NEW DEVELOPMENTS IN DIFFERENTIAL EQUATIONS
This Page Intentionally Left Blank
N 0 RTH- H0 LLAND MATHEMATICS STUDIES...
33 downloads
877 Views
8MB Size
Report
This content was uploaded by our users and we assume good faith they have the permission to share this book. If you own the copyright to this book and it is wrongfully on our website, we offer a simple DMCA procedure to remove your content from our site. Start by pressing the button below!
Report copyright / DMCA form
NEW DEVELOPMENTS IN DIFFERENTIAL EQUATIONS
This Page Intentionally Left Blank
N 0 RTH- H0 LLAND MATHEMATICS STUDIES
New Developments in Differential Equations Proceedings of the second Scheveningen conference on differential equations, The Netherlands, August 25-29, 1975 Editor
WIKTOR ECKHAUS University of Utrecht
1976
NORTH -HOLLAN D PU BLlSH ING COM PANY AMSTERDAM - NEW YORK - OXFORD
21
@ North-Holland Publishing Company - 1976
All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior permission of the copyright owner.
North-Holland ISBN: 0 7204 0466 5 American Elsevier ISBN: 0 444 11107 7
PUBLISHERS :
NORTH-HOLLAND PUBLISHING COMPANY AMSTERDAM, NEW YORK, OXFORD SOLE DISTRIBUTORS FOR THE U.S.A. AND CANADA:
AMERICAN ELSEVIER PUBLISHING COMPANY. INC. 52 VANDERBILT AVENUE, NEW YORK, N.Y. 10017
Library of Congress Cataloging in Publication Data
Scheveningen Conference on Differential Equations, 1975. New developments in differential equations. (North-Holland mathematics studies ; 21) 1. Differential equations--Congresses. 2. Nonlinear theories--Congresses. I. Eckhaus, Wiktor. 11. Title.
QA371.S38 1975 515 ' * 35 76-9754 ISBN 0-444-11107-7 (American Elsevier)
PRINTED IN THE NETHERLANDS
P R E F A C E The field of differential equations is an ever flourishing branch of matnematics, attracting research workers who, in the traditional terminology, range from very "pure" to very "applied". New problems continue to arise from applications or just from the mathematicians' curiosity, old problems give rise to new developments through application of new methods of analysis.
In 1973 a group of Dutch mathematicians consisting of B.L.J.Braaksma, W.Eckhaus, E.M.de Jager and H.Lemei, organized a conference on differential equations with the aim of promoting contacts and stimulating the exchange of ideas among a purposely limited, relatively small, number of invited participants. The success of that meeting, the favourable reactions afterwards and the favourable reception of the proceedings (published as North Holland Mathematics Studies Vo1.13), convinced the organizing committee of the usefulness of such conferences. This volume i s an account of the lectures delivered at the Second Scheveningen Conference on Differential Equations, held on Aug. 25-29,
1975. The organization
was in the hands of the same committee and the conference was again made possible through the generous financial support of the Minister of Education and Sciences of the Netherlands. There were 51 participants from 9 countries. (A list of participants can be found on page 249 of these proceedings.)
The emphasis in this second conference was on nonlinear analysis. This is reflected by the fact that approximately half of the volume is devoted
to
nonlinear
problems. However, linear problems in differential equations are still challenging and subject to new developments, as witnessed by the other half of the contributions. It is a pleasure to acknowledge the gratitude to all authors who have contributed such stimulating accounts of their research. Particular thanks are due to Professor J.L.Lions, who has accepted the invitation' to act as principal speaker and
delivered a series of four lectures. His contribution opens this volume. Wiktor Eckhaus, Editor Utrecht.
V
February 1 9 7 6 .
This Page Intentionally Left Blank
C O N T E N T S J.L.Lions
Some topics on variational inequalities and applications
G.Stampacchia Free boundary problems for Poisson's equation
.
H Amann
1
39
Nonlinear elliptic equations with nonlinear boundary conditions
43
H.Brezis
On the range of the sum of nonlinear operators
65
L.A.Peletier
On the asymptotic behaviour of solutions of an equation arising in population genetics
73
Optimal control o f a system governed by the Navier-Stokes equations coupled with the heat equation
81
Secondary o r direct bifurcation of a steady solution of the Navier-Stokes equations into an invariant torus
99
C.Cuvelier G.Iooss
W.A.Harris Jr. Application of the method of differential inequalities in singular perturbation problems
111
P.P.N.de Groen A singular perturbation problem of turning point type
117
.
B Kaper H.-D.Niessen
Asymptotics for a class of perturbed initial value problems
125
On the solutions of perturbed differential equations
135
On certain ordinary differential expressions W.N.Everitt and M.Giertz and associated integral inequalities 0
A.Pleije1
On Legendre's polynomials
161 175
W.Jurkat, D.A.Lutz and A.Peyerimhoff Invariants and canonical forms for meromorphic second order differential equations.
181
C.G.Lekkerkerker On generalized eigenfunctions and linear transport theory
189
A.Dijksma R.Martini
Integral-ordinary differential-boundary subspaces and spectral theory
199
Some degenerated differential operators on vector bundles
213
vii
LIST OF PARTICIPANTS Invited speakers H Amann Ruhr-Universitgt, Bochum, Germany H.Brezis Universitg P.et M.Cu~ie, Paris, France W.N.Everitt The University, Dundee, Scotland W.A.Harris Jr. University of Southern California, USA G.Iooss Universitg de Nice, France J.L.Lions Colllge de France, Paris, France D .A.Lutz University of Wisconsin, USA H.-D.Niessen University of Essen, Germany A.Pleije1 Uppsala University, Sweden A.Schneider Gesamthochschule Wuppertal, Germany G.Stampacchia Scuole Normale Superiore, Pisa, Italy
.
Participants from the Netherlands B.L.J.Braaksma T.M.T.Coolen C.Cuvelier B.R.Damst6 0.Diekmann A. Dijksma W.Eckhaus J.A.v.Gelderen B.Gilding J.de Graaf J. Grasman P.P.N.de Groen P.Habets A.v.Harten M.H.Hendriks A.J.Hermans H.W.Hoogstraten F.J. Jacobs E.M.de Jager M. Jansen B . Kaper E.W.M. Koper H.A.Lauwerier C.G.Lekkerkerker H.Lemei R.Mart ini H.G.Meijer G.Y.Nieuwland L.A.Peletier H.Pij 1s J. W. Reijn J.W.de Roever J.D.Siersma M.Sluij ter 1.G.SprinkhuizenKuyper M.N.Spijker J.Sijbrand N. M. Temme J.v.Tie1 P.J.Zandbergen
Rijksuniversiteit Groningen Mathematisch Centrum Amsterdam Technische Hogeschool Delft Landbouwhogeschool Wageningen Mathematisch Centrum Amsterdam Rijksuniversiteit Groningen Rijksuniversiteit Utrecht Technische Hogeschool Delft Technische Hogeschool Delft Rijksuniversiteit Groningen Mathematisch Centrum Amsterdam Vrije Universiteit Amsterdam Institut MathEmatique, Louvain-le-Neuve, Belgium Rijksuniversiteit Utrecht Landbouwhogeschool Wageningen Technische Hogeschool Delft Rijksuniversiteit Groningen Technische Hogeschool Eindhoven Universiteit van Amsterdam Vrije Universiteit Amsterdam Rijksuniversiteit Groningen Universiteit van Amsterdam Universiteit van Amsterdam Universiteit van Amsterdam Technische Hogeschool Delft Technische Hogeschool Delft Technische Hogeschool Delft Vrije Universiteit Amsterdam Technische Hogeschool Delft Universiteit van Amsterdam Technische Hogeschool Delft Mathematisch Centrum Amsterdam Rijksuniversiteit Groningen Mathematisch Centrum Amsterdam Mathematisch Centrum Amsterdam Rijksuniversiteit Leiden Rijksuniversiteit Utrecht Mathematisch Centrum Amsterdam Rijksuniversiteit Utrecht Technische Hogeschool Twente
viii
W . Eckhaus ( e d . ) ,
New Developments i n D i f f e r e n t i a l E q u a t i o n s
@ N o r t h - H o l l a n d P u b l i s h i n g Company (1976)
SOME
TOPICS ON VARIATIONAL INECUALITIES AND APPLICATIONS
J.L. LIONS College de France, Paris and IRIA-LABORIA, Le Chesnay, France.
INTRODUCTION. We present here some results on Variational Inequalities of stationary (elliptic) type (Chapters I and 11) and of evolution type (Chapter 111). Each Chapter gives a direct approach to me of the results (without any attempt to be exhaustive) and it also presents some open problems. The plan is as follows : .I,
CHAPTER I. INTRODUCTION TO SOME VARIATIONAL INEQUALITIES(.
'
I . An example of a V.I. 2 . Proof of existence by a penalty argument. 3. Some other V.I. 4 . An application of dynamic Programming.
Bibliography of Chapter I. CHAPTER 11. STOPPING TIMES AND SINGULAR PERTURBATIONS. 1.
Optimal stopping times and V.I.
2 . Singular perturbations and optimal stopping time. 3 . A direct proof. 4.
Singular perturbations in Visco-plasticity. Bibliography of Chapter 11.
CHAPTER 111. V.I. OF FVOLUTION. I . Optimal stopping times. 2 . Strong and weak formulations of the V.I.
3. Existence of a weak maximum solution.
Bibliography of Chapter 111.
(!) A programme somewhat similar to what is done here could be completed by replacing "0 timal sto in time" by "0 timal im ulse control" in the sense of this would d a : l to Quaze-V:riational InEqualitiez instead of V.I.. We refer to 1 2 1 ; we do not study these questions here.
[
I ] A. Bensoussan and J.L. Lions, C.R.A.S., 276 (1973), pp. 1189-1192, pp. 13331338 ; 278(1974), pp. 675-679, pp. 747-751 ; 280 (19751, pp. 1049-1053.
A. Bensoussan, M. Goursat and J.L. LIONS. C.R.A.S. 276 (1973), pp. 1279-1284. (21 A . Bensoussan and J.L. Lions. Book to appear. Hermann ed.
1
:
2
J.L.
I.
LIONS
I N T R O C U C T I O N TO SOME VARIATIONAL INEQUALITIES
ORIENTATION
We give an elementary i n t r o d u c t i o n t o of t h e methods of V . I . We prove t h e existence of a s o l u t i o n by t h e u s e of a n a l t y methods. We give some e r r o r e s t i m a t e s which l e a d t o a general an{ e p p a r e n t l y open problem ( c f . (3.13) ). We present i n Section 4 some remarks r e l a t e d t o t h e question of t h e v a r i a t i o n of .the s ~o l u t i o n of a V . I . w i t h resgect t o t h e geometrical domain. ~~
1 . AN EXAMFLE
~
OF A V . I .
r
Let 9 be a bounded open s e t i n Bn with a smooth boundary consicer t h e e l l i p t i c operator A defined by
. In
9
we
(1.1)
where aij8
(1.2)
qi,
.
E
a.
~ ~ 1 9 )
We shall s e t : 1
H (62) = Sobolev space of o r a e r 1 = f u n c t i o n s
v
v , E E L2(Q), dx .
such t h a t " )
provided with t h e usual h i l b e r t i a n s t r u c t u r e , \\vl\
= norm i n
H~(61) 2
= norm i n L ( 9 )
Iv/
1
u , v E H (62)
and f o r
:
, (f,v)
2
,
= s c a l t r product i n L (62)
we define :
(1.3)
We define H i ( B ) = closed subspace of
(1.4)
which a r e zero on
H'(62)
of f u n c t i o n s
v
r.
We assume t h a t ( e l l i p t i c i t y hypothesis)
(1.5)
a(v,V)
5
a i\vi\',
a7 0,
v
.
1
v E ~ ~ ( 6 2 )
The problem we want t o consider f i r s t c o n s i s t s i n f i n d i n g IJ
Au
------
where
f
i s given i n 1
( )
E
1
Ho(Q)
-f
< O
,
u GO
,
(nu-f)u = 0
in
--
L2(6a).
A l l f u n c t i o n s a r e supposed t o be r e a l valued.
62
u
such t h a t
3
SOME TOPICS ON VARIATIONAL INEQUALITIES
a(u.v-u) p (f,v-u)
T
1
v E H~(Q).
v
y .
u
s a t i s f i e s the " l l e r ?quation"
du" (4.26)
(' )
y
Actually
3
u*
has a compact support.
J .L.LIONS
26
I f we now multiply (4.26) by
- u"
, we
obtain :
X denotes t h e l e f t hand s i d e of ( 4 . 2 3 ) . w e o b t a i n
and i f
and
v
X
>0
m
hence (4.23) follows. 4
One i n t r o d u c e s next
w
obtained by, a Taylor expansion in (4.25) :
One v e r i f i e s next t h a t
From t h e d e f i n i t i o n s , i t follows t h a t JE(wE) 3 JE(uE) 3 H , ( t )
3 HE(wE)
+ O(E)
so that JE(uE) = HE(WE)
+ O(E)
and we f i n a l l y o b t a i n ( 1 ) : JE(uE) = L(1
(428)
+
e"
f:,
ds
f
O(E))
.
8
Remark 4.6. -------
I n [g], t h e A. give t h e second term hi t h e expansion of
I _ -
(')
I n [9],
t h e A. f i n d
= 3
i n s t e a d of
4 v 3 .
JE(uE).
8
SOME TOPICS ON VARIATIONAL INEQUALITIES
REFERENCES (CHAPTER i I )
21
. Book t o appear. H e r m a n n Ed. Vol.1
[l]
Bensoussan, A . and Lions, J . L . Vol.2 (1977).
(1976),
[2]
Bensoussan, A . and Lions, J . L . (1975). Problkmes de temps d ' a r r 8 t optimaux e t de p e r t u r b a t i o n s s i n g u l i b r e s dans les i n g q u a t i o n s Quasi Variationn e l l e s . Lecture Notes i n Economics and Mathematical Systems. Springer ( 1 0 7 1 , p p , 567-584. (1973). Problhmes de temps d f a r r 8 t optimal e t Ingquations V a r i a t i o n n e l l e s p a r a b o l i q u e s . Applicable Analysis. V O ~ . 31 PP. 267-294.
[3] Bensoussan, A . and Lions, J . L .
[4]
Deny, J. and Lions, J . L . 305-370.
[5]
Gagliardo, E. (1957) C a r a t t e r i z z a z i o n e ,
[6]
Huet, D.
[7]
Lions, J . L . ( l q T ) . P e r t u r b a t i o n s s i n g u l i h r e s dans l e s problbmes aux l i m i t e s e t e n c o n t r 8 l e optimal. Lecture Notes i n Mathematics. S p r i n g e r . Vol. 323.
[8]
f i g n o t , F. and Duel, J.?. (1976).&chive
[g]
Mosolov, P .P. and Miasnikov, V.P. Boundary layer i n t h e problem of l o n g i t u d i nal motion of a c y l i n d e r i n a v i s c o p l a s t i c medium. P.M.M. 38 ' (1974)r PP. 682-692.
Les espaces du t y p e de Beppo L e v i . 5 (195>1954),
. . Rend.
pp.
Sem.Mat. Padova 27,p.284-305.
(1968) P e r t u r b a t i o n s s i n g u l i h r e s d'In6q. V a r . 267, pp.932-934.
Rat. Mech.Analysis.
..
[lo] Bensoussan, A . and Lions, J.L. (1975) Diffusion Processes . i n Probabilistic Methods i n I Z i f f e r e n t i a l Equations , Lecture Notes i n Mathematics, ' S p r i n g e r , 451 , pp. E 2 5 . [ l l ] Brbzis, H.
P e r s o n a l communication.
28
J .L.LIONS
111. VARIATIONAL INEQUALITIES OF EVOLUTION
ORIENTATION [2], t h e s o l u t i o n of a s t o c h a s t i c optimal stopping time problem i n terms of V . I . of evolution. Section 2 d e f i n e s t h e strong and weak s o l u t i o n s o f t h e V.1. & e v o l u t i o n met i n Section 1 , and i n Section 3. we prove an important r e s u l t of fignot-
W e present i n Section 1. following [l],
Puel [6]. Other r e s u l t s along t h e l i n e s of this ( i n t r o d u c t o r y ) chapter w e given i n r21. 1 . OPTIMAL STOPPING TIMES.
We consider, a s i n Chapter 11, Section 1 , t h e s t o c h a s t i c I t o ’ s d i f f e r e n t i a l equation :
+
dy = g(y)ds
(1.1)
whose s o l u t i o n i s denoted by
, we
t
s
>t
f ( x , t ) i s a given -say continuous
,
+(x, ) is a given -say
- f u n c t i o n i n V x]~t
6-2.
We consider stopping times
..
,
s
Yxt(S)
where
y(t) = x
t h e smallest
S t a r t i n g from such t h a t :
(1
,
y (9). xt
,at
xE D
a(y)dw(s)
A
r w i l l be a s o l u t i o n o f ( 2 . 1 ) it BR S i n c e t h i s means t h a t (2.2) w i l l be a u t o m a t i c a l l y s a t i s a l s o a s o l u t i o n t o PtlobLem [ * J ,we s h a l l n o t d i s i s f i e d , s o t h a t R,wl EMr BR t i n g u i s h between w and i n t h e sequel. i l y i n v e s t i g a t e d . We n o t e t h a t t h e r e s t r i c t i o n o f
.
w"
We have t h e f o l l o w i n g THEOREM 1 - L e t S2,u y>O. Suppuhe t h a t r ud PkubLem (*J doh
be a b O & t h n 0 6 PtlubLem I whehe F 4 a t i b d i e n ( 1 . 1 ) and .in a brmoth cuhue. Then t h e h e e x h a a bullLtiun r , w EIKr f(z) = -
1 ^IP
F(z)
buch t h a t
R = (z:w(z)>log P }
and
U ( Z ) = T(1-pw
P
(2))
.
FREE BOUNDARY PROBLEMS FOR POISSON'S EQUATION
441
03 - According t o a w e l l known theorem, t h e r e i s a s o l u t i o n t o t h e v a r i a t i o n a l i n e q u a l i t y ( 2 . 1 ) f o r each r > O . To e s t a b l i s h i t s smoothness i n E r we s h a l l prove t h a t i t i s bounded. F o r once t h i s i s known, t h e o b s t a c l e l o p may be r e p l a c e d by a smooth o b s t a c l e which equals locy when
f u n c t i o n s which exceed \I, and ( 2 . 1 ) may be s o l v e d i n t h e convex IKb o f H'(Er) in B and s a t i s f y t h e boundary c o n d i t i o n v ( z ) = log r , ] z ] = r . The s o l u t i o n t o t h i s r a t t e r problem i s known t o be s u i t a b l y smooth and i s e a s i l y shown t o be t h e solution o f (2.1). By s t a n d a r d methods we have t h e f o l l o w i n g
LEIWA
-
Let
f E Lp(Br)
p.2
604 b o w
f O On t h e o t h e r hand we have t h e f o l l o w i n g
-
THEOREFI 2
Let
f E Lyoc(R2)
d o t a p>2 bdL566y
sup f R*
0
p (Br). r , w E IK t o P h u b f k c i I * ) . r The main s t e p i n t h e p r o o f i s t o c o n s t r u c t a s u p e r s o l u t i o n g ( z ) = h ( p ) t o
Then t h e h e e d L . 6 a
bV&U%fl
the form i
a(w,r) -
f o r some
I
fSdx
' Br
r > l , which s a t i s f i e s
h EKr
(3.2)
h = -1 ~r
(3.3).
I f ( 3 . 2 ) and ( 3 . 3 ) a r e f u l f i l l e d , then, from a we1
w < h i-loreover, s i n c e
Therefore
Br.
l o o pLwLh we conclude from ( 3 . 3 ) t h a t 1 w (z) = f0.r / z I : r: P
and, s i n c e
in
known p r o p e r t y
w=log r
for
r
Iz/ = r ,
w (z) = 0 for /zl = r 8 d e f i n e d by ( 2 . 2 ) i s i n C ' ( R ' ) .
04 - Next s t e p i s t o show t 3 a t t h e s e t where t h e s o l u t i o n t o Paobleni I*] exceeds l o g p i n starshaped under an assumption about f . Indeed we have t h e f o l l o w i n g :
GUIDO
42
-
THEOREM 3
r,w E Mr
Let
f E Lyoc(R’)
L&
STAilPACCHIA
hc~tA6y
denote t h e n o l d o n
06
SUP
and p - l ( ~ ’ f ) ~ ~ O .
f l o g p l
Then SL an n t u h a p e d w i t h h e 4 p e d t o
z = 0
doh
f
and n e t
.
