w r i t t e n down
Now a / W
- a+* (.recall W
= (1,s)
o r e q u i v a l e n t l y sat = a-t = a t ’ )
where, f o r $
+.
where sH = -H f o r
H
E
a
and one can w r i t e
L2 (B),
E
The q u a s i r e g u l a r r e p r e s e n t a t i o n L o f G on L2 (V), d e f i n e d by L ( g ) f ( v ) = f(g-’v)
(1.5)
L =
decomposes i n t h e form
r Ja*,W
‘plC(w)
dp
where L a c t s i n H by t h e same r u l e as L. L i s i n f a c t i r P Fc 1-1 r e d u c i b l e and u n i t a r y and i s e q u i v a l e n t t o t h e s o - c a l l e d c l a s s one p r i n c i p a l s e r i e s r e p r e s e n t a t i o n induced from t h e p a r a b o l i c subgroup MAN by means o f t h e c h a r a c t e r man
-f
a”
= exp i p l o g a.
We r e c a l l here a l s o t h e d e f i n i t i o n o f t h e mean value o f a f u n c t i o n @ over t h e o r b i t o f g r ( h ) = gu under t h e i s o t r o p y = gKg-’ subgroup Iv
Thus w r i t i n g MhC$ = Mu$ we have
a t v = .rr(g).
( r e c a l l i n g t h a t r ( h ) = u)
93
SINGULAR AND DEGENERATE CAUCHY PROBLEMS
( c f . Helgason [l]) when D
D(G/K).
E
The symbol D(G/K) denotes
t h e l e f t i n v a r i a n t d i f f e r e n t i a l o p e r a t o r s on G/K which a r e d e f i n e d as follows.
If $ : G
G i s a diffeomorphism and f i s
-+
a f u n c t i o n on G one sets f $ ( g ) = ( f
while i f D i s a -1 l i n e a r d i f f e r e n t i a l o p e r a t o r we w r i t e DQ : f + (Of$ )$. Thus
G o r on V = G/K.
$-')(g)
W r i t i n g akg = gk f o r g
G and k T
space of d i f f e r e n t i a l o p e r a t o r s on G such t h a t D D i s denoted by D(G/K);
E
K, t h e
= D and D"k =
t h i s i s i d e n t i c a l w i t h t h e space o f
l i n e a r d i f f e r e n t i a l o p e r a t o r s on G/K ( c a l l e d l e f t i n v a r i a n t ) such t h a t
T
D
= D (the condition
proof obvious).
Dak
D i s automatic on G/K
=
-
Now t h e zonal s p h e r i c a l f u n c t i o n s on G a r e
d e f i n e d by t h e formula
(u
E
*
a )
. ..,
and we can w r i t e by i n v a r i a n t i n t e g r a t i o n 41 (9) = $ (gK) (ac-
u
..,
u
..,
t u a l l y t h e $ (9) a r e K b i i n v a r i a n t i n t h e sense t h a t $ ( 9 ) =
u
* -
..,
$ ( k g k ) ) and K i s u n i m d u l a r
u
-
u
c f . Helgason [l], Maurin [l],
-
Nachbin [2], Wallach [l],o r Weil [l] and thus
.,K CI$
f(k-')dk
-u
f(ikk)dk = ..,
f(k)dk. I t i s known f u r t h e r t h a t $,(g) = K ( g - l ) ( c f . Harish-Chandra [2] o r Warner [l;21) w h i l e t h e
E
=
C"(G/K),
@u X ($ ) u ..,u of
1
I,
for D
and a r e c h a r a c t e r i z e d b y t h e i r eigenvalues AD = E
D(G/K), w i t h $u(n(e)) = 1, p l u s t h e b i i n v a r i a n c e
(I~.We now demonstrate a lemma which has some i n t e r e s t i n g
consequences.
94
CANONICAL SEQUENCES OF SINGULAR CAUCHY PROBLEMS
2.
h The F o u r i e r t r a n s f o r m o f 1.1 = MU
Lemma 1.1
.rr(h)) i s g i v e n b y FMh = (M‘f)(v)
-
A
1-I
(h) and i f h = kak w i t h a
Proof:
A+ then
o f u = .rr(h).
h The a c t i o n o f M o r MU as a d i s t r i b u t i o n i n E ’ ( V )
determined by ( c f . (1.6)) b e i n g t h e i d e n t i t y i n G).
<M h ,$>
=
h (14 $ ) ( w ) where w
Thus i n (1.6)
A
is
r ( e ) (e
=
take Q = exp (ill+ p )
A
where b = kM.
Since A(ekhK,kM)
t h e u n i m o d u l a r i t y o f K ( c f . (1.8) (1 -9)
E
(u =
( M a f ) ( v ) so t h a t (Muf)(v) depends o n l y on t h e r a d i a l
=
component a i n t h e p o l a r decomposition (kM,a)
A(w,b)
E’(V)
E
-
(Mh$)(w) =
I
= -H((kh)-’;)
one has by
and comments t h e r e a f t e r )
exp [-(ip+p)H(h -1 k -1,- k)dk
K
j
=
exp [-(ip+p)H(h -1 kk)dk =
K
,. = $
-1-1
(h-’)
=
-
0
1-I
j
e-(i’+p)H(h-lk)dk
K h (h) = FM
h One notes here t h a t M = MU works on f u n c t i o n s i n V such as exp (i’+p)A(v,b) placed by 14‘
= $
h
= M
h and <M ,I$> i s p r e c i s e l y (1.1) w i t h f r e -
and $ = exp (iu+p)A(w,b);
no i n t e g r a t i o n over
V i s i n v o l v e d s i n c e v = w i n $, o n l y a d i s t r i b u t i o n e v a l u a t i o n A
i s o f concern here.
-
F i n a l l y we n o t e t h a t w i t h h = kak as above
t h e r e f o l l o w s from (1.6) (1.10)
(Mhf)(v) =
J K f(gkiaK)dk
=
i,
f(gkaK)dk = (Maf)(v)
T h i s l e a d s t o a symmetric space v e r s i o n o f theorem o f Zalcman [l],g e n e r a l i z i n g an o l d formula o f P i z z e t t i ; here G/K
95
QED
SINGULAR AND DEGENERATE CAUCHY PROBLEMS
i s of a r b i t r a r y rank and we refer t o Helgason [l ; 23 o r Warner [ l ; 23 f o r general information on h i g h e r rank s i t u a t i o n s . Theorem 1.2
Suppose the o p e r a t o r s
(Pizzetti-Zalcman).
A k w i t h eigenvalues X k g e n e r a t e D(G/K) as an a l g e b r a ; then l o -
cal l y (1.11)
(MUf)(V)
=
bqu,
Ak)f)(V)
00
1 Pn(u,X k ( 4 v ) ) i s expressed i n terms o f i t s n=l eigenvalues a s $ (u,Xk).
where $,(u)
= 1 +
P
Zalcman [ l ] , i n an Rn c o n t e x t , has g e n e r a l i z e d t h i s kind o f theorem considerably Proof:
.
h
One n o t e s from Helgason [ l ] t h a t (M @,)(v)
-
i,
=
= z X ( g ) $ X ( h ) and the l o c a l expres$X(gk.rr(h))dk = ;Jgkh)dk K O3 sion (Muf)(v) = ([1 + 1 Pn(u,Ak)]f)(v) holds. Consequently n=l we o b t a i n $ X ( h ) = applying t h i s l o c a l e x p r e s s i o n t o
-
m
1 P , ( U , A ~ ( $ ~ ) =) $,(u) and (1.11) follows. One should n=l remark a l s o t h a t the polynomials P a r e without cons t a n t terms. 1 +
n
2.2
Resolvants.
The o b j e c t s o f interest n a g e n e r a l i z e d
EPD theory a r e the r a d i a l components o f a b a s i s f o r the H
v spaces
of (1.4), m u l t i p l i e d by a s u i t a b l e weight funct on, the r e s u l t
o f w h i c h we w i l l denote by @ ( t , u ) ( c f . C a r r o l l [21; 221, C a r r o l l S i l v e r [15; 16; 171, S i l v e r [ l ] ) and we mention t h a t m can denote
96
CANONICAL SEQUENCES OF SINGULAR CAUCHY PROBLEMS
2.
a m u l t i i n d e x here.
F i r s t we remark t h a t V i s endowed w i t h t h e
Riemannian s t r u c t u r e induced by t h e K i l l i n g form B ( * , * )
(but
t h e Riemannian s t r u c t u r e does n o t p l a y an i m p o r t a n t r o l e here, e.g.,
$(*,*)
c o u l d serve
-
see Example 3.2) and f o r rank
V
=
1, D(G/K) i s g e n e r a t e d by a s i n g l e Laplacian A, determined by
-
t h e standard Casimir o p e r a t o r C i n t h e enveloping algebra o f g ( c f . Remark 3.15 about C).
We l o o k a t t h e r a d i a l component AR o f
passing t h i s from t h e coord n a t e t i n at E A t o r ( A ) i n an a obvious manner, and s e t t i n g Mt = M w i t h Ro(t,p) = FMt = $,,(at) A,
-
A
one o b t a i n s an eigenvalue e q u a t i o n ( c f . Helgason [5]) (2.1)
[of. +
(ma+mZa)cotht Dt + mZatanht D
141
t u
The s o l u t i o n o f (2.1), a n a l y t i c a t t = 0, i s e.g., A
(2.2)
^Ro(tyu) = (I (exp t H o ) = F(G,B,y,
u
"0
"0
where e v i d e n t l y R ( 0 , ~ =) 1 and Rt(o,p)
B
=
-
2
sh t )
0 ( 6 = (ma+2m2a+ 2 i p ) / 4 ,
= ( m +2mZa-2iu)/4,
and y =(ma+m2a+1)/2). The general EPD a "0 s i t u a t i o n i n v o l v e s embedding R i n a "canonical" sequence o f "m l l r e s o l v a n t s " R (t,p), f o r m > 0 a p o s i t i v e i n t e g e r o r a m u l t i index, such t h a t t h e r e s o l v a n t i n i t i a l c o n d i t i o n s (2.3)
"m R ( 0 , ~ =) 1;
are s a t i s f i e d .
"m Rt(oYp) = 0
There w i l l a l s o be "canonical" r e c u r s i o n r e l a t i o n s
97
SINGULAR AND DEGENERATE CAUCHY PROBLEMS
“m between t h e R of i n d i c e s d i f f e r i n g by k1 o r k 2 which w i l l a r i s e group t h e o r e t i c a l l y f r o m c o n s i d e r i n g a f u l l s e t o f b a s i s elements i n the H
spaces.
1-I
2 Thus we must f i r s t determine a b a s i s f o r L ( B ) and t h i s i s w e l l known (thanks a r e due here t o R. Ranga Rao f o r some h e l p f u l information).
We l e t { I T ~ , V ~ Iw i t h dim VT = dT be a complete s e t
o f i n e q u i v a l e n t i r r e d u c i b l e u n i t a r y r e p r e s e n t a t i o n s o f K and l e t
M VT C V T be t h e s e t o f elements f i x e d by M.
One knows by a r e -
s u l t o f Kostant [l]t h a t dim V y = 1 o r 0 i n t h e rank one case ( c f . a l s o Helgason [5]) and f o r t h e s e t
T E
we l e t w i be a b a s i s v e c t o r f o r VTM w i t h
T where dim
wY(1
2
normal b a s i s f o r VT under a s c a l a r p r o d u c t < example t h e c o l l e c t i o n o f f u n c t i o n s
(T E
T)kM
= 1
i 5 dT) an o r t h o >T.
-+
!V
Then f o r
<wi,aT(k)w;>T
is
2 known t o be a b a s i s f o r L ( B ) and we d e f i n e ( c f . (1.4) w i t h v = atK and b = kM) n -
(2.4)
$;,(atK)
= El 9T(BT:-ip:a -1 )w T
t
where BT
E
j
HomM(VT9$) i s determined by t h e r u l e Bw‘:
= 61,s
(Kronecker symbol) so t h a t B T r T ( k )-1 wT. = <W:,IT (k)w;>T = J J T ~ (note t h a t $(k)dk = $(km)dm)db and K B M
J
I (I
CANONICAL SEQUENCES OF SINGULAR CAUCHY PROBLEMS
2.
H(g-’km)
= H(g-’k)
w i t h nT(km) = nT(k)nT(m)).
above
The
are E i s e n s t e i n i n t e g r a l s as d e f i n e d i n Wallach [l]and t h e A .
( a K) a r e t h e r a d i a l components o f b a s i s elements i n H t FC ( c f . below). I t i s p o s s i b l e t o o b t a i n an e x p l i c i t e v a l u a t i o n o f M,T
these f u n c t i o n s , u s i n g r e s u l t s o f Helgason [5;
111 as f o l l o w s .
One d e f i n e s ( c f . Helgason 151)
Then i t i s proved i n Helgason [5;
f o r x = gK.
111 ( t o whom we
g r a t e f u l l y acknowledge some conversation .on t h e m a t t e r ) t h a t
where Y
,T
( x ) i s d e f i n e d by
j
e (-iA-p)H(g-’k) < wl,T a T ( k ) w i >T dk K Thus r e c a l l i n g t h e p o l a r decomposition (kM,a) + kaK we have yA,T(x) =
(2.7)
Theorem 2.1
;’
(2.8)
MrT
h
General b a s i s elements i n
(iatK) =
1
K
e (iu-p)H(at k
dT- l l 2 fT,J -’.(katK)
I n p a r t i c u l a r one has s i n c e ti1
H me FC
-1^-1
A
=
A
G’
k ) <wi,aT(k)w1>., T
dk
A T = < w i , ~ ~ ( k ) w Y-FC,T(atK) ~>~
;’
-’
( a K) = ( a K) = Y (atK) t 1-I.T t ,T ( t h i s “ c o l l a p s e ” was observed i n VaT
= <W?,IT ( l ) w T > ,j J T 1 T s p e c i a l cases by S i l v e r [l]). A
Now l e t KO = {nT,VTl f o r T ET (i.e.,
dim VTM = 1 ) and f o l l o w -
i n g Helgason [5] we use a p a r a m e t r i z a t i o n due t o Johnson and
99
SINGULAR AND DEGENERATE CAUCHY PROBLEMS
Wallach (see Helgason [5] f o r references and c f . a l s o Kostant h
[l]).I f mZcl = 0, KO = {(p,q)> w i t h p ..
1, KO = {(p,q)> w i t h (p,q)
E
E
Z+ and q = 0; i f mZa =
Z+ x Z where p
f
q
E
i f mZa =
22,;
h
3 o r 7, KO = {(p,q)>
w i t h (p,q)
E
Z+ x Z+ where p
&
q
E
One
22.,
s e t s R = (ip+p)(HO) and then ( c f . Helgason [5; 111) Theorem 2.2
The r a d i a l components o f b a s i s elements i n
H
IJ
are given by
= cI.1'
STt h P t
ch-'tF(-,--p
R+
where t h = tanh, ch = cosh, and
+
R+p-q+l -m2a
+
ma+m
+1
,th2t)
F i s t h e standard hypergeometric
function. Note here t h a t o u r R i s t h e n e g a t i v e o f t h e R i n Helgason [5;
111 where R = ( i A - p ) ( H ) = ( - i p - p ) ( H
i n (1.1))
0
0
) ( c f . here o u r n o t a t i o n
and r e c a l l t h a t a(Ho) = 1 w i t h at = exp tHo.
now sets dZa = -4q(q+mZcl-1) and da = -p(p+m +mZa-l) i t i s shown i n Helgason [5] t h a t Y ( t ) = Y
the d i f f e r e n t i a l equation
100
-1-I ,T
I f one
+ q(q+mZcl-l)
(atK) a l s o s a t i s f i e s
2.
(2.11)
CANONICAL SEQUENCES OF SINGULAR CAUCHY PROBLEMS
ytt
+ (ma+in2a)coth t Yt + m2a tanh t i t
+
[d sh-2t + d2ash-22t + p(Ho) 2 + v(Ho) 2 19' = 0 a
where sh = s i n h , and we have used t h e i d e n t i t y c o t h 2 t = h c o t h t + tanh t). R e c a l l i n g t h a t p ."(-in21 a+m 2a ) one sees t h a t the t r i v i a l representation 1 g i v e s r i s e t o (2.1),
T, corresponding t o p = q = 0,
E
so t h a t (since c
R+I-M,
.w
1) we should have +I 2a , a 2a , t h t). This =
m +m
(atK) = +(at) = ch-'t F(L/2, ,1 i s borne o u t b y t h e Kummer r e l a t i o n F(a,b,c,z)
y-lJ
w i t h z = -sh 2 t, a = 6
c,z/z-1)
( l - ~ ) -= ~ch-'t
and c
-
b = y
B
=
(l-z)-aF(a,c-b,
b , = 8 , and c = y, so t h a t
= R/2,
-
=
(ma+2+2iv)/4 = (R+1-m2a)/2;
. "
hence Y
-v,1
2.3 h
(atK) = +(a ) as i n d i c a t e d . t = 0.
Examples w i t h
Now t o c o n s t r u c t r e s o l v a n t s
A
( a K) one m u l t i p l i e s t h e t by a s u i t a b l e f a c t o r i n o r d e r t o o b t a i n t h e r e s o l v a n t i n i -
Rm(t,v) = RPsq(t,p) from t h e Y
y-v ,T
-!J,T
. "
t i a l c o n d i t i o n s (2.3)
and t o produce I$ (a,)
v
when p = q = 0.
These requirements are n o t alone s u f f i c i e n t t o produce t h e "cano n i c a l " r e s o l v a n t s s i n c e one needs t o i n c o r p o r a t e c e r t a i n group t h e o r e t i c r e c u r s i o n r e l a t i o n s i n t o t h e t h e o r y which serve t o " s p l i t " t h e second o r d e r s i n g u l a r d i f f e r e n t i a l equation f o r t h e A
Rp,q
a r i s i n g from (2.11)
i n t o a composition o f two f i r s t o r d e r
equations ( c f . here I n f e l d - H u l l [ l ] ) .
Even then we remark t h a t
t h e r e s o l v a n t s w i l l n o t be unique since one can always m u l t i p l y ^Rm(t,p) by a f u n c t i o n $, I+
0
( t ) E 1.
E
C2 such t h a t $,(o)
=
1, $ i ( o ) = 0,
T h i s w i l l simply g i v e a d i f f e r e n t second o r d e r
101
SINGULAR AND DEGENERATE CAUCHY PROBLEMS
s i n g u l a r d i f f e r e n t i a l equation f o r t h e new r e s o l v a n t and d i f -
-
ferent s p l i t t i n g recursion r e l a t i o n s b u t the resolvant i n i t i a l c o n d i t i o n s w i l l remain v a l i d and t h e r e d u c t i o n t o 0 (a ) f o r m = U
0 i s unaltered.
t
Thus we w i l l choose the s i m p l e s t form o f r e s o l -
vant f u l f i l l i n g t h e s t i p u l a t i o n s imposed w h i l e r e f e r r i n g t o these r e s o l v a n t s and equations as canonical. Example 3.1 -p
2
, d2,
Take t h e case where ma = 1 , mZa = 0, da =
= 0, and p = 1/2.
T h i s corresponds t o G = SL(2,IR)
with
K = S O ( 2 ) and t h e r e s o l v a n t s a r e given i n C a r r o l l [18; 19; 223,
C a r r o l l - S i l v e r [15; 16; 171, S i l v e r [l]as
where 5 = c h t and P i m denotes t h e standard associated Legendre f u n c t i o n o f t h e f i r s t kind.
Now we r e c a l l a formula ( c f . Snow
111, P. 18)
2 S e t t i n g z = t h t i n (3.2) and n o t i n g t h a t 1 have fi = sech t and 1 as above.
-
-
2 2 t h t = sech t we
n / ( l + Z ) = 5-1/ 0) and m = 0, 21,
T)
k2,
....
In view of (3.20) and the d e f i n i t i o n (3.21) ( o r (3.22)) of
8 we
can write f o r m 2 0, z = ch
Using (3.30)
-
T,
and R = -1/2 + iu
(3.31) one can then e a s i l y prove ( c f . Carroll-
S i l v e r [16])
115
SINGULAR AND DEGENERATE CAUCHY PROBLEMS
Theorem 3.8
The " c a n o n i c a l " r e l a t i o n s (3.33)
a r e e q u i v a l e n t t o t h e r e c u r s i o n r e l a t i o n s (3.9)
and H-
f o r H,
"m for R
.
I n par-
t i c u l a r t h e v a r i a b l e t i n Example 3.1 can be i d e n t i f i e d w i t h t h e geodesic d i s t a n c e Proof:
T,
from w = K t o k a K. UJt
The r e c u r s i o n r e l a t i o n s (3.9)
low immediately from (3.30)
-
(3.31)
(and hence (3.8))
and (3.34)
suggests o f course t h e i d e n t i f i c a t i o n o f
T
-
(3.35).
A
n
This
and t b u t one can g i v e
We w r i t e k , a k 2 = pk
a computational p r o o f also.
fol-
n
so t h a t kla =
3
pk4 and f o r p = exp (aX+BY) we o b t a i n from Example 3.3 t h e d i s -
tance r 2 = a2 + B
2
n
between w = T ( e ) = K and pK t o g e t h e r w i t h
angle measurements cos
e
=
a/T
and s i n 8 =
B/T.
Set now kl =
a = aty and k4 = kg, u s i n g t h e n o t a t i o n o f (3.13); kJ, r e s u l t a number o f equations connecting t h e v a r i a b l e s
there
( T y t , $ y $ y a y f 3 ) y t h e s o l u t i o n o f which i n v o l v e s t h e i d e n t i f i c a t i o n o f t w i t h T.
Thus t h e r a d i a l v a r i a b l e t i n t h e n a t u r a l p o l a r
expression ( k M y a ) f o r u = T(h) = kUJatK i s i n f a c t t h e geoJ , t d e s i c r a d i u s T o f T ( h ) = pK. The a c t u a l c a l c u l a t i o n s a r e someA
what t e d i o u s b u t completely r o u t i n e so we w i l l o m i t them. We n o t e now t h a t t h e second e q u a t i o n i n (3.9) i n t h e form (shZmt @ ) ' = 2mshzm-lt.
QED
can be w r i t t e n
8-l which y i e l d s ,
i n view
of (2.3), (3.36)
shZmt i m ( t y u )= 2m
1:
sh2m-1
One r e w r i t e s t h e i n t e g r a n d i n (3.36),
116
rl
In-1 R (rlyil)drl
i n terms o f (3.36)m-1
and
2.
CANONICAL SEQUENCES OF SINGUALR CAUCHY PROBLEMS
i n t e g r a t e s o u t t h e rl v a r i a b l e ; i t e r a t i n g t h i s procedure we o b t a i n I f in 2 k 2 1 a r e i n t e g e r s t h e "Sonine" formula
Theorem 3.9 (3.37)
shZmt km(t,p) =
r
m-k+l
r
i s a d i r e c t consequence o f (3.9). have
jt(cht-chrl)k-'
I n p a r t i c u l a r f o r m = k we
t
(3.38)
shZmt @(t,p)
=
2%
0
(cht-chn)m-lsho
s p h e r i c a l f u n c t i o n s (1.8)
and u s i n g t h e n o t a t i o n o f (3.17)
we see t h a t ( c f . (3.19))
,.,
(3.39)
+v(at)
where P
=
&
I
ip-
27T
( c h t + s h t cose)
ip-
0
'
-
1
dB = P o
,(cht)
1v-7
1 -
i s t h e standard Legendre f u n c t i o n . ,.. a l s o by Lemma 1.1 t h a t FMat = FMt = ( a ), which
'p-7l ( c h t )
We r e c a l l
io(q,u)do
R e f e r r i n g back t o t h e formula f o r t h e zonal
Remark 3.10
(3.18)
I
k
= Po
SO
'(cht)
+U
t
equals i o ( t , p ) . Using (3.38)
we can d e f i n e Rm(t,-)
integrating i n E'(V)
=
F
-l*m
R (t,v)
E I ( V ) and
E
( c f . C a r r o l l 1141) we have f o r m
2
1 an
integer, (3.40)
shZmt Rm(t,-) = Zmm
(cht-cho)m-lshn
We r e c a l l t h a t i n t h e p r e s e n t s i t u a t i o n D(G/K)
M drl rl
i s generated by
t h e Laplace-Beltrami o p e r a t o r A which a r i s e s from t h e Casimir
117
SINGULAR AND DEGENERATE CAUCHY PROBLEMS
.
o p e r a t o r C i n t h e enveloping a l g e b r a E(g) (here A corresponds t o 2C due t o o u r choice o f Riemiannian s t r u c t u r e determined by 2 2 2 and C = 1/2(X +Y -Z )
1/2B(-,*)
-
see Remark 3.15 below on Casi-
-3
m i r operators).