.
-
85 I n t h i s s e c t i o n we s h a l l r e p o r t on t h e smoothness o f t h e f r e e boundary d e t e r mined by a s o l u t i o n t o PtrobLem ( * ) assuming t h a t f E C’(RZ) and t h a t
Then t h e f o l l o w i n g theorems h o l d THEOREM 4 - L e i f E C1(R2) 4&6y t o Phobteni (*) don f Lei
( 5 . 1 ) and L e t
.
a =tz:w(z),log r 06
Then t h e boundahy
in
THEOREH 5
-
denote t h e nolLLtion
.
PI
52 han the. hephebent&on
r when@ p
r,w E K p
: P = p(e)
, 0 9 ~ 2 ~
a cuw%nuow d u n c t i o n u6 bounded
vahiation.
L e t f E C1(R2) 4aLLn& ( 5 . 1 ) . Let r , w E M r denote t h e n o t a t i o n t o Phobteni (*I doh f and r t h e boundany 0 6 fl= t z : w ( z ) > l o g PI . Then r h a a c”7 pahanlethization, O < T < l .
A t t h i s p o i n t we can g i v e an answer i n t h e a f f e r m a t i v e t o t h e q u e s t i o n o f t h e e x i s t e n c e o f a s o l u t i o n t o Phobtem I w i t h t h e THEOREI.1 6
a
-
Let
and a 6unCtion
F E C1 (RZ ) 4 e A h 6 y conditionn (1.1 ) u E tiy:c(Rz ) nuch t h d
-A =
p-’F
P
in
.
Then thehe e . x h & a domLin
a
u(0) = Y
whehe
v
.LA t h e uLLtwatld d h e c t e d M O h n d !
vectotr
and
s
i 4
t h e ahc LenghL
06
and y>O A & w n . Denote by
1 Given F , d e f i n e f ( z ) = - 7F ( z ) z,w EM t h e s o l u t i o n t o ”PtrobLeni U ( Z ) = y(l-pwp(z))
I t can be proved t h a t
and observe t h a t f s a t i s f i e s (5.1). [ * ) f o r f and d e f i n e
z
E R2
.
u ( z ) , so d e f i n e d , s a t i s f i e s t h e c o n c l u s i o n s o f theorem 6 .
REFERENCES
For t h e b i b l i o g r a p h y and more d e t a i l s we r e f e r t o D . K i n d e r l e h r e r and G.Stampacchia. A Free Boundary Value Problem i n P o t e n t i a l T h e o y Ann.1m.t. FowLieh, 25 3/4 (1975) t o appeah.
W . Eckhaus ( e d . ) , New Developments i n D i f f e r e n t i a l Equations
@ North-Hol land Pub1 i s h i n g Company (1976)
NONLINEAR ELLIPTIC EQUATIOIJS WITH NONL I REAR BOUNDARY CONDITIONS
Herbert Amann Department o f Flathematics Ruhr- Uni v e r s i t a t Ecochum , Germany
INTRODUCTION I n t h i s paper we study m i l d l y nonlinear e l l i p t i c boundary value problems (BVPs) o f the form in R , (1.1) on r , where n i s a bounded domain i n R N w i t h s u f f i c i e n t l y smooth boundary r We suppose t h a t A i s a second order, s t r o n g l y u n i f o r m l y e l l i p t i c d i f f e r e n t i a l operator and B i s a boundary operator o f the form Au = f ( x , u ) Bu = g(x,u)
where
6
.
i s an outward pointing, nowhere tangent vector f i e l d on
r .
Problems o f t h i s type a r i s e , i n p a r t i c u l a r , i n the study o f steady s t a t e s o l u t i o n s o f n o n l i n e a r parabolic equations o f t h e form
* at
t AU = f(x,u)
in
n
(o,-)
,
Bu = g(x,u) u = u
on on
-r x
(0,-)
,
0
(1.2)
R
I n t h i s connection, n o n l i n e a r boundary c o n d i t i o n s seem t o b e ' o f p a r t i c u l a r i m portance. For the study o f t h e s t a b i l i t y o f t h e s o l u t i o n s o f t h e p a r a b o l i c i n i t i a l - b o u n d a r y value problem ( l . Z ) , one has t o have a aood knowledge o f t h e steady states, t h a t i s , o f the s o l u t i o n s o f t h e e l l i p t i c BVP (1.1). O f course, the most i n t e r e s t i n g case occurs i f t h e e l l i p t i c BVP has several d i s t i n c t solut i o n s . (For an i n t e r e s t i n g a n a l y s i s o f an i n i t i a l BVP o f the form (1.2) i n t h e case o f one space dimension and i n the presence o f several d i s t i n c t steady s t a t e s cf. [ 6 1 ). Unlike t o t h e s i t u a t i o n where Q i s independent o f u , n o t much seems t o be known i f the boundary c o n d i t i o n depends n o n l i n e a r l y on t h e unknown f u n c t i o n Recently, the theory o f monotone operators has been a p p l i e d t o BVPs o f t h e u form (1.1) ( c f . [ 7,B,10 I ) . However, i n a l l o f these papers t h e boundary c o n d i t i o n i s o f the special form
.
where g i s decreasing and v i s the co-normal w i t h respect t o the d i f f e r e n t i a l operator A Moreover, the theory o f monotone operators does n o t seem t o y i e l d proper mu1t i p 1 i c i t y r e s u l t s .
.
Besides these r e s u l t s , there are some s c a t t e r e d existence theorems f o r n o n l i n e a r Stecklov problems o f the form
4:i
44
HERBERT AMANN
where A i s supposed t o be f o r m a l l y s e l f - a d j o i n t such t h a t the homogeneous l i n e a r BVP possesses a n o n t r i v i a l s o l u t i o n ( c f . [ 9 , 1 2 ] ). This s i t u a t i o n w i l l a l s o be covered by our general r e s u l t s . So f a r , the o n l y general existence theorem f o r t h e BVP (1.1) seems t o be a r e s u l t due t o t h e author [ 2 I (see a l s o [ 261 ), namely t h e r e s u l t t h a t the s 0 existence o f a subsolution 7 and o f a supersolution ^v f o r (1.1) w i t h guarantees the existence o f a s o l u t i o n . I n t h i s paper we g i v e a new (and more e l e gant) proof f o r t h i s theorem by transforming t h e BVP (1.1) i n t o an e q u i v a l e n t This transformation has t h e advantage t h a t i t maf i x e d p o i n t equation i n C(E) kes the problem (1.1) accessible t o the powerful t o o l s o f n o n l i n e a r f u n c t i o n a l analysis. Some o f these tools, namely the theory o f increasing, completely c o n t i nuous maps i n ordered Banach spaces ( c f . [5,15] ) are then used t o enlarge the domain o f a p p l i c a b i l i t y o f t h e general existence theorem by d e r i v i n ? simple s u f f i c i e n t c r i t e r i a f o r the existence o f sub- and supersolutions. I n addition, we can d e r i v e a nonexistence and a general uniqueness theorem. Moreover, i n o r d e r t o dem n s t r a t e the power o f t h i s a b s t r a c t approach, we d e r i v e a m u l t i p l i c i t y r e s u l t , namely a c r i t e r i o n guaranteeing the existence o f a t l e a s t t h r e e d i s t i n c t s o l u t i o n s .
.
I n the f o l l o w i n g paragraph we s t a t e the main r e s u l t s f o r t h e n o n l i n e a r BVP (1) which a r e proved i n t h i s paper. I n Paragraph 3 we e s t a b l i s h a fundamental a p r i o r i estimate f o r the s o l u t i o n s o f the l i n e a r BVP Au = v i n 62 , Bu = w on r , i n v o l v i n g o n l y a L -norm o f t h e boundary term. P I n Paragraph 4 we d e r i v e t h e e q u i v a l e n t f i x e d p o i n t equation and we prove, besides o f the fundamental existence r e s u l t , t h e above mentioned m u l t i p l i c i t y theorem. The l a s t paragraph i s o f more a b s t r a c t nature. Namely, i t contains a d e r i v a t i o n o f the basic s p e c t r a l p r o p e r t i e s f o r p o s i t i v e l i n e a r operators i n an order e d Banach space which map every p o i n t o f t h e p o s i t i v e cone e i t h e r i n t o the i n t e r i o r o f t h e cone o r onto zero. These r e s u l t s generalize t h e known s p e c t r a l prope r t i e s f o r s t r o n g l y p o s i t i v e operators ( c f . [ 141 ). Moreover they are needed f o r the study o f the l i n e a r eigenvalue problem in n
Au =
mu
Bu =
r u on
r
,
.
This eigenvalue problem p l a y s a considerable r61e i n t h e s o l v a b i l i t y theory o f t h e nonlinear BVP (1.1). 2. STATEMENT
OF THE M A I N RESULT
Throughout t h i s paper a l l f u n c t i o n s a r e real-valued and a l l v e c t o r spaces a r e over the r e a l s . I n t h e f o l l o w i n g we s p e c i f y the hypotheses which a r e used throughout t h e remainder o f t h i s paper. We suppose t h a t n i s a nonempty bounded domain i n R N , N h 2 , o f class C3+U f o r some (X E o 1 , t h a t i s , t h e boundary , r , o f fi i s an (N-1)-dimensional compact CJ+A-!anifold such t h a t n l i e s l o c a l l y on one s i d e of r
.
We denote by
A
a l i n e a r d i f f e r e n t i a l operator o f t h e form
NONLINEAR BOUNDARY CONDITIONS
Au :=
N
N
-
aikDiDku
I:
+ au
z aiDiu
t
i.k=l
i=1
.
w i t h symmetric c o e f f i c i e n t m a t r i x ( a i k ) t!e suppose t h a t a i k E C2+a(z) , ai E Cl+a(;;i) , and a E P ( E ) Moreover, A i s suppose t o be s t r o n g l y uniforml y e l l i p t i c , that is, N i k 2 I: aik(x)c 6 2 aolEI i. k = l f o r some constant a0 > o and every x E 0 , and 5 = ( [ l , . . ,c N ) E lR N
.
.
We denote by t o r f i e l d on
CZta(r,IR N )
6E
r , and
:= zsiDiu
.
an outward p o i n t i n g , nowhere tangent vec-
r
denotes t h e d i r e c t i o n a l d e r i v a t i v e on
of
C1(z)
uE w i t h respect t o 6 . I t should be observed t h a t B i s n o t supposed t o be a u n i t vector f i e l d . Then we d e f i n e a ( r e g u l a r o b l i q u e d e r i v a t i v e ) boundary operator B by
BU where
b
E
C1+a(r)
au t as
:=
.
,
bU
-
.
L e t I be a nonempty s u b i n t e r v a l o f R We denote by f : 5 x I R a f u n c t i o n which i s a-Holder continuous i n the f i r s t v a r i a b l e and l o c a l l y L i p s c h i t z i n the second v a r i a b l e . More p r e c i s e l y , f o r every compact s u b i n t e r v a l I' o f I , t h e r e e x i s t s a constant y ( 1 ' ) such t h a t If(X,C) - f(y,n)l I Y ( I ' ) ( l x - y l a + 1 6 - r l l ) f o r every p a i r ( x , ~ ) , (y,?) E x I' . Moreover, we suppose t h a t i s a l o c a l l y L i p s c h i t z continuous function.
z
g :
r
x
I
-R
Then we consider m i l d l y n o n l i n e a r e l l i p t i c BVP's o f t h e form Au = f ( x , u )
in
R
,
Bu
on
r
.
= g(x,u)
(2.11
By a -elution u o f ( 2 . 1 ) we mean a c l a s s i c a l s o l u t i o n , t h a t i s , a f u n c t i o n u E C2(Q) * C1(E) such t h a t u(5) c I , Au(x) = f ( x , u ( x ) ) f o r x E R , and Bu(x) = g(x,u(x))
for
x E
r
.
A f u n c t i o n u i s c a l l e d a s u b s o l u t i o n f o r the BVP (2.1) i f C1(n) , u(H) c I , and Au(x) S f ( x , u ( x ) ) Bu(x)
(o,o) means (u,v) 2 (o,o) b u t (u,v) (o,o)
*
.
The second fundamental r e s u l t o f t h i s paper concerns the l i n e a r eigenvalue problem (EVP) Au = xmu i n R , (2.21 . . Bu = Xru on r , where, as a r u l e , we suppose t h a t the f o l l o w i n g hypothesis (H) i s s a t i s f i e d :
m
E
~ ( 5, )r
E cl-(r)
, and
There e x i s t s a constant (a + , b + u r ) 2 (o,o)
~1 2 0
.
(m,r)
>
(0.0)
. (HI
such t h a t
By means o f the above mentioned reduction t o a f i x e d p o i n t equation i n
C(5)
we
s h a l l prove the f o l l o w i n g important r e s u l t :
( 2 . 2 ) Theorem: Let t h e hypothesis (H) be s a t i s f i e d . Then t h e EVP ( 2 . 2 ) possesses a s m a l l e s t eigenvalue xo(m,r) , t h e p r i n c i p a l eigenvalue, a n d xo(m,r) i s p o s i t i v e i f (a,b) > (0.0) The EVP ( 2 . 2 ) has e x a c t l y one l i n e a r l y independent t o the eigenvalue X0(m,r) , and Uo can be eigenfunction Uo E c2(R) f- cl(T) chosen t o be everywhere p o s i t i v e . Moreover, Xo(m,r) is t h e only eigenvalue of ( 2 . 2 ) having a nonnegative e i g e n f u n c t i o n .
.
L a s t l y , xo(m,r) c i s e l y , suppose t h a t m l Then xo(m,r) > Xo(ml,rl)
i s a s t r i c t l y decre s i n g f u n c t i o n o f (m,r) ~ " ( 5 ) and r l E ~ f - ( r ) s a t i s f y (m1,rl)
E
.
. More pre.
> (m.r)
The next theorem contains some useful r e s u l t s concerning the s o l v a b i l i t y o f the l i n e a r BVP Au - xmu = c i n Q , (2.3) Bu - Xru = d on r , where
h E
R
,
( 2 . 3 ) Theorem:
(H) be s a t i s f i e d and suppose t h a t t h e l i n e a r BvP ( 2 . 3 ) has f o r every X < Xo(m,r) e x a c t l y one s o l u t i o n which i s everywhere p o s i t i v e i f (c,d) > (0,O) L e t t h e hypothesis
c"(z) m) . Then
(c,d)
E
(c,d)
The BvP ( 2 . 3 ) has no p o s i t i v e s o l u t i o n i f e i t h e r 2 (0.0) or = xo(m,r) and (c.d) > (0.0)
.
.
On the basis o f the theorems (2.1)
-
X
> ho(l",r)
and
(2.3), which w i l l be proved i n the
NONLINEAR BOUNDARY CONDITIONS
47
f o l l o w i n g paragraphs, i t i s easy t o d e r i v e the f o l l o w i n g general existence, nonexistence, and uniqueness r e s u l t s . (2.4) Theorem: L e t t h e h y p o t h e s i s (H) be s a t i s f i e d . Suppose t h a t t h e r e e x i s t n o n n e g a t i v e c o n s t a n t s y and 6 and a r e a l number X < xo(m,r) such t h a t f(.,c)
I Y +
x
m
g(.,c) 5 6 + f o r every
6 2 0
, and f ( - , c ) t -Y + g(.,c) t -6 +
f o r every
5
I 0
.
r
x x
c , c
m r
c ,
(2.4)
( 2 . 1 ) h a s a t l e a s t one s o l u t i o n .
Then t h e n o n l i n e a r BVP
P r o o f : I t f o l l o w s from Theorem (2.3) t h a t t h e l i n e a r BVP
Au = Xmu Bu = hru has e x a c t l y one s o l u t i o n l i n e a r BVP-
0
2 o
.
+ +
y
in R
6
on
Hence 7 := -0
Au = xmu Bu = x r u
-
,
r
(2.5)
i s the o n l y s o l u t i o n o f t h e
y
in a .
6
on
r .
7 i s a subsolution and 0 i s a s u p e r s o l u t i o n f o r the BVP (2.1). Hence Theorem (2.1) i m p l i e s t h e a s s e r t i o n . Q.E.D. I t i s obvious t h a t
I t should be observed t h a t the above p r o o f shows t h a t t h e r e e x i s t s a sol u t i o n i n the order i n t e r v a l [ -9.91 , where 0 i s t h e s o l u t i o n o f t h e BVP (2.5).
We emphasize the f a c t t h a t Theorem (2.4) imposes one-sided growth condit i o n s only. For example, Theorem (2.4) i m p l i e s t h e e x i s t e n c e o f a s o l u t i o n o f t h e BVP (2.1) provided (a,b) > (0.0) and f ( x , . ) , g(y,.) a r e nonincreasing f o r every x E Ci and every y E r , r e s p e c t i v e l y , w i t h o u t any growth r e s t r i c t i o n whatsoever.
Furthermore, i t i s i m p o r t a n t t o n o t i c e t h a t the f u n c t i o n s m and r can be chosen independently o f each other. For example, suppose t h a t a 2 o , b t o , and f ( x , . ) i s nonincreasing f o r every x E 5 Then we can take m = o and Theorem (2.4) i m p l i e s t h e existence o f a s o l u t i o n f o r t h e BVP (2.1) f o r every f u n c t i o n g s a t i s f y i n g one-sided estimates o f t h e above form w i t h x < xo(o,r) . These I t f o l l o w s from Theorem (2.2) t h a t A (o,r) > xo(m,r) , f o r every m > o considerations show, t h a t , by u s i n g s t a r p estimates f o r f , i t i s p o s s i b l e t o e n l a r g e t h e c l a s s of admissible f u n c t i o n s g , and v i c e versa.
.
.
C l e a r l y , i f (a,b) 2 (o,o) , then i t i s always p o s s i b l e t o such t h a t m(x) I 11 f u n c t i o n s m and r by constants 7 and f o r a l l x E 5 and y E r , r e s p e c t i v e l y . However, s i n c e xo(m,r) Theorem (2.2), i t may be advantageous t o use nonconstant f u n c t i o n s a given concrete s i t u a t i o n .
replace t h e and -rLy) I 2 ho(u,p) by m and r i n
O f course, t h e r e a r e o t h e r geometric c o n d i t i o n s f o r t h e f u n c t i o n s f and g which guarantee the existence o f subsolutions and supersolutions f o r t h e BVP (2.1). and t h a t t h e r e e x i s t s a p o s i t i v e conFor example, suppose t h a t (a,b) 2 (o,o) s t a n t el such t h a t
Then i t i s obvious t h a t t h e constant f u n c t i o n
x
+
el
i s a supersolution f o r
HEEBERT AMANN
48
(2.1). Consequently, the BVP (2.1) has a s o l u t i o n i f t h e i n e q u a l i t i e s (2.6) and, f o r 5 5 E~ , the i n e q u a l i t i e s (2.4) are s a t i s f i e d , f o r example. We leave i t t o t h e reader t o deduce s i m i l a r existence r e s u l t s by e s t a b l i s h i n g f u r t h e r geometric c o n d i t i o n s o f t h i s form. I n any case, we l i k e t o p o i n t o u t t h a t , up t o r e g u l a r i t y assumptions, t h e above existence theorems contain and considerably generalize most o f the known existence r e s u l t s f o r the BVP (2.1) which have been deduced by means o f the theory o f monotone operators ( c f . Paragraph 1). Next we prove a simple n o n e x i s t e n c e t h e o r e m which i m p l i e s t h a t , i n some sense, Theorem (2.4) cannot be improved upon. (2.5) Theorem: L e t t h e h y p o t h e s i s (H) be s a t i s f i e d . Suppose t h a t (c,d). > (o,o) , and l e t A 2 ho(m,r) .
CECa(z)
and d E C1-(r)-fy
Then t h e BVP (2.1) h a s no p o s i t i v e s o l u t i o n i f
,
f(*,5) 2 c t b m 5
f o r every
for every
(2.7)
g(-.c) 2 d t x r 5 5 2 0 , and ( 2 . 1 ) h a s no n e g a t i v e s o l u t i o n i f
5 I 0
f ( * , 5 ) 5 -c t g(a.5) 5 -d t
.
x x
,
m 5 r 5
Proof: L e t the c o n d i t i o n (2.7) be s a t i s f i e d and suppose t h a t u i s a p o s i t i v e s o l u t i o n o f (2.1). Then i t f o l l o w s from (2.7), t h a t u i s a p o s i t i v e supersolution f o r the l i n e a r BVP A u - x m u = c in R , (2.8) B u - x v u = d on r . Since zero i s a s t r i c t subsolution f o r t h i s BVP, Theorem (2.1) i m p l i e s t h e e x i s t ence o f a p o s i t i v e s o l u t i o n o f (2.8). But t h i s c o n t r a d i c t s Theorem (2.3). The p r o o f f o r the remaining case i s s i m i l a r . Q.E.D.