I n E(g ) we have then a l s o 2C = ($1)'
B
+ X-X,)
$X+X-
+
and i n geodesic p o l a r coordinates ( c f . Proposi-
t i o n 3.6) (3.41)
A = H:
+ $H+H-
+ H-H,)
a2
=
+ coth
T
a/a-c
+ csch2T a2/ a 0 2 We n o t e here t h a t an a l t e r n a t e expression f o r A i s given by A = H+H
.
+ H'3 -
-
H3 (cf.
Theorem 3.11
C a r r o l l - S i l v e r [15;
16; 171 and S i l v e r
Ill).
For v f i x e d (Muf)(v) depends o n l y on t h e geo-
d e s i c r a d i u s T = t o f u = n ( h ) = k a K and can be i d e n t i f i e d
9 t
w i t h t h e geodesic mean value M(v,t,f). Proof:
T h i s can be proved b y t e d i o u s c a l c u l a t i o n based on
t h e coordinates i n t r o d u c e d i n p r e v i o u s examples b u t i t i s simply a consequence o f Lemma 1.1 and Theorem 3.8. A
Indeed, s e t t i n g
AA
A
h
g = k ak w i t h h = k a k we have v = n ( g ) = k aK and u = n ( h ) = 1 2 1 t 2 1 klatK w i t h (3.42)
a ( M u f ) ( v ) = (M t f ) ( v ) = ( M t f ) ( v )
A
A
A
A
We n o t e t h a t t h e d i s t a n c e from v = klaK t o k aka K i s t h e same as 1 t
118
2. CANONICAL SEQUENCES OF SINGULAR CAUCHY PROBLEMS A
t h e d i s t a n c e from w = n ( e ) = K t o katK = (exp E ) w = pw, which as
n d i c a t e d i n Theorem 3.8, ~n
V '
Hence as k
d e s c r i b e a geodesic " c i r c l e " around
v a r es t h e p o i n t s klak.rr(at) n
w i l l s i m p l y be T = t.
h
k aw o f r a d i u s T = t; (3.42) i s , upon s u i t a b l e normal z a t i o n
1 a l r e a d y provided, t h e d e f i n i t i o n o f M(v,t,f). Remark 3.12
QED
The v a l i d i t y o f t h e Darboux equation ( .7) f o r and D = A i n spaces o f constant
geodesic mean values M(v,t,f)
n e g a t i v e curvature, harmonic spaces, etc. has been discussed, u s i n g p u r e l y geometrical arguments by G h t h e r [l],O l e v s k i j [l], and Willmore [l]( c f . a l s o Fusaro [l],G'u'nther [2;
31, Weinstein
and Remark 5.6).
[12],
We can now define, f o r m c o n v o l u t i o n ) o f Rm(t,.) means o f (3.40).
E
2 1 an i n t e g e r , a composition ( o r
E'(V) w i t h a f u n c t i o n f ( - ) on V by
Thus f o r v = n ( g ) we w r i t e
(3.43) where
denotes a d i s t r i b u t i o n p a i r i n g as i n Lemma
( c f . He gason [2] f o r a s i m i l a r n o t a t i o n ) . (Rm(t,=
1.1
Then, s e t t i n g
# f ( * ) ) ( v ) = um(t,v), we have from (3.40) a Sonine
formula (3.44)
sh2mt urn(t,v)
Since F
= R(R+l)FT w i h
Ehrenpre s-Mautner [1])
R
=
-1/2 + i u when T
I
E
t h e r e f o l l o w s from (3.8),
119
E (V) (cf. (3.9),
(3.43)
SINGULAR AND DEGENERATE CAUCHY PROBLEMS
and t h e d e f i n i t i o n o f u" = u"(t,v)
F o r m 2 1 an i n t e g e r and f
Theorem 3.13
= um(t,v)
t i o n um(t,v,f)
= um(t,v,f)
=
(Rm(t,-)
E
2 C ( V ) t h e func-
# f(*))(v) satisfies
qq)sh [A-m(m+l)]~"+~
(3.45)
u;
(3.46)
uy + 2m c o th t um = 2m csch t urn-'
(.3.47)
utt + (2m+l) c o th t uy + m(m+l)um = Aum
(3.48)
um(o,v) = f ( v ) ;
=
m
where (3.45),
and (3.48)
=
0
h o l d f o r m 2 0.
The n o t a t i o n here d i r e c t s t h a t (AMt # f ) ( v ) =
Proof: Av(Mtf)(v)
(3.47),
uT(o,v)
= (MtAf)(v)
(see e.g.
(1.7)
and r e c a l l a l s o t h a t A =
6d + d6 i n t h e n o t a t i o n o f de Rham [1] i s f o r m a l l y symmetric) w h i l e (3.48) example.
i s immediate fro m t h e d e f i n i t i o n s and (3.45)
for
One can a l s o prove Theorem 3.13 d i r e c t l y from t h e de-
f i n i t i o n (3.44)
and t h e Darboux e q u a ti o n (1.7) w i t h o u t u s i n g t h e
F o u r i e r t r a n s f o r m a t i o n ( c f . C a r r o l l - S i l v e r [15; (3.46)
f o l l o w s immediately from (3.44)
(3.45)
and (3.47)
16; 173).
by d i f f e r e n t i a t i o n w h i l e
r e s u l t then by i n d u c t i o n u s i n g (1.7)
r a d i a l Laplacian DU = A = m = 0 t oget her w i t h (3.46)
a 2/ a t 2 +
Indeed
coth t
and t h e
a / a t ( c f . (3.41)) f o r
(see C a r r o l l - S i l v e r [16] f o r d e t a i l s ) . QED
D e f i n i t i o n 3.14
The sequence o f s i n g u l a r Cauchy problems
120
2.
(3.47)
-
CANONICAL SEQUENCES OF SINGULAR CAUCHY PROBLEMS
sequence. (3.8)
f o r m 2 0 an i n t e g e r w i l l be c a l l e d a canonical
(3.48)
The canonical r e s o l v a n t sequence corresponding t o
under i n v e r s e F o u r i e r t r a n s f o r m a t i o n i s ( c f . a l s o (2.3))
+ ( 2 1 ~ 1 c) o t h
(3.49)
RTt
(3.50)
Rm(o,*) = 6 ;
t RY + rn(m+l)Rm = ARm m
= 0
Rt(0,*)
where 6 i s a D i r a c measure a t w = .rr(e). t i o n s (3.9)
S i m i l a r l y from t h e equa-
we have
2(m+l) sh [A-m(m+l)]Rm+’
(3.51)
Rt =
(3.52)
R Y + 2m c o t h t R” = 2m csch t Rm-’ Remark 3.15
We r e c a l l a t t h i s p o i n t a few f a c t s about .-.
Casimir o p e r a t o r s f o r a semisimple L i e algebra g ( c f . Warner [l]).
. . ., Xn i s any b a s i s f o r gE l e t gij .-.
I f X1,
= B(Xi,X.)
where J be t h e elements o f
B(=,*) denotes t h e K i l l i n g form and l e t gij the matrix (g’j)
i n v e r s e t o (g..). 1J
The Casimir o p e r a t o r i s then
d e f i n e d as
T h i s o p e r a t o r C i s independent o f t h e b a s i s {Xi} .-.
c e n t e r Z o f t h e enveloping algebra E(gt).
Another f o r m u l a t i o n
. “ -
can be based on a Cartan decomposition g = k
Cartan subalgebra j o f g
.-.
.
such j always e x i s t .
(i.e.,
. - . . - . - . . - .
j = j f) k +
L e t a b a s i s Hi
121
and l i e s i n t h e
+ p and a e s t a b l e
jfl p
= j- + jp);
k ( 1 0, xo > 0, then G : R4
+
R4.
The
LobaEevskij space V can be i d e n t i f i e d w i t h t h e p o i n t s o f t h e 2 "pseudosphere" r (x,x) = 1, xo > 0 (which l i e s w i t h i n R4). coordinates
122
If
2.
(3.55)
CANONICAL SEQUENCES OF SINGULAR CAUCHY PROBLEMS
x1 = r sh
e3
s i n €I2 s i n
-
e 1’
x3 = r sh €I3 cos €I
x2 = r sh €I3 s i n €I2 cos €I1; with 0
5
€I1
C $2nt1
A
= {a(+)>= {(ao(+),
al(+),
i s real.
- uA and A w i l l
where
A
The f u n c t i o n s ak w i l l then be be i s o m e t r i c a l l y isomorphic t o
C(QA) where C(QA) has t h e u n i f o r m norm.
I n o u r s i t u a t i o n above
7 and we a s s o c i a t e t h e complex v a r i a b l e s ( z
we have u A C$
z),
A
. . .,S
A
t o (I(+),B(+),
c1 = max(IIBII,IIRII,IISII) responds t o X C(U,)
-+
-+
m
A*
for k
2
1.
One notes t h a t zo
and we w i l l c a l l t h e map
A t h e Gelfand map.
0'
' *'
( + ) ) w i t h h = l / z o 2 c and l z k l 5
r
:
+
-+
0 cor-
a : C(QA) =
A f u r t h e r n o t i o n o f use here ( c f . also
149
SINGULAR AND DEGENERATE CAUCHY PROBLEMS
i s t h a t o f decomposing H by an i s o m e t r i c isomor-
Section 1.5) phism 0 : H
-+
H =
I@ H(z)dv which d i a g o n a l i z e s A and r may be uA
extended, as an i s o m e t r i c isomorphism, f o r example t o a l l bounded B a i r e f u n c t i o n s B(uA) w i t h values i n t h e von Neumann algebra A (here A ’ l C L(H
i s t h e bicommutant o f A and i s t h e c l o s u r e o f
A i n say Ls(H))
we w i l l use uA f o r convenience i n measure
t h e o r e t i c arguments i n s t e a d o f QA.
I I
The so c a l l e d b a s i c measure
v a r i s e s n a t u r a l l y ( c f . Dixmier [l]o r Maurin [l;31) and we w i l l n o t g i v e d e t a i l s here. A v-measurable f a m i l y o f H i l b e r t spaces H(z) on uA c o n s i s t s o f a c o l l e c t i o n o f f u n c t i o n s z E
h(z)
-+
H(z) such t h a t t h e r e i s a v e c t o r subspace F C I I H ( z ) w i t h z
1) h ( z ) Ilff(z)measurable all h
E
F implies g
E
(h
E
F), ( h ( * ) , g ( - ) )
H(z)
-+
measurable f o r
F, and t h e r e i s a fundamental sequence hn
E
F such t h a t t h e c l o s e d subspace o f H(z) generated by hn(z) i s H(z).
2 Now under t h e p r e s e n t circumstances ff = L,,(uA,ff(z))
H i l b e r t space w i t h norm
[I
-
I l h ( ~ ) 1 ) ~ d v ]=~ l’l ~ h l l H and a Lebesque
0-
Diagonalizable
dominated convergence t h e o r h w i l l be v a l i d . operators G
E
is a
..
L(H) are d e f i n e d i n t h e obvious manner ( c f . Section
1.5) w i t h G = 0-lG0
E
- G(z)
L(H) where G
E
L(H(z)), a l l argu-
ments being c a r r i e d o u t i n Ls(H) f o r example. Now i n connection w i t h (2.1)
we c o n s i d e r t h e equation (under
our assumptions S ( t ) = s ( t ) S , etc.),
o b t a i n e d by a p p l y i n g 0 t o
(2.1) (2.2)
A l
u
I
n
ni
+ haz3s(t)u + h b z 2 r ( t ) u + h[a(t)+zozlb(t)]u
150
n
= 0
3.
where
^u
DEGENERATE EQUATIONS WITH OPERATOR COEFFICIENTS
- z3'
= eu and f o r example
We w r i t e z = ( z
1' ' [l]( c f . a l s o
z6) and by Coddington-Levinson [l]o r Dieudonn; Chapter 1 ) t h e r e e x i s t unique s o l u t i o n s Z(t,-r,z,A) o f (2.2)
such t h a t Z(T,T,Z,X)
= 1, Z (T,T,z,A)
0, and Yt(TyTyzyX) = 1 f o r 0 5 c 5 X
, c and i f c < 1 a f u r t h e r argument see below).
If
= 0 then t h e A"
corporated i n t o t h e r i g h t hand s i d e o f (2.9) obvious manner.
term can be i n -
o r (2.10)
i n an
We have then ( s i n c e l z k l 5 c1 f o r k 2 1 )
152
3. DEGENERATE EQUATIONS WITH OPERATOR COEFFICIENTS
t
( Y t I 2 + Aa(.t)IYI2 5 1 + 2cl
(2.10)
1YS12dS
T
It
Aa(S)IY12dS t o t h e r g h t hand s i d e o f
u s i n g G r o n w a l l ' s lemma (Lemma 1.5.10)
E(t,S)
=
IYtI2
and
one obtains, s e t t ng
exp 2cl(t-S),
(2.12)
2.10)
+ Aa(t)lYI2 5 E(t,r) +
I
t
AP(S)E(t,S)lY12dt
T
The f o l l o w i n g lemma now g i v e s a somewhat sharper e s t i m a t e on
IYI2
than can be o b t a i n e d b y a d i r e c t a p p l i c a t i o n o f t h e Gronwall
lemma; t h e p r o o f however i s a simple v a r i a t i o n . Lemma 2.2 for 0
from which follows
T h i s may be written
w h i c h proves the lemna
QED
We observe from (2.11) t h a t P/a so t h a t exp
(2.17)
(I
t
T
[:Id 0 r e a l f o r t > 0,
28 5 1, y = max(a,l/2),
2 0, and f / QE Co(D(Ay)). and u
C'(D(Ay)),
E
2 C (H), u
(2.25)
E
C0(D(Ay+'I2)).
E
F o r uniqueness we w i l l g i v e s e v e r a l r e s u l t s . (2.6)
F i r s t from
one o b t a i n s ( c f . 1.3.17) Y
T
= -Z+A"Z~S(T)Y
Duplicating our esti-
and some bounds f o r Z must be e s t a b l i s h e d . mates l e a d i n g t o (2.9) we have (2.26)
t
IZtI2 +
2Re(Xaz3s(S
T
and under t h e same assumptions and procedures as b e f o r e i t f o l lows t h a t ( c f . (2.27)
2
Then t h e r e e x i s t s a s o l u t i o n o f
g i v e n b y (2.7) w i t h u ( o ) = u t ( o ) = 0, u
(2.1)
Re(z3s(t))
(2.9)
-
(2.12))
l
JT
157
1
t
A a ( t ) l Z 1 2 + lZtL2 5 Aa(-r) + 2c
T
1Z512dE
SINGULAR AND DEGENERATE CAUCHY PROBLEMS
Then by a v e r s i o n o f Lemmas 2.2 and 2 . 3 one o b t a i n s
Consequently we have proved Lemna 2.7
Under t h e hypotheses o f C o r o l l a r y 2.4,
@ ( T ) v J ( T ) l Z l 25 c2. We s e t q = (@vJ)”‘
with
V(t,T)
=
e-1q(T)Z(t,T,z,A)8
by Lebesque dominated convergence again V(t,T) V(t,T)
E
A
I I
1/2 w i t h
lY,l
Next we
again t h a t
so t h a t I Q ( T ) Z ~5 ~c7A1l2 w h i l e from (2.25)
5
L(H) (and
can be shown b u t again t h i s i s n o t needed).
observe from (2.6)
a
E
so t h a t
.“
5 c6 f o r a 2 1/2.
lYTl
5 c&
a-7
for
The case a 5 1/2 i s essen-
t i a l l y t r i v i a l and w i l l n o t be discussed f u r t h e r .
Thus u s i n g
Lebesque dominated convergence again we can s t a t e t h a t f o r T > 0, a- 1 1 T +. Y(t,.r) = e”Ye E C (Ls(D(A z),H)) while T + Z(t,r) = e-lZ0
E
C 1 (Ls(D(A 1/2 ),H)).
Therefore i f u i s a s o l u t i o n o f (2.1)
w i t h data p r e s c r i b e d a t T > 0 we operate on (2.1) w i t h Y(t,C) (changing t t o 5 i n (2.1)) and i n t e g r a t e t o o b t a i n f o r u y t -.1 Co(D(A ’I), u E C’(D(Ay)), and u E C 2 ( H )
158
E
DEGENERATE EQUATIONSWITH OPERATOR COEFFICIENTS
3.
JT
JT
Our hypotheses and lemmas i n s u r e t h a t e v e r y t h i n g makes sense and u s i n g (2.25) w i t h (2.30)
p l u s a n o t h e r i n t e g r a t i o n by p a r t s t h e r e
r e s u l t s f r o m (2.31) u ( t ) = Z(t,-clu(r) + Y ( t y T ) u t ( T ) +
(2.32)
i:
Y(t,S)f(S)dS
As i n S e c t i o n 1.5 f o r example we a r e making use i n (2.31) and
o f t h e h y p o c o n t i n u i t y o f s e p a r a t e l y c o n t i n u o u s maps E
(2.32)
F
+.
F i s barreled.
G when
Now as T
+.
x
0, b y h y p o c o n t i n u i t y again
we have ( n o t e t h a t t h i s i s s t a t e d b a d l y i n Carroll-Wang [12]) Theorem 2.8 t i n u o u s u/q (2.1)
Assume t h e hypotheses of Theorem 2.6 w i t h conI
+
0 and u /Q
-f
0 as
T
-f
0.
Then t h e s o l u t i o n o f
g i v e n b y Theorem 2.6 i s unique.
I n t h e e v e n t t h a t q ( t ) > 0 f o r 0 5 t 2 T a somewhat s t r o n g e r uniqueness theorem f o r (2.1)
can be p r o v e d ( c f . Carroll-Wang
I
1121).
Assume u ( o ) = u ( 0 ) = f = 0, w i t h u s a t i s f y i n g t h e con-
c l u s i o n s o f Theorem 2.6,
and t h e n we d e f i n e t h e o p e r a t o r i n
L(H) by (2.33)
1
t
L(t,.r)
O p e r a t i n g on (2.1)
= 0 - l exp(-Aaz3
w i t h L(t,S),
T
s(S)d 0 means Z(t,T)u(T)
0
-+
lr21 a dC
O f o r O < t < T < m and (ra(E)dE)6/a(r)’/2
bounded ( 6 < 1/2).
160
Then u i s unique.
DEGENERATE EQUATIONS WITH OPERATOR COEFFICIENTS
3.
Remark 2.10
We emphasize here t h a t a ( - ) i s n o t r e q u i r e d t o
be monotone, i n c o n t r a s t t o P r o t t e r [2] o r Krasnov [l],since by P(t) 2 0 for a l l
(2.111,
x
merely i m p l i e s a
I
2
-cllrl
2
(examples
o f nonmonotone a are given i n Carroll-Wang 1123 and Wang [l;23). The c o n d i t i o n o f P r o t t e r [2]
( c f . a l s o Berezin 111, Bers [ l ] ) ,
phrased here i n t h e form t r ( t ) / a ( t ) ’ / ’
-+
0 as t
-+
0 (with a(-)
monotone), f o r s o l u t i o n s o f a numerical version o f (2.1)
t o be
w e l l posed i n a l o c a l uniform m e t r i c , has i t s analogue here i n t h e c o n d i t i o n s on + ( t ) .
Thus f o r example i f A
tm,A B R ( t ) u = r ( t ) u x = tnux, and A”S(t)
A
-.-32/ax 2 , a ( t )
= s(x,t)
- b ( t ) then i t f o l l o w s t h a t t rT( t ) / a ( t ) ” ‘
1 B(t)
0 if n > m (?n-m+l
-
:1
1 whereas j;lr,‘/a
t2n-ni+l
-.21
d< =
) so t h a t + ( t )
=
with = t
=
n + l ~ -+
2n-mdS = (1/2n-m+l)
k exp (c,t 2n -m+l /Zn-m+l) > 0
2 c o n d i t i o n o f Krasnov [l]involvesmonotone -- 1 a ( t ) = O(tm) and r ( t ) = O ( t 2 0 but y ( t ) ) where y ( t ) * 0 as t
when n >
Tm -
The L
-+
Krasnov i s d e a l i n g w i t h weak solutions.
Now our existence and
uniqueness c o n d i t i o n s are phrased i n terms o f u/q, u-/Q, and f/Q where q2 = +$ and Q
2
= $$a which permits a r a t h e r p r e c i s e com-
parison between t h e behavior o f f, u, and u
I
as t
+
The 9
0.
term seems somewhat curious however since f o r example i f h
Ibl
2 k
1 which i m -
poses growth c o n d i t i o n s on b ( t ) i n order t h a t $ ( t ) > 0. seems t o i n d i c a t e t h a t t h e r o l e o f the
i)
This
term should be i n v e s t i -
gated f u r t h e r and some comnents on t h i s a r e made i n Remark 2.12.
161
SINGULAR AND DEGENERATE CAUCHY PROBLEMS
Walker, i n as y e t unpublished work, i n v e s t i g a t e s b y s p e c t r a l methods ( u s i n g Riesz o p e r a t o r s Rk d e f i n e d by a/axk = i ( - A ) 1/ 2
Rk)
t h e q u e s t i o n o f d i r e c t i o n a l o s c i l l a t i o n s and w e l l posedness ( c f . a l s o Walker 11; 2; 31) I n Wang [2] i t i s p o i n t e d o u t t h a t i f a ( - )
Remark 2.11
i s monotone near t = 0 ( l o c a l l y monotone) then (2.35)
can be
*
r e p l a c e d by I l u l ( t ) l I 5 c 1 0 t 1 / Z a ( t ) 1 / 2 f o r small t so t h a t i n (2.36)
a1/2(T)Y(t,.r)-l/2 with (TI a T h i s produces a somewhat s t r o n g e r
one c o u l d w r i t e Y(t,T)ul(T)
ul(s)/a1/2(T)
+.
0 as
T
+.
0.
=
v e r s i o n o f Theorem 2.9 when a ( - ) i s l o c a l l y monotone. a r e given i n Wang [2] t o show t h a t A ( t ) =
Examples
a ( S ) d S / a ( t ) may n o t
even be bounded f o r nonmonotone a ( t ) = O ( t m ) . Remark 2.12 (2.9)
F i r s t we assume z1 and b ( t ) a r e r e a l so t h a t
becomes, a f t e r e l i m i n a t i o n o f t h e A" term as before, and
adding a term clX
I
Now assume I b ( t ) / b ( t ) l
5 c 3 on [o,T]
162
h
and s e t c = max(cl,c3);
3. DEGENERATE EQUATIONSWITH OPERATOR COEFFICIENTS
then, i f zlb(S)
?
0 one can w r i t e (for 2f3
A
A
S e t t i n g E(t, 0, and suppose we are g i v e n a f u n c t i o n f : I ,x I
.
A s o l u t i o n o f (3.1)
i s a f u n c t i o n u : 1,
y ) a b s o l u t e l y continuous, d i f f e r e n t i a b e a.e.
166
-f
V which on I,,
DEGENERATE EQUATIONS WITH OPERATOR COEFFICIENTS
3.
has i t s range i n B ( u ) , and s a t i s f i e s (3.1) b o
a.e.
on I,.
f i c i e n t c o n d i t i o n s f o r t h e Cauchy Problem f o r (3.1)
Suf-
t o be w e l l -
posed a r e g i v e n i n t h e f o l l o w i n g . Theorem 3.1 k ( - ) : 1,
+
Assume t h e r e i s a p a i r o f measurable f u n c t i o n s 1 ( o , m ) and Q ( - ) : Ia [ l , m ) w i t h Q ( * ) / k ( - ) E L ( I a , R ) -f
such t h a t
Then any two s o l u t i o n s u 1 ( * ) , u 2 ( - ) o f (3.1) w i l l s a t i s f y t h e estimate
I n p a r t i c u l a r , t h e Cauchy problem o f (3.1) w i t h u ( o ) = uo has
a t most one s o l u t i o n . Proof: M(t) : V
+
Since k ( t ) > 0, t h e c o e r c i v e e s t i m a t e (3.2) i m p l i e s V I i s a b i j e c t i o n and
11
M(t)-’
Thus, we o b t a i n from (3.1) llu;(t)-u;(t)
11
f(t,ul(t))-f(t,u2(t))
t
E
11
V
I
IIL(,,lYv)
/Iv
S k(t1-l.
5 k(t1-l
5 k ( t ) - l Q ( t ) ~ ~ U l ( t ) - u 2 ( t ) ~ ~,, ,a.e.
I,. Since u l ( * )
-
u
2
(
0
)
i s absolutely continuous w i t h a 167
SINGULAR AND DEGENERATE CAUCHY PROBLEMS
I
summable d e r i v a t i v e , we have Ilul(t)-u2(t)llV 5 t l ~ / U ~ ( O ) - U ~+ ( O I() ul(s)-u;(S)IIVds. ~~~ Therefore t h e bounded function Z ( t ) Z(o)
+
0
= Ilul(t)-u2(t)llv
Itk(s)-’g(s)Z(s)dsy
t
s a t i s f i e s the i n e q u a l i t y Z ( t ) 5 E
Ia. The e s t i m a t e (3.4)
follows
0
i n m e d i a t e l y by t h e Gronwall i n e q u a l i t y .
I n a d d i t i o n t o t h e hypotheses o f Theorem 3.1
Theorem 3.2 assume t h a t t and t h a t t
+
+.
QED
<M(t)x,y>
i s measurable f o r each p a i r x, y
i s measurable f o r x
Bb(u0) and y
E
F i n a l l y , l e t bo > 0 and c > 0 s a t i s f y Ilf(t,uo)II
V’
rC
t
E
Icy and
k ( t ) - ’ Q ( t ) d t 5 b/(bo+b).
E
V
E
V.
,
5 Q ( t ) b o y a.e.
Then t h e r e e x i s t s a
0
which s a t i s f i e s u ( o ) = uo. (unique) s o l u t i o n u ( * ) of (3.1) on Ic Proof:
We f i r s t show t h a t f o r every f u n c t i o n u : Ic +
Bb(uo) which i s s t r o n g l y (= weakly) measurable i n V, t h e f u n c t i o n t
+
Ll(t)-l
0
f(t,u(t))
: Ic V i s measurable.
we have < I $ , M ( t ) - l f ( t , u ( t ) ) >
For each I$
-+
= .
E
VI
The i n d i -
i s , and so i t s u f f i c e s
t o show t h e m e a s u r a b i l i t y o f each f a c t o r .
To show f ( t , u ( t ) )
i s measurable we consider f i r s t t h e case where u ( * ) i s countablywhere { G : j 2 1) i s a valued; thus u . ( t ) = x . f o r t E G J J j’ j measurable p a r t i t i o n o f I,. L e t t i n g @ . denote t h e c h a r a c t e r i s t i c J function o f G we o b t a i n f ( t , u ( t ) ) = l { f ( t , x . ) I $ . ( t ) : j f 1) jy J J on I and t h i s i s c l e a r l y measurable. The case o f general C’
u ( * ) f o l l o w s from t h e s t r o n g c o n t i n u i t y o f x
+
P(t,x),
s i n c e any
measurable u i s t h e s t r o n g l i r r i t o f countably-valued rlieasurable
168
3. DEGENERATE EQUATIONS WITH OPERATOR COEFFICIENTS
functions.
To show t h e second f a c t o r i s measurable, l e t m 2 1
and c o n s i d e r t h e r e s t r i c t i o n o f M : 1,
+
L(V,V’)
t o t h e s e t Jm o f
Since M i s measurable on t h i s s e t ,
t h o s e t €IC w i t h k ( t ) 2 l/m.
t h e r e i s a sequence o f c o u n t a b l y - v a l u e d f u n c t i o n s Mk : Jm M(J,)
C L(V,V’)
such t h a t
Lz
-+
Mk(t) = M(t) strongly f o r t V
I
Jm.
E
we have
on o f 14-l
to
desired re-
The p r o o f o f Theorem 3.2 now f o l l o w s s t a n d a r d arguments. L e t X be t h e s e t o f u
E
C(Ic,V)
w i t h range i n Bb(uo).
the function t * M ( t ) - l f ( t , u ( t ) ) and i t s a t i s f i e s t h e e s t i m a t e
For u
E
i s measurable Ic + V ( f r o m above)
IIM(t)-’
0
2 k(t1-l
f(t,u(t))lIv
-
Q ( t ) (bo+b) on I,. Hence, we can d e f i n e F : X + X by [ F u ] ( t ) t M(s)-lf(s,u(s))ds, t E I,, and i t s a t i s f i e s t h e e s t i m a t e uo + 0
(3.5)
11 [ F u l ( t ) - [ F v l ( t ) l l v
X
I
=
t
5
0
k ( s ) - l Q ( ~ ) l ~ U ( S ) - V ( S ) I ~ ~ds,
u,v
E
x,
t
E
B u t any map F o f a c l o s e d and bounded subset X o f Lm(Ic,V)
I,. into
i t s e l f which s a t i s f i e s (3.5)
i s known t o have a unique f i x e d -
p o i n t , u ( c f . C a r r o l l [14]).
This fixed-point i s c l e a r l y the
OED
d e s i r e d s o l u t i on.
A s o l u t i o n e x i s t s on t h e e n t i r e i n t e r v a l 1, i n c e r t a i n 169
SINGULAR AND DEGENERATE CAUCHY PROBLEMS
Two such s i t u a t i o n s a r e g i v e n below.
situations.
Theorem 3.3 w i t h Bb(uo) = V.
Assume a l l t h e hypotheses o f Theorem 3.1 h o l d Also, suppose t
+
<M(t)x,y>
a r e measurable Ia + R f o r e v e r y p a i r x,y bo > 0 be g i v e n and s a t i s f y I l f ( t , u o ) l l V I
functions u I,.
If u
E
Set g ( t ) = bo exp{ E
C(Ia,V)
rt
5 Q ( t ) b o a.e.
f o r which ]lu(t)-udl,
E
V,
on I,.
and l e t X be t h o s e
5 g(t)
X, t h e e s t i m a t e I l ~ l ( s ) - l f ( s , u ( s ) ) I I V
k(s)-’Q(s)(llu(s)-udlv
on I, w i t h u ( o ) = uo.
k(s)-’Q(s)dsI
JO
+
V, and l e t uo
E
Then t h e r e e x i s t s a unique s o l u t i o n o f (3.1) Proof:
and t
-
bo f o r t
E
5
+ bo) shows t h a t Fu d e f i n e d as above belongs
t o C(Ia,V) and f u r t h e r m o r e s a t i s f i e s 11 [ F u ] ( t ) - u d l v 5 t k ( s ) - l Q ( s ) g ( s ) d s = g ( t ) - b o , so Fu E X. Thus F has a unique 0
fixed-point u
E
X which i s t h e s o l u t i o n o f t h e Cauchy problem
f o r (3.1).
QED
Remark 3.4
The p r e c e d i n g r e s u l t s p e r m i t t h e l e a d i n g opera-
That i s , t o r s t o degenerate i n a v e r y weak sense a t any to E I a’ 1 (3.2) must be m a i n t a i n e d w i t h k ( * ) - ’ E L ( I a , R ) On t h e o t h e r
.
hand we have p l a c e d no upper bounds on t h e f a m i l y { M ( t ) t h e y may be s i n g u l a r ( c f .
I n t r o d u c t i o n ) on a s u i t a b l e
: t
E
Ial;
s e t of
p o i n t s i n Ia. The L i p s c h i t z c o n d i t i o n (3.3) f(t,uo)
t o g e t h e r w i t h t h e e s t i m a t e on
i n Theorem 3.3 impose a g r o w t h r a t e on f ( t , u )
l i n e a r i n u.
which i s
Such hypotheses a r e f r e q u e n t l y a p p r o p r i a t e , espec-
i a l l y i n l i n e a r problems such as Example 1.1 which o c c u r
170
DEGENERATE EQUATIONS WITH OPERATOR COEFFICIENTS
3.
However they a r e not appropriate f o r
frequently in practice.
nonlinear wave propagation models, and these applications c a l l f o r solutions global in time. The t h i r d order nonlinear equation
Example 3.5 (3.6)
Ut
-
+ ux + uux = 0
Uxxt
provides a model f o r the propagation of long waves of sma 1 amplitude.
The interest here i s i n solutions of (3.6) f o r a l l t 2 0.
Similarly, models with other forms of n.on1inearity i n add tion t o dispersion and dissipation occur i n the form (3.7)
u
t
-
u
xxt
l u l q + buPux = cu
+ a sgn(u)
xx'
W e r e f e r t o Benjamin [ l , 2 3 , Dona [2,3, 4, 51, and Showalter [ l o , 19, 211 f o r discussion of such models and r e l a t e d systems. I n i t i a l boundary value problems f o r (3.6) and (3.7) can be resolved in the following a b s t r a c t forni. Theorem 3.6
Let a > 0 and assume given f o r each t
[O,a] an operator M ( t ) Banach space.
E
L(V,V
I
+
1,
Assume there i s a k > 0 such t h a t
E
V the function
E
I,,
<M(t)x,y> i s absolutely continuous w i t h
(3.9)
d dt <M(t)x,x>
=
), where V i s a separable reflexive
each M(t) i s symmetric, and f o r each p a i r x,y
t
E
5 m(t)llxll
2
,
171
a.e. t
x
E
V,
SINGULAR AND DEGENERATE CAUCHY PROBLEMS
f o r some m ( - )
E
1 L (Ia,R).
f o r each p a i r x,y
E
L e t f: 1, x V
V the function t
-+
-+
v~
Ilf(t,x)-f(t,y))I
f o r x,y
B and a.e.
E
t
I
be g i v e n such t h a t
and f o r each bounded s e t B i n V t h e r e i s a that
V
a(-)
E
i s measurable, 1 L ( I a y R ) such
5 Q ( t ) Ilx-Alv, and I l f ( t , x ) I I I,.
E
5 O(t),
F i n a l l y , assume t h e r e e x i s t K ( - )
and L ( * )
E
1 L (Ia) with L ( t )
2 0 a.e. such t h a t < f ( t , x ) , x > 5
K(t)/Ixlf
+
L(t)llXll, x
Then f o r each uo
unique s o l u t i o n u : 1, Proof:
E
V.
+
V there e x i s t s a
E
V o f ( 3 1 ) w i t h u ( o ) = uo.
Uniqueness and l o c a
from Theorem 3.1 and Theorem 3.2,
e x i s t e n c e f o l l o w immediately respectively.
The e x i s t e n c e
o f a ( g l o b a l ) s o l u t i o n on 1, can be o b t a i n e d by standard c o n t i n u a t i o n arguments i f we can e s t a b l i s h an a p r i o r i bound on a s o l u tion.
But i n t h e present s i t u a t i o n t h e a b s o l u t e l y continuous
function a ( t )
I
5
Z C2L(t)/k’/2]
< M ( t ) u ( t ) , u ( t ) > s a t i s f i e s ( c f . Lemma 5.1) a ( t ) =
+ <Ml(t)u(t),u(t)> 5 [(ZK(t)+m(t))/k] a ( t ) +
o(t)”*;
hence we o b t a i n
’0
with H ( t )
=
’0
(K+(t) + (1/2)m(t))/k.
The estimates (3.10) and
(3.8) p r o v i d e t h e d e s i r e d bound on a l l s o l u t i o n s o f (3.1) w i t h given i n i t i a l data uo. Remark 3.7
on 1, OED
Equations s i m i l a r i n form t o (3.G) have been
used t o “ r e g u l a r i z e ” h i g h e r o r d e r equations as a f i r s t s t e p i n s o l v i n g them ( c f . Bona [4],
Showalter [17]).
172
The advantage over
3. DEGENERATE EQUATIONS WITH OPERATOR COEFFICIENTS
standard (e.g.
, parabolic)
r e g u l a r i z a t i o n s i s t h a t the order o f
t h e problem i s n o t increased by t h e r e g u l a r i z a t i o n . case o f t h i s i s t h e "Yosida
An i m p o r t a n t
approximation" AE o f a maximal ac-
c r e t i v e ( l i n e a r ) A i n Banach space g i v e n by AE = ( I + E A ) - ~ A , E > I
Thereby, one approximates t h e e q u a t i o n u ( t ) + A u ( t ) = 0 by
0.
one w i t h A r e p l a c e d by t h e bbunded A E y or, e q u i v a l e n t l y , solves I
t h e equation (I+EA)uE(t) + AuE(t) = 0 and then l o o k s f o r u(t) =
1i m
E4+
T h i s technique was used f o r a
u E ( t ) i n some sense.
n o n l i n e a r time-dependent e q u a t i o n by Kato [2] and t o study t h e nonwell-posed backward Cauchy problem i n Showalter [ l G ,
201;
a l s o see B r e z i s [3]. The preceding r e s u l t s immediately y i e l d corresponding r e s u l t s f o r second o r d e r e v o l u t i o n equations o f t h e form (3.11)
M(t)u
I 1
( t ) = F(t,u(t),ul(t)) L e t t h e f a m i l y o f operators ( M ( t ) > and t h e
Theorem 3.8
L e t uoyul
f u n c t i o n s k ( - ) and Q(*) be g i v e n as i n Theorem 3.2. V and t h e f u n c t i o n F : 1, x Bb(uo)
t V.
1)
+
measurable f o r x1
Suppose we have IIF(t,x)-F(t,y)ll F(t,uoyul)l~v,
f o r x = (xl,x2)
x
VI
Bb(ul)
+
V
I
be given w i t h
Bb(ul). y E 2 2 5 Q(t)[llxl-YIIlv + IIx2-YdIv1~
E
Bb(u0), x2
E
E
5 c l ( t ) , and
a n d y = (y1,y2) i n B ( u ) x Bb(ul) b o
and a.e.
t
E
Then t h e r e e x i s t s c > 0 and a unique u : Ic + V which i s Ia. 173
SINGULAR AND DEGENERATE CAUCHY PROBLEMS
continuously d i f f e r e n t i a b l e w i t h u ( t ) u
I
I
E
Bb(uo). u ( t )
E
Bb(ul),
,
i s ( s t r o n g l y ) a b s o l u t e l y continuous and d i f f e r e n t i a b l e a.e.
(3.11)
I
holds a.e.
on Ic and u ( o ) = u o y u ( 0 ) = ul.
The preceding f o l l o w s from Theorem 3.2 a p p l i e d t o t h e equation M(t)U'(t) = f(t,U(t))
in V z
[M(t)xlyM(t)x2]
5
and f ( t , x )
V x V w i t h M(t)x
[M(t)x2,F(t,xl
!
The e s t i m a t e
,x,)].
l i m i t s t h e growth r a t e o f t h e l e a d i n g o p e r a t o r ( c f . Re-
(3.12)
T h i s hypothesis can be d e l e t e d when V i s a H i l b e r t
mark 3.4).
space; we need o n l y r e p l a c e M ( t ) by t h e Riesz map V
-+
V ' i n the
f i r s t f a c t o r o f each o f M ( t ) and f. I n i t i a l boundary value problems f o r p a r t i a l d i f f e r e n t i a l equations a r i s e i n t h e form (3.11)
i n various applications.
Tww
c l a s s i c a l cases a r e g i v e n below; see L i g h t h i l l [l]o r Whitham
[l]f o r a d d i t i o n a l examples o f t h i s type. The equation o f S. Sobolev [l]
Example 3.9 (3.13)
+ uzz = 0
A3utt
describes t h e f l u i d motion i n a r o t a t i n g vessel.
I t i s on ac-
count o f t h i s equation t h a t t h e term "Sobolev equation" i s used i n the Russian l i t e r a t u r e t o r e f e r t o any equation w i t h s p a t i a l d e r i v a t i v e s on t h e h i g h e s t o r d e r t i m e d e r i v a t i v e (cf., Example 1 .2). Example 3.10 (3.14)
u
tt
-
The equation
aA3utt
-
A3u = f ( x , t )
174
however,
DEGENERATE EQUATIONS WITH OPERATOR COEFFICIENTS
3.
was i n t r o d u c e d by A.E.H. t i o n s i n a beam.
Love [l]t o d e s c r i b e t r a n s v e r s e v i b r a -
The second t e r m i n (3.14)
represents r a d i a l
i n e r t i a i n t h e model. F i n a l l y , we d e s c r i b e how c e r t a i n d o u b l y - n o n l i n e a r e v o l u t i o n equations (3.15)
M(t,u’(t))
=
f(t,u(t)),
0 5 t < _ T ,
i n H i l b e r t space can be s o l v e d d i r e c t l y b y s t a n d a r d r e s u l t s on monotone n o n l i n e a r o p e r a t o r s .
The t e c h n i q u e a p p l i e s as w e l l t o
t h e corresponding v a r i a t i o n a l i n e q u a l i t y I
<M(t,u ( t ) )
(3.16)
-
f(t,u(t)),
v
-
I
u (t).
2 0,
v
E
K(t),
w i t h a p r e s c r bed fami y o f c l o s e d convex subsets K ( t ) c V, so we c o n s i d e r t h i s more general s i t u a t i o n . L e t V be r e a l H i l b e r t space and f o r each a 2 0 we l e t Va be t h e H i l b e r t space o f square summable f u n c t i o n s from [O,T] i n t o T V w i t h t h e norm I l v l l a = I l v ( t ) l l i e - 2 a t dt)”2. F o r each t E
(1
[O,T]
0
we a r e g i v e n a p a i r o f ( p o s s i b l y n o n l i n e a r ) f u n c t i o n s f ( t , * ) ,
M(t,*) from V i n t o V
I
.
The f o l l o w i n g elementary r e s u l t i s funda-
mental f o r t h i s technique. Lemma 3.11 uous M : Va
Assume [Mv](t)
E M(t,v(t))
I
+
d e f i n e s a hemicontin-
V a and t h a t t h e r e i s a k > 0 w i t h
175
SINGULAR AND DEGENERATE CAUCHY PROBLEMS
(3.17) L e t uo Va
+
V
2 2 kllx-yllv,
<M(t,x)-M(t,y),x-y> E I
a
=
V and assume [ f ( v ) ] ( t )
f(t,uo+
a.e.
t
E
[O,T],
x,y
E
V.
v(s)ds) defines f :
and t h a t t h e r e i s a Q > 0 w i t h
(3.18)
11 f ( t , x ) - f ( t > Y ) l l
Then t h e o p e r a t o r T
=M-
f : Va
a.e.
t
E
[O,T],
x,y
E
V.
I
-+
Va i s hemicontinuous and
s t r o n g l y monotone f o r "a" s u f f i c i e n t l y large. Proof:
For u,v
E
Va we have
The c o n d i t i o n o f u n i f o r m s t r o n g m o n o t o n i c i t y on M ( t , - ) <Mu-Mv,u-v>
2 c IIu-vII,
2
, u,v
E
Va,
gives
2
hence we obtai'n
E c - Q ( T / ~ ~ ) ~ / ~ I ~u,v ~ u E- vV a' ~ ~ ~The , d e s i r e d r e s u l t holds f o r a >
QED
TQ~/ZC~.
176
3. DEGENERATE EQUATIONS WITH OPERATOR COEFFICIENTS
Theorem 3.12
L e t t h e hypotheses o f Lemma 3.11 h o l d and as-
sume i n a d d i t i o n t h a t we a r e g i v e n a f a m i l y o f nonempty c l o s e d convex subsets K ( t ) o f V f o r w h i c h t h e c o r r e s p o n d i n g p r o j e c t i o n s P(t) : V
+
K ( t ) a r e a measurable f a m i l y i n L(V) ( c f . C a r r o l l
Then t h e r e e x i s t s a unique a b s o l u t e l y c o n t i n u o u s u [14]). I 2 L (0,T;V) w i t h u E Vo, u ( o ) = uo, and s a t i s f y i n g (3.16). Proof: Va * V {v
E
I
a
W i t h a chosen as i n Lemma 3.11
E
K ( t ) , a.e.
and nonempty i n Va,
t
E
[O,T]).
(3.19)
Define K 5
Then K i s closed, convex
so i t f o l l o w s f r o m Browder [5]
t h a t t h e r e e x i s t s a unique w
E
2 0,
Vo =
operator T :
i s hemicontinuous and s t r o n g l y monotone.
Va : v ( t )
[14])
, the
E
(cf. Carroll
K such t h a t v
E
K.
F i n a l l y , t h e measurable f a m i l y o f p r o j e c t i o n s ( P ( t ) > i s used t o show t h a t (3.19)
i s e q u i v a l e n t t o (3.16) w i t h u ( t ) = uo
rt
J o w ( s ) d s , 0 5 t 5 T.
i.
QED
The p r e c e d i n g r e s u l t i s f a r from b e i n g b e s t p o s s i b l e , b u t s e r v e s t o i l l u s t r a t e t h e a p p l i c a b i l i t y o f t h e t h e o r y o f monotone nonlinear operators t o t h e situation. t h e doubly-nonl i n e a r e q u a t i o n
and more genera l y , t h e
neq ua it y
177
A s i m i l a r r e s u l t holds f o r
SINGULAR AND DEGENERATE CAUCHY PROBLEMS
(3.21)
-
<M(t,u(t)) u(t)
E
'0
K(t),
-
rt
1, f(S,u(S))dS,
v-u(t)> a.e.
2 0, v
M(t,-)
K(t),
t E COYTI,
under e s s e n t i a l l y t h e same hypotheses as Theorem 3.12. be d e s i r a b l e t o have such r e s u l t s on e i t h e r (3.16)
E
I t would
o r (3.21) w i t h
p e r m i t t e d t o be degenerate, o r a t l e a s t n o t u n i f o r m l y
bounded ( c f . Remark 3.4).
We s h a l l r e t u r n t o s i m i l a r problems
i n Banach space w i t h a s i n g l e l i n e a r o p e r a t o r ( p o s s i b l y degenera t e ) i n S e c t i o n 3.6 below. ng Theorem 3.12)
A l s o , a v a r i a t i o n on Theorem 3.8
g i v e s e x i s t e n c e r e s u l t s f o r second o r d e r
u t i o n e q u a t i o n s and i n e q u a l i t i e s . Remark 3.13
Theorems 3.1,
3.2 and 3.3 a r e f r o m S h o w a l t e r
and Theorem 3.4 i s f r o m S h o w a l t e r [18,
191.
Theorem 3.8 was
u n p u b l i s h e d and Theorem 3.12 i s due t o Kluge and Bruckner [l]. F o r a d d i t i o n a l m a t e r i a l on s p e c i f i c 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 s and on general e v o l u t i o n e q u a t i o n s o f t h e t y p e c o n s i d e r e d i n t h i s s e c t i o n we c i t e t h e above r e f e r e n c e s and Amos [l],Bardos-BrezisB r e z i s [Z],
Bhatnager [l],Bochner-von Nuemann [l],Bona [l],
B r i l l [l],Brown [l],C a l v e r t [l],Coleman-Duffin-Plizel
[23,
C o l t o n [l,2, 31, Davis [l,23, Derguzov [l],Dunninger-Levine 111, E s k i n [l],Ewing [l,2, 31, W. H. F o r d [l,21, Fox [31, Gajewski-Zacharius [1,
2, 31, Galpern [l,2, 3, 4, 51, Grabmuller
[l],tiorgan-Wheeler [l],I l i n [l],Kostyuzenko-Eskin [l],Lagnese
[l,2, 3, 4, 5, 6, 7, 8, 91, Lebedev [l],L e v i n e [3, 5, 6, 7, 8,
91,
Lezhnev [l],L i o n s [5,
9,
lo], 178
Maslennikova [l,2, 3, 4, 5,
3. DEGENERATE EQUATIONS WITH OPERATOR COEFFICIENTS
61, tkdeiros-Menzala [l],M i k h l i n [Z],
Fliranda [l],Neves [l],
Prokopenko [l],Rao [l,2, 4, 51, Rundell [l,2, 31, Selezneva
[l],Showalter [l,7, 8, 9, 11, 12, 13, 141, S i g i l l i t o [l,2, 31, Stecher-Rundell [l],T i n g [l,31, Ton [l],Wahlbin [l],Zalenyak One c o u l d a l s o i n c l u d e c e r t a i n r e f e r e n c e s f r o m
[l,2, 3, 41. Remarks 4.10,
5.18,
6.24 and 7.10 t o which we r e f e r f o r a d d i t i o n -
a l material. 3.4
Weakly r e g u l a r e q u a t i o n s
We r e s t r i c t o u r a t t e n t i o n
t o a c l a s s o f l i n e a r e v o l u t i o n e q u a t i o n s o f t h e form Mul(t) t L u ( t ) = f ( t ) ,
(4.1)
and o b t a i n we1 1-posedness r e s u l t s when as s t r o n g as L.
t > 0,
M i s not (necessarily)
A l t h o u g h t h e semigroup techniques employed h e r e
w i l l be used f o r n o n l i n e a r degenerate problems i n S e c t i o n 3.6 below, we i n t r o d u c e them i n t h e s i m p l e r s i t u a t i o n o f (4.1) where we s h a l l be a b l e t o d i s t i n g u i s h t h e a n a l y t i c s i t u a t i o n from t h e s t r o n g 1y-continuous case. L e t Vm be a complex H i l b e r t space w i t h s c a l a r - p r o d u c t (-,*), and d e f i n e t h e isomorphism ( o f R i e s z ) f r o m V , o n t o i t s a n t i d u a l V,
I
x,y
of c o n j u g a t e - l i n e a r continuous f u n c t i o n a l s by <Mx,y> E
I
-f
(x,y),,
Suppose D i s a subspace o f Vm and t h e l i n e a r map
. ,V
I f u E V and f o m c o n s i d e r t h e problem o f f i n d i n g a u ( * )
L :D
=
Vm i s given.