I n the f o l l o w i n g we g i v e an existence and u n iq u e n e s s theorem which general i z e s the main r e s u l t o f 1 4 1 ( c f . a l s o [11,25] ). I n t h a t paper the uniqueness be dea s s e r t i o n has been proved under the assumption t h a t the f u n c t i o n g(y,.) creasing f o r every y E r
.
Ca@)
x
(2.6) Theorem: L e t t h e h y p o t h e s i s C l - ( r ) s a t i s f i e s (m,r) > (0.0)
lution i f
€or every
5.n E
with
s a t i s f i e d . Suppose t h a t (m,r) E t h e BVP ( 2 . 1 ) h a s e x a c t l y one so-
f(-,5)
-
f ( * , n ) 5 Arn(5-n)
g(*,c)
-
g(*,n) s xr(5-n)
ri
1 and k = 1,2 p > 1 , there e x i s t s a constant y such t h a t
f o r every
u
E ~'(0)
51
I n p a r t i c u l a r , f o r every
.
P We denote by A ' A'u =
N
-
A
t h e a d j o i n t operator o f
N
D.D (a. u) i,k = l 1 k 1k C
- c
Di(aiu)
i.1
, that
+
is
au
A'
I t should be observed t h a t the u n i f o r m l y e l l i p t i c d i f f e r e n t i a l operator a-Holder continuous c o e f f i c i e n t s on 5
.
has
I t can be shown ( c f . 1 2 0 1 ) t h a t t h e r e e x i s t a f u n c t i o n co E C2+'(r) w i t h c o ( x ) > o f o r very x E F and an outward p o i n t i n g , nowhere tangent v e c t o r f i e l d 0 ' E C2+a(r,R ) such t h a t Green's formula takes t h e form
A
/(uAv
n
f o r every
u,v E
-
vA'u)dx = / ( v E'u
C2(x) , where
B'u :=
r
$,+
-
c0 u Bv)do
b'u
(3.3)
and b ' E C1+a(r)
w: L e t g E c ( F ) . Then t h e r e e x i s t s a f u n c t i o n
(3.2) uIr = o
and
satisfying
where t h e c o n s t a n t
BU =
U E
g , such t h a t
i s independent of
y
.
g
cl@)
,
.
N P r o o f : ( a ) We f i r s t consider l o c a l coordinates. The general p o i n t y E R w i l l be denoted by y = (7,t) , w i t h RN-1 and t E IR Moreover,
.
YE
N N Q : = I y E B + U C 1 ;j + t := ( y l Let
J ~ G C(C
N)
be given and d e f i n e v ( y ) := t
2-N
v : Q + R
+
t,.
. . ,yN-l + t ) E
zN1
by
-
- I-y+t *(q)dn Y
w i t h an obvious a b b r e v i a t i o n f o r t h e (N-1)-fold i t e r a t e d i n t e g r a l . Then i t i s e a s i l y seen t h a t v E C l ( Q ) , t h a t v l x N = o , and t h a t (DNv)IzN = JI
.
By means o f H o l d e r ' s i n e q u a l i t y i t i s e a s i l y v e r i f i e d t h a t b-t x+t b It-' f ( c ) d c l d x I1 I f ( x ) l d x a X a
-
f o r every f E q a , b ] , -m < a < b < , and every t h i s i n e q u a l i t y repeatedly, one e a s i l y proves t h a t
/
Q
i = l,.,.,N
, where
y
lDivlP dy 5 y l N I + I P 6 C i s independent o f J,
t E (o,b-a)
,
.
By a p p l y i n g
(3.4)
HERBERT AMANN
52
set of j = 1,
( b ) For each j = 1, U j such t h a t v j n
...,M
,
...,M r
,
= Uj
o?J (-)au a8
$ j 1 ( Q ) . Thfn-Vj FF .Vioi'every u C (n) E
i s an open suband every
N
z
=
k=l
m
k.j
D [ $?(u) 1 k J
and i t i s easy t o see t h a t
.
i k - @( N x 6 Di $j ) mk,j - J i=l ) due t o the f a c t t h a t 8 i s nowhere tangent t o r Hence, mk,j E C 2 + C C ( ~ Nand, does nowhere vanish. the function m j mN.j L e t g E C ( r ) . For every j = 1 M d e f i n e q. E C(CN ) by q j := J 1 T j ( g ) Denote by v t h e f u n c t i o n i n C1(Q) defined i n p a r t ( a ) by means j au N bu. = o f q j E C ( Z ) , and l e t u j := 'j @,j Then u. = o and Bu. = J + J as J J
'=
,...,
.
.
'
au.
A aa =g set
r
,
.
U. n r 3 Let e l , e be a smooth p a r t i t i o n o f u n i t y w i t h respect t o t h e compact i n JRij , s u b o r j i n a t e t o be open covering {vl,..,,vM} , and l e t on
...,
m
M
u :=
.
Then u E C1(z) , u l r = o , and Bu = g on r L a s t l y , by using PoincarG's ineq u a l i t y and t h e estimate (3.4). i t i s e a s i l y v e r i f i e d t h a t u s a t i s f i e s the asserted estimate. Q.E.D. A f t e r these preparations we are ready f o r t h e p r o o f o f the f o l l o w i n g a p r i o r i estimate.
( 3 . 3 ) P r o p o s i t i o n : Suppose t h a t (a,b) Then t h e r e e x i s t s a c o n s t a n t y such t h a t
> ( o , o ) , and
let
1< p
N This i m p l i e s i n p a r t i c u l a r t h a t i s compactly imbedded i n C(5) Here T can be considered as a mapping o f - i n t o C(n) C(n) The f o l l o w i n g lemma i s o f fundamental importance f o r o u r considerations.
.
d
e:
.
.
.
-
is (4.1) Let (a,b) > ( 0 , o ) Suppose t h a t f : 'rz x I -. l o c a l l y r H o l d e r continuous and 9 : r x 1 i s l o c a l l y L i p s c h i t z continuous. Then t h e BVP (4.5) i s e z u i v a l e n t t o t h e f i x e d p o i n t e q u a t i o n U = T ( U ) i n c(5). The map T : 1~6;) + C(n) i s c o m p l e t e l y c o n t i n u o u s , t h a t i s , T i s continuous and maps bounde s e t s i n t o compact s e t s .
of
.
T
Proof: I t i s obvious t h a t every s o l u t i o n o f t h e BVP (4.5)
i s a f i x e d point
.
Conversely, suppose t h a t u E C(5) i s a f i x e d p o i n t o f T Then u belongs t o the range o f S , hence t o Wi(n) Consequently, t ( u ) E Wb-l/P(r) , Since F(u) E C(1) c Lp(n) , and Lemma (3.1) i m p l i e s t h a t G 0 t ( u ) E Wi-'lp(r) 2 and since, by (4.3). S maps L (n) x W1-l/P(r) i n t o b' (n) , i t f o l l o w s t h a t P P P u E Wi(n) I t i s well-known ( c f . 211 ) t h a t Wi(a) is continuously imbedded
.
.
.
.
.
Cl+'(E) , where u := I-N/p Hence u E C1@) n W2(n) This i m p l i e s t h a t F ( u ) E C"(n) and onsequently, by the i n t e r i o r r e g u l a r i l y theory f o r e l l i p t i c equat'ons, u E C5tu(n') f o r every subdomain R' such t h a t n ' C a Hence, u E C (n) n C1(5) , and u i s a s o l u t i o n o f the BVP (4.7). in
h
.
NONLINEAR BOUNDARY CONDITIONS
51
I t i s e a s i l y seen t h a t the maps
F : Ic(a) + C(5) and G : I c ( r ) + C ( r ) := { u E c ( r ) I u ( r ) C I1 . Moreover, t C(r)i n t o C(r) such t h a t t ( I c ( a ) ) C i s a continuous l i n e a r operator from C(a) I c ( r ) . Since S E L(C(a) x C(r),$(a)) , and s i n c e W1(a) i s compactly imbedded
are bounded and continuous, where
i n C(a) f o r p > !I i n t o C(E) Q.E.D.
.
,it
I
follows t h a t
T
P i s completely continuous from
L e t X be a nonempty compact Hausdorff space. We denote by C+(X) the s e t o f a l l nonnegative continuous f u n c t i o n s on X , t h a t i s , C+(X) := { u E C(X)I u 2 o 3. Clearly, C+(X) i s a closed convex p r o p e r cone i n C(X) w i t h nonempty i n t e r i o r . I n f a c t , u E i n t C+(X) i f and o n l y i f u ( x ) > o f o r every x E X Observe t h a t u -< v i f and o n l y i f v - u E C+(X) I n the f o l l o w i n g we w r i t e u >> v i f u - v E i n t C+(X)
.
.
.
.
L e t D be a nonempty subset o f C(X) Then a map h : D + C(X) i s s a i d t o be i n c r e a s i n g i f h ( u ) _< h ( v ) f o r every p a i r u,v E D s a t i s f y i n g u -< v
.
s:
(4.2) L e t t h e h y p o t h e s e s o f Lemma (4.1) b e - s a t i s f i e d and s u p p o s e t h a t f(X,*) and g ( y , * ) a r e i n c r e a s i n g f o r e v e r y X E fl and y E I' , r e s p e c t i v e l y . Then t h e map T : + C(a) i s i n c r e a s i n g . Suppose i n a d d i t i o n t h a t
for some
X E 8
and e v e r y
f o r some y E r for every p a i r
and every U,V
E,n
f(X,S) < f(x,n) 1 w i t h F < n , or
E
!3(YSO < CJ(Y*n) S,rl E I w i t h 5 < n E I c ( a ) such t h a t v
0
r ( S ) < r(T) Sv E P \ k e r ( T ) Hence t h e a s s e r t i o n f o l l o w s from t h e f a c t t h a t
. . I.E.D.
be an a r b i t r a r y OBS and l e t e > o be a p o s i t i v e element i n T o f E i s s a i d t o be e - p o s i t i v e if, f o r every u E P , there e x i s t p o s i t i v e numbers a and 5 such t h a t ae 5 Tu I Be This d e f i n i t i o n i s a special case o f the more general d e f i n i t i o n o f an e - p o s i t i v e l i n e a r operator due t o M. A. K r a s n o s e l ' s k i i [ 1 5 1 The reason f o r t h i s r e s t r i c t e d d e f i n i t i o n o f an e - p o s i t i v e endomorphism l i e s i n t h e f a c t , t h a t , besides their importance f o r a p p l i c a t i o n s , these operators t u r n out t o be c l o s e l y r e l a t e d t o t h e class o f s t r o n g l y p o s i t i v e operators. I n f a c t , by endowing t h e v e c t o r subspace Ee := A[-e,e] o f E w i t h t h e order u n i t topology ( t h a t i s , w i t h the eLet
E
(E,P)
. An endomorphism
.
.
norm), i t can be shown (e.g. 15,151 ) t h a t Ee becomes an ordered normed v e c t o r space whose p o s i t i v e cone P has nonempty i n t e r i o r . Moreover, T i s e - p o s i t i v e i f f T(P) c Pe ( f o r more d e t a i l s c f . 1 5 1 ). This f a c t suggests the d e f i n i t i o n o f an almost e - p o s i t i v e endomorphism. Namely, a l i n e a r operator T : E E i s s a i d t o be a l m o s t e - p o s i t i v e , i f P ker(T) d and i f , f o r every u E P \ k e r ( T ) , t h e r e e x i s t p o s i t i v e numbers a and 5 such t h a t ue 5 Tu I Be Then i t can be shown by a c a r e f u l a n a l y s i s o f the proofs i n [ 151 , t h a t almost e - p o s i t i v e endomorphisms o f an OBS have, roughly speaking, the same spectral p r o p e r t i e s as s t r o n g l y p o s i t i v e endomorphisms ( c f . [I31 ). I n a much more elegant way, these r e s u l t s can be e s t a b l i s h e d by u s i n g t h e above i n d i c a t e d connection w i t h almost s t r o n g l y p o s i t i v e endomorphisms.
*
-+
.
Moreover, by modifying t h e corresponding p r o o f i n [ 1 5 ] i t can be shown t h a t the s p e c t r a l r a d i u s o f an almost s t r o n g l y p o s i t i v e endomorphism T i s t h e o n l y eigenvalue o f the c o m p l e x i f i c a t i o n o f T l y i n g on t h e s p e c t r a l c i r c l e . F i n a l l y i t should be observed t h a t i n the above theorems (5.1) - (5.5), the completeness o f E and the compactness o f T have o n l y been used ( v i a t h e KreinHence Rutman theorem) i n order t o guarantee t h a t r ( T ) i s an eigenvalue o f T these theorems can considerably be generalized.
.
Proof o f Theorem ( 2 . 2 ) and Theorem (2.3): A t t h e end o f the preceding paragraph i t has already been observed t h a t the BVP (2.3) i s e q u i v a l e n t t o t h e equation p~ - TU = v
HERBERT AMANN
62
i n C(E) , where P := ( A + W ) -1 , v := PSW(c,d) , and T i s an almost s t r o n g l y p o s i t i v e compact endomrphism o f C(3i) such t h a t i n t C+(E) n k e r ( T ) = @ Hence t h e a s s e r t i o n s f o l l o w f r o m t h e above general r e s u l t s b y o b s e r v i n g t h a t S (c,d) E p r i n c i p l e , t h e BVP (2.3) i s u n i q u e l y Y o l v a b l e
.
BIBLIOGRAPHY
11 1 S. AGNON, A. DOUGLIS, and L. NIRENEERG: Estimates n e a r t h e boundary f o r so-
(2 1 [ 3 1. .
(41 [51 [6 1 17 1 18
I
[9 1 1101
[11 1 [12 I [ 13
I
[141 115 I [16 1 [171 [181
[19 1 [20 1
[21 I
s o l u t i o n s of e l l i p t 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 I..Conun.Pure s a t i s f v i n a aeneral boundarv c o n d i t i o n s .~. Appl .Mith: iII (1959), 623-727. H. AMANN: On t h e e x i s t e n c e o f p o s i t i v e s o l u t i o n s o f n o n l i n e a r e l l i p t i c bound a r y v a l u e problems. I n d i a n a Univ. Math. J., 2 (1971). 125-146. : On t h e number o f s o l u t i o n s o f n o n l i n e a r e a u a t i o n s i n o r d e r e d Banach spaces. J . F u n c t i o n a l Anal., 11 ( 1 9 7 2 ) , 346-384. : A uniqueness theorem f o r n o n l i n e a r e l l i p t i c boundary v a l u e problems. Arch. Rat. Mech. Anal., 44 (1972), 178-181. : F i x e d p o i n t e q u a t i o n s and n o n l i n e a r e i g e n v T u e problems i n o r d e r e d Banach spaces. S I A M Review, t o appear. D.G. ARONSON and L.A. PELETIER: Global s t a b i l i t y o f symmetric and a s y m t r i c c o n c e n t r a t i o n p r o f i l e s i n c a t a l y s t p a r t i c l e s . Arch. Rat. Rech. Anal., 54 (1974), 175-204. H. BREZIS: Probldmes u n i l a t e r a u x , J. F l a t K Pures Appl., 51 (1372), 1-168. H. BRILL: Eine s t a r k n i c h t l i n e a r e l l i p t i s c h e G l e i c h u n g u z e r e i n e r n i c h t l i n e a r e n Randbedingung, Z e i t s c h r . Angew. Math. Mech. t o appear. J.M. CUSHING: N o n l i n e a r S t e k l o v problems on t h e u n i t c i r c l e . J. Math. Anal. Appl., 38 (1972), 766-783. J.-P. D I A Z : Un theordine de S t u r m - L i o u v i l l e pour une c l a s s d ' o p h a t e u r s non l i n e a i r e s m x i m a u x monotones. J. Math. Anal. Appl., 47 (1974). 400-405. J.P.G. EWER and L.A. PKETIER: On t h e a s y m p t o t i c b e h a v i o u r o f s o l u t i o n s o f s e m i l i n e a r parabo1;c'equations. SIN{ J. Appl. Math., 28 (1975), 43-53. K. KLINGELHUFER: N o n l i n e a r harmonic boundarv v a l u e Droblems. I.Arch. Rat. Mech. Anal., 31 ( i m j , 364-371. W. KNOKE: P o s i t i v e Losunaen f u r ZweFPunkt-RandwertDrobleme. D i p l o m a r b e i t . R u h - U n i v e r s i t a t Bochum, 1974'. M.G. KREIN and M.A. RUTMAN: L i n e a r o p e r a t o r s l e a v i n g i n v a r i a n t a c c x i n a Banach soace. Amer. Math. SOC. T r a n s l . . Ser. 1. l o (1962). i99-325, M.A. KRASNOSEL'SKII: " P o s i t i v e S o l u t i o n s o f O p e r a t o r Equations". Noordhoff. Groningen, 1964. O.A. LADYZHENSKAYA and N.N. URAL'TSEVA: " L i n e a r and O u a s i l i n e a r E l l i p t i c Equations". Academic Press, New York, 1968. J.L. LIONS and E. MAGENES: Problemi a i l i m i t i non omogenei (111). Ann. Sc. Norm. Sup. P i s a 15 (1961), 39-101. : Problemi a i 7 i m i t i non omogenei ( V ) . Ann. Sc. (1962), 1-44. Norm. Sup. P i s a : "Non-Homogeneous Boundary Value Problems and A p p l i c a t i o n s I." S p r i n g e r Verlag, B e r l i n - H e i d e l b e r g New York, 1972. C. MIRANDA: " P a r t i a l D i f f e r e n t i a l Equations o f E l l i p t i c Type", S p r i n g e r Verl a g , Berlin-Heidelberg-New York, 1970. J. NECAS: "Les methodes d i r e c t e s e n t h e o r i e des e q u a t i o n s e l l i p t i q u e s " . Academia, E d i t i o n s de 1'Academie Tchecoslovaque des Sciences, Prague, 1967.
-
-
-
16
NONLINEAR BOUNDARY CONDITIONS
[22 1 M.H.
[23 1 [24 ]
[25 ] [261
63
PROTTER and H.F. WEINBERGER: "Maximum P r i n c i p l e s i n D i f f e r e n t i a l Equat i o n s " . Prentice-Hall, Englewood C l i f f s , N.Y., 1967. H.H. SCHAEFER: "Topological Vector Spaces". Springer Verlag, B e r l i n - H e i d e l berg-New York, 1971. M. SCHECHTER: On Lp estimates and r e g u l a r i t y , I. Amer. J. Math. (1963), 1-13. J. SERRIN: A remark on the preceding paper o f Amann. Arch. Rat. Mech. Anal. 44 (1972), 182-186. P. HESS: On the s o l v a b i l i t y o f n o n l i n e a r e l l i p t i c boundary value problems. t o appear: Indiana U n i v e r s i t y Math. Journal.
This Page Intentionally Left Blank
W . Eckhaus ( e d . ) , New Developments i n D i f f e r e n t i a l Equations
@ N o r t h - H o l l a n d P u b l i s h i n g Company (1976) ON THE RANGE OF THE SUM
OF NONLINEAR OPERATORS
H. BREZIS Dept. de Mathematiques U n i v e r s i t e P. e t ) I . C u r i e 4 place Jussieu 75230 PARIS 5"
INTRODUCTLON
-
Let A and B be two continuous functions on R. Clearly we have : R(A+B)
u A~ +
BU
c R(A) + R(B)
t.
u
AV
+ B~
,
V,We(R
M!R
and in general R(A+B) is much smaLler than R(A) + R(B).
However equelity
holds in two simple cases : CASE I : A and B are both non decreasing
-
CASE I1 : A is linear and B is (non decreasing and) bounded. Our purpose is to extend this observation to mappings in infinite dimension a l spaces and t o discuss some applications.
In 9 1 (the extension of case I) we present some results from a joint paper with A. muUX(Image d'une somme d'opErateurs monotones et applications,
to
appear in Israel J. of Math.).
In 02 (The extension of case II) we present a preliminary version of a joint work with L. NIRENBERG,
91.
THE MONOTONE + MONOTONE CASE Let H be a real Hilbert space and let A and B be maximal monotone
-
operators in H. In general,R(A+B) could be much smaller than R(A) + R(B) ; consider for example in H -R2, A
a rotation by + n / 2 and B = a rotation by
- n/2.
Hovewer it turns out that in "many" important cases, R(A+B) and R(A) + R(B) are almost equal in the following sense. We say that two sets Sl and S2 are almost
65
66
H.BREZIS
equal
(S
1
2:
THEOREM 1 .
S ) 2
A = %
Assume
f u n c t i o n s such t h a t Then
-S 1
provided
-
= S
and
2
- a$
B
and
I n t S, = I n t S2
.
are s u b d i f f e r e n t i a l s o f convex 1.s.c.
.
a @ + $ ) = % + a$
R(A+B) = R(A) + R(B).
THEOREM 2 .
Assume
is any maximal monotone o p e r a t o r and
A
B =
a$
with
D(B) = H Then
R(A+B) = R(A) + R(B).
Proof. -
By a w e l l know r e s u l t of R.T. ROCKAFELLAR
We p r o v e f i r s t t h a t
f
E
Av + Bw ( f o r some Let
(I)
R(A) + R ( B ) C
u E E D(A) EU
v
and
+ AuE + BuE 3
a b l e t o conclude t h a t By t h e m o n o t o n i c i t y of (AuE
-
AV
Let
f
E
,A+B
i s maximal monotone.
R(A) + R(B), so t h a t
w).
be t h e s o l u t i o n o f
We are g o i n g t o p r o v e t h a t
(2)
R(A+B).
f
,
E >
& luEl .
f o R(A+B)
we h a v e
A
,u -
v)
>
0
.
.
0
remains bounded as
E +
0
. So,
w e w i l l be
67
ON THE RANGE OF THE S u t l OF NONLINEAR OPERATORS
On the other hand we have (BuE
-
(BuE
- Bw, uE - V)
(3)
-
Bw,uE
(Bw,~) - $(v)
v)
-
JI*(Bw)
so
-
$(u,)
$(v)
(Bw,~) - $(v)
+ (Bw, v
-
uE)
- $*(Bw).
Adding ( 2 ) and ( 3 ) we find (f
-
i.e.
-
f, u
E U ~
-
3
V)
- C, C
4 E / U E [ !vI +
EIUJ2
c
independent of
E.
.
Therefore Next we prove that
Int [R(A) + RCB)]
remains bounded as
E
*
C R(A+B)
.
; we are going to show that the solution uE of ( I )
Let f e Int [R(A) + R(B)]
0. This wilL enable us to infer that us&
(weak convern
gence) with Au + Bu 3 f. By assumption, there is some r > , O such that for a l l h f + h
E
E
H with Ihl< r,
Avh + Bwh (for some vh and wh ) .
Using now ( 2 ) and ( 3 ) with v and w replaced by vh and w we get h - f h , u - vh) > C(h) (f
-
-
where C(h) is independent of
E
(but depends qn h).
Thus (h.UE)
(h,vh)
+
C(h)
1
+
7 cIVhl2
.
Applying the uniform boundedness principle, it follows that uE remains bounded as E
-+
0.
Let B C RN be a bounded smooth domain.
EXAMPLE 1 Given f
E
L2(n),
the equation -Au+
1
T i 3
- €
onB
on
% - O
an
68
H.BREZIS
11
1
h a s a (unique) s o l u t i o n u e H2 (a) i f f
f (x) d x / < 1
n
.
F i r s t o b s e r v e t h a t i f u e x i s t s we have from ( 4 )
-
Next we can w r i t e ( 4 ) i n H Au + Bu
-
Au --Au
-
D(A) with
-
{u e H2(Q) ;
-
D(B)
H
and
2 we s e e t h a t
R(A
B
-
0 on
anl
a$
/= dx)
Applying Theorem 1
f,]l
-
where
with
- J-iT
Bu (Jl(u)
f
L2'(Q) a s
+ B)z R ( A ) + R(B). But, t h e a s s u m p t i o n
I R I i m p l i e s f e I n t [R(A) + R(B)],
dxl
/hlL2 < r
( r s m a l l enough)
f + h
-
[ (f+h)
-
we have
m 1n 1
s i n c e f o r every h e L2(n) with
(f+h)dx] +
[-p!q-
(f*h) dxl E R ( A ) + R(B) 52
By a s i m i l a r argument one can t r e a t the f o l l o w i n g EXAMPLE 2
Given f
8
L2(Q), the equation 1
\
- b ~ - X u + - - f I m
/ u - o where X
on0
onan
is t h e f i r s t e i g e n v a l u e of -A ( w i t h homogenous D i r i c h l e t B.C.)
1
and v1
(> 0 on Q) i s t h e c o r r e s p o n d i n g e i g e n f u n c t i o n , h a s a u n i q u e s o l u t i o n i f f
T h i s k i n d of e q u a t i o n can a l s o b e s o l v e d by t h e "semi-coercive"
REMARK
methods ( s e e t h e p a p e r s o f SCHATZMAN and H5SS)
-e 11 , - -
Given f
EXAMPLE 3
1I
du + dt
h
L2(0,T), th e equation on ( 0 , ~ )
JIG7
u(T)
u(0)
has a s o l u t i o n i f f
f\ < 1
.
0
PROOF AU
Bu
-
u1
Use Theorem 2 i n H
, D(A)
J i T
IU
D(B)
H(B
a$)
-
L2(0,T) w i t h
e H * ( o , T ) ; U(O)
.
u(T)~
69
ON THE RANGE OF THE Sm4 OF NONLINEAR OPERATORS
THE LINEAR NON MONOTONE + MONOTONE NONLINEAR CASE.
2
§
THEOREM 3.
Let
b e a l i n e a r (unbounded) o p e r a t o r i n
A
H
D(A) = H
with
and c l o s e d g r a p h . Assume N(A) = N(A*)
(5)
f u e D(A)
(6) Let
B
Proof. -
H
+ u2 with
u = u = A
IR(A)
R(A)
-
1
1
f
i n t o i t s e l f . Given
, u2
with
R(B)
R(A) @ N(A)
and
H
- ..
By ( S ) ,
E N(A).
R(A)
and o n t o
6
H
dim N(A)
0
+
B,(uE) = f l
a
u
D(A)
E
satisfying
(7)
EU
2E
+ AuC + BuE = f
I n d e e d (7) c a n b e w r i t t e n a s a s y s t e m
1
1 or
1 1
AulE
€u2€ + B2(uE) = f z
-- 1 [ f l
uIc = A
[ f2
u2€
-
B1(uE)I
- B2(uE)1
which h a s a s o l u t i o n by Schauder f i x e d p o i n t theorem ( n o t e t h a t
-
Let
f
,
R(A) + R(B)
E
i s continuous
H). F i r s t we p r o v e t h a t R(A) + R(B)
from t h e s t r o n g t o t h e weak t o p o l o g y i n C R(A+B).
B
f = Av + Bw.
so t h a t
C l e a r l y w e have IuIE/
(8)
C
,
lAulEl
By t h e m o n o t o n i c i t y of
B
C
we o b t a i n (BuE
-
Bw
,u
- W) > 0
ideu
(f
-
c u Z E - Au
-
- w)
f + Av,u
0
Hence
E I u 2~ ~ ~ Q
and t h e r e f o r e
6~
IwI
+
(C
+
IAvI)(C
+
u Z Er e~m a i n s bounded as
Iwl) E
-+
0,Consequently f
E
R(A+B).
70
H.BREZIS
I n t [ R(A) + R(B)] C R(A+B)
Next we show t h a t Let
f
[h/
< r
E
which i m p l i e s u
+
f + h = Avh + Bwh , f o r a l l h
that
2a
- Au 6
( h , u Z E ) 4 C(h)
-h
+ Avh
u
-
Au + Bu
>
0
remains bounded as
2f
n Remark
- Wh)
,u
and t h e r e f o r e
which i s a s o l u t i o n of
u
with
now (- f u
Hence
, so
I n t [ R(A) + R(B)]
. We have
.
E
+
0
.
f
By a s l i g h t m o d i f i c a t i o n of t h e p r o o f , one can show t h a t , under
t h e assumptions of Theorem 3
, R(A) + conv R(B) = R(A+B).
TREOREM 4.
Let
f
E
(9)
Let
H
and
A
B
.
b e a s i n Theorem 3
be such t h a t
lim ( B ( t v ) , v ) > ( f , v ) t++-
for a l l
v
E
Then t h e r e i s a
(lo)
,v #
N(A) u
E
Au + Bu = f
0
D(A)
. s o l u t i o n of
.
Conversely i f (10) h a s a s o l u t i o n . t h e n
l i m (B(tv) ,v) 2 ( f , v ) t++-
(11)
Sketch of t h e p r o o f .
for a l l
exists.
.
i s monotone, t
B
Observe t h a t s i n c e
l i m B(tv),v)
nondecreasing. and
v e N(A)
I+
(B(tv),v)
F i r s t assume t h a t (10) h a s a
t++-
s o l u t i o n . We t h e n have (B(tv)
-
Bu
(B(tv)
-
f + Au
,
tv
- u)
>
0
- u)
a
or
Therefore, f o r (B(tv),v) and ( 1 1 )
v
E
,
tv
.
0
N(A) = N(A*)
a
+ (f,v)
-
(Au,u)
-+
t
follows.
Assume now t h a t ( 9 ) h o l d s . We have
lim
(B(tv),v)
t++-
Since (12)
dim N(A) < 6(v)
-
-
,
0
such t h a t
.
From (12) we deduce by a s e p a r a t i o n argument ( r e l y i n g o n t h e f a c t t h a t
is
.
THE RANGE OF THE
ON
dim N(A)
u2 on IR, y 1 < y2 on (0, 1).
Because fx(x,u) =
= -at(x)u(l-u) > 0 on (0, 1 ) this implies that
It follows that problem I can only have one solution.
POPULATION GENETICS
77
3. S t a b i l i t y We now i n v e s t i g a t e t h e s t a b i l i t y of t h e t r a n s i t i o n l a y e r $ ( x ) , we constructed i n t h e previous s e c t i o n . Thus, w e consider t h e Cauchy problem
U(X,O)
= *(x)
-m<x<m
i n which $ e C ( I R ) , and t a k e s on v a l u e s i n t h e i n t e r v a l LO,
(6)
11. The e x i s t e n c e
and uniqueness of a s o l u t i o n of t h i s problem was e s t a b l i s h e d i n C61; we denote
it by u ( x , t ; $ ) . Consider t h e one parameter family o f functions v(x, h ) = $(x
+
h)
hem.
because f x > 0. Therefore, if h > 0 , v ( x , h ) i s a s u p e r s o l u t i o n of problem I. S i m i l a r l y , i f h < 0, v ( x , h ) i s a subsolution of problem I. This family of sub and supersolutions enables us t o prove t h e following r e s u l t . THEOREM 2.
&
(2),
Let u ( x , t ; + )
be t h e s o l u t i o n of problem (11, ( 6 ) i n which f i s niven
a s a t i s f i e s t h e assumptions ( i ) - ( i i i ) .Suppose t h e r e e x i s t numbers
h , , h2cIR such t h a t hl < 0 < h2,
Then uniformly on 1R.
&
78
L.A.PELETIER
Proof. Since
v(x, h,)
5
$(XI 2 v(x, h2)
-m<x<m
it follows from t h e m a x i m p r i n c i p l e t h a t u(x,t;v(. ,h,))
5 u ( x , t ; $ ) 5 u ( x , t ; v ( .,h2) 1.
However, it can be shown by means of an argument, similar t o one used by Aronson and Weinberger [ l ] t h a t t h e f u n c t i o n s u ( x , t ; v ( . , h i ) )
(i =
1, 2 ) both
tend t o a s o l u t i o n of problem I. Since 4 is t h e only s o l u t i o n of problem I t h e r e s u l t follows. The basic t o o l i n t h e proof of Theorem 2 was t h e family v ( x , h ) of sub and super s o l u t i o n s of problem I. By using more s u b t l e f a m i l i e s of sub and super s o l u t i o n s , we can prove t h e following r e s u l t . THEOREM 3. &J
&
(2) & a
u ( x , t ; $ ) be t h e s o l u t i o n of problem ( 1 1, ( 6 ) i n which f i s Riven s a t i s f i e s assumptions ( i ) - ( i i i ) . Suppose
l i m inf X”
+(XI
> l i m a(x), X”
uniformly on IR.
The proof of t h i s theorem, as well as t h e d e t a i l s of t h e proofs of t h e previous theorems w i l l appear i n a subsequent paper.
POPULATION GENETICS
REFERENCES
El1 Aronson, D.G. and Weinberger, H.F., Nonlinear diffusion in population genetics, combustion, and nerve propogation, Proc. W a n e Progr. in Partial Differential Eqns., Springer Lecture Notes in Mathematics,
(446), 1975. Conley, C., An application of Wazewski's method to a nonlinear boundary value problem which arises in population genetics, Univ. of Wisconsin Math. Research Center Tech. Sumnary Report No. 1444, 1975. C31 Fleming, W.H., A selection-migration model in population genetics, Journal Math. Biology (to appear). C41 Hoppensteadt, F.C., Analysis of a stable polymorphism arising in a selection-migration model in population genetics.
C51 Kanel', Ja.I., Stabilization of solutions of the Cauchy problem for equations encountered in combustion theory, Mat. Sbornik (N.S. ) 59 ( 1 0 1 ) (1962), supplement, 245
- 288.
C61 Kolmogoroff, A., Petrovsky, I., and Piscounoff, N., Etude de 1'6quation de la diffusion avec croissance de la quantit6 de matisre et son application 5 un problzme biologique, Bull. Univ. Moskou, Ser. Internat., See. A, 1 (1937) # 6 ,
1
- 25.
79
This Page Intentionally Left Blank
W. Eckhaus ( e d . ) , New Developments i n D i f f e r e n t i a l Equations @ North-Holland P u b l i s h i n g Company (1976)
OPTIMAL CONTROL OF A SYSTEM GOVERNED BY THE NAVIER-STOKES EQUATIONS COUPLED WITH THE HEAT EQUATION. C. CUVELIER university of Delft Delft, The Netherlands.
Introduction.*)
'.
Let 0 be an open set in R The boundary r of n, which is assumed to be regular, is divided into two parts: rl and r2. We consider the following problem of free convection (cf. LANDAU, LIFCHITZ [ I ] ) : Find two scalar functions Te = Te(x,t) = temperature, p-p( x,t )=pressure axid one vector function u=u( x,t ) = [ul(x,t ) ,u2( x,t ) ) = velocity, defined on 'i x [ O , T [ , (x is the space variable, t is the time variable) such that
-
(3)
div u
-
0
,
in Q x ]o,T[ where n = thermal diffusivity, V - kinematic viscosity, u = constant two-component vector, g = heat source6 anf f = external force. The functions T e , u, p boundary conditions:
(4)
Te(x,o)
-
Teo(x),
should satisfy the following initial and
-
. ( X , . )
uo(x),
x E
n,
a denotes the normal derivative at r, directed towards the exterior of 0. The function v is at our disposal and is called the control of the problem. Frequently, in practice, the control is subject to some constraints and we express this fact by requiring that v belungs to a set Uad of admissible controls.
*) For standard notation see LIONS [l], 81
[2]
.
C.
82
CWELIER
We are particularly interested in the temperature distribution at time t-T and our objective is to determine the function v in such a way that the temperature distribution at time t=T is as close as possible (in scme definite sense) to a desired distribution zd(x), x E Q
.
F o r v belonging to the set of admissible controls Uad, the system
(1),..,,(7)
defines Te(x,T) in a unique way: Te(x,T) = Te(x,T;v).
Next, we define a functional v
-
J ( v ) by
-
Zd(x)I2dx
T
J(v)
&(
I
Find
inf 'ad
=
Te(x,T;v)
+ a
/
Iv(x,t)12dr Q O rl where a is a positive constant. The problem of optimal control can be stated as follows: (8)
dt )
J(v).
Any element v+ E Uad such that J(v*)
=
inf 'ad
J ( v ) is termed an optimal
+=
control.
The contents of this paper are as follows:
...,
In chapter 1 we study the well-posedness of problem ( 1 ), ( 7 ) for fixed V. Next we prove the existence of an optimal control and we stqte a necessary cvndition for an element v E Uad to be an optimal control. In order to write this necessary condition in an simpler way we introduce an adjaint system of equations, With the aid of this system we construct an iterative method which provides us, in the limit, with an element of Uad which satisfies the necessary condition for optimality.
...
Because system ( 1 ), ,( 3 ) is not of Cauchy-Kowaleska type (i.e. div u=O does not contain a term it is not easy t o solve numerically.
g),
In order to overcome, in some sensetthis difficulty, we introduce a perturbed system of equations (depending on a parameter e
>
0) which is of
,
Cauchy-Kowaleska type (cf. LIONS [2], TEMAM [ I ] [2]). In chapter 2 we study the well-posedness of this pertuybed system and the existence of an optimal control. As in the unperturbed case we introduce an adjoint system and define an iterative method for the construction o f an element satisfying the necessary condition for optimalily, Chapter 3 is devoted to a convergence theorem in which we prove that the solution (Te , u,, p,) of the perturbed system corresponding to an optimal control veopt converges, in some topological sense, as the parameter E tends to zero, to the solution (Te,u,p) of the unperturbed system corresponding to one of its optimal controls vopt.
83
OPTIMAL CONTROL
In chapter
4
w e g i v e , v e r y b r i e f l y , some of t h e n u m e r i c a l r e s u l t s ,
0PTIi.iAL CONk'ROL OF TKE UlII'ERTUHBEU SYSTEM OF EQUATIONS.
1. 1 .I
THE UlC?ERTURBED SYSTFX. N O T A ' I I O N P ( T e , u , p ; v l .
F3RI~iULATlON O F
B e f o r e we g i v e t h e e x a c t f o r m u l a t i o n o f o u r problem we i n t r o d u c e some f u n c t i o n s p a c e s :
v=Iu E
2
1
d i v u=O) 2 H = c l o s u r e of V i n ( L ) ? 2 V = c l o s u r e of $pin (Ho) (D(D))
E L2, uo E H,
"1
E L2(0,T;L2),
f E L2(0,T;(L2)
2
) , w ,v 0, u = { 0 1 , u 2 } c o n s t a n t v e c t o r , v E L ~ ( c =~ ) L 2 ( o , T ; L 2 ( r l ) ) , we can s t a t e o u r problem as f o l l o w s : Given Teo
g
>
positive canstants, T
Find a s c a l a r f u n c t i o n Te and a v e c t o r f u n c t i o n u s u c h t h a t Te E L2(o,T;H 1 ),
Te' E L 1 ( o , T ; ( H 1 ) . ) ,
u E L2(o,1';V),
U'
**)
T e ( o ) = Teo,
E L~(O,T;V'),
=
4 0 )
uo,
and
(9)
+
Te' +wC1[Te;v]
+ vC,[u]
u' where C1,
C2,
- uTe
B [u,u]
= f
ind'(]o,T[
1 ; (H
i n &(]o,T[
; V'),
I,),
A a n d B a r e d e f i n e d as f o l l o w s :
C1 [Te;v]
C2
+
A [u,Te] = g
[u]
A[u,Te]
B[u,w]
1
: Y(EH )
-
: V(E(H;)~)
: Y(€H1)
:
-
( g r a d Te, g r a d Y )
-
-g -
(v,
( g r a d u, g r a d T )
( LLO P
- ui
i=l
'
YOYIL
au -
Te axi
2
)dx
(r1
***I 1 9
,
We w i l l c a l l t h i s problem: P ( T e , u , p ; v ) .
#=
The f o l l o w i n g theorem d e a l s w i t h t h e well-posedness
of P ( T e , u , p ; v ) .
THEOREM 1.1. Problem P ( T e , u , p ; v )
admits, f o r f i x e d v E L2(C1),
( T e , u ) which i s c o n t i n u o u s from [o,T] u n i q u e element o f L,
*) Pi6
*",
We w r i t e L2, HA,
) N o t a t i o n 7'
-
-
a unique s o l u t i o n
L2 x H. The p r e s s u r e p i s t h e
( o , T ; L ~ ) / Xi a a t i s f y i n g : 1 1 H' i n s t e a d of L 2 ( Q ) , Ho(0), H (a).
; X'
d e n o t e s t h e d u a l s p a c e of
yo d e n o t e s t h e t r a c e o p e r a t o r , yo : H 1 ( Q )
-8
x.
H (P).
C. CUVELIER
84
< u ’ ( t ) + vc2[u(t)] f o r a l l p E (HA)2,
+ B[u(t),u(t)]
where D[p]:
- oTe(t) - D
1; q( E (Ho)
)
-
[ ~ ( t )- f]( t ) , p >
p d i v p dx and
= 0
0
1 2 denotes t h e d u a l i t y p a i r i n g of (HA)2 and ( ( H o ) )’. PROOF
7
The proof i s c l a s s i c a l and can be found i n LIONS[2],TEMAM[2]or
CUVELIER [ I ] .