E
C((o,m),Vm)
E
C([o,m),
I
a r e g i v e n , we 1 Vm) f ) C ((o,m),
Vm) which s a t i s f i e s (4.1) and u ( o ) = uo.
To o b t a i n an elementary uniqueness r e s u l t , l e t u ( * ) be a 179
SINGULAR AND DEGENERATE CAUCHY PROBEMS
s o l u t i o n o f (4.1)
with f
-2Re,
t > 0.
( u ( t ) , u(t))A’2 tained.
0 and n o t e t h a t D,(u(t),u(t)),,,
=
Hence, if L i s monotone then I l u ( t ) I l m =
i s nonincreasing and a uniqueness r e s u l t i s ob-
Recall t h a t L monotone means Re
Z
0, x
E
D.
This
computation suggests t h a t Vm i s an a p p r o p r i a t e space i n which t o The equation i n V
seek well-posedness r e s u l t s .
-1 u l ( t ) + M oLu(t) = M-lf(t),
(4.2)
lil
t > O
i s e q u i v a l e n t t o (4.1) and suggests c o n s i d e r a t i o n o f t h e o p e r a t o r -1 A = M OL w i t h domain D ( A ) = D. We see from (4.3)
(Ax.Y),
,
=
x
E
D, y
E
Vm
t h a t L i s monotone i f and o n l y i f A i s a c c r e t i v e i n t h e t l i l b e r t space Vm.
I n t h i s case, -A generates a c o n t r a c t i o n semigroup
on Vm i f I + A i s s u r j e c t i v e ; then t h e Cauchy problem i s w e l l posed f o r (4.2). C a r r o l l [14],
These o b s e r v a t i o n s prove t h e f o l l o w i n g ( c f .
H i l l e [2],
Theorem 4.1 H i l b e r t space Vm,
o r Kato [l]).
L e t M : Vn, L : D
-t
V,
I
* Vm be t h e Riesz map o f t h e complex
I
be g i v e n monotone and l i n e a r w i t h I
domain i n V m y and assume M + L : D each f
E
C([O,m),V,)
I
and uo
E
+
Vm i s s u r j e c t i v e .
Then f o r
D t h e r e i s a unique s o l u t i o n u ( - )
o f (4.1) w i t h u(o) = uo. The Cauchy problem f o r (4.1)
i s solved above by a s t r o n g l y
continuous semigroup o f c o n t r a c t i o n s .
T h i s semigroup i s a n a l y t i c
when t h e o p e r a t o r A i s s e c t o r i a l (Kato [1]) 180
and we describe t h i s
3. DEGENERATE EQUATIONS WITH OPERATOR COEFFICIENTS
s i t u a t i o n i n the following. Theorem 4.2
Suppose M i s t h e Riesz map o f t h e H i l b e r t space
Vm, V i s a H i l b e r t space dense and continuous i n Vm, and L : V V ' i s continuous, l i n e a r and coercive: f o r some c > 0. and uo
E
V,
Re L cllxIl;,
x
Then f o r e v e r y Hb'lder continuous f: O[).,
-+
-+
V,
E
V,
I
w i t h u(o) =
t h e r e i s aunique, s o l u t i o n u ( = ) o f (4.1)
uO*
Each o f t h e two preceding r e s u l t s i m p l i e s well-posedness of t h e Cauchy problem f o r a l i n e a r second-order e v o l u t i o n equation (4.4)
Cu
I 1
I
( t ) + Bu ( t ) + A u ( t ) = f ( t ) ,
The i d h a i s t o w r i t e (4.4)
Theorem 4.3
as a f i r s t - o r d e r system i n t h e form
L e t Va and Vc be complex H i l b e r t spaces w i t h
Va dense and continuous i n Vc,
Vc
and denote by A : Va
I
-f
' 0.
t
I
Va and C :
-+
L e t B be l i n e a r and monotone
Vc t h e r e s p e c t i v e Riesz maps. I
from D(B) C Va i n t o V a and assume A + B + C : D(B) -+ V i i s sur1 j e c t i v e . Then f o r each f E C ( C O , ~ ) , V L ) and p a i r uo E Va, u1 D(B) w i t h Auo + Bul c([o,m),va)
E
Vc t h e r e i s a unique s o l u t i o n u ( * )
n c 1((oym),va) n
cl([o,m),vc)
n c2((o,m),vc)
E
of (4.4)
I
w i t h u ( o ) ) = u o and u ( 0 ) = ul. Proof:
E
I
Define Vm = Va x Vc;
then we have V,
181
I
I
I
= Va x Vc and
SINGULAR AND DEGENERATE CAUCHY PROBLEMS I
t h e Riesz map M : Vm [xl,x2]
E
I
I
[xl,x2]
I
+
Vm by L([xl ,x2])
I n o r d e r t o a p p l y Theorem 4.1 we need o n l y t o v e r i f y t h a t
L i s monotone.
E
Va and C : Vc
fly x2
E E
I
Va and f2 E Vc, D(B),
then s e t x1 = x2 + A - ’ f l
D w i t h (M+L)[xl ,x2] = [fl ,f2].
I
-+
then we can
QE D
L e t t h e H i l b e r t spaces and operators A: Va
Theorem 4.4 I
-
so B monotone i m p l i e s
I
F i n a l l y , i f fl
t o o b t a i n [x, ,x2]
But an easy computation
[x, ,x2]> = Re
s o l v e (A+B+C)x2 = f2
L e t B : Va
Vc be g i v e n as i n Theorem 4.3.
be continuous and l i n e a r w i t h B + AC : Va
E
=
I
shows Re 0.
I
E
With M and L so defined, t h e
I).
E
= [Axl,Cx2],
V a x D(B) : Axl + Bx2
E
and d e f i n e L : D
+ Bx2],
system (4.5) t > 0.
Vm i s g i v e n by M[xl,x2]
L e t D E {[x,,x,]
Vm.
r e c a l l i n g Vc C Va, [-Ax2,Axl
+
u1
E
Vc,
n c2 ((O,m),Vc)
Va
Va c o e r c i v e f o r some I
+
t h e r e i s a unique s o l u t i o n u ( = )
c 1 ((0,m),Va) f l c 1 “O,m),Vc)
I
+
I
+
Then f o r e v e r y H6lder continuous f : [O,m)
v a’
+
of (4.4)
E
Vc and p a i r
C([O,m),Va)n w i t h u(0) =
I
uo and u ( 0 ) = ul. Proof:
F o l l o w i n g t h e p r o o f o f Theorem 4.3,
can a p p l y Theorem 4.2 w i t h V i s V - e l l i p t i c f o r some X > 0.
=
we f i n d t h a t we
Va x Va i f we v e r i f y t h a t AM + L But AB
p r e c i s e l y what i s needed.
+ C being V a - e l l i p t i c i s QED
B r i e f l y we i n d i c a t e some examples o f p a r t i a l d i f f e r e n t i a l equations f o r which corresponding i n i t i a l - b o u n d a r y value problems 182
3.
DEGENERATE EQUATIONS WITH OPERATOR COEFFICIENTS
a r e s o l v e d by t h e preceding r e s u l t s .
The examples a r e f a r from
b e s t p o s s i b l e i n any sense and a r e d i s p l a y e d here o n l y t o suggest t h e types o f equations t o which t h e r e s u l t s apply. Let m(*)
Example 4.5
Lm(O,l) w i t h m(x) > 0 f o r a.e.
E
and d e f i n e Vm t o be t h e completion o f C;(O,l)
(0,l)
s c a l a r - p r o d u c t (u,v) s e t
=
rl
I
m
m(x)u(x)mdx.
E
0
u.(x)vl(x)dx,
u,v
V.
E
x
E
with the
L e t V = HA(0,l) and
Then Theorem 4.2 shows
JO
t h a t t h e i n i t i a l -boundary va 1ue problem m(x)Dtu(x,t) u(0,t)
=
-
u(1,t)
Dxu(x,t) z =
=
t > 0,
f(x,t),
0,
U(X,O) = uo(x),
O < X < l ,
i s well-posed f o r a p p r o p r i a t e f(-,.) and measurable uo w i t h m1/2uo
E
2 L (0,l).
The i n i t i a l c o n d i t i o n above means
lim"m(x) l u ( x , t ) - u o ( x )
ta+
Jo
I 2 dx
= 0 and t h e equation holds a t each
I
t > 0 i n t h e space Vm o f measurable f u n c t i o n s v on ( 0 , l ) w i t h
m-1'2v
E
2 L (0,l).
I 2 Note t h a t Vm C L ( 0 , l ) C V m and t h e equation
may be e l l i p t i c on a n u l l s e t (where m(x) = 0). 1 L e t Vm = Ho(O,l) w i t h t h e scalar-product
-5--J o l u ( x ) W + m u l ( x ) v l ( x ) ) d x where Exam l e 4.6
(u,~),,,
=
m > 0.
(More general
3 Set Lu = Dxu on D E c o e f f i c i e n t s c o u l d be added as above.) 2 1 {u E H (0,l)fl Ho(O,l) : c u ' ( 0 ) = ~ ' ( 1 ) )where c i s given w i t h IcI
5
1.
Then 2 Re = J uI ( o ) l
-
lu1(1)I2
2 0,u
E
D, s o
L i s monotone; we can show M + L i s s u r j e c t i v e , so Theorem 4.1 shows t h e problem
183
SINGULAR AND DEGENERATE CAUCHY PROBLEMS
+ D,u(x,t) 3
(Dt-mDxDt)u(x,t) 2 u(0,t)
= u(1,t)
u(x,o)
= U,(X)
= 0,
c
0 < x < 1,
= f(x,t),
ux(O,t)
=
i s well-posed f o r a p p r o p r i a t e f ( * , * ) and u
0'
t
ux(l,t),
2
0,
T h i s equation i s a
1 i n e a r r e g u l a r i z e d Korteweg-deVries equation ( c f . Bona [4]).
2 Take Vm as above and l e t V = Ho(O,l)
Example 4.7 D:
: V
-+
1;
V ' d e f i n e d by =
2 Dxu(x)Dxv(x)dx, u,v
with E
L
=
V.
Then Theorem 4.2 g i v e s existence-uniqueness r e s u l t s f o r t h e metap a r a b o l i c (Brown [l]) problem (Dt
- mDxDt)u(x,t) 2 = u(1,t)
when uo
E
H:(0,1)
and t
+ Dxu(x,t) 4
= ux(O,t)
+
=
=
f(x,t),
ux(l,t)
f ( * , t ) : [O,m)
+
O < X < l , t > 0,
= 0,
H-'(O,l)
i s H6lder
con t i nuous. Example 4.8
2 L (0,l)
1 L e t Va = Ho(O,l)
added as i n Example 4.5.
+ bul, u
E
rl
J",
dxy v c = uydx; p o s i t i v e c o e f f i c i e n t s c o u l d be
w i t h =
by Bu = r u
11,
w i t h =
Let r
Va.
monotone whenever Re(r)
E
$, b
E
R and d e f i n e B: Va
Since Re<Bu,u> = Re(r)
2
0.
The o p e r a t o r A + B
,B
Ilully'
+ C : Va
+
C
Vc
is I
+
Va i s
continuous, l i n e a r and coercive, hence s u r j e c t i v e , so Theorem 4.3 shows t h e problem
184
DEGENERATE EQUATIONS WITH OPERATOR COEFFICIENTS
3.
2 x,t)-Dxu( x,t)
t)+rDtu( x,t)+bDxDtu( y 0 , t )
(4.9)
= u(1,t) =
U(X,O)
= 0,
damped wave equation.
u1
E
0 < x < 1, 1 Ho(O,l)
and t
-+
T h i s i s a c l a s s i c a l problem f o r t h e
f(*,t) E C ([0,~),L~(0,1)).
E
= u 1( x ) ,
1 2 (O,l), Ho"H
E
1
s e t B = EA,
t > 0,
u0(x), Dtu(x,o)
i s w e l l posed w i t h uo
,
= f ( x ,t)
I f i n s t e a d o f t h e above choice f o r B we
> 0, then from Theorem 4.4
f o l l o w s well-posedness
f o r t h e " p a r a b o l i c " problem rDtU(X,t)-EDXDtU( 2 2 u(0,t)
X,t)-Dxu( 2 x,t)
= u(1,t)
= f (x,t)
,
O < X < l ,
= 0,
u(x,o) = u0(x),
t > 0,
Dtu(x,o)
=
u,(x),
o f c l a s s i c a l v i s c o e l a s t i c i t y ( c f . Albertoni-Cercignani t h e data i s chosen w i t h u
0
f(*,t)
:
[O,m) -+
L'(0,l)
E
1 H o ( O Y l ) , u1
E
2 L (O,l),
[l]).Here
and t
-t
H6lder continuous.
We r e t u r n t o consider t h e p a r a b o l i c s i t u a t i o n o f Theorem 4.2 and d e s c r i b e a b s t r a c t r e g u l a r i t y r e s u l t s on t h e s o l u t i o n . a d d i t i o n t o t h e hypotheses o f Theorem 4.2,
assume given a H i l b e r t
space H which i s i d e n t i f i e d w i t h i t s dual and i n which V, and c o n t i n u o u s l y embedded. I
I
H = H C VmCV
I
.
: Mu
E
HI.
i s dense
Thus we have t h e i n c l u s i o n s V c V m C
1 L e t M be t h e (unbounded) o p e r a t o r on H ob-
t a i n e d as t h e r e s t r i c t i o n o f M : Vm {u E V,,
In
-t
I 1 Vm t o t h e domain D(M ) =
Similarly, the r e s t r i c t i o n o f L : V
1 t h e domain D ( L ) = { u
E
V : Lu
E
185
-t
V
I
to
H I w i l l be denoted by L1. Each
SINGULAR AND DEGENERATE CAUCHY PROBLEMS
o f D(M 1) and D(L 1 ) i s a H i l b e r t space w i t h t h e induced graph-norm and t h e y a r e dense and continuous i n Vm and V, r e s p e c t i v e l y .
We
s h a l l describe a s u f f i c i e n t condition f o r the s o l u t i o n u ( * ) o f w i t h f ( * ) = 0 t o belong t o C([O,m),Vm)/)Cm((O,m),D(L
(4.1)
1) ) .
The arguments depend on t h e t h e o r y o f i n t e r p o l a t i o n spaces and f r a c t i o n a l powers o f o p e r a t o r s ( c f . C a r r o l l [14] f o r references). S p e c i f i c a l l y , t h e o p e r a t o r L1 i s r e g u l a r l y a c c r e t i v e and c o r r e s ponding f r a c t i o n a l powers Le, 0
5
L
0
1, can be defined.
Their
corresponding domains a r e r e l a t e d t o i n t e r p o l a t i o n spaces between 1 0 5 8 5 1, D(L ) and H by t h e i d e n t i t i e s D(Llme) = [D(L),H;O], and corresponding norms a r e e q u i v a l e n t , Theorem 4.9
I n a d d i t i o n t o t h e hypotheses o f Theorem 4.2,
we assume D(L 1-e ) C D(M 1 ) f o r some 8 , 0 < 8 5 1. uo
E
V,
cm( (0,m)
t h e s o l u t i o n u ( * ) o f (4.1)
with f(*)
, ~1)d
Remark 4.10
= 0 belongs
Theorem 4.1 i s new i n t h e form given.
c l o s e l y r e l a t e d Theorem 4.2
i s from Showalter [6];
[7, 111 f o r an e a r l i e r s p e c i a l case.
Theorem
to
The
c f . Showalter
4.3 i s t h e l i n e a r
case o f a r e s u l t i n Showalter [5] and Theorem 4.4 Theorem 4.9
Then f o r each
i s new.
i s presented i n Showalter [6, p t . 111 w i t h a l a r g e
c l a s s o f a p p l i c a t i o n s t o i n i t i a l -boundary value problems,
The
a d d i t i o n a l hypothesis i n Theorem 4.9 makes L s t r i c t l y s t r o n g e r than M; t h e r e s u l t i s f a l s e i n general when
e
= 0.
M i k h l i n [2], work we r e f e r t o C o i r i e r [l],K r e i n [l],
[l]and Showalter [17]. 186
For r e a t e d Phil ips
3.
3.5
DEGENERATE EQUATIONS WITH OPERATOR COEFFICIENTS
L i n e a r degenerate equations.
We s h a l l consider appro-
p r i a t e Cauchy problems f o r t h e equation
whose c o e f f i c i e n t s { M ( t ) )
and { L ( t ) ) a r e bounded and measurable
f a m i l i e s o f l i n e a r operators between H i l b e r t spaces.
We show
( e s s e n t i a l l y ) t h a t t h e problem i s well-posed when t h e { M ( t ) ) are nonnegative, (L(t)+XM(t)) are c o e r c i v e f o r some X > 0, and t h e o p e r a t o r s depend smoothly on t.
Some elementary a p p l i c a t i o n s
t o i n i t i a l - b o u n d a r y value problems are presented i n o r d e r t o i n d i c a t e t h e l a r g e c l a s s o f problems t o which t h e r e s u l t s can be a p p l i e d (cf.
Examples 1.1,
case of (5.1)
1.4 and 1.6).
Consideration o f t h e
on a product space leads t o a p a r a b o l i c system
which c o n t a i n s t h e second o r d e r equation ( c f . Example 1.2) (5.2)
2 %C(t)u(t)) dt
+ $B(t)u(t))
+ A(t)u(t)
= f(t).
C e r t a i n h i g h e r o r d e r equations and systems can be handled s i m i l a r ly.
I n o r d e r t o describe t h e a b s t r a c t Cauchy problem, l e t V be a separable complex H i l b e r t space w i t h norm I l v l I ; V dual, and t h e a n t i d u a l i t y i s denoted by .
I
i s the anti-
W i s a complex
H i l b e r t space c o n t a i n i n g V and t h e i n j e c t i o n i s assumed continuous w i t h norm 5 1.
L(V,W)
denotes t h e space o f continuous l i n e a r
o p e r a t o r s from V i n t o W and t h a t f o r each t
E
T i s t h e u n i t i n t e r v a l [0,1].
Assume
T we are g i v e n a continuous s e s q u i l i n e a r form
187
SINGULAR AND DEGENERATE CAUCHY PROBLEMS
on V and t h a t f o r each p a i r x,y
a(t;*,*)
i s bounded and measurable.
E
E
2
2
L (T,V) t h e f u n c t i o n t
g r a b l e ( c f . C a r r o l l [14], f o r each t
E
measurable.
p. 168).
E
E
V
E
Let {V(t) : t
2 L (T,V)
W and f
+
a(t;u(t),v(t))
i s inte-
S i m i l a r l y , we assume g i v e n
W t h e map t E
E
+.
m(t;x,y)
on W such
i s bounded and
T I be a f a m i l y o f c l o s e d subspaces
2 o f V and denote by L ( T , V ( t ) )
uo
K R l l x l l - llyll f o r a l l x,y
T a continuous s e s q u i l i n e a r form m(t;*,*)
t h a t f o r each p a i r x,y
E
R(t;x,y)
-+
Standard m e a s u r a b i l i t y arguments then show t h a t f o r
T.
any p a i r u,v
those (I
V t h e map t
By u n i f o r m boundedness t h e r e i s a
number KR > 0 such t h a t IR(t;x,y)l and t
E
t h e H i l b e r t space c o n s i s t i n g o f
f o r which (I(t)
E
L 2 (T,V l ) be given.
V ( t ) a.e.
A
on T.
Finally, l e t
s o l u t i o n o f t h e Cauchy prob-
lem (determined by t h e preceding data) i s a u
E
L 2 (T,V(t)) such
that
=
for all v
E
lo
1 < f ( t ) , v ( t ) > d t .+ m(O;uo,v(o)
L2(T,V(t))/)
A family {a(t;-,*)
1
H (T,W) w i t h v ( 1 ) :t
E
T I o f continuous s e s q u i l i n e a r forms
on V i s c a l l e d r e g u l a r i f f o r each p a i r x,y a(t;x,y)
= 0.
E
V the function t
i s a b s o l u t e l y continuous and t h e r e i s an M ( - )
such t h a t f o r x,y
E
1 L (T)
E
V we have a.e.
188
t
E
T.
-t
3.
DEGENERATE EQUATIONS WITH OPERATOR COEFFICIENTS
The f o l l o w i n g " c h a i n r u l e " w i l l be used below. Lemma 5.1
Let {a(t;*,-)
Then f o r each p a i r u,v
: t
1 H (T,V)
E
E
T} be a r e g u l a r f a m i l y on V.
the function t
a(t;u(t),v(t))
-+
i s a b s o l u t e l y continuous and i t s d e r i v a t i v e i s g i v e n by I
a(t;u(t),v
( t ) ) , a.e.
Proof: Fix x
V.
E
=
E
L(V,V)
by (a(t)x,y),,
Dt(a(t)x,yn)v,
t
E
T.)
a.e:t
E
T.
a(t;x,y),
absolute c o n t i n u i t y o f t I t & ( s ) x ds, t
E
T.
+
(The e s t i m a t e
*
V
The weak
Thus a ( - ) x i s s t r o n g l y a b s o l u t e l y continuous
1 H (T,V).
( u ( t ) ,a ( t ) v I v
E
a ( t ) x t h e n shows a ( t ) x = a ( o ) x +
and s t r o n g l y d i f f e r e n t i a b l e a.e. E
E
F o r each n 2 1
shows i t i s i n L'(T,V).
'0
Let u
x,y
The map & ( * ) x i s weakly, hence s t r o n g l y , mea-
s u r a b l e and t h e e s t i m a t e (5.4)
on T.
F o r each v
E
V we have ( a ( t ) u ( t ) , v ) v
=
a b s o l u t e l y continuous, s i n c e t h e above d i s c u s -
s i o n a p p l i e s as w e l l t o t h e a d j o i n t a " ( t ) . i s weakly a b s o l u t e l y continuous.
Hence, t
-+
a(t)u(t)
The s t r o n g d i f f e r e n t i a b i l i t y o f
a ( - )f r o m above i m p l i e s D t [ a ( t ) u ( t ) ] E
=
shows ( & ( t ) x y y ) v i s d e f i n e d and continuous a t e v e r y y
and a.e.
t
+
T.
V and l e t { yn : n Z 1) be dense i n V.
define (&(t)x,yn)v (5.4)
t
Define a ( t )
E
+ a(t;ul(t),v(t))
= al(t;u(t),v(t))
uta(t;u(t),v(t))
=
i ( t ) u ( t ) + a ( t ) u l ( t ) , a.e.
T, hence t h e i n d i c a t e d s t r o n g d e r i v a t i v e i s i n L 1 (T,V).
From
t h i s i t f o l l o w s t h a t a ( * ) u ( - ) i s s t r o n g l y a b s o l u t e l y continuous and t h e d e s i r e d r e s u l t now f o l l o w s e a s i l y .
189
QED
SINGULAR AND DEGENERATE CAUCHY PROBLEMS
Our f i r s t r e s u l t g i v e s e x i s t e n c e o f a s o l u t i o n w i t h an a p r i o r i estimate. Theorem 5.2
L e t t h e H i l b e r t spaces V ( t ) C V C W , sesqui-
l i n e a r forms m(t;*,-) uo
W and f
E
: t
{m(t;-,=) m(t;x,y) x
E
W.