4
Concerning t h e dependence of (Te,u) on v we have PROPOSITION 1.1. Let {Te(v), u ( v ) ) be t h e s o l u t i o n o f P(Te,u,p;v).
Then:
00
with 0
0. When i t c r o s s e s a f i r s t c r i t i -
i t can be o b s e r v e d an o t h e r f l o w w h i c h i s
o f c e l l u l a r t y p e and p e r i o d i c i n ' t i m e . I f we i n c r e a s e a g a i n
t h e n a f t e r t h e o c c u r e n c e of
wl,
some more and more
complicated flows, a t r u e t u r b u l e n t f l o w occurs. T h i s i s j u s t an example t o j u s t i f y t h e f o r m a l t h e o r i e s developped by E. HOPF
[2]
i n 1942 and L. LANOAU
i n certain situations,
b i f u r c a t i o n s of s o l u t i o n s , when a p a r a m e t e r ( a s
[I23
i n 1944. To e x p l a i n t u r b u l e n c e
t h e y had f o r m u l a t e d a n e x p l a n a t i o n b s s e d on s u c c e s s i v e
0,)
o f t h e N a v i e r - S t o k e s e q u a t i o n s . becoming u n s t a b l e increases.
21 N a v i e r - S t o k e s e q u a t i o n s . We have i n g e n e r a l a system o f t h e f o r m
+
VP = uAV
+
f ,
G . IOOSS
100
where V is t h e v e l o c i t y o f t h e f l u i d a t t h e p o i n t ( x . t l c Q
x R + , p is t h e
p r e s u r e . f i s a g i v e n e x t e r n a l f o r c e , and a i s g i v e n on t h e boundary 2 ' 3
bounded r e g u l a r domain il o f t R
an
of a
.
or R
I n f a c t , o t h e r phenomenon such as t h e % n a r d c o n v e c t i o n , o r f l o w s w i t h t h e obeys systems o f e q u a t i o n s w h i c h have a
occurence o f e l e c t r o m a g n e t i c f i e l d , s i m i l a r s t r u c t u r e as (11,
( s e e [I] I
. I n t h e example c i t e d above, we have t o
t a k e an R w h i c h i s a bounded domain o f p e r i o d i c i t y o f t h e f l o w i n z ( p e r i o d 2n/(rI,
i
as t h e e x p e r i m e n t s suggest u s . The t h e o r y , t h a t we s h a l l d e v e l o p p , r u n s
w e l l w i t h t h i s s o r t o f domain s l s o . Moreover,
o t h e r boundary c o n d i t i o n s a r e
.
p o s s i b l e [3]
31 B a s i c f l o w ,
perturbed equations.
L e t us assume t h a t we know a s t e a d y s o l u t i o n (Vo,pol I n fact,
c a l l t h i s s o l u t i o n "the basic flow".
u-'
c h a r a c t e r i s t i c p a r a m e t e r such as i t by A and assume t h a t V
v
o f [I]. We s h a l l
f o l l o w i n g t h e problem, we have a
o r w1 i n t h e example o f 11. L e t us d e n o t e A. Now we pose
i s analytic i n
= Vo[A1 + u.
Hence, t h e p e r t u r b a t i o n u s a t i s f i e s a system o f t h e form du dt
[2)
LAu - H r u l = 0 .
+
where we l o o k f o r t w u ( t 1 as a c o n t i n u o u s f u n c t i o n t a k i n g v a l u e s i n t h e domain g o f t h e l i n e a r o p e r a t o r
LA, w i t h
a c o n t i n u o u s d e r i v a t i v e i n an H i l b e r t
space H. I n t h e case o f t h e s y s t e m (11. we t a k e H = { ~ E [ L ~ [ ~ I: - J ~v . u = G .
g = { U E [ H ~ [ S I I ] ~i
x
=
D}# o n t o H.
uEd
L A u = n[-uA u
H(u1
ulan
0.u = 0,
GI,
Il t h e o r t h o g o n a l p r o j e c t i o n * i n [L2(Rl]3
and i f we denote by we have V
u.n/an =
=
+
(u.0)
Vo[A1
+
[Vo[A).V)u]
I
-ll[[u.Vlu].
I t i s known,
following
[Ill , [IS]
, t h a t H I = I u = V(f.(PEH1[Q)}.
SECONDARY
These d e f i n i t i o n s
BIFURCATION
101
ensure us t h e l o c a l e x i s t e n c e and uniqueness o f t h e Cauchy
problem ( 2 1 , w i t h U(OI
= u o ~ . 9 ( s e e [4]1.
The p r o o f i s based on t h e f a c t s t h a t
il
oo i s an h o l o m o r p h i c f a m i l y o f c l o s e d o p e r a t o r s i n H.
o f domain
9 ,
where Oo c C.
V h E Do,
i i )
i t can be d e f i n e d an h o l o m o r p h i c semi-group o f o p e r a t o r s i n
-LXt H : [e
} t 2 0'
iiil There e x i s t s C 2 D such t h a t VXE Do and V u e g / / e- L A t M
[UI/[~S C
t - a l ( u l l . 92,
with
Q
< 1. t E ] O . T ] .
T
0.
I l9
1 J v ( ~ -I
V,,(X.~+~~I
:
/ I V [ O l - V1LA,aoll/9 +
-
o
XI), ;
a certain coefficient,
- PAlg,
neighbourhood either there
an i n v a r i a n t a t t r a c t i n g " c i r c l e "
o r t h e r e e x i s t s an i n v a r i a n t r e p e l l i n g " c i r c l e "
t h e " r a d i u s " of
Th i s o f orderlX
-
A,,
1/2
.
.
G IOOSS
106
T h i s g i v e s us t h e
Theorem.
Let u s assume r e a l i s e d t h e a s s u m p t i o n s H.1 and H.2, t h e n i n g e n e r a l t h e r e e x i s t s a neighbourhood o f a*circle”r i n
X
X 1 such t h a t there i s [ o n l y on o n e s i d e of
[I - p X ] g s u c h t h a t t h e s e t
9-{ V c t l
=
V1[X,tl
+
u[t)
XI) ;
T ( X , U ~ I ] , uo€rX , t H u [ t l i s t h e s o l u t i o n o f (31 c o n t i n u o u s i n 9 , w i t h u ( 0 ) = u 1 i s i n v a r i a n t by t h e dynamical s y s t e m ( 1 ) . Following t h e s i g n of a c e r t a i n c o e f f i c i e n t t h i s t o r u s F o c c u r s f o r X > XI and i s a t t r a c t i v e o r it o c c u r s f o r X < XI and i s r e p e l l i n g . t€[O,
A detailed
p r o o f can be found i n
[I31
f o r t h e RUELLE
For t h e e x p l i c i t c a l c u l a t i o n o f c o e f f i c i e n t s s e e [6]
-
TAKENS theorem.
.
111. D i r e c t b i f u r c a t i o n i n t o an i n v a r i a n t t o r u s .
Let u s c o n s i d e r t h e c a s e when we have
There a r e only 4 s i m p l e e i g e n v a l u e s
( z i w o and + i w , l
of
eigenvalues c r o s s t h e imaginary a x i s w h i l e
LA on t h e i m a g i n a r y X cFosses
Xo.
t h e s p e c t r u m s t a y on t h e r i g h t s i d e of t h e complex p l a n e f o r
11 B i f u r c a t i o n i n t o p e r i o d i c s o l u t i o n s .
Let u s n o t e T t h e unknown p e r i o d , and r e s c a l e t : T = 2n T - I t . u [ t l = :(TI.
then
We have now t h e s y s t e m
w h e r e H m [ T , E ) d e n o t e s t h e Sobolev s p a c e of n e a r l y everywhere 2 n - p e r i o d i c
f u n c t i o n s such t h a t
SECONOARY
%
L
J
H
t o t h e p r o p e r t i e s o f LA it is shown t h a t t h e l i n e a r o p e r a t o r 1 -dT' a d m i t s a bounded i n v e r s e i n H ( T , H ) , i f and o n l y i f qLX i s n o t an e i g e n v a l u e o f L, , V n E Z . T h e n f o r X n e a r Ao, we s h a l l o b t a i n
Thanks du -
ni/q
a bifurcation point f o r f o l l o w i n g we assume
*I
107
BIFURCATION
1
Z p y ,
v
q
near
p/wl o r q/wo f o r a c e r t a i n p o r qEN. I n t h e
o1 > wo > 0 [ n o l o s s o f g e n e r a l i t y ) .
p€N.
We can do e x a c t l y t h e same c a l c u l a t i o n s a s i n t h e c l a s s i c a l c a s e . w h e n
( o r f i w l ) on t h e imaginary a x i s . T h i s l e a d s t o t h e e x i s t e n c e
t h e r e a r e only f i w o
o f two p e r i o d i c s o l u t i o n s :
-
a A-
e
AoI
-
a, e
e a c h b i f u r c a t i o n i s o n l y on one s i d e of
ir
u
(01
..
1/2
i~
u
(1)
f o l l o w i n g t h e s i g n of a c e r t a i n
Ao,
coefficient. Moreover, i t can be shown t h a t these two p e r i o d i c s o l u t i o n s X a r e t h e a n l y o n e s b i f u r c a t i n g from
Xo
(see [7]
1.
I n t h e o t h e r c a s e s we have t h e f o l l o w i n g r e s u l t s (see [7] If,
I
:
=wo E i t h e r t h e r e e x i s t s only t h e s o l u t i o n
(&,
t h e r e e x i s t s 3 s o l u t i o n s : %, and 2 s o l u t i o n s b i f u r c a t i n g from
or
( o f o r d e r l h - Ao11/21,
a2 and
o f o r d e r [X
-
Xo),
ho.
If w , do E i t h e r t h e r e e x i s t s 2 o r 4 o r 6 o r 8 s o l u t i o n s o f o r d e r Ih- Xo(1/2. c a t i n g from If
w:
=
P wo.
Xo
(one o f t h e s e s o l u t i o n s i s p
z
bifur-
8).
4
There e x i s t s two s o l u t i o n s o f o r d e r ( A
x We mean p e r i o d i c s o l u t i o n of ( 2 ) .
-
Ao11/2,
b i f u r c a t i n g from
Ao.
G . moss
108
One o f t h e s e s o l u t i o n s i s %, previous
t h e o t h e r has t h e p r i n c i p a l p a r t a s t h e
"21,.
21 B i f u r c a t i o n i n t o a t o r u s . Let u s assume
iL wl/wo
#Q or,
"1=
wl/woE 0 t h e n
if
> 1
with p
f
q b 5.
wo
Remark. = pw,
w,.
T h i s contains t h e case when
p >I 4. and a l l c a s e s when
w,
# pwo.
WpEN (see t h e r e s u l t s o f 11.
Let u s now c o n s i d e r
~ ~ €i n9a , neighbourhood
o f 0 , a s an i n i t i a l c o n d i t i o n
f o r t h e s y s t e m ( 2 1 . We can t h e n d e f i n e t h e s o l u t i o n t continuous i n g f o r t
c-1 ' & ( t . X , u o l which i s , w h e r e T is chosen a r b i t r a r i l y f o r t h e moment,
E [D,T]
b u t f i n i t e p o s i t i v e . The map (111
C-,
Uo
$5Ix(u0l = @ ~ T , h . u o J
can t h e n be d e f i n e d i n a neighbourhood o f 0 i n 9 and ( h , u o ) w $ h ( u o l
is
a n a l y t i c . Moreover we have t h e d e r i b a t i v e a t 0 :
-LA T
o
(0) =
e
O
0
and t h i s compact o p e r a t o r i n 9 h a s 4 s i m p l e e i g e n v a l u e s o f moduli 1 : +iwoT
e
+iwlT
. e
. The
o t h e r e i g e n v a l u e s a r e o f moduli l e s s t h a n 1.
Hence, we can use t h e " c e n t e r - m a n i f o l d theorem" [ s e e [I31 problem t o a 4-dimensional
I t o reduce t h e
one.
T h e n , i n t h e aim t o u s e t h e work o f R. JDST and E . ZEHNOER [ I D 1 f o r t h e
new map i n a 4 - d i m e n s i o n a l s p a c e , we have t o choose T s u c h t h a t (S1w0
f
S2 w l l T
= 2r
m, S i E E , mEZ,
(121
Is1/ leads t o S
1
=
S
2
=
+
Isz/ 0, where u = u(t) is the solution of
a(t) = O(exp{-kt/E)),
the reduced problem u u '
k
- u = 0,
clearly,
u ( 1 ) = B.
Case 2.
O O
X 0,
& v(+l) =
T ) i s a s follows:
LIE
v < 1, then the Zargest eigenvaZue of VTE tends to
If v > 1 then
01,
u(
Define W(x) := $
T ) becomes dense i n ( - m , O )
"F
for
E
+
-m
for
+O.
( t l t \ v - l / a ( t ) ) d t . A f u n c t i o n u i s an e i g e n f u n c t i o n
-
0
of
(4)
if and o n l y i f v : = u e x p ( W ( . ) / E ) i s a n e i g e n f u n c t i o n o f t h e o p e r a t o r
V T E ~: = =
exp(w/E) v ~ E exp(-w/E)) { ~ = ETv
-
(ilxlV-'
+ ixlxlv-l
b/a
+ tlxl
2v
/Ea)v.
v
T
E'
p r o v i d e d ReX > - E ( v - ' ) ' ( v + l )
if
E
If
119
PROBLEM
A SINGULAR PERTURBATION
+ C . T h i s proves t h a t t h e spectrum d i s a p p e a r s a t
-m
+O.
-t
w >
D( ?
1 we d e f i n e f o r each v E
v
the function w , E.
~
Analogous t o
(4) t h i s i n d u c e s t h e t r a n s f o r m a t i o n o f wTE i n t o
1w
(5)
v E
= -d2w 7- - P(S,E)W,
w(tE-9)
dc
"tE, D(~F~),
= 0, f o r all w c
which is s e l f a d j o i n t , The " p o t e n t i a l " f u n c t i o n P s a t i s f i e s
+
P(5,E) = o(E$v-; T
uniformly f o r a l l
5
E
E
+
E v - ' I p )
(E
-+
+O)
i
(-E-~,E-'),
s o it i s m a j o r i z e d by t h e "square-well''
potential V,
It i s e a s i l y seen t h a t t h e e i g e n v a l u e s of d 2 / d c 2 for E
-t
-
V
become d e n s e on
(-m,O)
+O and by well-known comparison theorems t h i s is t r a n s f e r r e d t o q.e.d.
and hence t o U("T,)%
REMARK 1 :
The o p e r a t o r a s s o c i a t e d w i t h (2-1 s a t i s f i e s t h e same r e s u l t a s VTE
d o e s ; t h e o n l y d i f f e r e n c e i n t h e proof i s t h a t w h a s t o b e r e p l a c e d by -w i n formula
(4).
From h e r e on we s h a l l d e a l w i t h t h e i n t e r m e d i a t e c a s e drop t h e s u b s c r ip t
v of
T
V E
v =
1
o n l y , so we w i l l
f o r V = 1 . T h i s c a s e i s t h e most i n t e r e s t i n g , b o t h
p a r t s , E T and xd/dx o f t h e o p e r a t o r T
have an i n f l u e n c e o f e q u a l s t r e n g t h on t h e
spectrum o f t h e o p e r a t o r and t h e s e i n f l u e n c e s a r e i n b a l a n c e , such t h a t t h e spectrum n e i t h e r v a n i s h e s nor t e n d s t o a dense s e t :
THEOREM 2 :
Let { A k ( € )
I
k c N} be the s e t of eigenvalues of TE, arranged i n
decreasing order, i.e. Ak+, < X k , then a l l eigenualues s a t i s f y :
(6)
A
k
(E)
= -k + I ) ( € '
uniformly with respect t o k.
k3l2)
(E +
+0)
P.P.N.DE GROEN
120
We w i l l merely s k e t c h t h e p r o o f ; f o r a d e t a i l e d v e r s i o n we r e f e r t o C31 o r C41.
-
i n t o T,
Again we t r a n s f o r m T
nE
operator
:= Ed2/dxZ
-
as i n ( 4 ) and we c o n s i d e r t h e s p e c i a l ( H e r m i t e - )
x 2 / 4 E on t h e same domain o f d e f i n i t i o n . By computing
X
t h e s o l u t i o n s o f Il v = hv we c a n v e r i f y d i r e c t l y t h a t t h e e i g e n v a l u e s o f satisfy
( 6 ) and t h a t t h e f u n c t i o n xn’
(7)
x,(x,E)
:= e x p ( - ~ x 2 / E ) H k - , ( x / ~ ) ,
( H k i s t h e k-th Hermite p o l y n o m i a l )
approximates t h e e i g e n f u n c t i o n o f RE a t hk up t o L)(E-ne-1/2E) f o r E
I n o r d e r t o compare TE and Il
+O.
-+
we connect them by t h e c o n t i n u o u s c h a i n
..
R
:= ( l - t ) I I E + t T E , which s a t i s f i e s
.t
1 IRE,t
(8)
u
uniformly f o r a l l s , t
-R
E,S
u ( / = (‘((s-t)(l/RE,t u / I
Ilu(1))
+
c 0 , 1 1 and a l l u i n t h e domain o f d e f i n i t i o n . By s p e c t r a l
E
C51 c h . 5 , t h i s i m p l i e s t h a t t h e e i g e n v a l u e s o f R
p e r t u r b a t i o n theorems, c f .
since R
E .t
a( (s-t ) / 1 XI ) ,
depend c o n t i n u o u s l y on t and t h a t t h e i r v a r i a t i o n i s o f o r d e r i s (nearly) selfadjoint.
E .t
Next we prove t h a t t h e approximate e i g e n f u n c t i o n s o f
I I (RE,t -
(9)
uniformly f o r a l l k t h e n h (E,O) k
E
k)Xkl
I
Iand t
satisfies
’
= E
3/2
satisfy
I IX,l I ) ,
(E
-+
+o),
i s t h e k-th e i g e n v a l u e o f R
C0,ll. I f \ ( E , t )
(6). By ( 8 )
n
we can f i n d numbers
E
> 0 and t n
E,t’
d/n
2
( w i t h d > 0 and independent o f E ) such t h a t t h e c i r c l e C ( - j , $ ) around - j w i t h
4
radius E E
i s contained i n t h e r e so lv e n t s e t o f R
1 and t
[O,E
E
[O,tnl
E ,t
for a l l j
S
n (j,n
E
U), a l l
and t h a t t h i s c i r c l e c o n t a i n s o n l y t h e e i g e n v a l u e
h . ( E , t ) and no o t h e r . With t h e a i d o f t h e p r o j e c t i o n o n t o t h e e i g e n f u n c t i o n o f
J
X.(E,t)
J and j
5
s a t i s f i e s (6) f o r a l l t E C O , t n l J n . Now we can r e p e a t t h e argument, s t a r t i n g from t h e p o i n t t = t
it f o l l o w s from ( 9 ) t h a t
h.(E,t)
instead
o f t = 0 ; however, s i n c e we d i d n o t prove a n y t i n g a b o u t X n + , ( e , t ) , t h i s p o s s i b l y can e n t e r C ( - n , ; ) satisfies
(6) f o r
f o r some t > t , . So we p r o v e i n t h e n e x t s t e p t h a t h . ( E , t ) J a l l j 5 n-1 and t
choose n s o l a r g e t h a t
7
j=k+l a f t e r n-k s t e p s , q . e . d .
t.
5
d
E
Ct
1
n’
j=k+l
t
n
+ t
n-1
1
and so on. S i n c e we can
l / j 2 1 , we have proved f o r m u l a
(6)
121
A SINGULAR PERTURBATION PROBLEM REMARK 2: From the proof of theorem 2 we obtain the inequality
I lGEu -
(10)
Xu1
I
2
I lul I{dist(A,{-n
I
n
E
Wj)
- DE')
for some constant D > 0. REMARK 3: By taking the adjoint of T we find that the eigenvalues of the E
operator connected with ( 1 - )
converge to the nonpositive integers for
with the same asymptotic estimate as in
E
-+
+O
(6).
REMARK 4: Theorem 2 can be generalized to elliptic problems in a higher dimensional space which degenerate to a first order operator with a (simple) critical point for
E
+
+o, cf. C41.
REMARK 5 :
The spectrum of the limit operator T
U(To) = {?I
E
E
I
ReA
5
-$I;
in L z ( - l , l ) is the set
we see that there is an apparent lack of spectral con-
tinuity in L2-sense. However, in distributional Sense there is spectral continuity: the only (Schwartz-) distributions whose support is contained in ( - 1 , l )
and which
satisfy the equation xu' = Au (in distributional sense), are Dirac's &-distribution
6(n-l) = -n6 (n-1), n
and its derivatives. They satisfy x
E
W; moreover, the
(approximate) eigenfunctions of T converge to them in distributional sense.