E
2
L (T,V
' ) be given as above.
Assume t h a t
T I i s a r e g u l a r f a m i l y o f H e r m i t i a n forms on W:
E
'my x,y W,
=
on W and V, r e s p e c t i v e l y ,
and k ( t ; - , - )
E
a.e.
t
E
T, and m(o;x,x)
2 0 for
Assume t h a t f o r some r e a l X and c > 0
(5.5)
Rek(t;x,x)
2
+ hm(t;x,x)
+ m-(t;x,x)
2
x
CllXIl"'
E
V(t),
a.e.
t
E
T.
Then t h e r e e x i s t s a s o l u t i o n u o f t h e Cauchy problem, and i t s a t i s f i e s IIuII
5 const.(IIfII
2
I
L (T,V) L (T,V where t h e constant depends on A and c. Proof:
1
+
m(o;uo,uo))
1/2
,
Note f i r s t t h a t by a standard change-of-variable
argument we may r e p l a c e k ( t ; - , - )
by !L(t;-,*)
+ Xrn(t;*,=).
There-
f o r e we may assume w i t h o u t l o s s o f g e n e r a l i t y t h a t A = 0 i n (5.5):
+ m'(t;x,x)
2 ~ 1 1 x 1 1 ~x.
ale. t E T. Now 1 2 I l u ( t ) / 1 2 d t and l e t define H = L2(T,V(t)) w i t h t h e norm IIuIIH = 0 2 F E {$ E H : $ I E L2(T,W), $ ( 1 ) = 0) w i t h t h e norm 11011, =
2Rek(t;x,x)
190
E
V(t),
3. DEGENERATE EQUATIONS WITH OEPRATOR COEFFICIENTS
sesquilinear, L : F
-+
I$ i s c o n j u g a t e - l i n e a r , and we have t h e
continuous. Finally, f o r
0
i n F we have f r o m Lemna 5.1 and 5.5 w i t h A =
These e s t i m a t e s show t h a t t h e p r o j e c t i o n theorem o f L i o n s [5, p. Thus, t h e r e i s a u
371 a p p l i e s here. L(@) f o r a l l @
E
E
H such t h a t E(u,@) =
F and i t s a t i s f i e s l l u l l H 5 ( 2 / n i i n ( ~ . l ) ) I I L ~ ~ ~ ~ .
But t h i s i s p r e c i s e l y t h e desired r e s u l t . Remark 5.3 m(t;x,x)
QED
The hypotheses o f Theorem 5.2 n o t o n l y p e r m i t
t o v a n i s h b u t a l s o t o a c t u a l l y t a k e on n e g a t i v e values.
T h i s w i l l be i l l u s t r a t e d i n t h e examples. Remark 5.4 r e p l a c e d b y 2X
(5.7)
An easy e s t i m a t e shows t h a t (5.5)
+ a) i f
ReR(t;x,x)
we assume t h a t f o r
some c > 0 and a 2 0
2 2 cllxlIv,
+ Xm(t;x,x)
T h i s p a i r of c o n d i t i o n s i s t h u s s t r o n g e r t h a n (5.5) frequently i n applications. i m p l y m(t;x,x)
2 0 for all t
Note t h a t (5.6) E
holds ( w i t h X
T.
191
b u t occurs
and m(o;x,x)
2 0
SINGULAR AND DEGENERATE CAUCHY PROBLEMS
We p r e s e n t an elementary uniqueness r e s u l t w i t h a r a t h e r severe r e s t r i c t i o n o f m o n o t o n i c i t y on t h e subspaces { V ( t ) > .
This
h y p o t h e s i s i s c l e a r l y "ad hoc" and i s a l i m i t a t i o n o f t h e t e c h n i que which i s n o t i n d i c a t i v e o f t h e b e s t p o s s i b l e r e s u l t s .
In
p a r t i c u l a r , much b e t t e r uniqueness and r e g u l a r i t y r e s u l t s f o r t h e very
s p e c i a l case o f m(t;x,x)
C a r r o l l [14],
= (x,x),
C a r r o l l - C o o p e r [35],
have been o b t a i n e d ( c f .
C a r r o l l - S t a t e [36],
t h e n o v e l t y here i s i n t h e forms m ( t ; - , - )
L i o n s [5]);
determining t h e leading
( p o s s i b l y degenerate) o p e r a t o r s ( M ( t ) ) . L e t t h e H i l b e r t spaces V ( t ) C V C W,
Theorem 5.5
l i n e a r forms !L(t;*,-) uo
E
W and f
E
2
L (T,V
2 0, x
Rem(t;x,x)
'
) be g i v e n as above.
V ( t ) , a.e.
E
on V and W,
and m ( t ; * , * )
t
E
sesqui-
respectively,
Assume t h a t
T, t h a t {!L(t;=,*)
: t
E
T) i s
a r e g u l a r f a m i l y o f H e r m i t i a n forms on V ( t ) : x,y and f o r some r e a l A and c > 0, R(t;x,x) ~ 1 1 x 1 1 ~x.
V ( t ) , a.e.
t
E
o f subspaces { V ( t ) : t
E
TI
(5.9)
E
T
E
t >
T,
t,T
T.
E
V(t),
+ ARem(t;x,x)
a.e.
t
E
2
F i n a l l y , assume t h a t t h e f a m i l y
i s decreasing: imply V ( t ) C V ( - r ) .
Then t h e r e i s a t most one s o l u t i o n o f t h e Cauchy problem. Proof: uO
L e t u ( - ) be a s o l u t i o n o f t h e Cauchy problem w i t h
= 0 and f ( * ) = 0.
By l i n e a r i t y i t s u f f i c e s t o show t h a t
192
T
3.
u ( - ) = 0.
[o,s]
DEGENERATE EQUATIONS WITH OPERATOR COEFFICIENTS
Let s
E
(0,l)
and v ( t ) = 0 f o r t
shows v ( t )
E
2 L (T,W)
V ( t ) f o r each t
and v ( 1 ) = 0.
,flR(t;v'(t),v(t))dt (5.8)
rs
E
T.
(O,l),
E
S i n c e u(.)
-
1:
E
['u(r)dr
9
L (T,V)
for t
E
and (5.9)
t
Also, v ( t ) = - u ( t ) f o r t so we have v '
E
E
L2(TyV(t))C
i s a s o l u t i o n we have by (5.3)
rn(t;u(t),u(t))dt
= 0.
Lemma 5.1 and
g i v e us t h e i d e n t i t i e s
j 1 2 Rem( t ;u ( t ) ,u ( t ) ) d t =
I
Then v
[s,l].
E
and v ' ( t ) = 0 f o r s
(0,s)
-
and d e f i n e v ( t ) =
1:
Dt%( t ;v ( t ) ,v ( t ) )
+ R'(t;v(t),v(t))}dt
{ZRem(t;u(t),u(t))
- R ' ( t ;v ( t ) ,v ( t ) ) 1dt ,
+ Q(o;v(o),v(o))
0.
=
'0
L ~
As b e f o r e , we may assume %(t;x,x) D e f i n e W(t) = for t
E
cllW(s)
(0,s)
11'
1:
u(.r)d-r; t h e n W(s) = - v ( o )
1 1 x ~ E ~V ( t )~ , t~ E . T. and W(t) - W(s) = v ( t )
and we o b t a i n t h e e s t i m a t e
5 %(o;W(s),W(s))
,f M ( t ) { l l w ( t ) l 1 2
+
:j
ZRem(t;u(t),u(t))dt
S
5 2
+
I/W(s)/121dt.
0
Choose s
0
> 0 so t h a t 2
have IIW(s) 11'
-
5 (2/(c
1
S
2
M ( t ) d t < c.
0rS
Then f o r s 8 [o,so]
O M ( t ) d t ) > j S 1 4 ( t ) l l W ( t ) 1I2dt.
t h e Gronwall i n e q u a l i t y we conclude t h a t W(s) = 0 f o r s hence u ( s ) = 0 f o r s
E
[o.so].
From
0
JO
Since
E
[oyso].
M(*) i s i n t e g r a b l e on T we
c o u l d use t h e a b s o l u t e c o n t i n u i t y o f t h e i n t e g r a l
193
we
SINGULAR AND DEGENERATE CAUCHY PROBLEMS
I
T+ So
choose so > 0 i n t h e above so t h a t T 2
0 with
T
M ( t ) d t < c/2 f o r every
T
+ so 5 1.
Then we a p p l y t h e above argument a f i n i t e
T.
number o f times t o o b t a i n u ( - ) = 0 on
QED
A fundamental problem w i t h degenerate e v o l u t i o n equations i s t o determine t h e p r e c i s e sense i n which t h e i n i t i a l c o n d i t i o n i s
A c o n s i d e r a t i o n o f t h e two cases o f m(t,x,y)
attained.
Lzt
=0
and m(t;x,y) of
:( X , Y ) ~
shows (see below) t h a t we may have t h e case
u(t) = u
0
i n an a p p r o p r i a t e space or, r e s p e c t i v e l y , t h a t These remarks
lim+ u ( t ) does n o t n e c e s s a r i l y e x i s t i n any sense. t-4 i l l u s t r a t e t h e importance o f t h e f o l l o w i n g . Theorem 5.6
L e t u ( * ) be a s o l u t i o n o f t h e Cauchy problem
and assume t h e subspaces { V ( t ) t h e r e i s a to E (0,1] Then f o r e v e r y
T E
: t
T I a r e i n i t i a l l y decreasing:
E
such t h a t (5.9)
(o,to)
and v
holds f o r
V(T) we have t
E
t,T +
E
[o,to].
m(t;u(t),v)
1 i s i n H ( 0 , ~ )(hence i s a b s o l u t e l y continuous w i t h d i s t r i b u t i o n ) m(o;u(o) d e r i v a t i v e i n L2 ( 0 , ~ ) and
:
(){V(T)
T
Proof: t
E
CT,~].
-
uo,v)
> 0) i s dense i n W, then m(o;u(o)
Define v ( t ) = Then v ( * )
E
for t
2 L (T,V(t))f)
E
= 0.
-
R(*;u(-),v)
t
Dtm(*;u(*),v)
194
uo,u(o)
-
uo) = 0.
(o,T) and v ( t ) = 0 f o r
H1(T,W)
and v ( 1 ) = 0, so
That i s , we have t h e i d e n t i t y (5.10)
Thus, i f
=
3.
DEGENERATE EQUATIONS WITH OPERATOR COEFFICIENTS
i n t h e space o f d i s t r i b u t i o n s on ( 0 , ~ f)o r e v e r y v
f i r s t and l a s t terms o f (5.10)
E
V(T).
The
2
) so, then, i s a r e i n L ( 0 , ~ and
t h e second and we t h u s have p o i n t w i s e v a l u e s a.e.
i n (5.10)
and
hence t h e e q u a t i o n
=
for g
E
j l < f ( t ) ,v>g( t ) d t
L2 ( 0 , ~ ) and v
E
V(r).
1 Suppose now t h a t $ E H ( 0 , ~ i)s g i v e n w i t h v ( * ) as above so as t o o b t a i n f r o m (5.3) /)(t;u(t),v)$'(t)dt v E V(T).
=
S i n c e (5.11)
these r { D t m ( t ; u ( t )
:1
:1
$(T)
0.
=
Define
-
R(t;u(t),v)$(t)dt
< f ( t ) , v > $ ( t ) d t t m(o;uo,v)$(o)
where
h o l d s w i t h g ( t ) = $ ( t ) , we o b t a i n from
,v)$(t)
t m(t;u(t),v)$'(t)Idt
= -m(o;uo,v)$(o).
0
The i n t e g r a n d i s t h e d e r i v a t i v e o f t h e f u n c t i o n t m(t;u(t),v)$(t)
-+
1 i n H ( 0 , ~ and ) $(r) = 0, so t h e d e s i r e d r e s u l t
QED
f o l l ows.
Another s i t u a t i o n which a r i s e s f r e q u e n t l y i n a p p l i c a t i o n s and t o which t h e above t e c h n i q u e i s a p p l i c a b l e i s t h e f o l l o w i n g . Theorem 5.7
L e t u ( * ) be a s o l u t i o n o f t h e Cauchy problem
and assume t h e r e i s a c l o s e d subspace Vo i n V w i t h V o C f ) { V ( t ) t
E
TI.
L(V,Vi)
Vo,
t
D e f i n e t h e two f a m i l i e s o f l i n e a r o p e r a t o r s { L ( t ) ) C and { M ( t ) I C L ( W , V i )
E
:
by
T; <M(t)x,y> = m(t;x,y),
x
195
E
W, y
= R(t;x,y), E
Vo,
t
E
x T.
E
V, y
E
Then i n
SINGULAR AND DEGENERATE CAUCHY PROBLEMS I
t h e space of Vo-valued d i s t r i b u t i o n s on T we have (5.1) function t
-f
M(t)u(t) : T
and t h e
I
-t
Vo i s continuous.
The p r o o f o f Theorem 5.7 f o l l o w s from (5.10) t h e remark t h a t each term i n (5.1),
with v
as w e l l as l . I ( - ) u ( - ) ,
E
Vo and
is in
Our l a s t r e s u l t concerns v a r i a t i o n a l boundary c o n d i t i o n s . Suppose we have t h e s i t u a t i o n of Theorem 5.7 and a l s o t h a t (5.9)
L e t H be a H i l b e r t space i n which
holds.
imbedded and Vo i s dense.
We i d e n t i f y H
E
H, v
Then from (5.11) rT
E
Vo.
and (5.1)
Dtm(t;u(t) , v ) $ ( t ) d t =
JO
each @
E
w i t h H by t h e Riesz
H 4 Vo and t h e i d e n t i t y = 2 Assume f E L (T,H), T > 0 and v E V ( T ) . we obtair:
:1
R(t;u(t),v)@(t)dt +
( L ( t ) u ( t ) + Dt(M(t)u(t)) ,V)H$(t)dt for
This g i v e s us t h e f o l l o w i n g .
C;(O,T).
Theorem 5.8 (5.9)
Assume t h e s i t u a t i o n o f Theorem 5,7 and t h a t
holds.
L e t H be g i v e n as above and f E 2 each T > 0 and v E V(T) we have i n L ( 0 , ~ ) (5.12)
V i s continuously
I
theorem and hence o b t a i n V o + (h,v)H f o r h
I
R(t;u(t),v)
L 2 (T,H).
Then f o r
+ Dtm(t;u(t),v)
We b r i e f l y i n d i c a t e how t h e preceding theorems can g i v e c o r responding r e s u l t s f o r second o r d e r e v o l u t i o n equations which a r e "parabol i c " .
These r e s u l t s are t i m e dependent analogues o f Theo-
rem 4.4.
196
3. DEGENERATE EQUATIONS WITH OPERATOR COEFFICIENTS
L e t Va and Vc be separable complex H i l b e r t
Theorem 5.9
spaces w i t h Va dense and c o n t i n u o u s l y imbedded i n Vc. {a(t;=,-)
:t
E
TI, {b(t;-,-)
: t
E
{ A ( t ) l and {B(t)IcL(Va,V:)
and { C ( t ) > CL(Vc.Vi)
5.7.
: t
Assume t h a t { a ( t ; * , - )
t 0 for x
E
I
Xa(t;x,x)
+
(5.14)
2Reb(t;x,x)
f(-),g(.)
E
TI
V a y and V c y
E
as i n Theorem
T I and ( c ( t ; * , * )
a (t;x,x)
T I are
E
Assume t h a t f o r some X and c > 0
Va.
2
2 cllxllvay
+ Xc(t;x,x)
Va and a.e. 2 l L (T,Va)
: t
2 0 and
r e g u l a r f a m i l i e s o f H e r m i t i a n forms w i t h a(o;x,x)
E
E
Define corresponding fami J i e s o f 1 i n e a r operators
respectively.
f o r each x
:t
T I and { c ( t ; * , - )
be f a m i l i e s of continuous s e s q u i l i n e a r forms on Va,
c(o;x,x)
Let
t
T.
E
I
t
+ c (t;x,x) Then f o r uo
E
cIIxII 2 , 'a
Va,
vo
E
V c y and
given, t h e r e e x i s t s a p a i r u ( * ) , v ( * )
E
2
L (T,Va) which s a t i s f i e s t h e system
I
i n t h e sense o f Va-valued d i s t r i b u t i o n s on T and s a t i s f i e s t h e i n i t i a l c o n d i t i o n s A(o)u(o) = A(o)uo, C(o)v(o) = C(o)vo. Proof:
T h i s f o l l o w s d i r e c t l y from Theorem 5.2 and Theorem
5.7 w i t h m(t;X,?) and !L(t;Y,a
v
V(t)
f
a(t;xl
= a(t;x,,y2)
v0
5
va
x
,yl)
-
+ c(t;x2,y2)
a(t;x2,yl)
va.
for
X,y E
+ b(t;x2,y2)
W E Va x Vc
for
~,TE QED
197
SINGULAR AND DEGENERATE CAUCHY PROBLEMS
There are two n a t u r a l r e d u c t i o n s o f t h e f i r s t o r d e r system (5.15)
t o a second o r d e r equation, and we s h a l l d e s c r i b e these.
F i r s t , we s e t g(
0 )
=
0 i n Theorem, 5.9 and n o t e than t h a t b o t h
C ( - ) v ( * ) and D t ( C ( * ) v ( * ) ) + B(*)v (.)
1
'
belong t o H (T,Va).
Thus
v ( = ) i s a s o l u t i o n o f t h e second o r d e r e q u a t i o n
and t h e i n i t i a l c o n d i t i o n s (Cv)(o) = C(o)vo, (Dt(Cv) + B v ) ( o ) = -A(o)uo.
Second, we s e t f ( - )
A(t)
i s independent of t, i.e., and Hermit ian A 1 H (T,Va) (5.17)
I
E
=0
L(Va,Va).
5
i n Theorem 5.9 and assume A ( t )
A f o r some n e c e s s a r i l y c o e r c i v e
From (5.15)
i t follows that u(-)
E
and i s a s o l u t i o n o f t h e second o r d e r e q u a t i o n Dt(C(t)u'(t))
+ B(t)ul(t)
t Au(t) = g ( t ) I
and t h e i n i t i a l c o n d i t i o n s u ( o ) = uo, ( C ( * ) u ) ( o ) = C(o)vo. We s h a l l pre s e n t some elementary a p p l i c a t i o n s o f t h e precedi n g r e s u l t s t o boundary value problems f o r p a r t i a l d i f f e r e n t i a l equations i n t he form (5.2).
These examples w i l l i l l u s t r a t e
some l i m i t a t i o n s on th e types o f problems t o which t h e theorems apply. Example 5.10 For o u r f i r s t example we choose Vo = V ( t ) = 1 1 V = Ho(T), W = { @ E H (T) : @ (1 ) = 03 and H = L2(T). Recall t h e est imat e
198
3. DEGENERATE EQUATIONS WITH OPERATOR COEFFICIENTS
Let a : T
T be a b s o l u t e l y c o n t i n u o u s and d e f i n e t h e H e r m i t i a n
+.
forms m(t;@,$)
= ~ ~ ( t ) @ ( x ) ~ @,$ d x Ey
W,
t
E
T.
Then m(*;@,$)
I
i s a b s o l u t e l y c o n t i n u o u s and s a t i s f i e s m (t;@,$) = ' 1 I @ ( a ( t ) ) 1 2 a ' ( t ) , @ E W, a.e. t E T. S i n c e a E L (T), t h e e s t i m a t e (5.18) F o r @,$
shows t h a t t h i s i s a r e g u l a r f a m i l y o f forms. 1 1 V, we s e t !L(t;@,$) = @ ( x ) $ ' ( x ) d x , t E T, From t h e
E
1
0
we1 1 -known i n e q u a l i t y
w i t h A = 0.
i t f o l l o w s t h a t t h e p r e c e d i n g f a m i l y s a t i s f i e s (5.7)
W and F
F i n a l l y , l e t uo
E
F(-,t),
Then t h e hypotheses o f Theorem 5.5
t
T.
E
ness) a r e s a t i s f i e d .
E
L 2 ( T x T) be g i v e n and s e t f ( t ) = (on unique-
Suppose a d d i t i o n a l l y t h e r e i s a number a,
0 < a < 2, such t h a t I
(5.20)
(t) 2 a
-
2,
a.e. I
Then (5.18) @
E
i t f o l l o w s t h a t (5.5)
hence, t h e hypotheses o f Theorem 5.2 Let u(*,t)
T.
I 2 1) 2 , L (TI h o l d s w i t h A = 0,
(on e x i s t e n c e ) a r e s a t i s f i e d .
= u ( t ) be t h e s o l u t i o n o f t h e Cauchy problem.
i s t h e unique g e n e r a l i z e d s o l u t i o n o f t h e e l l i p t i c -
p a r a b o l i c boundary v a l u e problem, ut a ( t ) ; -uXx = F ( x , t ) , u(x,o)
E
2 (2-0)11@ shows t h a t 2!L(t;@,@) + m (t;@,@)
V, so by (5.19)
Then u ( x , t )
t
E
T, so u ( * , t )
0 < x
i n o r d e r t o o b t a i n Theorem 5.2.
a s s e r t i o n as f o l l o w s :
E
on t h e f a m i l y
One can p r o v e t h i s
( 1 ) use t h e c l o s e d graph and u n i f o r m
boundedness theorems t o o b t a i n an e s t i m a t e
'0
( 2 ) approximate u
I
2 i n L (T,V)
Lebesgue theorem w i t h (5.21) case o f c o n s t a n t v
E
by s i m p l e f u n c t i o n s and use t h e t o obtain the r e s u l t f o r the special
V; approximate v
I
by s t e p f u n c t i o n s and use
t h e r e s u l t s o f ( 1 ) and ( 2 ) t o o b t a i n t h e g e n e r a l r e s u l t .
The de-
t a i l s a r e s t a n d a r d b u t i n v o l v e some l e n g t h y computations. Example 5.13
Our second example i s s i m i l a r b u t a l l o w s t h e
e q u a t i o n t o be o f Sobolev t y p e i n p o r t i o n s o f T
x
T.
Choose t h e
spaces and t h e forms a(t;-,*)
E
W, d e f i n e
m(t;Qy+3)
E
T, where
as b e f o r e . F o r Q,$ I r@(t) I = r ( t ) Q ( x ) m d x + Jo 4 (x)+ (x)dx, t 0
200
ci
3.
DEGENERATE EQUATIONS WITH OPERATOR COEFFICIENTS
and B map T i n t o i t s e l f , b o t h a r e a b s o l u t e l y continuous, and B i s nondecreasing.
For $I,$
W , m(-;$I,$)
E
uous s o Remark 5.12 a p p l i e s here. sup{l$I(s)12 : 0 5 s 5 B ( t ) )
i s absolutely contin-
Using t h e e s t i m a t e
2 ~ ~ ( t ) l $ I ' ( x ) 1 2 d x ,$I E V,
show t h a t t h e e s t i m a t e (5.5)
holds i f a
I
one can
has an e s s e n t i a l lower
( 0 )
bound ( p o s s i b l y n e g a t i v e ) and i f t h e r e i s a numberu, 0 < u < 2, I
-
such t h a t a ( t ) 2 u W and F
2 where a ( t ) > B ( t ) .
Thus, f o r each uo
E
L2(T x T) i t f o l l o w s from Theorems 5.2 and 5.5 t h a t
E
t h e r e e x i s t s a unique g e n e r a l i z e d s o l u t i o n o f t h e e l l i p t i c parabolic-Sobolev boundary value problem, ut 0 < x < a ( t ) , x < B(t);
-
-Uxxt
= 0, t E T;
= u(1,t)
B(o)).
uxxt
-
uxx = F,
uxx = F, a ( t ) < x < B ( t ) ;
uXx = F, B ( t ) < x < a ( t ) ; -uXx = F, a ( t ) < u(0,t)
-
u(x,O)
X,
-
B ( t ) < x < 1;
= uo(x). 0 < x < max{a(o),
a ( t ) E 0) show t h a t i f B i s p e r m i t t e d
Examples (e.g.,
t o decrease, then a s o l u t i o n e x i s t s o n l y i f t h e i n i t i a l data uo s a t i s f i e s a c o m p a t i b i l i t y condition.