3. CONVERGENCE OF THE SOLUTIONS In order to be able to prove convergence of a formal approximation of the solution of (1') Lemma 3:
we can use the folbwing lemma:
Constants C > 0 and K > 0 exist, such that TE
-
is invertible and
E E
C0,ll.
satisfies
for all u
E
D(T ) , A
E
Q with ReA >
-;+
KE and for all
For the proof we refer to C31 o r 141. Let u
be the solution of the full problem (1')
reduced equation xu' 1.e. :
-
hu = f of ( l ' ) ,
and u the solution of the
which satisfies both boundary conditions,
122
P.P.N.DE GROEN
u o ( x , h ) :=
(11)
xilxl I f f"
E
and i f Re1
L2(-1,1)
II(ET + xd/dx
since u ( C 1 , h )
-
-
u (i1,h) =
= L)(E/
then : u
f(I
-
h)uo
t + dt
f(t)(t)
>.3/2,
=
E
IlETflI
$ ( A + B)/xI' + $ ( A
-
A- 1
.
B)x/x/
and s a t i s f i e s
L2(-1,1)
= L)(El(flI + E l I f " ( / ) ,
+o);
(E
-+
(E
+
0 t h i s i m p l i e s by lemma 3:
I / u E - uoII 5 (Reh
(12)
I"
-
+
If1 I
K E ) - ~ I I ( T -~ h)(uE
-
uo)lI =
I
+ E l If") ),
By S o b o l e v ' s i n e q u a l i t y c u l z 5
+o).
+ 21 lull IIu'II we i n f e r from ( 1 2 ) and
lemma 3: 1
(13)
CUE-
provided Reh >
Uol
=
+ \If''(l)),
O(E'(llflI
(E
+o),
-+
312.
From ( 1 3 ) we s e e t h a t ( i n f i r s t a p p r o x i m a t i o n ) t h e r e a r e no boundary l a y e r s i n t h e a p p r o x i m a t i o n , i f Fie1 > 3 / 2 . From ( 1 1 ) however, we s e e t h a t a non-uniformity can be expected n e a r t h e p o i n t x = 0 and t h a t it w i l l grow l a r g e r a s -ReX grows larger. For Reh 5
6
u;(.,h)
a proof o f convergence becomes more d i f f i c u l t , s i n c e
L 2 ( - l , l ) i n t h a t c a s e . T h e r e f o r e we extend t h e boundary v a l u e problem equiped w i t h t h e norm u
t o t h e l a r g e r space H - n ( - l , l ) ,
(1')
Iu[-n' w i t h n
-+
E
fN,
C81 c h . 1.13. I n t h i s s p a c e we can prove t h e analogue o f ( 1 2 ) p r o v i d e d
cf.
Reh > -n + 3 1 2 . By i n t e r p o l a t i o n we can o b t a i n convergence i n s t r o n g e r norms: LEMMA
4: A positive function Ck(X) e x i s t s such t h a t
(14)
1 lxkul I
I Ixk+'utI I
+
+
€1
Ixku"(
k
f o r a l l functions u, for which ( I x u"II PROOF:
Cf.
+
5
k
81 Ix
(ET + xdidx
*, and f o r every k
-
h)ul E
I
+ Ck(h)Iul-k
1.
C41 lemma 3 . 1 3 .
We o b s e r v e t h a t u g ( * , h ) Reh > -n
I
3/2.
E
H i n ( - l , l ) and xnu:(.,h)
With t h e a i d o f lemma
ixk+ i w 1 2
5
E L 2 ( - l , l ) provided
4 and t h e i n e q u a l i t y
(2k+2)[]~~w + I2]) ~/ x k w ) l I [ x k + ' w '
11,
(kEW),
123
A SINGULAR PERTURBATION PROBLEM
we obtain the final result: THEOREM 5: I f n
E [N
and i f h
(a) satisfies
E C \
Reh > -n +
312,
then
We point out that the weight factors xn and xn+' in the norms of the estimates (15a) and (15b) smoothe down the non-uniformity of uE at x = 0. If h is a negative integer, we cannot expect convergence because of a neighbouring eigenvalue.
The solution of problem ( 1 - ) converges also to a solution of the reduced equation -xu'
-
hu = f in the major part of the interval, as the solution of
does. However, in this case we cannot impose boundary conditions at x = f l
)'1(
on the set of solutions of the reduced equation. We have to select the right solution from this set by a smoothness condition at x = 0; this smoothness condition arises in a very natural way from the choice of suitable domains for the operator connected with ( 1 - ) and for its limit (for
E
+
+O). At the points
x = 21 (ordinary) boundary layers of width O ( E ) arise. We will merely state the final result; for a proof we refer to [ b l . For any f
E
cn(-l ,1) we
provided h
E
define the function w
(n
E
W u {O}) by
4 u {O} and Reh > -n. Clearly this function is a solution of the
reduced equation and is smooth at x = 0. It satisfies:
for
E
-t
+O and uniformly f o r a l l x
E
C-1
,I1.
124
P.P.N.DE GROEN
REMARK 6:
The method by which theorem
6 i s p r o v e d , i s i n some s e n s e d u a l t o t h e
one o f theorem 5 . I n t h i s p r o o f we have t o r e s t r i c t t h e boundary v a l u e problem (1-)
t o t h e smaller spaces H
+n
(-1,l)
(n
E
W ) i n o r d e r t o be a b l e t o e n l a r g e t h e
p a r t o f t h e 1 - p l a n e i n which an analogue o f lemma 3 is t r u e and i n which we hence can prove v a l i d i t y o f t h e a s y m p t o t i c formula ( 1 6 ) .
REFERENCES 1. Ackerberg, R.C.
& R.E.
S t u d i e s i n Appl. Math.,
O'Malley, Boundary Layer problems e x h i b i t i n g resonance,
9 (1970),
p.277-295.
a boundary value problem of singular perturbation type, S t u d i e s i n Appl. Math. , 52 ( 1973 ) , p , 129- 139. 2. Cook, L. Pamela
W. Eckhaus, Resonance i n
! i
3. Groen, P.P.N. d e , Spectral properties of second order singularly perturbed boundary value problems with turning points, p r e p r i n t : r e p o r t 39 (May 1975) o f t h e "Wiskundig Seminarium d e r V r i j e U n i v e r s i t e i t " , Amsterdam, t o appear i n t h e J o u r n a l of Math. Anal. and A p p l i c a t i o n s .
4.
Groen, P.P.N.
d e , SingularZy
perturbed
d i f f e r e n t i a l operators o f second
order, Mathematisch Centrum Amsterdam, t r a c t 68 ( t o appear 1976). 5. Kato, T . , Perturbation theory of linear operators, S p r i n g e r V e r l a g , B e r l i n e t c . , 1966.
6 . Rubenfeld, L . A . and B. W i l l n e r , The general second order turning point problem and the question of resonance f o r a singularly perturbed second order ordinary d i f f e r e n t i a l equation, t o a p p e a r .
7 . L i o n s , J . L . , Perturbations singulikres duns lea problPmes a m l i m i t e s e t en Contr6te optimal, L e c t u r e n o t e s i n Math. 323, S p r i n g e r V e r l a g , B e r l i n e t c . , 1973. 8 . L i o n s , J . L . & E. Magenes, Problkmes a m l i m i t e s m n homogPnes, Dunod, P a r i s , 1968.
W . Eckhaus (ed.), New Developments in Differential Equations
@ North-Holland Publishing Company (1976)
ASYMPTOTICS FOR A CLASS OF PERTURBED INITIAL VALUE PROBLEMS Bob Kaper Department of Mathematics, University of Groningen, Groningen, the Netherlands
INTRODUCTION In this paper we are dealing with initial value problems containing asmall nonnegative perturbation parameter
E.
On time intervals initiating the origin we
will approximate the exact solution (provided it exists) asymptotically with respect to
E
as
6
+
0. The asymptotic solutions could be derived from so-called
formal asymptotic 50lutions, i.e., functions which satisfy the differential equation and the initial conditions up to an asymptotic accuracy of certain order. These formal asymptotic solutions should then be compared asymptotically with the exact solution. The question arises whether such a function approximates the exact solution up to an asymptotic accuracy of the same order as it approximates the equation and the initial conditions. Or at least whether there exists a relation between these two orders. This question will be answered in connection with the type of the interval to be considered. As an application we consider a class of perturbed oscillations described by the nonlinear second order ordinary differential equation with slowly varying coefficients w"
+ F(w,~t) + E~(W,W',E~,E)= 0 ,
t
1. 0 ,
( 1 .a)
( I .b) W(0,E) = C I 1 ( E ) , W'(0,E) = a 2 ( E ) . The force term F, onwhich Ef is to be consideredas a small perturbation, is assumed
to be the derivative of apotential function whichhas an absoluteminimum atthe origin. Let u s briefly recall the concepts of (formal) asymptotic solution in dealingwith the vector differential equation in R": x' = f(X,Et,E),
-
x (x
-
t
E I,
(2.a)
x(0, E ) = a(E) A function Ti is called an asymptotic s o l u t i o n of order E C 0) if
-3=
O(K)
(2.b) K
(K(E)
=
o ( i ) as
on I
represents the exact solution of problem ( 2 ) ) . A function u is called a formal asymptotic s o l u t i o n df o r d e r q as E C 0 ) if g and 0 ,
g(t,E) = 'J(t,E) - f(u(t,E), are O ( q ) on I,
Et,E), 8 ( E ) = a ( € ) -
(call g and 8 the residuaZs of u for problem ( 2 ) ) .
125
(q(6) =
U ( ~ , E ) ,
o(l)
126
B. KAPER
An obvious modification leads to similar concepts for second order problems. Note that without loss of generality a slowly varying dependence of f on t is assumed in view of
the application.
The order symbols 0 and o are understood to be related to the limitprocess E
t 0 uniformly in t on I.
I n section I we will give a brief summary of the form of an Nth order formal
assymptotic solution $N ( whose residuals are O(E~+')) of problem ( 1 ) .
For a
complete description I refer to [ I ] . In section 2 we will treat the remainder problem x
-
u in dealing with the vector
problem ( 2 ) . It includes the existence and uniqueness of the exact solution. Connected to this section we will construct an improved Nth order formal asymptotic solution 0, in section 3 . In a final section 4 we will state some results that hold on the infinite interval [ 0 , - ) .
ON FORMAL ASYMPTOTIC SOLUTIONS OF OSCILLATION PROBLEMS
81.
We may expect the solutionsof problem ( 1 )
to be oscillating functions with
slowly varying amplitude and frequency. In order to include these large-scale variations in the approximation we consider for the moment intervals of order E
-1
.
, i.e., intervals of the type
[O,
E
-1
L] where L is independent of
E.
Asymp-
totically we may distinguish two different time scales, a local- ( o r f a s t - ) time
scale
on which the solution is periodic with a period of order one and a slow-
(or stretched-) time scale, characterized by the slow variable
T =
ct, which
accounts for the slow modulation of the oscillations. Both scales are made explicitely in the form of the formal asymptotic solution which technique is known as the two variable method. An Nth order formal asymptotic solution 0 N problem ( I ) is given by
with residuals
Of
N+ 1
gN+](t,E), gN+I = O ( I ) , and E N+ I Bi(~), Bi = 0 ( 1 ) , i = I , ? . contribution Uo,Uo = n + A Q to the expansion of $N is the even,
The @ ( I )
E
0 0'
2n-periodic solution of the nonlinear conservative system
in which
T
is t o be considered as a fixed parameter. The function n is the alge-
braic average of the extreme values of Uo, which makes it possible to introduce
127
A CLASS OF PERTURBED INITlAL VALUE PROBLEMS
an amplitude function A o ( r ) .
The function w follows from the normalization of the
period of Uo to 2n and will therefore depend on A system ( 1 . 2 ) .
in the case of a nonlinear
The higher order contributions U
Uv = A
V'
v
Z*
2
+
QV,
to the expansion
of $N are determined by linear, second order equations whose homogeneous part is
the first variational equation of ( 1 . 2 ) with respect to U
One homogeneous solution, z;(P,T),
0'
follows direcly by differentiating Uo with
respect to p. A second solution, z*(p,r) could be found by the variation of con2
j = 0,..., N - I , follow from houndedness
stants method. Equations for A . and S j ,
. J+l
requirements of U . 277
J
They are of the type
yj+, (p,r)z:(p,~)dp
0, i
=
.. . ., N
I , 2, j = 0 ,
=
(known a s suppression of secular terms in Yj+]).
- I.
A s an immediate consequence of
the determination of ON we have for the residual function g gN(t,E)
=
P
Y ~ + ~ ( P , T )+ O ( E ) ,
=
E
-1
N
S(T,E,N),T = Et.
At this stage we may draw the following conclusions.
- From the expansions (i)
(1.1)
we see that
ON(t9E) = '$G(P,T,E),
P
= E
-I
s(I,E;N),
T =
Et,
where (ii)
$:
is defined o n R + x [O,L]
x
[O,E~],
Let u s call such functions satisfying (it, (ii) and (iii) funceions of the ?eriodic two variable type.
- In consequence gN is of the periodic two variable type. - No equations for AN and SN have been determined yet. We introduced the quantities anticipating a question on the order of asymptotic accuracy of $N conceiving it as an asymptotic solution on
[~,E-'L].
We will
treat this problem in the next section when dealing with the vector differential problem (2) on arbitrary intervals I.
82.
PROOF OF ASYMPTOTIC CORRECTNESS I n this section we consider the vector differential problem ( 2 ) in Rn:
x'
=
X(0,E)
f(X,Et,E), = CY(E)
I
t
E I,
(2.la) (2.lh)
.
B KAPER
128
where I be some finite o r infinite interval, possibly depending on 6 . Let 1. I denote the vector- and matrix norm and llxll = sup Ix(t) 1 , t E I. Let u be a formal asymptotic solution of (2.1) of order n, i.e., the residuals g and 8 are O ( r l ) .
In
order to compare the formal asymptotic solution u with the exact solution x of (2.1),
whose existence and uniqueness should be established, we apply the change
of variables
x=u+p. Then the remainder function
should satisfy the nonlinear vector differential
p
problem (2.2a) (2.2b) where
Let Y(t,s;E), Y(t,t;E) = E , be a fundamental matrix solution of the linear equation 2' =
A(t,E)z,
Y(t,S;E) = VO(t,E) y o- 1
(S,E),
(2.3) Vo(t,E) = 'Y(t,O;E).
The initial value problem (2.2) for
may be transformed into the nonlinear
p
Volterra integral equation
k(t,E) = Yo(t,E) B ( E )
+
t J Y(t,S;E)g(s,E)dS.
(2.5)
0
Provided f is sufficiently smooth we can show by means of a contraction mapping principle the existence and uniqueness of the solution
p
of (2.4)
within a ball B(R) with radius R if Y(E)K(E)
5 t with R
=
2K(E),
where K(E)
=
IIK(t.E)II
,
t Y(E)
=
II/IY(t,s;E)ldsll 0
Let
K
.
be an asymptotic order function, i.e.,
following
K
=
O(1)
as
E
4 0. Then we have the
A CLASS OF PERTURBED INITIAL VALUE PROBLEMS
129
Theorem I . Let u be a formal asymptotic solution of (2.1) with residuals g and 8 of order q . If = o(K-I)
Y(E)
then for sufficiently small
0 problem (2.1) has an exact solution x
E
=
u + 0
with x - u
(uniformly in t on I)
U(K)
=
(Hence u is an asymptotic solution of order The relation between
K
K).
and I-,given by the function k(t,E) defined in
( 2 . 5 ) , depends on the order of magnitude of the interval I and the behaviour of
the fundamental matrix 1 . Let u s assume throughout this section that I=[O,t for some integer m and that I't'(t,s;E)/
2 K,
0
5 s 5 t, t
-m L]
E I.
In this case y = O(E-~) and the condition on y . of ~ Theorem1 impses a minimum order of asymptotic accuracy of $N as an asymptotic solution and hence also as a formal asymptotic solution. The relation between
K
and
I-
is simply
K
=
E - ~ I -which , means a reduction of the
order of asymptotic accuracy of u as an asymptotic solution when the order of magnitude of I(m)
increases. On intervals of order
E
-I
this means a loss of one
in the order of u as a asymptotic solution compared to u as a formal asymptotic solution. For a subclass of initial value problems related to the oscillation problem ( I ) we may improve the order with one
E
by the application of a partial
integration rule and by imposing a condition on the residual function g (hence on u ) . Therefore we need the concept of
- f i r s t order formal asymptotic m a t r i x s o l u t i o n function for which 8
-I
A(t,s)
0 of
(2.3), i.e., a matrix valued
@(t,E) - 8'(t,E) =cG(t,E), G = O ( 1 ) .
det 8
>
0 (hence
exists).
Apply the change 't'(t,s;E) = O(t,E) Y(t,s;E) of z =
-E
@
-1
@-'(s,E),
thenY is a fundamental matrix
(t,E)G(t,E)z.
In view of the oscillation problems we have the following Theorem 2. Let u be a formal asymptotic solution of (2.1) periodic two variable type, u(~,E) = u*(P,T,E), Let
@
of order q , which is of the -I
S(T,E), T = ~ t . be a first order formal asymptotic matrix solution of(2.3) of the periodic
two variable type, O(t,E)
=
O*(P,T,E), p =
E
-I
p
= E
S(T,E),
T
=
et. If
(2.6)
130
B .KAPER
where g* is the two variable counterpart of g,thenproblem ( 2 . 1 ) has an exact solution x = u + p with
Proof. The theorem is almost an immediate consequence of Theorem I except f o r the improved relation between
K
and q. Substitute the above assumptions on u , Y and
d in k (defined in (2.5))
t k(t,E) = YO(t,c)8(~) + @(t,c)
Y(t,s;dO
*- 1 (p(s),Es,E)g*(p(s),Es,E)dS,
0
p(s)
=
E- 1 S ( E S , E ) .
The variable s of integration appears in the integrand in two different ways: via the periodicity variable p and elsewhere characterized by a derivative of order O ( E )
(9 is of
a gain of one
E
O ( E ) since 0 and G are O ( I ) ) .
in the integral which means
K
=
A partial integration rule gives
E.E
-m
n.
Till now we made the assumption of uniform boundedness of Y . In a corrolory we will replace the assymption by a condition on the matrix A in the special case of intervals of order
E-'.
Corollary I . Let U and 0 be formal asymptotic solutions in the sense of Theorem 2 . If
(2.7)
then u is an asymptotic solution of ( 2 . 1 ) of order 11 on intervals of order
E
-I
.
Proof. Onceyo is bounded, condition ( 2 . 7 ) (ii) assures the boundedness of the inverse -I Yo
which means lY(t,s;E)I
5 K, 0 5
s
2
t
I
i=l
n+l) nonnegative. Suppose t h a t
ko5,kjt(J-1)V,kn,.,pn~(n)V€L1(I)
€ o r a l l s o l u t i o n s 5 , of ~ (2.38).
51,...,62n
(j=I,...
,n)
Then €or any fundamental system
o€ s o l u t i o n s o€ (2.38) and €or any s o l u t i o n
...,5&)
6 ('1 ( t ) = ( y !i) ( t >, (2.40) THEOREM.
,...,52n & G I ,...,62n
resp..
Then € o r any s o l u t i o n
be fundamental systems of s o l u t i o n s
C
o€ (2.41)
and 5 of (2.381
( t ) 9 . . '52n ( i ) ( t ) ) o(1)
3-
€or i=O,
)... 1 ,. .,ti;) ., c p
s(i)(t)=(c;i)(t)
6 (n) ( t ) = p ( t
5 ( n ) ( t )=(C,(n)( t ) 9 . .
,6$;)(t))O(l
...,
n-1
,
1
P,W
(t)) O ( $ T ) , n qn(t) ( t >) O ( p ( t 7 . ) n
The p r o o f s proceed a s t h o s e of theorems 2.5 and 2.16 transformations:
2
Suppose t h a t € o r a l l s o l u t i o n s 5.q
Let s1
c ( i ) ( t )=(5!i)
of (2.391
..,n).
(i=o,.
( t ) )0(1)
C
except €or t h e
148
NIESSEN
(2.38) i s transformed by
to x '=Ax with
and
- as i t s
- by
(real) adjoint
to )LA*Y.