T h i s i s t o be expected
s i n c e t h e l i n e s "x=constant" a r e c h a r a c t e r i s t i c f o r t h e t h i r d o r d e r Sobolev equation. Example 5.14
We s h a l l seek c o n d i t i o n s on t h e f u n c t i o n
m(x,t) which a r e s u f f i c i e n t t o apply o u r a b s t r a c t r e s u l t s above t o t h e boundary v a l u e problem a/at(m(x,t)u(x,t)) (x,t) a.e.
E
x
E
T
x
T.
T; u(O,t)
uxx = F(x,t),
= 0; ~ ( x , o ) ( u ( x , o ) - u ~ ( x ) ) = 0,
1 We choose t h e spaces Vo = V ( t ) = V = Ho(T) and t h e
forms k ( t ; = , - ) valued m ( - , * )
= u(1,t)
-
2 as above and l e t H = W = L (T). E
Lm(T x
Assume t h e r e a l
T) i s such t h a t m ( x , - ) i s a b s o l u t e l y 201
SINGULAR AND DEGENERATE CAUCHY PROBLEMS
continuous f o r a.e. t
E
T, where
M(-)
x
E
1 L (T).
E
H e r m i t i a n forms m(t;$,$) gular.
L e t uo
E
H, F
Then i t f o l l o w s t h a t t h e f a m i l y o f
1:
: m(x,t)$(x)mdx,
E
5 M ( t ) f o r a.e.
T and s a t i s f i e s Imt(x,t)I
2 L (T
x
$,$
E
W, i s re-
T), and s e t f ( t ) = F ( * , t ) .
Theorems 5.6 and 5.7 show t h a t t h e boundary value problem above i s t h e r e a l i z a t i o n o f our a b s t r a c t Cauchy problem. I n o r d e r t o o b t a i n uniqueness o f a g e n e r a l i z e d s o l u t i o n from Theorem 5.5, n o t e t h a t t h e forms !L(t;*,*)
are c o e r c i v e over V,
so Theorem 5.5 i s a p p l i c a b l e i f and o n l y i f a d d i t i o n a l l y we ass ume
(5.22)
m(x,t)
L
0
x,t
E
T,
2 0 for a l l
since t h i s i s equivalent t o m(t;=,-)
$
E
W, t
E
T.
To o b t a i n e x i s t e n c e from Theorem 5.2 i t s u f f i c e s t o assume m(x,o)
a.e.
2 0,
x
E
T, and
T
x
T I > -2~r
(5.23) ess i n f { m t ( x , t )
: (x,t)
E
f o r then we o b t a i n t h e e s t i m a t e m'(t;4,$)
41
E
V, f o r some number
0,
2
0 < a < ZIT
2
.
t a i n (5.5) w i t h X = 0 and c = a / 2 ~
.
2
2
(a
-
,
2~r')Il 0
Then from (5.19)
I$l 2, we ob-
Thus, (5.22) g i v e s unique-
ness and (5.23) i m p l i e s e x i s t e n c e o f a g e n e r a l i z e d s o l u t i o n o f t h e problem. C l e a r l y n e i t h e r o f t h e c o n d i t i o n s (5.22), (5.23) i m p l i e s t h e other.
I n p a r t i c u l a r (5.23) p e r m i t s m(x,t)
202
t o be negative;
3.
DEGENERATE EQUATIONS WITH OPERATOR COEFFICIENTS
then t h e e q u a t i o n i s backward p a r a b o l i c .
This i s t h e case f o r
t h e equation ~ ( ~ ‘ / 4 ) ( 1 - 2 t ) ~ u ( x , t ) )= uxx
(5.24)
= - ( 3 2~/2)(1-2t)‘;
f o r which mt(x,t)
and e x i s t e n c e f o l l o w s . 1/2,
0 i t terval.
But m(=,.)
hence, (5.23)
i s satisfied
i s nonnegative o n l y when
so uniqueness i s claimed o n l y on t h i s s m a l l e r i n -
To see t h a t uniqueness does n o t h o l d beyond t = 1/2, = ( ~ 2 t ) e - ~~ p { l - ( 1 - 2 t ) - ~ s} i n ( T X ) i s a solu-
n o t e t h a t u(x,t) t i o n o f (5.24);
a second s o l u t i o n i s o b t a i n e d by changing t h e t o zero f o r a l l t > 1/2.
values o f u ( x , t )
The l e a d i n g c o e f f i c i e n t i n t h e e q u a t i o n (5.25)
$$(T
2/ N ) ( 1 - t 2 ) ’ / 2 u ( x , t ) )
s a t i s f i e s (5.22),
= uxx
( N > 0) 2 > -27~
hence uniqueness f o l l o w s , b u t mt(x,t)
o n l y on t h e i n t e r v a l [0, 1-(4N)-‘).
Hence, t h e e x i s t e n c e o f a
s o l u t i o n f o l l o w s on t h e i n t e r v a l [O, 1-(4N)-2-u]
f o r any u > 0;
choosing N l a r g e makes t h e i n t e r v a l o f e x i s t e n c e as c l o s e as However t h e f u n c t i o n u(x,t)
d e s i r e d t o [O,l]. exp(2N(1-t)’/2)
sin
w i t h uo(x) = s i n
(TX)
(TX)
-1/2
= (l-t)
i s t h e unique s o l u t i o n o f t h e problem
on any i n t e r v a l [0, 1-u], u > 0, and t h i s
f u n c t i o n does n o t belong t o L
2
(T x T).
Hence, t h e r e i s no s o l u -
t i o n on t h e e n t i r e i n t e r v a l . Example 5.15
L e t G be a smooth bounded domain i n Rn and
r ( x ) be a smooth p o s i t i v e f u n c t i o n on G such t h a t f o r those x 20 3
SINGULAR AND DEGENERATE CAUCHY PROBLEMS
near the boundary a G , r ( x ) equals t h e distance from x t o aG. Parabolic equations of the form
t u t + r(x)L(t)u
(5.26)
=
F(x,t),
X E G ,
t E T
a r i s e i n mathematical genetics ( c f . Kimura and Ohta [ l ] , Levikson They a r e t y p i c a l l y degenerate i n t ( a t t
and Schuss [ l ] ) .
and degenerate i n x ( a t x e l l i p t i c operator.
E
=
0)
3G); L ( t ) i s a strongly uniformly
W e i n d i c a t e how such problems can be inSuppose we a r e given r e a l -
cluded i n the preceding theorems.
valued absolutely continuous R. . ( t ) ( l < i, j S n ) and R . ( t ) ( O < j i n ) 1J J n 2 2 with R o ( t ) 2 0 and 1 Rij(t)ziyj 2 c ( l z l I + + I Z n / 1, i ,j=l c > 0, z E qn. Take Vo = V = H 1o ( G ) and define a regular family
...
of coercive forms by R ( t ; @ , $ )
n
1 R.(t)Dj$ J
$)dx, @,I) E V.
j=o
n
1 R i j ( t ) D i @ D . $J + G i ,j=1 The corresponding e l l i p t i c operators =
{
a r e given by
-
L(t) =
(5.27)
n
1
Rij(t)DjDi i ,j=l
+
n
1 R.(t)D j'
j=o
J
t
Let m(x,t) be given a s i n Example 5.14, set W = H 1o ( G ) , m(t;@,$)
=
1
G
ity (5.28)
( m ( x , t ) / r ( x ) 2 ) @ ( x ) m d x , $,$
II@/rll 2 5 C l l @ l l W ,
W.
T.
and define
From the inequal-
$
L
i t follows t h a t m ( t ; * , * )
E
E
E
W,
i s a regular family of Hermitian forms
T ) and define f ( t ) = F ( m , t ) / r ( * ) f o r t E T. From (5.28) i t follows t h a t f E L 2 (T,V ' ). Finally, assume uo E
on W.
Let F
E
L2(G
x
204
3.
DEGENERATE EQUATIONS WITH OPERATOR COEFFICIENTS
From Theorems 5.6 and 5.7 we see t h a t o u r a b s t r a c t
Hi(G).
Cauchy problem i s e q u i v a l e n t t o t h e i n i t i a l - b o u n d a r y v a l u e prob1em La t( w rl (x x , t ) )
+ r(x)L(t)u(x,t)
m(x,o)(u(x,o)-uo(x))
= F(x,t),
0,
=
x
E
G,t
s
E
aG,
x
E
G.
E
T
(The f i r s t e q u a t i o n has been m u l t i p l i e d by r ( x ) and t h e l a s t by r(x)'
E x i s t e n c e and uniqueness f o l l o w f r o m t h e a d d i t i o n a l
> 0.)
2 0, mt(x,t)
assumptions m(x,t)
2 0, a.e.
x
E
G, t
E
T.
These
a r e n o t b e s t p o s s i b l e f o r e x i s t e n c e ; t h e y can be weakened as i n Example 5.14. Remark 5.16
The e q u a t i o n (5.29)
permits t h e leading operat-
o r t o be s i n g u l a r and t h e second t o be degenerate i n t h e s p a t i a l variable,
w h i l e t h e l e a d i n g o p e r a t o r may be degenerate i n time.
Remark 5.17
A m i n o r t e c h n i c a l d i f f i c u l t y a r i s e s from t h e i s g i v e n w i t h t h e t i m e d e r i v a t i v e on u, n o t
f o r m i n which (5.26) on ( t u ) .
We f i r s t n o t e t h a t t u t = ( t u ) ,
r e s u l t i n g second o p e r a t o r L ( t )
-
C~TU
T
=
"(TU),
-
u b u t see t h a t t h e
u / r m i g h t n o t be coercive.
However, i n t h e change of v a r i a b l e given by tut =
-
T =
t" t h e l e a d i n g t e r m i s
a u and t h e second o p e r a t o r r e s u l t -
i n g as above i s g i v e n by L ( T ~ / " ) - au/r. t h a t t h i s l a s t operator i s coercive f o r
205
From (5.28) CI
i t follows
> 0 s u f f i c i e n t l y small.
SINGULAR AND DEGENERATE CAUCHY PROBLEMS
Remark 5.18 Showalter [2].
Theorems 5.2,
5.6,
5.7 and 5.8 a r e from
Theorem 5.9 which c o n t a i n s (5.16)
See Showalter [Z]
new.
5.5,
and (5.17)
is
f o r examples i n h i g h e r dimension w i t h
e l l i p t i c - p a r a b o l i c i n t e r f a c e o r boundary c o n d i t i o n s and o t h e r c o n s t r a i n t s on t h e t i m e - d i f f e r e n t i a l along a submanifold o r p o r t i o n o f t h e boundary.
For a d d i t i o n a l m a t e r i a l on s p e c i f i c par-
t i a l d i f f e r e n t i a l equations o r e v o l u t i o n equations o f t h e t y p e considered i n t h i s s e c t i o n we r e f e r t o Baiocchi [l], Browder [4], Cannon-Hi 11 [Z],
F i chera [2]
, Ford [l] , Ford-Wai d
[2]
, Friedman-
Schuss [l],Gagneux [l],Glusko-Krein [l],Kohn-Nirenberg [l], L i o n s [5, 91, O l e i n i k [2],
Schuss [l],V i c i k [110 A l s o see
references g i v e n i n preceding s e c t i o n s f o r r e l a t e d l i n e a r problems and i n t h e f o l l o w i n g s e c t i o n s f o r n o n l i n e a r problems. 3.6
S e m i l i n e a r degenerate equations.
We s h a l l present some
methods f o r o b t a i n i n g existence-uniqueness r e s u l t s f o r t h e a b s t r a c t Cauchy problem f o r t h e equation
where M i s continuous, s e l f - a d j o i n t and monotone.
The m s t we
s h a l l assume on t h e n o n l i n e a r N i s t h a t i t i s monotone, hemicontinuous, bounded and coercive.
S p e c i f i c a l l y ; 1 e t . V be a r e f l e x -
i v e r e a l Banach space w i t h dual V
V to V (6.2)
I
.
I
and l e t N be a f u n c t i o n from
Then N i s s a i d t o be monotone i f 2 0,
206
u,v
E
v,
3. DEGENERATE EQUATIONS WITH OPERATOR COEFFICIENTS
hemicontinuous i f f o r each t r i p l e u,v,w
E
V the function s
-+
R t o R , bounded i f t h e image o f
i s continuous from
each bounded s e t i n V i s a bounded s e t i n V - , and c o e r c i v e i f f o r each v
E
V l i m (/llull)
(6.3)
=
II u ll-
.
+a0
L e t H be a H i l b e r t space i n which V i s dense and continuously I d e n t i f y H and i t s dual and thereby o b t a i n t h e i n c l u -
imbedded.
sions V C H C V
I
.
We suppose
M i s a continuous, s e l f - a d j o i n t and
monotone o p e r a t o r o f H i n t o H; thus t h e square r o o t M112 o f M i s defined.
H = L2 (o,T;H),
LP(o,T;V), u
E
D(Lo) = { u
closure o f L
0
u
E
,> 2, l / p + l / q
Let p
V
V : du/dt
E
in V
x
D(L) then M1/2u
V'.
=
00,
and d e f i n e V =
2 = L (o,T;VI).
E
H I and denote by L : D(L)
L e t Lou = dMu/dt f o r E
V
I
the
A standard computation shows t h a t i f
C(o,T;H),
E
1, 0 < T
1
so A ( t ) i s a c c r e t i v e i f N ( t ) i s monotone. D ( t ) t h e r e i s a unique p a i r w1,w2 1,2.
Choose u1 ,u2
Then [q(uj),wj]
E
E
Conversely, i f u1,u2
*
= q Mowj, j J Hence N ( t ) i s
W w i t h N(t,u.)
A ( t ) and (6.24)
holds.
monotone i f A ( t ) i s a c c r e t i v e . Remark 6.14
E
=
QED
N ( t ) i s s t r i c t l y monotone i f and o n l y i f i t i s
i n j e c t i v e and A ( t ) i s a s t r i c t l y a c c r e t i v e f u n c t i o n . Theorem 6.15 monotone f o r each t
L e t N ( t ) be monotone and M + N ( t ) s t r i c t l y E
[o,T].
Then f o r each uo
a t most one s o l u t i o n u ( - ) o f (6.21) Proof:
E
D(o) t h e r e i s
w i t h (Mu)(o) = Mu.,
Since A ( t ) i s a c c r e t i v e f o r each t by Lemma 6.13,
uniqueness h o l d s f o r (6.23).
The r e s u l t t h e n f o l l o w s f r o m QED
C o r o l l a r y 6.12. We c o n s i d e r now t h e e x i s t e n c e o f s o l u t i o n s .
220
Specifically,
3. DEGENERATE EQUATIONS WITH OPERATOR COEFFICIENTS
we s h a l l r e c a l l t h e e x i s t e n c e r e s u l t s o f Crandall and Pazy [l] as t h e y a p p l y t o t h e r a t h e r s p e c i a l H i l b e r t space s i t u a t i o n o f (6.23).
(The monograph o f H. B r e z i s [3] p r o v i d e s an e x c e l l e n t
i n t r o d u c t i o n t o t h i s t o p i c w i t h an e x t e n s i v e b i b l i o g r a p h y ; c f . a l s o t h e r e c e n t monograph o f Browder [7].!
The equation (6.23)
has a (unique) s o l u t i o n v w i t h v ( o ) s p e c i f i e d i n q(D(o)) under t h e f o l l o w i n g hypotheses: Each A ( t ) i s a c c r e t i v e and I + A A ( t ) i s s u r j e c t i v e f o r a l l A > 0. It follows that J A ( t )
of
(I+AA(t))-’
i s a f u n c t i o n d e f i n e d on a l l
w. i s independent o f t;
The domain o f A ( t ) , q[D(t)], we denote i t by q[D]. There i s a monotone g : [o,m)
[ o , ~ ) such t h a t
+
11 JA(t,x)-JA(s,x) 5 A [ t - s [ g ( I[xllw) .(l+ i n f rllyllw : y E A ( s , x ) I ) ,
t,s 2 0, x
E
W,
O 5 1).
For each t
(possibly nonlinear) N ( t ) : D ( t )
sup {II
[o,T] assume we a r e given a
E
+
E
I
w i t h domain D ( t ) C E.
Assume f u r t h e r t h a t f o r each t, N ( t ) i s monotone and D(t)
+
E
I
i s surjective; M[D(t)]
:
M + AN(t) :
i s independent o f t; and f o r
221
SINGULAR AND DEGENERATE CAUCHY PROBLEMS
some monotone i n c r e a s i n g f u n c t i o n g : [o,m)
Then f o r each uo
E
+
[o,m) we have
D(o) t h e r e e x i s t s a s o l u t i o n u o f (6.21)
with
(Mu)(o) = Muo. From C o r o l l a r y 6.12 we see t h a t i t s u f f i c e s t o show
Proof:
has a s o l u t i o n v w i t h v ( o ) = q(uo), and t h i s w i l l f o l l o w
(6.23)
i f we v e r i f y t h e hypotheses on CA(t)) above.
Also, M [ D ( t ) ]
each A ( t ) i s a c c r e t i v e by Lemma 6.13. o f t i m p l i e s t h e same f o r q[D(t)].
*
F i r s t note t h a t independent
To determine t h e range o f
*
*
I + A A ( t ) we have ( q Mo)(I+AA(t)) = q Mo + A N o ( t ) = q Mo + AN(t)q-’ I
=
*
(M+AN(t))q-l on q[D(t)].
The range o f t h e above i s
E and q Mo i s a b i j e c t i o n , so I + A A ( t ) i s n e c e s s a r i l y onto The e s t i m a t e on J A ( t , x ) f o l l o w s from (6.25)
and we r e f e r t o
Showalter [5] f o r t h e r a t h e r l e n g t h y computation. Remark 6.17
W.
QED
I n c o n t r a s t t o preceding r e s u l t s o f t h i s sec-
t i o n , N need n o t be c o e r c i v e o r even d e f i n e d everywhere on
V.
Thus i t may be a p p l i e d t o examples where N corresponds t o a n o t n e c e s s a r i l y e l l i p t i c d i f f e r e n t i a l o p e r a t o r ( c f . Theorem 4.1 and Example 4.6). Second o r d e r e v o l u t i o n equations w i t h ( p o s s i b l y degenerate) o p e r a t o r c o e f f i c i e n t s on time d e r i v a t i v e s can be r e s o l v e d by t h e preceding r e s u l t s .
These c o n t a i n as a s p e c i a l case a f i r s t 222
3. DEGENERATE EQUATIONS WITH OPERATOR COEFFICIENTS
o r d e r s e m i l i n e a r e q u a t i o n which i s n o n l i n e a r i n t h e time d e r i v a tive. Theorem
6.18
L e t A and C be symmetric continuous l i n e a r I
o p e r a t o r s f r o m a r e f l e x i v e Banach space V i n t o i t s dual V ; asl I
Denote by Vc t h e ( H i l b e r t
sume C i s monotone and A i s coercive.
Let B : V
space) dual o f V w i t h t h e seminorm '".
-+
V I be
a ( p o s s i b l y n o n l i n e a r ) monotone and hemicontinuous function. I
Then f o r each p a i r u1,u2 E V w i t h Aul + B(u2) E Vc t h e r e e x i s t s I 1 a unique v E L (o,T;V) such t h a t Cv : [o,T] -+ Vc i s a b s o l u t e l y d continuous, &Cv)
+ B(v) : [o,T]
-+
V
I
i s (a.e.
equal t o ) an ab-
s o l u t e l y continuous f u n c t i o n , Cv(o) = Cu,,[$Cv)+B(v)](o)
= Aul,
and
Proof: [Ax,Cy], Ax + B(y)
On t h e product space E E V x V consider M[x,y] I
hence, E l = V ' x Vc. I
E
Vc) and N :
D
+
Define D = ([x,yl E
I
by N[x,y]
E
=
V x D(B) :
= [-Ay,Ax+B(y)].
Then
a p p l y Theorem 6.16 t o o b t a i n e x i s t e n c e o f a s o l u t i o n u = [w,v] 2 of (6.21). Note t h a t C + AB t h A i s monotone, hemicontinuous I
and coercive, hence onto V ' ,
and t h i s shows M
( c f . p r o o f o f Theorem 4.3).
The second component v o f t h i s solu-
t i o n u w i t h Mu = M[u,,u2]
=
N[x2,y21
AN i s s u r j e c t i v e
i s t h e s o l u t i o n o f (6.26).
f o l l o w s from C o r o l l a r y 6.12 s i n c e M[xl,yl] N[xl,~ll
t
i m p l y [xl,yll
= [x2,y21.
223
= M[x2,y2]
Uniqueness and
QED
SINGULAR AND DEGENERATE CAUCHY PROBEMS
Remark 6.19
The f i r s t component o f t h e s o l u t i o n u o f (6.21)
i n t h e preceding p r o o f i s t h e unique a b s o l u t e l y continuous
w : [o,T]
-f
dw V w i t h B ( a ) : [o,T]
s a t i s f y i n g w(o) = ul,
(6.28)
+
Remark 6.20
I
V c a b s o l u t e l y continuous and
-f
I
Bw ( 0 ) = Bu2,
dw B ( a ) + Aw(t) = 0,
a.e.
E
[o,Tl.
The s p e c i a l case t h a t r e s u l t s from choosing C =
0 i s o f i n t e r e s t and should be compared w i t h (6.21). c a l l y , (6.26)
t
and (6.27)
Specifi-
become f i r s t o r d e r equations i n which
t h e time d e r i v a t i v e a c t s on t h e n o n l i n e a r term. v e r t i b i l i t y o f A was never used i n s o l v i n g (6.26). f i c i e n t f o r e x i s t e n c e o f a s o l u t i o n o f (6.26)
Also, t h e i n It i s suf-
t h a t B + AA be
c o e r c i v e f o r each A > 0; however, uniqueness may then be l o s t take A = 0).
(e.g.,
We b r i e f l y sketch some a p p l i c a t i o n s o f t h e r e s u l t s o f t h i s s e c t i o n t o i n i t i a l - b o u n d a r y value problems.
L e t G be a bounded
domain i n Rn w i t h smooth boundary aG, Wp(G) be t h e Sobolev space of u
E
Lp(G) w i t h each D.u = au/ax.
D u
= u.
g
Lq, q = p / ( p - l ) ,
J
J
E
Lp(G), 1 5 j 5 n, and
L e t f u n c t i o n s N.(x,y) be given, measurable i n x E G J Suppose f o r some C > 0, c > 0, and and continuous i n y E Rn+’ 0
E
.
p
L 2, we have
224
DEGENERATE EQUATIONS WITH OPERATOR COEFFICIENTS
3.
n +
for x
n}, l e t V N :
V + V
(6.32)
n+l
, and
0 5 k 5 n.
L e t Du = {D.u : 0 j J be a subspace o f Wp(G) c o n t a i n i n g Ci(G), and d e f i n e
G, y,z
E
g ( x ) 2 CIYIP,
I
R
E
by
=
1
1
j=o
N.(x,Du(x))D.v(x)dx, G J J
The r e s t r i c t i o n o f Nu t o C:(G)
-
-
E
v.
i s t h e d i s t r i b u t i o n on G g i v e n by
n
- 1 D.N.(.,Du)
NU =
(6.33)
u,v
+ No(*,D$).
j=1 J J
The d i v e r g e n c e theorem g i v e s t h e ( f o r m a l ) Green' s f o r m u l a
lam
v ( s ) d s , v E V, where t h e conormal d e r i v a t i v e n au on aG i s g i v e n by - = 1 N.(*.Du)nj and (nly nn) i s t h e a' j=1 J I u n i t outward normal. The o p e r a t o r N : V -t V i s bounded, hemicon-
=
. . .,
t i n u o u s , monotone and c o e r c i v e ; c f . Browder [3, 61, C a r r o l l [14] o r Lions
[lo]
f o r d e t a i l s and c o n s t r u c t i o n o f more general op-
e r at o r s . Example 6.21 define
M :V
(6.34) L e t uo
Let m(-)
-t
Lm(G), mo(x) 2 0, a.e.
x
E
G, and
E
v.
V by
<Mu,v> = E
E
I
m(x)u(x)v(x)ds, l G
V w i t h N(uo)
=
ml"h
f o r some h
225
u,v E
2 L (G).
Then each of
SINGULAR AND DEGENERATE CAUCHY PROBLEMS
Theorems 6.2,
6.6,
and 6.16 g i v e s e x i s t e n c e o f t h e s o l u t i o n t o
the e l l i p t i c - p a r a b o l i c equation (6.35)
-
+ N(u(x,t))
D,(m(x)u(x,t))
= 0
w i t h i n i t i a l c o n d i t i o n m(x)(u(x,o)-uo(x)) boundary c o n d i t i o n depending on V.