I n t h e c a s e of theorem 2.37,
(2.39) i s transformed by (2.42)
to
z' =Az+me2n; i n t h e case of theorem 2.40 t h e d i f f e r e n t i a l e q u a t i o n (2.41) is transformed by
to z'=A
1
z
PERTURBED DIFFERENTIAL EQUATIONS
149
The rest of the proofs is then easily carried out. Finally, applying theorem 1.16 to the systems arrising from (2.1)
and (2.181,
we get the following theorem which is a
generalized and strenghtened version of a theorem due to Bellman [ 21 t (2.43)
THEOREM. Let the real part of
Pn-1 (r 1 dr Pn
t0
bounded above and let
If t(i)
is bounded for all solutions 5
(2.44)
and for any i=O,...*n-l,
then for any solution
C of
(2.45)
and for any fundamental system
Especially, c(i)(i=O,...,n-l)
c,, ...*tn
of solutions of (2.44)
is bounded for any solution of (2.45).
3.0~ THE LIMIT CIRCLE CASE FOR WEIGHTED EIGENVALUE PROBLEMS. Let the differential operator
be formally selfadjoint on IcR with a,(t)#O
(tEI), aiEC1(I) and
let w be a nonnegative continuous function on I. Then the (possibly singular) eigenvalue problem (3.2)
l(t)=XN,
considered in the space
150
NIESSEN
i s s a i d t o be i n t h e l i m i t c i r c l e case i f f o r every AEC every
s o l u t i o n of (3.2) belongs t o L2(w,I).
Recently Walker proved t h e
following (3.3)
TH30REM (Wslkcr r161). I f f o r some A o € C
s o l u t i o n s of
=x oWc
1( f 1
(3.4)
and
1 ( c )=X,wf
(3.5)
belong t o L 2 ( w , I ) , thcn (3.2) i s i n t h e l i m i t c i r c l e case. For ur,l and n even E v e r i t t [9 ] p r m e d t h i s thcorem assumi n g only t h a t f o r some x0EC a l l s o l u t i o n s of (3.4)
are square-
i n t e g r a b l e . The same r e s u l t follows € o r a r b i t r a r y n > 2 and for an a r b i t r a r y weight f u n c t i o n w from theorem 9.11.2 forming (3.2)
of [ 1 ] by t r a n s -
t o an n-th order system u s i n g t h e transformations
given i n [ 141. As s t a t e d above, t h e theorem i s a simple consequence of theorem 2.16 applied t o
x
L :=1- ow,
L+ :=1- Tow
(whepe L+ denotes t h e complex a d j o i n t ) pO:=ao-~ow,qo:=ao-~w,pj:=qj:=aj(j=l
,...
,n).
Theorem 3.3 may e a s i l y be generalized t o ei'genvalue problems of t h e form (3.6)
m(f )=In( s 1
w i t h formally s e l f a d j o i n t d i f f e r e n t i a l o p e r a t o r s m,n such t h a t n
i s p o s i t i v e on a s u i t a b l e f u n c t i o n space.
More g e n e r a l l y , w e get an analogous r e s u l t f o r s i n g u l a r r i g h t - d e f i n i t e 6-hermitian eigenvalue problems. Such eigenvalue
PERTURBEI) DIFFERENTIAL EQUATIONS problems (for definition and properties compare e.g.
151
[12])
may be
reduced to systems of the form (3.7)
F1x'+F2x=AGx
with (n,n)-matrix-valued functions F1,F2,G,defined and continuous on some interval IcR such that Fl(t) is nonsingular for tEI.
A
rightdefinite S-hermitian eigenvalue problem (3.7) especially has the following properties: There exists a continuously differentiable (n,n)-matrixvalued function H and a positive-semidefinite- valued continuous function W such that €or every AEC (3.8)
H'=HFT1 (F2-AG)+[Fy1 (F2- yG)]*H,
(3.9)
G=WF;-~H*.
Let K denote the continuous and positive-semidefinite square-root of W and define
u : = K F -~l ~ * , L ( u ,I
:=( x U ~ L~ E ( I) 1
.
Then the eigenvalue problem (3.7) considered in the space L2(U,X) is said to be in the limit circle case if for every AEC every solution of (3.7) belongs to L2(U,I). Then the following theorem holds : (3.10) THEOREM. If for some A o € C all solutions of F~ X' + F ~ X = A.GX
(3.11)
and of
F,x'+F~x= TOG.
(3.12;
belong to L2(U,I),
Proof. Let
then (3.7) is in the limit circle case.
l € C be fixed. Defining
A:= -F;'
( F ~ -x,G),
"A=(A-x,)F;~G
152
NIESSEN
we have
-
A+A= -FYI (F2-kG). T h e r e f o r e , (3.11) and (3.7) (3.13)
x’=Ax
-
and (3.14) resp..
+re equivalent t o
x’=(A+A)x, Furthermore, by (3.9)
and t h e d e f i n i t i o n of K and U w e
obtain
Z=( A-I o ) F;’
KKF;-” H*= ( A -A o ) H-I
Now l e t x and y be s o l u t i o n s of (3.13)
u*u. and
y’= -A*y resp..
2
Then xEL ( U , I ) by assumption, and u s i n g (3.8) i s a s o l u t i o n of
seen t h a t H*-’y
it is easily
3.12) and t h e r e f o r e belongs t o
L2(U,I), too. Thus y*”A= ( A -I o ) uH* -1 y ) * UxE L
Now theorem 1 .I2 i m p l i e s t h a t a l l s o l u t i o n s of (3.141,.
i.e.,
all
s o l u t i o n s of (3.7) belong t o L 2 ( u , I ) . S i n c e t h e e i g e n v a l u e problem (3.6)
may be transformed t o a
r i g h t d e f i n i t e S-hermitian e i g e n v a l u e problem, theorem 3.10 i m p l i e s an analogous r e s u l t f o r (3.6). 4,GENERALIZATIONS The preceding r e s u l t s may be g e n e r a l i z e d i n v a r i o u s d i r e c t i o n s . E.g.,
i P I = [ t o , b ) , i n t h e c a s e of a l i n e a r p e r t u r b a -
t i o n w e may r e p l a c e t h e i n t e g r a b i l i t y c o n d i t i o n (1.7) by t h e assumption t h a t t h e i n t e g r a l of t h e l a r g e s t e i g e n v a l u e of t h e r e a l p a r t of X’%X
i s bounded above. More g e n e r a l l y , w e g e t
(4.1) THEOREM. L e t X d e n o t e a fundamental m a t r i x of (1.2) and z any s o l u t i o n of (1.13).
Then
PERTURBED DIFFERENTIAL EQUATIONS
153
Here
For p r o p e r t i e s of p compare [ 8
3 ,p.41.
E s p e c i a l l y l p (M)j llMl
, and
i f t h e v e c t o r norm i s taken t o be t h e Euclidean norm, then v(M) is t h e l a r g e s t eigenvalue of t h e r e a l p a r t of M. Theorem 4.1 i s an easy a p p l i c a t i o n of a theorem due t o Lozinskil' (compare
[ a ],p.581
to UI
AS
=x -1 kxu.
an example we c o n s i d e r f o r a#O
which a r r i s e s from
611+qc=o
(4.2) by t h e t r a n s f o r m a t i o n
Choosing A:=O
, X :=E ,
and t h e Euclidean norm, theorem 4.1 i m p l i e s
f o r a l l s o l u t i o n s 6 of (4.2). theorem I of [ 111.
For a > 0 and i = O t h i s result i s
154
NIESY EN
Another a p p l i c a t i o n t o d i f f e r e n t i a l equations gives t h e following theorem, which i s an improved v e r s i o n of [ 101, theorem 2:
(4.3) THEOREM.
Let p,q
g r a b l e , and denote by f , ,
a r e a l fundamental system of
c "+p5=0
(4.4) such t h a t f ,
5,
be realvalued and l o c a l l y i n t e -
fd
- 5,'
c 2 = l . Then (4.2)
(and (4.4)) a r e of l i m i t
c i r c l e type provided
-
Proof. Let
( '), 0
A:=
-P
(, ,). 0
"A=(p-q)
0
0
Then (4.4) i s equivalent t o x'=Ax with
(4.5)
x=(zt),
(4.2) i s equivalent t o z'=(A+I
(4.6)
Furthermore,
z with z=(:t).
If,
i s a fundamental matrix of (4.5) and t h e r e a l p a r t of X-%X eigenvalues
& ~1 J P - q ) (2C + I 5 22) ' Thus, by theorem 4.1,
implying ~ C L , ( I )
.
f o r any s o l u t i o n z=
has
PERTURBED DIFFERENTIAL EQUATIONS
155
Theorem 1.4 may be g e n e r a l i z e d t o
let
(4.7) THEOREM. X1,X2
Let X be a fundamental m a t r i x of x‘=Ax
&
be l o c a l l v a b s o l u t e l y c o n t i n u o u s and such t h a t X=X1X2.
For f i x e d UE
To,i1 l e t
f ( t ,z )=B ( t ) g (t ,z ) +C ( t ) ID ( t ) z ‘h ( t ,z) , where B is l o c a l l y i n t e g r a b l e and I g ( t , z ) ] , ] h ( t , z ) \ S 1. (4.8)
I x 2 ( t ) x - l (T)C(T)llD(T)X1 ( T I ! ‘5 kq ( t ) k 2 ( T )
-
( T E b O , t l 5 ) jP
where k l , k 2 a r e l o c a l l y i n t e g r a b l e nonnegative f u n c t i o n s , t h e n €or
any s o l u t i o n z o€ (4.9)
z’=A(t)z+f(t,z)
we have a)
(4.13)
51
f o r t c t o , [to,t]:=[t,to]
156
NI ES S EN
then
(4.14) Proof. Using a% l - a + u a f o r a 20,we o b t a i n w i t h u:=X-’z
I
similarly
t o t h e proof of theorem 1.4 I X 2 ( t ) u ( t ) l I I X 2 ( t ) u ( t o ) l+
t ( T ) B ( T ) I dT
IX2(t)X-’
I+
and by a m o d i f i c a t i o n of t h e Gronwall
a
product k e r n e l K) t h i s y i e l d s (4.10).
To prove b) choose kl:=y,k2:=IX;’ holds by (4.11)
CllDx11‘.
X 2 is bounded by (4.11),
and (4.12),
Then (4.8) kl i s bounded
and k2 i s i n t e g r a b l e by assumption. Thus (4.14) follows from (4.10).
As s p e c i a l c a s e s we mention and u=l
1 ) Choosing X1 :=X,X2:=E
1.4.
, theorem
4.7 b ) reduces t o theorem
I n t h i s case, p a r t i a l i n t e g r a t i o n of
(4.10)
with kl :=l, k2:=lX’1C((DXI
)=x( t ) o(e
2( t
If
Ix-’(7)C(7)((D(T)X(7)1
to t
+IS
yields
I f I X-’
+
(s )C (s )/ID(s > X( s >I ds I
I X - ~ ( ~ > B ( ~e ) 7I
d71)-
to 2) Choosing X2:=X,X1:=E
(4.15)
theorem 4.7
THEOREM. If f o r some
(4.16)
IX(t)X-1(7>1s y
(4.17)
ICIIDIU,
b) i m p l i e s 2 0, OE[O,I]
(7Wo,tl),
I B I E L (~I ) ,
then a l l s o l u t i o n s of (4.9)
are bounded.
PERTURBED D I F F E R E N T I A L EQUATIONS
157
(4.16) i m p l i e s t h a t X i s bounded. By F l o q u e t ' s .theory t h e converse is t r u e i f A i s periodic: i 4 . 1 8 ) REMARK.
-
Let A be p e r i o d i c and X bounded. Then f o r
some y 2 0 (4.16) i s v a l i d . Theorem 4.15 may be c o n s i d e r e d a s a g e n e r a l i z a t i o n of t h e
following (4.19) THEOREM ( C e s a r i r 6
I).
Suppose t h a t
fi-aicL1 [ to,m) ,Pi( t )+ a i ( t + m ) , ( i = o , .
..
,n-l)
and t h a t a l l s o l u t i o n s of (4.20) d e r i v a t i v e s are bounded.
t o g e t h e r with t h e i r f i r s t n-I Then a l l s o l u t i o n s of (4.21)
d e r i v a t i v e s are bounded.
t o g e t h e r with t h e i r f i r s t n-I
A:=(
0 1*
-ao (4.20),
(4.21)
(4.22)
) v
...,
,
-a n-1
a r e equivalent t o
x'=Ax,
x=($ ( i - 1 ) 1
and z' =Az+f ( t , z )
resp..
, z=( 6 ( i - 1 ) ) .
By assumption any fundamental m a t r i x X of (4.22)
S i n c e A i s c o n s t a n t , remark 4.18 i m p l i e s (4.16).
i s bounded.
Finally the intc-
g r a b i l i t y of C shows t h a t (4.17) holds. Thus t h e boundedness oP c(i)(i=O,
...
,n-I ) f o l l o w s from theorem 4.15.
N I ES S EN
158
Obviously t h i s proof does n o t need t h e converqence assumption on t h e Pi.
Furthermore, t h e ai may b e allowed t o be
p e r i o d i c f u n c t i o n s . I t may b e mentioned t h a t t h e o r i g i n a l proof of theorem 4.19 took about 16 pages. The proof given h e r e a l s o seems t o be more s i m p l e t h a n t h a t i n d i c a t e d i n [ 2
1.
Finally we remark
t h a t theorem 4.15 i m p l i e s t h e second p a r t of theorem 1.16 and t h e
theorem of [ 2
1.
R e c e n t l y Wong announced t h e f o l l o w i n g (4.23)
THEOREM (Wonq 1171). Suppose t h a t a l l s o l u t i o n s of
(4.24) belong t o L2rO,m)nLm[Op) and t h a t f o r some a € r O . l l Irn(t,C)lS A ( t ) 1 6 I u w i t h X€LprO,m) f o r some p , l 5 p 5 2 , Then, a l l s o l u t i o n s of
L 6 = d t ,6 1
(4.25)
belong t o L2r O,m)nLmr 0.m). T h i s theorem i s a l s o a c o r o l l a r y t o theorem 4.7 b ) : Transforming (4.24) theorem 2.37,
and (4.25) t o systems as i n t h e proof of and u s i n g a s i m i l a r argument
choosing X1:=X,X2:=E
a s remark 1.9 t h e assumptions of theorem 4.7 b) are seen t o be fulfilled i f (4.26)
for a l l solutions
A !5\'q f,q
ELq[ 0 , ~ )
of (4.24).
T o prove (4.26) w e remark t h a t by
2 5 + 1 < m 1- 1 P
-
Thus
PERTURBED DIFFERENTIAL EQUATIONS
159
Therefore
i m p l i e s (4.26).
Now t h e a s s e r t i o n f o l l o w s from theorem 4.7 b ) .
T h i s proof shows t h a t (4.26) h o l d s i f w e o n l y suppose t h a t a l l s o l u t i o n s of (4.24) belong t o L a r b i t r a r y p 2 1).
9 ’-’ (with
ut[O,l]
and
Then by theorem 4.7 b)
f o r any s o l u t i o n 6 of (4.25) and any fundamental system
T1,...,52n
of (4.24).
REFERENCES
[I]
Atkinson, F.V.: Discrete and continuous boundary problems. Academic ?less, New York 1964.
[2)
Bellman, R.: The s t a b i l i t y OP s o l u t i o n s of l i n e a r d i f f e r e n t i a l e q u a t i o n s . Duke Math. J. 10 ( 1 943). 643-647.
[3]
Bellman, R.: A s t a b i l i t y p r o p e r t y of s o l u t i o n s of l i n e a r d i f f e r e n t i a l e q u a t i o n s . Duke Math.J. 1 1 (1944). 51 3-51 6.
[4] Bellman, R.: S t a b i l i t y t h e o r y of d i f f e r e n t i a l e q u a t i o n s . Dover, New York 1953.
[ 51
Bradley, J.S. : Comparison theorems f o r t h e s q u a r e i n t e g r a b i l i t y of s o l u t i o n s of ( r ( t ) y ’ ) ’ + q ( t ) y = f ( t , y ) . Glasgow Math.J. 13 (1972), 75-79.
[ 61
C e s a r i , L. : S u l l a s t a b i l i t d d e l l e s o l u z i o n i d e l l e e q u a z i o n i d i f f e r e n z i a l i l i n e a r i . Ann. S c u o l a Norm. Sup. P i s a (2) 8 (1939), 131-1460
[7] Chu,S.C.
and F.T.
M e t c a l f : On Gronwall’s i n e q u a l i t y . Proc.
AMS 18 (19671, 439440. 181
Coppel, W.A.:
S t a b i l i t y and asymptotic b e h a v i o r of d i f f e r e n -
NIESSEN
160
tial equations. Heath, Boston 1965.
[ 91
Everitt, W.N.: Singular differential equations I: The even order case. Math. Ann. 156 (1964). 9-24.
[ 101 Halvorsen,S.: On the quadratic integrability of solutions of d2x/dt2+f (t)x=O.
Math. Scand. 14 (1 964), 1 1 1-1 19.
[TI] Levinson, N.:
The growth of the solutions of a differential equation. Duke Math. J. 8 (1941), 1-10.
[ 121 Niessen, H .D. : Singulare S-hermitesche Rand-Eigenwertprobleme. manuscripta math. 3 (1970), 35-68. [13] Patula, W.T. and J.S.W. Wong: An LP-analogue of the Weyl alternative. Math. Ann. 197 (19721, 9-28.
[ 141 Schneider, A. : Zur Einordnung selbstadjungierter Rand- Eigenwertprobleme bei gewi5hnlichen Dilferentialgleichungen in die Theorie S-hermitescher Rand-Eigenwertproblcme. Math. Ann. 178 (19681, 277-294. [ 153 Shin, D.: Existence theorems for the quasi-differential equation of the n-th order. C.R. Acad. Sci. URSS 18 (19381, 515-518. [16] Walker, Ph. W.: Weighted singular differential operators in the limit circle case. J. London Math. S O C . ( ~ ) . 4 (1972). 741-744.
[ 171 Wong, J.S.W.: Square integrable solutions of Lp perturbations of second order linear differential equations. Springer lecture notes no. 145 (19741, 282-292. [IS] Zettl, A.:
Square integrable solutions of Ly=f(t,y). AMS 26 (1970), 635-639.
Proc.
W . Eckhaus ( e d . ) , New Developments i n D i f f e r e n t i a l Equations @ North-Holland P u b l i s h i n g Company (1976)
ON CERTAIN ORDINARY DIFFERENTIAL EXPRESSIONS AND ASSOCIATED INTEGRAL INEQUALITIES
W N Everitt aud M Giertz
1.
T h i s paper is concerned with inequalities of the form
for functions f in certain linear manifolds of the integrable2
squsre function space L (a,-) vith the usual norm A,B
and
c
-
11 11.
Here a,B,y,
(with c to be taken as ‘ s m a l l ’ ) are non-negative real numbers,
and the differential expression M is defined, in terms of the positive valued coefficients p and q, by Mfl = -(pf’)’ + qf on [a,-)
( ’ :d/dx).
(1.3)
In C7, Chapter VI, Sections 6 and 81 Goldberg has given certain a priori estimates of the form (1.2) for differential expressions of arbitrary order with bounded coefficients. The inequalities considered in this paper give an extension of this type of estimate to second-order expressions with. in general, unbounded coefficients. The inequality
(1.1 )
is considered by Everitt and Giertz in
C61. The results given here avoid one of the difficulties met in determining conditions on the coefficients p and q for C6; Theorem 31 to hold. Aa in C61 separation results for the differential expression M
follow from inequalities of the form (1.1);
161
for the definition of separation
362
W.N.EVERITT A N D M.GIERTZ
see
C4; Section 11 o r C6; Section 61. Additionally a compete separation
r e s u l t i s obtained here; f o r t h i s concept see t h e definition i n
(5; Section 11 Inequalities of the form (1.1) and (1.2) a l s o y i e l d r e s u l t s i n the theory o f r e l a t i v e l y bounded perturbations of t h e d i f f e r e n t i a l 2
operators generated by M i n L (a,-).
For results i n t h i s direction see
[6; Section 91, which a l s o depend on thc work of Kato, see C8; Chapter
V,
Section 41. There are also connections with i n e q u a l i t i e s considered by Everitt i n C31. In section 2 of t h i s paper there is a statement of t h e r e s u l t s t o be proved; following sections contain t h e proof of these r e s u l t s together w i t h some coments on t h e i r consequences.
There i s a l i s t
o f references.
2.
R and C denote the r e a l and complex number f i e l d s respectively.
For a E R t h e h d f - l i n e [a,-) is closed a t a and open a t
m.