= 0, x
I f V = W:(G),
E
G, and a the D i r i c h l e t
boundary c o n d i t i o n i s obtained, w h i l e t h e ( n o n l i n e a r ) Neumann c o n d i t i o n au/aN = 0 r e s u l t s when
V
= Wp(G).
c o n d i t i o n i s o b t a i n e d by adding t o (6.32)
The t h i r d boundary
a boundary i n t e g r a l .
The assumption t h a t m ( * ) i s bounded may be relaxed; c f . BardosB r e z i s [l]o r Showalter [5].
r
J aGu ( s ) v ( s ) d s Wp(G)
t o (6.34)
and i f
I f we add t h e boundary i n t e g r a l
V i s t h e space o f f u n c t i o n s i n
which are c o n s t a n t on aG, we o b t a i n t h e boundary c o n d i t i o n
o f f o u r t h t y p e ( c f . A d l e r [l])
This problem i s s o l v e d by Theorems 6.9 and 6.16. p r o p r i a t e terms t o (6.32)
and (6.34)
By adding ap-
i n v o l v i n g i n t e g r a l s over
p o r t i o n s o f t h e boundary ( o r over a m a n i f o l d S o f dimension n-1 in
c) one o b t a i n s
s o l u t i o n s o f e q u a t i o n (6.35)
s u b j e c t t o degen-
erate parabolic constraints D,(m(s)u(s,t))
-
div(a(s)grad u(s,t))
- -
s
E
s,
where t h e i n d i c a t e d divergence and g r a d i e n t are i n l o c a l
226
3. DEGENERATE EQUATIONS WITH OPERATOR COEFFICIENTS
c o o r d i n a t e s on S and t h e c o e f f i c i e n t s a r e nonnegative ( c f . Showalter [2,
51). 2 Choose V = Wo(G)
Example 6.22 =
n
r
j=1
G
and d e f i n e
Dju(x)D.v(x)dx, J
u,v
E
v,
where each c . E Lm(G) i s nonnegative, 0 5 j 5 n. L e t B = N be J given by (6.32) and assume (6.29), (6.30) w i t h 1 < p 5 2. Then Theorem 6.18 g i v e s a unique s o l u t i o n o f ' t ' ? e equation (6.36)
n
- j =1l Dj(cjDjv)]
Dt{Dt[cov
."
+ N(v))
n
- j =11D?v = J
0,
w i t h D i r i c h l e t boundary c o n d i t i o n and a p p r o p r i a t e i n i t i a l condi. "
tions.
Note t h a t
N i s g i v e n by (6.33).
i n c l u d e t h e wave equation (4.9), (4.10),
Special cases o f (6.35)
t h e v i s c o e l a s t i c i t y equation
p a r a b o l i c equations (4.6),
and Sobolev equations w i t h
f i r s t o r second o r d e r t i m e d e r i v a t i v e s .
A s p e c i a l case o f (6.36)
Example 6.23 erable interest.
has been o f eonsid-
j S n, and NO(x,s) = m ( x ) I s I P - l sgn ( s ) where m
n e g a t i v e and 1 < p (6.37)
N
S p e c i f i c a l l y , we choose C 5 0,
I 2.
E
= 0 for 1
j
Lm(G) i s non-
This gives t h e equation
Dt(m(x)Iv(x,t)IP-l
sgn v ( x , t ) )
-
Av(x,t)
=
0.
The change o f v a r i a b l e u = I v l P - ' sgn(v) p u t s t h i s i n t h e form
227
5
SINGULAR AND DEGENERATE CAUCHY PROBLEMS
(6.38)
D,(m(x)u(x,t))
-2=
with q
(2-p)/(.p-l)
-
n
(4-1)
2 0.
1 D.()u)q-2Dju(x,t)) j=1 J
= 0
This e q u a t i o n a r i s e s i n c e r t a i n
d i f f u s i o n processes; i t i s "doubly-degenerate"
since t h e leading
c o e f f i c i e n t may vanish and t h e power o f u may vanish independently.
Note t h a t t h e second o p e r a t o r i s n o t monotone so t h e equa-
t i o n i s n o t i n t h e form o f (6.21). Remark 6.24
[lo]
Theorem 6.2
i s from B r e z i s [l]. See Lagnese
f o r a v a r i a t i o n on Theorem 6.6 and corresponding p e r t u r b a -
t i o n r e s u l t s on such problems.
Theorem 6.9 i s contained i n t h e
r e s u l t s o f Bardos-Brezis [1] along w i t h o t h e r types o f degenerate e v o l u t i o n equations n o t covered here ( c f . Lagnese [5] f o r c o r responding p e r t u r b a t i o n r e s u l t s ) . a r e from Showalter [5]. Lions
[lo]
Theorems 6.15,
We r e f e r t o Aronson
6.16 and 6.18
[l],Dubinsky [l],
and R a v i a r t [l]f o r a d d i t i o n a l r e s u l t s on (6.38)
and, s p e c i f i c a l l y , t o Strauss [l]f o r a d i r e c t i n t e g r a t i o n o f s e m i l i n e a r equations o f t h e form ( c f . (6.37)) (6.39)
Dt(N(u)) + A ( t ) u ( t ) = f ( t )
w i t h n o n l i n e a r N and time-dependent l i n e a r { A ( t ) ) i n Banach spaces.
Cf.
B r e z i s [l],Kamenomotskaya [l],and Ladyienskaya-
Solonnikov-Uralceva [2] f o r a r e d u c t i o n o f c e r t a i n ( S t e f a n ) freeboundary value problems t o t h e form (6.39), Lions-Strauss [8] and Strauss [2, (6.28).
hence, (6.36),
and
31 f o r a d d i t i o n a l r e s u l t s on
Crandall [3] and Konishi [l]have s t u d i e d s p e c i a l cases
228
3. DEGENERATE EQUATIONS WITH OPERATOR COEFFICIENTS
o f (6.39)
by d i f f e r e n t methods.
3.7
Doubly n o n l i n e a r equations.
We b r i e f l y consider some
f i r s t o r d e r e v o l u t i o n equations o f t h e form (7.1)
$(u(t))
t N(u(t)) = 0
where ( p o s s i b l y ) b o t h operators are nonlinear.
Some s o r t o f de-
generacy i s d e s i r a b l e (and necessary f o r a p p l i c a t i o n s ) so we s p e c i f i c a l l y do n o t make assumptions on M o f s t r o n g monotonicity as was done i n Theorem 3.12.
Moreover, we can i n c l u d e c e r t a i n
r e l a t e d v a r i a t i o n a l i n e q u a l i t i e s (e.g.,
by l e t t i n g one o f
t h e operators be a s u b d i f f e r e n t i a l ) so we p e r m i t t h e operators t o be m u l t i v a l u e d f u n c t i o n s o r r e l a t i o n s as considered i n Section 3.6. L e t B be a r e a l r e f l e x i v e Banach space, l e t D(M) and D(N) be subsets o f B and suppose M : D(M) m u l t i v a l u e d functions.
-+
That i s , M
B and N : D(N)
c D(M)
x
-+
B are
B and N C D(N) x B.
(When M and PI a r e f u n c t i o n s , we i d e n t i f y them w i t h t h e i r respect i v e graphs.)
We c a l l t h e p a i r o f f u n c t i o n s u,v a s o l u t i o n t o
t h e d i f f e r e n t i a l " i n c l usion"
if u(t)
E
D(M)fl
D(N), v : [o,m)
-+
hence, s t r o n g l y - d i f f e r e n t i a b l e a.e.,
N are f u n c t i o n s , then (7.1) whereas i f o n l y
B i s L i p s c h i t z continuous, and (7.2)
holds.
i s c l e a r l y s a t i s f i e d a.e.
I f M and on [o,m),
M i s a f u n c t i o n we r e p l a c e t h e e q u a l i t y symbol i n 229
SINGULAR AND DEGENERATE CAUCHY PROBLEMS
(7.1)
M[D(M) f l D ( N ) ] denote t h e
L e t D( A)
by an i n c l u s i o n .
i n d i c a t e d image i n B and d e f i n e a r e l a t i o n A : D (A ) composition A = N some z
E
D(M)
fl
t h e p a i r (u,v)
0
M- 1
.
T h a t i s , (x,y)
D ( N ) we have (z,x)
E
E
o f (7.3),
a u(t)
E
U ( A ) and (7.3)
holds.
E
N.
Thus, i f
then i t f o l l o w s t h a t
so we c a l l a L i p s c h i t z f u n c t i o n v : [ o , ~ ) if v ( t )
B by t h e
A i f and o n l y i f f o r
E
M and (z,y)
i s a s o l u t i o n o f (7.2),
-f
-f
B a s o l u t i o n o f (7.3)
Conversely, i f v i s a s o l u t i o n
t h e d e f i n i t i o n o f A shows t h e r e e x i s t s f o r each t L 0 D(M) fl D ( N ) f o r which t h e p a i r u,v
i s a solution of
(7.2). L e t M and N be m u l t i v a l u e d o p e r a t o r s on t h e
Theorem 7.1
r e a l r e f l e x i v e Banach space B; assume M + N i s o n t o B and t h a t
II (x1-x2)+s(Y1-Y*)11
2
II
x p 2
II
f o r s > 0,
Y
(7.4) and (z.,Y.) J J Then f o r each uo
E
E N , (z.,x.) J J
D ( M ) n D(N) and vo
s o l u t i o n p a i r u,v o f (7.2) Proof:
E
From (7.4)
E
j
=
1,2.
M(uo), t h e r e e x i s t s a
w i t h v ( o ) = vo.
By t h e p r e c e d i n g remarks
has a s o l u t i o n .
M,
t s u f f i c e s t o show (7.3)
i t f o l l o w s t h a t t h e sane e s t i m a t e
h o l d s f o r s > 0 whenever ( x . , y . ) E A; t h i s i s p r e c i s e l y t h e J J statement t h a t A i s a c c r e t i v e ( c f . C r a n d a l l - L i g g e t t [2]).
230
3. DEGENERATE EQUATIONS WITH OPERATOR COEFFICIENTS
Furthermore, i t i s easy t o check t h a t (x,y) i f f o r some z
E
D(M) n D ( N ) we have (z,y-x)
E
I + A i f and o n l y
E
N and (z,x)
E
M,
M + N.
so i t f ollows t h a t t h e range o f I + A equals t h e range o f
Thus I + A i s h y p e ra c c re ti v e and i t f o l l o w s from r e s u l t s o f C r a n d a l l - L i g g e t t [2] t h a t (7.3) vo
E
has a unique s o l u t i o n f o r each
D(A).
QED
Remark 7.2 then i t f ollows
11
If u ,v and u2,v2 a re s o l u t i o n s o f (7.2), 1 1
v l (t)-v 2 (t)
11
5 Ilv,(o)-v,(o)
11.
Ifvl(o)
=
v2(o) and e i t h e r M - l o r N - l i s a fu n c ti o n , then t h e s o l u t i o n p a i r u,v i s unique. Remark 7.3
I n a H i l b e r t space H w i t h i n n e r product (*,*),,
th e c o n d i t i o n (7.4)
i s equivalent t o whenever ( z .,x .) J J
(x1-x2Y Y1-YZ)H
E
M
(7.5) and ( z
y ) j yj
E
N
f o r j = 1,2.
For l i n e a r f u n c t i o n s t h i s i s t h e r i g h t angle c o n d i t i o n ( c f . Grabmuller [l],Lagnese [2, 4,
81,
and Showalter [6, 14, 15, 16,
201). The d i f f i c u l t y i n a p p l y i n g Theorem 7.1 t o i n i t i a l - b o u n d a r y value problems a r i s e s from t h e assumption (7.4) which r e l a t e s t h e operat ors M and N t o each other.
A desirable alternative
which we s h a l l d e s c ri b e i s t o p l a c e hypotheses on each o f t h e o perat ors independently.
The r e s u l t s o f S e c t i o n 3.6 were
231
SINGULAR AND DEGENERATE CAUCHY PROBLEMS
o b t a i n e d when t h e l e a d i n g ( l i n e a r ) o p e r a t o r was symnetric.
The
n o n l i n e a r analogue o f t h i s i s t h a t t h i s o p e r a t o r be a d i f f e r e n t i a l o r , more g e n e r a l l y , a s u b d i f f e r e n t i a l . subdifferential a t x
E
Recall t h a t t h e
E o f an e x t e n d e d - r e a l - v a l u e d l o w e r semi-
c o n t i n u o u s convex f u n c t i o n $ on t h e l o c a l l y convex space E i s t h e s e t a$(x) o f those u
(Cf. B r e z i s [3],
E
E
I
verifying
Ekeland-Ternan [l],L i o n s
w i l l be g i v e n below.
Some examples
I n g e n e r a l , t h e s u b g r a d i e n t a$ i s a m u l t i -
valued operator from (a subset o f ) E i n t o Theorem 7.4
[lo].) E
I
.
L e t V1 and V2 be r e a l separable r e f l e x i v e
Banach spaces w i t h V1 C V2, i n c l u s i o n i s compact.
Let
V1 i s dense i n V2, (I
1
and $z be convex c o n t i n u o u s (ex-
tended) r e a l - v a l u e d f u n c t i o n s on V1 and V2, sume t h e i r s u b d i f f e r e n t i a l s
N
and assume t h e
=
a$l and M E
respectively.
As-
a$z a r e bounded ( c f .
and t h a t N i s " c o e r c i v e " i n t h e sense t h a t
S e c t i o n 3.6)
: l u l v -+ -1 > o f o r some p, 1 < p < m. v1 1 Suppose we a r e g i v e n uo E V1, vo E M(uo), and f E Lm(0,T;V;) l i m i n f I$~(U)/IUIP
with df/dt
E
a p a i r o f functions u L~(O,T;V;)
a.e.
where l / p + l / q = 1.
Lq(0,T;V;) E
Lm(o,T;V1),
satisfying
on [o,T]
and v ( o ) = vo. 232
v
E
Then t h e r e e x i s t s
Lm(0,T;V;)
w i t h dv/dt
E
3. DEGENERATE EQUATIONS WITH OPERATOR COEFFICIENTS
Remark 7.5
The compactness assumption above l i m i t s a p p l i c a -
t i o n o f Theorem 7.4 t o p a r a b o l i c problems; c f . examples below. We i l l u s t r a t e some a p p l i c a t i o n s o f t h e p r e c e d i n g r e s u l t s t h r o u g h some examples o f i n i t i a l - b o u n d a r y v a l u e problems.
p
W1’p(G),
T : V
-f
(6.30)
V
I
2
2 , and d e f i n e t h e n o n l i n e a r e l l i p t i c o p e r a t o r
by (6.32),
and (6.31).
2
mj(x)
V
I
, and
the closure o f Ci(G) i n
L e t V = W;”(G),
Example 7.6
0, a.e.
x
E
d e f i n e D(M)
m2T(u); D(N)
{u
2
E
where t h e f u n c t i o n s N.(x,y) s a t i s f y (6.29), J F o r j = 1,2, l e t m . ( * ) E Lm(G) be g i v e n w i t h J G, and l e t k > 0. S e t H = L 2 ( G ) so V C H
c
2
{u
E
V : mlu + m2T(u)
V : (k/ml(*))Tu
E
E
H I , M(u)
mlu +
H I , N(u) :( k / m l ( * ) ) T ( u ) .
M and N a r e ( s i n g l e - v a l u e d ) n o n l i n e a r o p e r a t o r s i n t h e
Then
H i l b e r t space H.
To a p p l y Theorem 7.1,
i n i t s e q u i v a l e n t form (7.5).
( M U - M V ~ N U - N V =) ~ k + dx 2 0, so (7.4)
holds.
I
F o r u,v
we f i r s t v e r i f y (7.4) E
D(M) f l D(N) we have
( km2(x)/ml ( x ) ) ( T ( u ) - T ( v ) G To show M + N i s o n t o H, l e t w E H and
consider the equation
The c o e f f i c i e n t s a l l belong t o Lm(G), so t h e o p e r a t o r on t h e l e f t s i d e o f (7.7) to V
I
, hence,
Since m,(*)
u
E
i s monotone, hemicontinuous and c o e r c i v e from V s u r j e c t i v e , so t h e r e i s a s o l u t i o n u
E
V o f (7.7).
and m 2 ( - ) a r e bounded, i t f o l l o w s from (7.7)
D ( M ) n D(N) and t h a t M(u) + N(u) = w.
that
Thus, Theorem 7.1
g i v e s e x i s t e n c e o f a weak s o l u t i o n o f an a p p r o p r i a t e i n i t i a l 233
SINGULAR AND DEGENERATE CAUCHY PROBLEMS
boundary value problem c o n t a i n i n g t h e equation
where T[u]
i s g i v e n by (6.33).
Flote t h a t t h e c o e f f i c i e n t s m1
and m2 may vanish over c e r t a i n subsets o f G. We a p p l y Theorem 7.4 w i t h V1 = WAYp(G) and V2 = n La(G), ct 5 P, Q1(u) = l / p 1 IDju(X)IPdx, @,(u) = G j=l lu(x)j"dx, and a p p ro p ri a te f : [o,T] -+ W-lgq(G). The c o r l/ct IG responding s u b d i f f e r e n t i a l s a re g i v e n by 34, ( u ) = Example 7.7
we o b t a i n exist e n c e o f a s o l u t i o n o f an i n i t i a l - b o u n d a r y value problem f o r t h e equation
Example 7.8 define @,(u)
= l/a
@1and f be given as above b u t -c u (x)l"dx, u E V2, where u ( x ) = 0 when
L e t V1,V2,
iGl
+
u ( x) < 0 and u+(x) = u ( x ) when u ( x ) >, 0.
Theorem 7.4 g i v e s
e x i st ence f o r an i n i t i a l - b o u n d a r y value problem f o r
Example 7.9
2 max {1/2,1ulH/21.
L e t V2 be a H i l b e r t space and s e t @,(u) The s u b d i f f e r e n t i a l i s g i v e n by
lulH
1 and t h e " e l l i p t i c " equa-
N ( u ( t ) ) where l u ( t ) l < l.
Remark 7.10
Theorem 7.1 i s an e x te n s i o n o f t h e correspond-
i n g r e s u l t o f Showalter [4] when M and N a r e f u n c t i o n s . 7.4 and Examples 7.7,
Theorem
7.8 and 7.9 a re from Grange-Mignot [l].
See Barbu [l]f o r r e l a t e d a b s t r a c t r e s u l t s and examples where M and N are s u b d i f f e r e n t i a l s which a re assumed r e l a t e d by an assumption s i m i l a r t o (7.4)
b u t d i s t i n c t from it.
(7.9) was s t u d i e d d i r e c t l y i n R a v i a r t [2]; L i o ns
[lo].
The equation
c f . Chapter IV.1.3
of
A doubly-nonl i n e a r parabolic-hyperbol i c system i s
discussed by Volpert-Hudjaev [l],and p a r a b o l i c systems a r e considered by Cannon-Ford-Lair [3].
These l a s t two references
i n d i c a t e how c e r t a i n doubly-nonl i n e a r equations a r i s e i n a p p l i c a t i o n s ; f o r a p p l i c a t i o n s o f equations o f t h e type (7.8), Gajewski-Zachari us [2].
235
cf.
This page intentionally left blank
Chapter 4 Selected Topics 4.1
Introduction.
I n t h i s chapter we w i l l discuss
b r i e f l y some f u r t h e r questions a r i s i n g i n t h e study o f s i n g u l a r o r degenerate equations.
Some o f t h e p r e s e n t a t i o n w i l l be i n a
more " c l a s s i c a l " s p i r i t and we w i l l f o l l o w t h e o r i g i n a l papers i n d e s c r i b i n g problems and r e s u l t s .
Proofs w i l l o f t e n o n l y be
sketched o r even o m i t t e d e n t i r e l y , m a i n l y f o r reasons o f space and time, b u t s u i t a b l e references w i l l be p r o v i d e d ( w i t h o u t a t tempting t o be exhaustive i n t h i s r e s p e c t ) ; f u r t h e r r e l e v a n t i n formation can o f t e n be o b t a i n e d from t h e b i b l i o g r a p h i e s i n these references.
The m a t e r i a l considered by no means covers a l l pos-
s i b l e questions and has been s e l e c t e d b a s i c a l l y t o be represent a t i v e o f t h e h i s t o r i c a l development and t o r e f l e c t some c u r r e n t research trends i n t h e area.
Thus we w i l l deal f o r example w i t h
Huygens' p r i n c i p l e , r a d i a t i o n problems, i n i t i a l - b o u n d a r y value problems, nonwellposed problems, e t c . , among o t h e r questions; t h e s e c t i o n t i t l e w i l l i n d i c a t e t h e problem under c o n s i d e r a t i o n . 4.2 and (1.4.10)
Huygens' p r i n c i p l e . with A
X
R e f e r r i n g back t o (1.4.8)
= -A l e t us suppose m t p =;
-
1 with
Ta
f u n c t i o n so t h a t wm+P(t) i s g i v e n i n terms o f a s u r f a c e mean value u x ( t ) want m 2 n
-
-
*
T o f T( A0 and o S t 5 b (ko and ho a r e independent o f ( y , t ) ) . The corresponding unique r e s o l v a n t Z!(t)
E l has support con-
E
*
t a i n e d i n t h e b a l l 1x1 5 t and u m ( t ) = Z!(t) s o l u t i o n o f (2.2) i n D Remark 2.2
I
.
The e s t i m a t e (2.5) i s u s e f u l i n determining t h e i n p a r t i c u l a r f o r Rem + 1/2 >
o r d e r o f t h e d i s t r i b u t i o n Z!(t); n
-
T i s t h e unique
1 i t f o l l o w s t h a t Z!(t)
w i l l be continuous i n x f o r 0 < t i b.
We w i l l say now t h a t (2.2) i s o f Huygens' t y p e i f and o n l y i f t h e support o f Z!(t)
i s concentrated on t h e sphere 1x1 = t.
Note here t h a t t h i s d e f i n i t i o n depends on t h e choice o f i n i t i a l data when m =
-
1 / 2 and n = 1 f o r example s i n c e t h e one dimen-
s i o n a l wave o p e r a t o r i s n o t a Huygens' o p e r a t o r i n t h e sense p r e v i o u s l y d e l i m i t e d f o r a r b i t r a r y Cauchy data u(x,o) ut(x,o).
0, 1, 2,
Solomon uses t h e d i s t r i b u t i o n s 6 ( ' ) ( \ x l
..., d e f i n e d
2
and
2 -t 1, v = V
f o r t > 0 and n > 1 by ( c f . Gelfand-Silov
[51)
Here 1x1 = p , 8 denotes t h e n
-
-
1 angular v a r i a b l e s (el,
...,
i n p o l a r s p h e r i c a l coordinates w i t h + ( p , B ) = + ( x ) , and Rn i s t h e s u r f a c e o f t h e u n i t sphere i n Rn.
240
For n = 1 one w r i t e s
4.
dV)(1 x
(2.7)
SELECTED TOPICS V
=
L[& wax)
[6(x+t) + 6(x-t)]
21tl
c l e a r l y the d i s t r butions
&'((XI 2- t2 )
a r e c o n c e n t r a t e d on t h e
sphere 1x1 = t.
Then t h e main r e s u l t i n Solomon [ll i s g i v e n by
Theorem 2.3
A necessary and s u f f i c i e n t c o n d i t i o n f o r ( 2 . 2 )
-
t o be o f Huygens ' t y p e i s t h a t n
(2m+l) = 2N + 1 w i t h N L 0 an
i n t e g e r w h i l e t h e r e e x i s t s a r e s o l v a n t Z:(t)
where c,
= (-1 )Nn-n'2r(m+l),
a.
=
o f the form
1 , and f o r v = 1 ,
. . ., N
(if
N > 0 ) one has E-m(aN) = 0 w i t h 4(ta\: + vav) = E-m(av-l)
(2.9)
Remark 2.4
0, 1, 2, (2.10)
V
We r e c a l l from G e l f a n d - S i l o v [51 t h a t f o r v
... FS(')(
where c(v,n)
2v-n+221/2(n-2)-vJ 1 (2) -(2 n-2) -v and z = t l y l . Then examples t o
1x12-t2) = ;(v,n)lyl
= 7(-1/2)v(2n)n/2 1
i l l u s t r a t e Theorem 2.3 can be g i v e n as f o l l o w s . t h a t Z:(t), a
V
=
Suppose f i r s t
expressed b y (2.8), has p r e c i s e l y one t e r m ( i . e . ,
= 0 f o r v _> 1 ) .
Then by ( 2 . 9 ) E-m(ao) = c ( t ) = 0 s i n c e a.
1 and we a r e i n t h e EPD s i t u a t i o n .
i n (2.8) w i t h o f n e c e s s i t y n i n t e g e r as i n d i c a t e d .
-
=
L e t N be t h e i n t e g e r i n v o l v e d
(2m+l) = 2N + 1 an odd p o s i t i v e
To check t h e f o r m o f Z:(t)
g i v e n by ( 2 . 8 )
i n t h i s case we use (2.10) t o o b t a i n Zm(y,t) = cmt-ZmaoFG(N)
241
SINGULAR AND DEGENERATE CAUCHY PROBLEMS
2 2 1 (1x1 - t ) and s i n c e $n-2)
- N
Am
= m t h i s y i e l d s Z (y,t)
"m 2mr(m+l)z-mJm(z) = R (y,t) as r e q u i r e d by (1.3.6).
E-m(ao) = c ( t ) o r c ( t ) = ( t a , ) '
+ al
= 0 we have ta;
and E-m(al)
+;
=
+ 1 2m)
= a;'
Combining t h e s e two e q u a t i o n s one has al
c ( t ) al = 0.
2mal
N e x t suppose
av = 0 f o r v _> 2 ) .
t h a t ( 2 . 8 ) has no more t h a n two terms ( i . e . , From (2.9) and t h e r e q u i r e m e n t E-m(al)
=
I '
t
+
9
+t2 2 al=
ct 0
for
cto
any c o n s t a n t .
' " w /w t h i s becomes w 2 (log w)' with c ( t ) = 2 (log Now i n terms o f w we have a l ( t ) = ables al(t)
=
2
Making a change o f v a r i ct 2m+l ' o 7 -2 w = 0.
w)" and s u i t a b l e w can be found as f o l l o w s .
If
= 0 we g e t
cto
w ( t ) = cr + f3t2m+2 where a and B a r e a r b i t r a r y c o n s t a n t s b u t may depend on m.
I n o r d e r t h a t t h e r e s u l t i n g a,
be such t h a t (2.8)
s a t i s f i e s t h e r e s o l v a n t i n i t i a l c o n d i t i o n s we must have
-
while f o r m =
B = 0.
1 / 2 i t i s necessary t h a t
If
cto
c1
j o
f 0 the
s o l u t i o n s w o f t h e d i f f e r e n t i a l e q u a t i o n f o r w w i l l be o f t h e k f o r m w = exp ( 4 2 ) F ( - ;T; - k;
T)
where
T
= tal/',
k = 2m+l
, and
F (y; 6; t ) i s any s o l u t i o n o f Kunnner's c o n f l u e n t hypergeometric equation tF"
+
(6
-
t ) F' + y F = 0.
Upon e x a m i n a t i o n o f t h e be-
h a v i o r o f such F(y;6; t ) as t + o i t f o l l o w s t h a t f o r t h e r e s u l t i n g a, t o s a t i s f y t h e a p p r o p r i a t e i n i t i a l c o n d i t i o n s F must be k k t a k e n i n t h e f o r m F = alF1 ( - 2; - k; T ) + f3 F2 ( - 2; - k; T) where lF1y
F2 i s any s o l u t i o n o f t h e Kumner e q u a t i o n independent o f a f
0,
and when m =
-
1/2 ( i . e .
k =
s e r v a t i o n s can be f o u n d i n Solomon [1;21.
242
0)
6 =
0.
F u r t h e r ob-
SELECTEDTOPICS
4.
We w i l l deal here e x p l i c i t l y o n l y w i t h some work
Remark 2.5
o f Fox [l]on t h e s o l u t i o n and Huygens' p r i n c i p l e f o r s i n g u l a r Cauchy problems o f t h e t y p e
w i t h data um(x,o) = T ( x ) and u;(x,o)
= 0 g i v e n on t h e s i n g u l a r
Except f o r some r e s u l t s o f Gunther ( l o c . c i t . )
hyperplane t = 0.
most o f t h e o t h e r work on Huygens' p r i n c i p l e mentioned e a r l i e r deals w i t h n o n s i n g u l a r Cauchy problems so we w i l l n o t discuss i t here; Gunther's work on t h e o t h e r hand i s expressed i n a more geometric language which r e q u i r e s some background n o t assumed o r developed i n t h i s book and hence t h e d e t a i l s w i l l be omitted. Thus l e t
r
=
r
(x,t;
C)
= t
2
- n1 ( x k -
Ck)
k= 1 L o r e n t z i a n d i s t a n c e between a p o i n t ( x , t )
(5,o) i n t h e s i n g u l a r hyperplane t =
0.
2 E
be t h e square o f t h e
Rn x R
and a p o i n t
We denote by (2.11)-,,
t h e e q u a t i o n (2.11) w i t h index -m and observe t h a t i f u - ~ satisf i e s (2.11)-,, 1.4.7).
( 5k/xk)
then um =
t-2m
-m u s a t i s f i e s (2.11) ( c f . Remark
Fox composes t h e t r a n s f o r m a t i o n s zk =
r
( k = 1,
. . ., n )
1
/4xkCk and yk =
t o produce new v a r i a b l e s (z,r,y)
p l a c e o f (x,t,C)
( s u i t a b l e regions a r e o f course d e l i n e a t e d ) . n m- s o l u t i o n o f (2.11)-,, i s sought i n t h e form u-" = r 2 V(z) where V(z) i s t o be determined.
in
A
A f t e r some computation one shows
t h a t such umma r e s o l u t i o n s i f V ( z ) s a t i s f i e s a c e r t a i n system o f n o r d i n a r y d i f f e r e n t i a l equations and such a V(z) can be found i n t h e form V = FB where FB i s t h e L a u r i c e l l a f u n c t i o n given f o r
243
SINGULAR AND DEGENERATE CAUCHY PROBLEMS
l z k l < 1 by ( c f . Appell (2.12)
-
Kamp6 de F 6 r i e t 111)
where ak =
1
+
1
n
-
FB = FB (a, 1-a, m
+ 1, z )
and (8,s) = r(R+q)/I'(R).
(1-4Ak)'/2
The s e r i e s
d e f i n i n g FB converges u n i f o r m l y i n ( a , m , z ) on closed bounded regions where l z k l < 1 and m t h a t s i n c e ak and 1
-
-
n
. . ..
+ 1 # 0, -1, -2,
a k a r e complex conjugates f o r
Xk r e a l ,
t h e products ( a k ,p k ) ( l - a k y p k ) a r e r e a l , and thus FB
m and t h e Ak a r e r e a l .
I f Rem >
n 2 -1
and l x k l >
One notes
It1
s r e a l when > 0 it
makes sense then t o c o n s i d e r as a p o s s i b l e s o l u t i o n o
the sin-
g u l a r Cauchy problem f o r (2.11) t h e f u n c t i o n . (2.13)
m uX(x,t,T)
= k,
where km = r(mc1)/vn/2r(m-"+l) 2 i s a s o l u t i o n o f (2.11).
n
and v!(x,t,C)
=
Itl-2mrm-T FB
We assume here t h a t T ( - ) i s s u i t a b l y
d i f f e r e n t i a b l e and denote by S ( x , t )
the region o f the singular
hyperplane t = 0 c u t o u t by t h e c h a r a c t e r i s t i c conoid w i t h vertex (x,t)
( t h u s S(x,t)
i s an n-dimensional sphere o f r a d i u s
It1 and c e n t e r x i n t h i s hyperplane determined b y r ( x , t , S ) 2 0 ) . As described, w i t h l x k l >
It1
2 0, S ( x , t )
does n o t i n t e r s e c t any
o f t h e hyperplanes xk = 0 and l i e s i n an o c t a n t o f t h e hyperplane t = 0.
Note here a l s o t h a t t h e f a c t o r r(m-;+l)-l
n p l a y e d o f f a g a i n s t (m - ? +
S 1, Ipk)-'
i n k,
can be
i n FB t o remove any d i f 1 n t 1 = -p. Now f o r Rem > + 1 one checks f i c u l t i e s when m244
5
4.
e a s i l y t h a t Lyu:
SELECTEDTOPICS
T(
1/2
m
(3.5)
v (t,r) =
~n r z(z2-r2)m-1/2 2 m+1/2
m
um(t,z)dz
Then, i f u" s a t i s f i e s (3.3) w i t h u m ( t , r ) = 0 f o r t 5 r, i t m f o l l o w s t h a t vTt = vrr w i t h v m ( t , r ) = 0 f o r t 5 r.
v m ( t , r ) = F m ( t - r ) where Fm(s) = 0 f o r s i 0.
Consequently
Now ( c f . below)
(3.5) can be i n v e r t e d i n t h e form
For Rem
= <s(E,), 1
5-1/2
for
$
E
qnx)).
c a l c u l a t i o n y i e l d s M f T = ( 4 5 I? + 2D ) MT f o r
5
Dx = d/dx, e t c . ) .
5
T
A simple E
D'(Rx) ( h e r e
Now r e f e r r i n g t o t h e s t a n d a r d spaces D 248
I
,
SELECTEDTOPICS
4.
D i , and D: o v e r (l/r(p))
R as i n Schwartz [ll, we s e t f o r p
Pf(xP-’)lx>b
Schwartz [ l ] ) .
Then Y
E
P Y
E, Y
E
P D i ( t h e n o t a t i o n P f i s explained i n
$.a
= o f o r x < 0, Y-R =
=
f o r R a non-
I
: Q: -+ D+ i s a n a l y t i c and Y P P * y q = I S e t now Z ( x ) = Y ( - x ) so t h a t Z E D 9 Z = 0 f o r x > 0, yP+q P P P P Z-, = (-I)‘#& f o r R a n o n n e g a t i v e i n t e g e r , etc.; i n p a r t i c u l a r
negative integer, p
-+
.
DZ
= -Zp-l and XZ = - P Z ~ + ~ .F i n a l l y one n o t e s t h a t T S*T i s P P I a c o n t i n u o u s l i n e a r map Di(n) D-(fi) when S E D- i s z e r o f o r -+
I
-+
1 S(x-y)T(y)dy) m
x > 0 ( f o r m a l l y (S*T) ( x ) =
*
Zp
JX
L (D:(a),
E
DL(S2)) t h e map T
Now l e t Lm = DE t @-$
,L
that
D-(n) I
:
-f
Dx (L,
i n t o Lm on D
(a)
For m
(i.e.,
DH ,
2
H,
= M-
-+
1 O
E
Q: L i o n s con-
* z-ln-1/2
0
M o f D2
= LmHm) such t h a t Hm:
I
I
D-(O)
= L i i n Chapter 1 ) and n o t e
D (S2) i s continuous.
s t r u c t s a transmutation operator I
Zp*T.
I
-+
and we denote by
D ( Q ) i s an isomorphism and m
-+
H, i s an e n t i r e a n a l y t i c
The i n v e r s e H, = Hi’ = 2 M, which n e c e s s a r i l y s a t i s f i e s D H, = HmL, on
f u n c t i o n w i t h v a l u e s i n L(Dl(S2), D l ( 0 ) ) .
* zm+1/2
v-’ D- ( a ) , e n j o y s 0
0
s i m i l a r p r o p e r t i e s and i n f u n c t i o n n o t a t i o n one
can w r i t e (3.11)
ff,T(x)
(3.12)
HmT(x) =
=
mr ~x ( y ~ - x ~ ) - ~ - ~ / ~ T (Rem y ) d
-
1/2
$ one extends t h e s e formulas by a n a l y t i c
I t i s important t o note t h a t i f T
249
E
Dl(n) i s
SINGULAR AND DEGENERATE CAUCHY PROBLEMS
L i o n s phrases
zero f o r x > aT t h e n HmT = HmT = 0 f o r x > aT. t h e g e n e r a l i z e d r a d i a t i o n problem as f o l l o w s . Let P
Problem 3.1
c R x R be t h e open h a l f p l a n e ( t , r )
where r > 0 and Q
cP
Rem < 0 f i n d urn
D l ( P ) s a t i s f y i n g (3.3) w i t h supp u m c Q and
um(-,r)
+
E
be t h e s e t Q = { ( t , r )
f i n D I as r
+
0 where f
E
E
P; t 2 r ) .
:D i s g i v e n w i t h f
For
= 0 for
t < 0.
The problem i s r e s t r i c t e d t o Rem < 0 s i n c e i t i s shown n o t t o be m e a n i n g f u l f o r o t h e r m Theorem 3.2
4
E
There r e s u l t s
( c f . L i o n s [4]).
Problem 3.1 has a unique s o l u t i o n urn depending I
continuously i n D (P) 0 n . f
D,
E
I
( f as d e s c r i b e d ) .
F o r m a l l y t h e p r o o f goes as f o l l o w s .
One w r i t e s vm =
( 1 @Hm)um so t h a t vm i s g i v e n by (3.5) f o r Rem > - 1/2 say ( c f . (3.12)). v"
E
m t o (3.3) one has vtt = vrr
A p p l y i n g 1 @ H,
D ' ( P ) has i t s s u p p o r t i n Q.
as b e f o r e w i t h Fm
E
and
Consequently v m ( t , r ) = F m ( t - r )
D I and Fm(s) = 0 f o r s < 0.
Since H,
i s an
m isomorphism w i t h H i ' = Hm we have u = ( 1 C9 Hm)vm and f o r Rem
= ( Z / r ( - m - l / Z ) ) Rem
z0,
The f u n c t i o n m
-+
S i can i n f a c t be
extended by a n a l y t i c c o n t i n u a t i o n t o be e n t i r e w i t h v a l u e s i n
250
4.
SELECTED TOPICS
D i and, f o r Rem < 0, a s r + O , i n .D,
I
Sk
Hence f r o m (3.13) as r
+
+
(Zr(-2m-l)/r(-m-l/Z))
Y-2m-l
0
and t h i s l i m i t must equal f ( c f . ( 3 . 9 ) ) .
Consequently t h e ex-
p l i c i t s o l u t i o n o f Problem 3.1 f o r Rem < 0 i s (3.15)
um(t,r)
w h i c h can be r e w r t t e n i n a f o r m e q u i v a l e n t t o (3.10)
(cf
Lions [41). 4.4
I m p r o p e r l y posed problems.
I m p r o p e r l y posed problems
such as t h e D i r i c h l e t problem f o r t h e wave e q u a t i o n or t h e Cauchy problem f o r t h e Laplace e q u a t i o n have been s t u d i e d f o r some t i m e and have r e a l i s t i c a p p l i c a t i o n s i n p h y s i c s .
For a
n o n e x h a u s t i v e l i s t o f r e f e r e n c e s see e. g., A b d u l - L a t i f - D i a z [lI, Agmon-Nirenberg [31, B o u r g i n [ll, B o u r g i n - D u f f i n [21, B r e z i s - G o l d s t e i n [ 5 ] , Dunninger-Zachmanoglou [21, Fox-Pucci [21 , John [23, L a v r e n t i e v [ Z ] , [121, Ogawa [1;23,
L e v i n e [l;5; 10; 111, Levine-Murray
Pucci [21 , S i g i l l i t o 141
by Payne [31 ( c f . a l s o
-
and a r e c e n t survey
Symposium on nonwellposed problems and
l o g a r i t h m i c c o n v e x i t y , S p r i n g e r , L e c t u r e notes i n mathematics, Vol. 316, 1973).
I n p a r t i c u l a r t h e r e has been c o n s i d e r a b l e
i n v e s t i g a t i o n o f i m p r o p e r l y posed s i n g u l a r problems by DiazYoung
[lo]
Dunninger-Levine [lI, Dunninger-Weinacht
[31
T r a v i s [ll, Young [4; 5; 7; 8; 113, e t c . and we w i l l d i s c u s s 251
SINGULAR AND DEGENERATE CAUCHY PROBLEMS
some o f t h i s l a t t e r work. The most general a b s t r a c t r e s u l t s seem t o appear i n Dunninger-Levine [l]and we w i l l sketch t h e i r technique f i r s t . L e t E be a r e a l Banach space and A a closed densely d e f i n e d l i n e a r o p e r a t o r i n E (as i s well-known t h e use o f spaces
R instead o f
i n v o l v e s no l o s s o f g e n e r a l i t y ) .
E over
We s h a l l denote
by cs ( A ) t h e p o i n t spectrum o f A. P D e f i n i t i o n 4.1
The E valued f u n c t i o n urn(-) i s s a i d here t o
be a s t r o n g s o l u t i o n o f
on (o,b) i f urn(-) i s norm continuous on (o,b), weakly, and f o r any e
I
E
E
urn(*)
E
C2(E)
I
m I 2m+1 rn I m ' + - + = 0
(4.2)
t
m on ( o , b ) , where = - am, e l > , etc., and w r i t i n g dt2 p ( t ) = t2m+1 11 A u m ( t ) l l f o r m > - 1/2 w i t h p ( t ) = t [I A u m ( t ) l l 1 f o r m < - 1/2, one has p ( * ) E L (o,b).
The n o n s i n g u l a r case m = 1/2 i s encompassed i n forthcoming work o f Dunninger-Levine and w i l l n o t be discussed here. now p = I m l and l e t (An)
{An)
denote t h e p o s i t i v e r o o t s o f
b) = 0 f o r m 2 0 o r m
- 1 / 2 assume
D e f i n i t i o n 4.1 w i t h um(b) = 0.
11 tlI
t2m+1(
u y l l + t l I u m I I ) - + 0 as t
that
u y l l + Ilum 11
t h e sequence {An}
-+
-+
0 as t
s a t i s f i e s An
0 while f o r m < +
0.
4
S I
-
1/2 assume
Then urn :0 i f and o n l y i f
( A ) f o r a l l n. P The p r o o f can be sketched as f o l l o w s . Consider t h e eigen-
v a l u e p r o b l em
(4.3)
$ ( b ) = 0; t 1 I 2 $ ( t ) bounded
and t a k e f o r m 2 0 o r m
0,
Ro ( r e s p .
T
(ttT)2
-
Go)
denotes t h e 2 (x+E)2 - (y-rl) y
> 0, bounded by t h e r e t r o g r a d e
*
D C D), and
cone w i t h v e r t e x a t ( - x , y , t ) ( t h u s
T i s the region
i n t e r c e p t e d on t h e p l a n e 5 = 0 by t h e cone R = 0
*
that for x 2 t > 0 D g r a l s o v e r D and
Theorem 5.4
*
and T a r e empty w h i l e as x
D cancel each o t h e r s i n c e D
*
(T +
Note
> 0).
0 the inteh
+
D and (R,R)
+
The s o l u t i o n o f Problem 5.3 i s g i v e n by (5.5).
I t i s i n t e r e s t i n g t o n o t e what happens when a l l o f t h e d a t a
i n an EPD t y p e problem a r e a n a l y t i c .
Thus f o r example Fusaro
[3] c o n s i d e r s f o r t > 0, 1 5 k 5 n, and x = (xl,
(5.6)
. . .,
u"tt + 2m+l t um t = F(x,t,um,Dku",u~,Dku~,DkDeum); m u (x,o)
=
f(x);
m
Ut(X,O)
266
= 0
Xn)
SELECTEDTOPICS
4.
where Dk = a/axk w h i l e F and f a r e a n a l y t i c i n a l l arguments. Theorem 5.5
-1, -3/2,
for m
The problem (5.6)
....
-2,
C o r o l l a r y 5.6
has a unique a n a l y t i c s o l u t i o n
. . .)
I f F(x,t,u,
=
G(x,u,Dku,D
k Dau ) +
+ h ( t ) u t w i t h g even i n t and h odd i n t, t h e n t h e problem
g(x,t)
has f o r a l l m f -1,
(5.6)
-3/2,
...a
-2,
unique a n a l y t i c s o l u -
. . . there exists
I f m = -3/2,-5/2,
t i o n and i t i s even i n t.
a unique s o l u t i o n i n t h e c l a s s o f a n a l y t i c f u n c t i o n s which a r e even i n t. Fusaro [3] uses a Cauchy-Kowalewski t e c h n i q u e t o p r o v e these r e s u l t s and we r e f e r t o h i s paper f o r d e t a i l s .
One s h o u l d n o t e
a l s o t h e f o l l o w i n g r e s u l t s o f L i o n s [l],o b t a i n e d by transmutat i o n , which i l l u s t r a t e f u r t h e r t h e r o l e p l a y e d b y even and Cm f u n c t i o n s o f t i n t h e uniqueness q u e s t i o n f o r EPD equations. L e t E,
be t h e space o f Cm f u n c t i o n s f on [o,m)
f(2n+1)(o) = 0 (thus f (-0,m))
and g i v e E,
E
E,
such t h a t
admits an even Cm e x t e n s i o n t o
t h e topology induced by
E on
(-,a).
Let H
be a H i l b e r t space and A a s e l f a d j o i n t o p e r a t o r i n H such t h a t f o r a l l h c D(A) (Ah,h) Lm=D
*'+'
+-
t
+ allh112 Z 0 f o r some r e a l a. S e t t i n g
D and p u t t i n g t h e graph t o p o l o g y on D(A) L i o n s
proves Theorem 5.7
For any m
e x i s t s a unique s o l u t i o n um
E E
f, m =f -1, -2, E,
267
h
BT D(A)
-3,
. . ., t h e r e
o f Lmum + hum = 0 f o r
SINGULAR AND DEGENERATECAUCHY PROBLEMS
t 2 0 w i t h um(o) = h
Here
T
and uT(o) = 0.
D(A")
denotes t h e p r o j e c t i v e l i m i t t o p o l o g y o f Grothendieck
( c f . Schwartz [5]) i s nuclear.
E
which c o i n c i d e s w i t h t h e
I f m f -1,
Coo f u n c t i o n s on [o,m)
-3/2,
w i t h t h e standard E-type topology then any
s o l u t i o n urn E E + B ~ D(A) o f E
topology s i n c e E,
. . . and E+ i s t h e space o f
-2,
A
h
E
L~U"'
+ hum = o s a t i s f y i n g um(o)
=
A
D(A")
and uT(o) = 0 i s a u t o m a t i c a l l y i n E * B T D ( A ) so t h e *
corresponding problem i n E+ BT D ( A ) has a unique s o l u t i o n . t h e e x c e p t i o n a l values m = -1, -2, Theorem 5.8
For
. . . Lions [l]proves
-3,
Let m = -(p+l) with p
2
0 an i n t e g e r so t h a t h
2m + 1
There e x i s t s a unique s o l u t i o n urn E E + q D ( A ) -(2p+l). m m 2p+l m + Au = 0, um(o) = uY(0) = = Ut u ( 0 ) = 0,
of u L,
...
DZP+*um(o) = h
E
D(Am), and DZP+3um(o) = 0.
I n connection w i t h t h e uniqueness q u e s t i o n we mention a l s o t h e f o l l o w i n g r e c e n t r e s u l t o f Donaldson-Goldstein [Z].
Let H
be a H i l b e r t space and S a s e l f a d j o i n t o p e r a t o r i n H w i t h no n o n t r i v i a l n u l l space.
L e t Lm = D2
+ 2m+l D again and d e f i n e urn t
t o be a t y p e A s o l u t i o n o f m LmU + S2um = 0;
(5.7)
if urn
E
2
um(o) = h
E
D(S2);
m ut(o) = 0
and a t y p e B s o l u t i o n i f urn E C1 ([o,m),H)n
C ([o,m),H)
C2( (oYrn),HI. Theorem 5.9 solution f o r m
The problem (5.7)
2 -1. 268
can have a t most one type' A
4.
SELECTED TOPICS
F u r t h e r , by v i r t u e o f (1.4.10) ( c f . Remark 1.4.7),
and t h e formula um =
t-2mu-m
f o r m < -1/2 n o t a n e g a t i v e i n t e g e r t h e r e i s
no uniqueness f o r t y p e B s o l u t i o n s .
We n o t e here t h a t f o r -1
as t
-+
- 21 < m 5
0 and assume (5.16)
holds.
If
R 1 i s a c l a s s i c a l s o l u t i o n o f Lmu = f ( u ) w i t h
u(x,o) = u ( x ) , ut(x,o) 0
C2(E)
Levine proves
Theorem 5.12
u : R x [o,T)
E
1 2 7 ~ R I V u o ldx,
= 0, and u = 0 on
then n e c e s s a r i l y T