AC denotes
absolute continuity and L Lebesgue integration; 'loc' f o r local, i.e. of la,-).
a property s a t i s f i e d on a l l compact subintervals
2 L (a,-)
denotes the c l a s s i c a l Lebesgue complex function
space, which i s a l s o indentified w i t h t h e Hilbert function apace of' equivalence classes.
Let the coefficients p and q s a t i s b t h e following basic conditions : p, q : [a,-)
+
R and
These conditions on p and q imply that t h e d i f f e r e n t i a l expression M, given by
(1.3), is regtdar a t a l l points of [a,-) but has a singular
163
DIFFERENTIAL EXPRESSIONS AND INTEGRAL INEQUALITIES
p i n t a t -; see C9; Section 15.13.
it i s knovn t h a t
Since q i s bounded below on [a,-)
M is i n t h e limit-point condition a t t h e singular
17.5 and 23.61 and C21.
point -; see C9; Sections
Following t h e notation i n C6; Section 32 we define t h e l i n e a r 2 of L (a,-) by
manifold D1 L D,(p.q)
D~ = D
E
and
: f l c A C ~ ~ ~ C ~ , -M )
L2(a,-)
2
C ~EI L
(2.3)
(a,-)).
2
1
i s t h e domain of the maximal oFerstor T1 generated by M i n L (a,-)
and defined by T,f = N f I
(f
2 i s not synnnetric i n L (a,-);
E
D
1
1;
it is known t h a t T is closed but 1
for these results see C9; Sections 17.4 and 51,
also the remarks i n C6; Section 31.
The basic operator theoretic
definitions a r e given i n C1; Sections 39 and 411. 2 We say t h a t M i s _separated i n L (a,-)
i f (see
C4; Section 11
and C6; Section 61) qf
E
2
L (a,-;
f o r all f
2 and completely separated i n L (a,-) pf", p ' f ' ,
-
qf
E
(2.4)
D1;
i f (see r5; Section 11) 2
L (a,-) f o r a l l f
a r e all i n
E
D1.
Note t h a t both (2.4) and (2.5) are conditions t o be s a t i s f i e d only at
since the basic conditions (2.1) and (2.2) on the coefficients 2
p and q imply t h a t all terms i n (2.4) and (2.5) are i n LlOc[a,-).
We mey now s t a t e ( r e c a l l t h a t
11 (1
denotes the usual norm i n
L2(a.-)) Theorem 1 Let t h e coefficients p and q satisfy t h e basic conditions (2.1)
(2.2); l e t addition-
p
q s a t i s f y the conditions;
(2.5)
164
W.N.EVERITT AND M.GIERT2
l e t K E @,m) -
where
C,
be given;
lat 11
b e chosen so t h a t 0
2 1
< 11 < min(1, 4C q ( a ) )
= max(7, 3K)
(b)
and (2) f o r
let
all
Q
E (0,n) t h e following i n e q u a l i t y i s v a l i d
where t h e p o s i t i v e number A depends only on t h e c o e f f i c i e n t s p A i s d v e n e x p l i c i t y i n (3.17)
9;
below.
Proof This i s given i n t h e s e c t i o n s which follov. Notes The e x p l i c i t dependence of t h e number A is given i n t h e s e c t i o n s devoted t o t h e proof of (2.9). The i n e q u a l i t y (2.9) has some s i m i l a r i t y with t h e i n e q u a l i t y i n
C6; (8.3) of Theorem 31 but t h e condition ( 2 . 7 ) ebove avoids t h e u n s a t i s f a c t o r y nature of t h e condition C6; (8.213 depending as it does on t h e v e r i f i c a t i o n of another inequality.
As i n C6; Theorem 31 t h e c o n t r o l condision (2.6) is necessary t o t h e proof o f (2.9); it prevents t o o much o s c i l l a t i o n i n t h e c o e f f i c i e n t s p and q i n t h e neighbourhood of
m.
165
DIFFERENTIAL EXPKESSIONS AND INTEGRAL INEQUALITIES
Under the conditions of Theorem 1 the differential 2 expression M is separated in L (a,-). Corollary 1
It follows at once from the inequality (2.9) that condition
h-oof
(2.4) is satisfied.
Corollary 2 If in addition to all the conditions of Theorem 1 coefficients p
the
q satisfy
(2.10)
Proof It follows from the inequality (2.9) that (pf')'
E
L2(a,-)
we obtain Ip'f'I f
E
(f E D1).
5 L{pq}1'2f'
Now (pf')' = pf" + p'f' and from (2.10) E
2 I, (a,-) on using (2.9) again and this for nl1
D1. It now follows that condition (2.5) is satisfied.
Note that in general the conditions (2.7) and (2.10) are independent of each other; howevsr if lower bound on [a,-)
the
ccefficient q has a positive
then (2.10) implies (2.7).
Finally we have Theorem 2
Under the basic conditions (2.1)
(2.2) on t h e coefficients
pandq
under the additional conditions (2.6)
and
(2.7) of Theorem 1
following inequality is valid for all
E E
(0,l)
the
166
W.N.EVERITT A N D M.GIERTZ
where t h e positive number B depends only on t h e c o e f f i c i e n t s p
Proof
and q.
This i s given i n t h e sections which follow. Note t h a t the inequality (2.12) should be compared with t h e
-a p r i o r i
estimates given by Goldberg i n C7; Chapter VI, Sections
6 and 83;
see i n particular
Theorem VI 8.1
However i n (2.12) t h e
coefficients are, i n general, unbounded on [a,-).
3.
In t h i s section we give t h e proof of Theorem 1. The l i n e a r manifold D1 i s defined i n (2.3).
Now define
D1,O I D1,O (p,q) as t h e collection of all f i n D1 which vanish i n
some neighbowhood, which may change Kith f, of
D1,O = {f
E
D1
: f o r some X E X ( f )
m,
> a, f ( x ) = 0
h. ( x c CX,-)).
(3.1)
no r e s t r i c t i o n i s placed on t h e values 190 taken by f a t the regular end-point a. Note t h a t i n defining D
The reason f o r t h e introduction of D
1,o inequality (2.9) i s f i r s t established on D1,3.
i s as follows.
The
From known r e s u l t s
i n the theory of d i f f e r e n t i a l operators it may be claimed t h a t given
any f
E
D
1
there is a sequence {f
n
: fn L D
such that t h e sequences {fn) and {MCf,]) t o f and Lfl respectively.
1,O
for n
= 1,2,3, ...I
a r e convergent i n
2
L (a,-)
Furthermore t h e tenas on t h e left-hand
2 side of (2.9), with f replaced by f n , also converge i n L (a,-)
t o the
corresponding terms with f i n D1. I n t h i s way the inequality is established i n D
1’
DIFFERENTIAL EXPRESSIONS AND INTEGRAL INEOUALITIES
This c l o s u r e argument l e a n s heavily on t h e f a c t t h a t t h e
maximal operator T (2.31, is closed. Sf = M C f l
(f
D
E
1
introduced i n t h a previous s e c t i o n , following
I n f a c t i f S : D,,O
+
2 L (a,m) is defined by 2
1to
) then S i s closeable i n L (a,-) ,and t h e c l o s u r e
= T,. This result depends on no r e s t r i c t i o n being placed on t h e values of f E D
1S O
a t t h e rfgular end-point a ( s e e above) and t h e
property o f M being limit-point at following (2.2)).
m
( s e e t h e previous s e c t i o n
Some a d d i t i o n a l a n a l y t i c a l d e t a i l s may be found
i n C6; Sections 4 and 81. We now prove t h a t (2.9) holds f o r a l l f
E
It i s c l e a r l y
D,,O.
s u f f i c i e n t t o prove t h e r e s u l t for real-valued f i n D
1S O
t h e r e is an immediate extension t o complex-valued f i n D
s i n c e then 1 ,O'
For t h e remainder of t h i s s e c t i o n we t a k e f t o be real-valued and i n DISO; s i n c e then f vanishes i n some neighbourhood of
there
are no convergence problems i n t h e i n t e g r a l s concerned.
As i n 16; Section 81 we have t h e following i d e n t i t y obtained on i n t e g r a t i o n by p a r t s (we r e c a l l from (2.1) m d (2.2) t h a t p and q
are p o s i t i v e on C a p ) )
167
168
W.N.EVERITT AND M.GIERTZ
Thus (3.2) may be r e w r i t t e n i n t h e form
(3.3)
+ 2(pqJff') )(a).
W e now f i n d
&1
estimate f o r t h e expression ( p q ( f f ' l ) ( a )
11 N,fl)I2
i n terms of
and
11 f1I2 ; it
i s f u r t h i s reason t h a t we
have t o have a v a i l a b l e t h e condition (2.7) on p and q.
Note t h a t we cannot t a k e q ( a ) = 0 t o avoid t h i s estimate f o r t h e
l a s t term of (3.3), unless w e t a k e q(x) = 0 e s t a b l i s h e d in
( x E [a,-)).
It was
C6; Section 101 t h a t wfienevw a condition of t h e
form o f (2.6) holds then e i t h e r q i s p o s i t i v e o r i d e n t i c a l l y zero on [a,"). q(x) >
o
It i s f o r t h i s reason t h a t t h e condition (x
E [a,m))
forms p a r t of (2.2).
The main d i f f i c u l t y i n e s t a b l i s h i n g i n e q u a l i t i e s o f t h e kind considered i n t h i s paper i s t o &ow
f o r t h e e f f e c t of having
no boundary condition on t h e elements of D1 at t h e r e g u l a r end-point a.
"his d i f f i c u l t y also occurs, for example, i n consideration o f an i n e q u a l i t y of t h e form
DIFFERENTIAL EXPRESSIONS AND INTEGRAL INEQUALITIES
which is closely related to the inequality (2.12) of this paper.
169
This
last inequality is discussed in detail in C31; see in particular the rermrks in C3; Sections 2, 3 and 41. In the analysis which follows we make use of the following inequalities 2ab 5
+ (b/tl2
valid for all positive a, b, t and
On integration by IP' I
5
p a r t s , use
1/2 2 8 p q, and then
% (k =
(3.5 1
,...,n).
1,2
of the condition (2.71,
(3.5) we obtain for f
E
b.
D 1S O
(3.7)
where t is an arbitrary positive number.
170
W.N.EVERITT AND M.GIERTZ
Multiplying (3.7) by (3.8) we obtain, a f t e r applying the first i n eq u al i t y
in (3.6) with n = 3, and then t ak i n g square r o o t s on both
sides (3.9) where we have p u t
Prom t h e i d e n t i t y
'a
ve
Ja
obtain, using (3.9) and with T = t S + R/2t
After
multiplying both s i d e s by t h e f a c t o r 2 and applying t h e
i n eq u al i t y (3.5) t o the last two terms we o b t ai n , f o r any real p o s i t i v e nunbers h and k, ( [ l p 1 / 2 f f ( (- T)2 < T2 + (hR)'
+ (S/h)?
+
k211 ( p f 1 ) ' [ 1 2 + k-*11
ill
'.
Again using t h e first in e q u a li ty i n (3.6), t ak i n g square r o o t s on both s i d es , s u b s t i t u t i n g f i r s t l y T = t S + R/2t and secondly f o r R and S from (3.101,
11 p'/2f11( -< ( 2 t
+ h-l)S + (t-' + h)R + kll ( p f ' ) ' l l
= ( 4 t 2 + 2th-'
+
(t-2+ ht-'
+ k)ll (pfl)'ll +
+ k-l)II
fll
( 2 t + h-l)K(I (pq)'''f'II
+ (t-' + h)KII q1/2flL
(3.13)
DIFFERENTIAL EXPRESSIONS AND INTEGRAL INEQUALITIES
I n t h i s l a s t result t , k and h are, thus far, a r b i t r a r y p o s i t i v e numbers.
We now choose t
&
(0.1) and put h
-' =
k = tli2
I
and estimate t h e l a s t term i n (3.13) by, using again (3.51,
Since s E (0,l) it follows from (3.13) t h a t
and then, with
= s3 and d e f i n i n g C1 = max{'l, 3K1
C2
= 3 + 2K,
we o b t a i n
where 2 2 5 C3 = 4C2 C16C1 q ( a ) l
This l a s t i n e q u a l i t y (3.14) i s v a l i d f o r a l l
E
.
satisfying
s (say)
171
172
W.N.EVERITT A N D M.GIERTZ
and f o r all real-valued f
E
D1,O.
Returning now t o (3.11) we obtain, on using (3.14) and then (3.5) f o r all real-valued f
again,
E
D
1 .O
where t h e positive number A i s defined by
on taking i n t o account t h e upperbound f o r
E
given i n (3.15).
Returning now t o (3.31, with rl chosen as i n the statement of Theorem 1, we s u b s t i t u t e t h e e s t i m t e (3.17) t o obtain t h e required inequality (2.9) but valid f o r all real-valued f
&
D1,O.
To complete
t h e proof of Theorem 1 it only remains t o r e c a l l the extension t o complex-valued f
and then t o invoke t h e closure argument given at 190 the start of t h i s section t o extend t o the maximal domain D1. E D
A detailed e d n a t i o n of the above analysis shows t h a t t h e
order of the term
reduced.
E-~,
for small
L,
i s best possible and cannot be
DIFFERENTIAL EXPRESSIONS AND INTEGRAL INEQUALITIES
4.
In t h i s section w e give t h e proof of Theorem 2. W e outline the proof of the Theorem only and omit t h e d e t a i l s which
determine an estimate f o r the positive number B, since t h i s follows the pattern of t h e analysis i n section 3.
The proof of (2.11) follows from known r e s u l t s f o r t h e d i f f e r e n t i a l expression M; see, f o r example, C2; Section 51.
To prove (2.12) we start with (3.14) but now extended by t h e closure argument t o t h e maximal domain D1; t h i s extension follows from the argument given a t t h e start of section 3 above.
From (2.9) and (3.14) it follows t h a t we may e s t a b l i s h a r e s u l t of t h e form
valid for a l l
L E
(O,l), where B
1
i s positive and depends only on the
coefficients p and q.
Also we have f o r any positive k and f E D,
Turning again t o (2.9), using only the t h i r d term on t h e left-hand side we obtain an inequality of t h e form, putting
E
1 =,rlr
Combining (4.2) and (4.3) we obtain, on chousing k t o be small,
v a l i d f o r all
L E
(O,l), where B is positive and depends only on t h e 2
coefficients p and q.
17 3
174
W.N.EVERITT AND M.GIERTZ
Taken together (4.1) and (4.4) give t h e required i n e q u a l i t y (2.12) with B
=
B1 + B2.
This completes t h e proof of Theorem 2. References 1.
N. I. Akhiezer and I. M. GlRzman, Theory of l i n e a r o p e r a t o r s i n (Ungar, New York, 1961; t r a n s l a t e d from t h e Russian
f i l b e r t space edition). 2.
W. N. E v e r i t t , 'On t h e limit-point c l a s s i f i c a t i o n of second-order d i f f e r e n t i a l operators', J. London Math. SOC. 41 (1966), 531-534.
3.
W. N. E v e r i t t , 'On an extension t o an i n t e g r o - d i f f e r e n t i a l i n e q u a l i t y
of Hardy, Littlewood and Polya', Proc. Royal SOC. Edinburuh (A)
69 ( 1971/72 1, 295-333. h.
W. N. E v e r i t t and M G i e r t z , 'Some p r o p e r t i e s of t h e domains of c e r t a i n d i f f e r e n t i a l operators', FToc. London Math. SOC. ( 3 ) (19711, 301-24.
5.
W. N. E v e r i t t and M. Giertz, 'On some p r o p e r t i e s o f t h e domains of
powers of certain d i f f e r e n t i a l o p e r a t o r s ' , Proc. London Msth. SOC. ( 3 ) 24 (1972), 756-768.
6.
W. N. h v e r i t t and M. Giertz, 'Some i n e q u a l i t i e s a s s o c i a t e d with c e r t a i n ordinary d i f f e r e n t i a l o p e r a t o r s ' , Math. Zeit. 126 (1972), 308-326.
7. 8.
S. Goldberg, Unbounded l i n e a r o p e r a t o r s (HcGraw-Hill
New York, 1966).
T. Kato, Perturbation theory f o r l i n e a r opcrators (Springer-Verlag, Heidelberg, 1966).
9.
M. A. Naimark, Linear d i f f e r e n t i a l operators; P a r t I1 (Ungar, New York, 1968; t r a n s l a t e d from t h e Russian editiczi)
W N Everitt Department of Mathematics The University Dundee, Scotland
.
M Giertz Department of Mathematics The Royal Institute o f Technology Stockholm, Sweden
W . Eckhaus ( e d . ) , New Developments i n D i f f e r e n t i a l Equations @ North-Holland Publishing Company (1976)
LEGENDRE’S POLYNOMIALS
ON
by Ake P l e i j e l .
Su = DpDu
With
1.
Su = A T u
>
p(x)
+
qu,
D = id/&,
i s considered on
I = {x: a
0 , 0 < r(x)
_
1 , and no o t h e r z e r o s i n t h e extended complex p l a n e c u t a l o n g t h e
segment I . It i s t h e n e a s y t o show t h a t
(6)
N = I u {zeros of A ( x ) )
= I u I+ vo}.
The spectrum N is s i m p l e ; a s a c y c l i c element f o r t h e o p e r a t o r A - l T , eo
one may t a k e
= (I-C)~~. By t h e f o r e g o i n g , t h e s p e c t r a l theorem may b e a p p l i e d t o A-lT c o n s i d e r e d
a s an o p e r a t o r in L2(I,A). T h i s g i v e s rise t o a u n i t a r y map F from L 2 ( I , A ) o n t o some space L 2 ( N , 0 ) such t h a t F d i a g o n a l i z e s A-lT,
w h i l e Feo = 1"
The c o n d i t i o n s
o f theorem 1 are f u l f i l l e d i f one t a k e s B = A-lT and L = L i p ( 1 ) ( c o n f e r chapter
63.
Here, t h e space L i p ( 1 )
i s d e f i n e d a s f o l l o w s . For 0 < a
5
C5,
1, let
L i p a ( I ) denote t h e l i n e a r s p a c e o f f u n c t i o n s f on I t h a t a r e ( u n i f o r m l y ) H6lder c o n t i n u o u s w i t h exponent a . I t i s a Banach s p a c e i n t h e norm
193
EIGENFUNCTIONS AND LINEAR TRANSPORT THEORY
We p u t L i p ( 1 ) = u o < a s l L i p a ( I ) and g i v e t h i s s p a c e t h e i n d u c t i v e l i m i t topo1og-y [observe t h a t L ip a ( I )
Lip6(i) if a
3
61.
We have t o t a k e some c a r e i f we want t o e x t e n d B = A-lT
t o t h e space
L i p ( I ) ' , s i n c e A-lT i s s e l f - a d j o i n t i n L 2 ( I , A ) , b u t n o t i n L 2 ( I ) . For any $ L i p ( I ) ' , l e t u s d e f i n e a new f u n c t i o n a l
($,flA =
(f
($,Af)
( $ , a )
E
by p u t t i n g
A
Lip(1)).
E
Then t h e e x t e n s i o n o f A-lT t o L i p ( 1 ) '
i s g i v e n by
= ($,A-lTf)A.
(A-'T$,f),
[We would have o b t a i n e d t h e same e x t e n s i o n i f we had t a k e n t h e second a d j o i n t where (A-lT)* i s t h e a d j o i n t o f AdT
((A-lT)*)',
i n L 2 ( i ) and t h e second a d j o i n t
, )I.
i s taken with r e sp e c t t o (
Applying theorem 1 we f i n d t h a t , on t h e subspace L i p ( i ) , t h e t r a n s f o r m a t i o n F i s g i v e n by ( F f ) ( v ) = ?(")
,
($u,Bf)A = v($v,f)A
($v,eO)A = 1
(u
N).
E
t h e n we g e t ? ( v ) = ( @ , , f ) , where
If we d e f i n e @ vby (@",f) = ( $ v , f ) A = ( $ " , A f ) ,
t h e $,
satisfies the relations
= ( $ , f ) * , where $
are d e t e r m i n e d by
(7)
(@,,Bf)
,
= v(@v,f)
From ( 7 ) we deduce ( $ , ( T - v ) f ) =
4" = aSv
- 52 -k , where p-v
-
2
(gv,eo) = 1 . ( T f 1 ) . T h i s r e l a t i o n i s s a t i s f i e d by ' I
a i s a constant, 6
denotes t h e delta-function centered
&y
a t t h e p o i n t v and stands f o r t h e functronal with values f(u)dp. u -v -1 P-V Taking i n t o account t h e second r e l a t i o n ( 7 ) we g e t a = X ( v ) . The r e s u l t o b t a i n e d may b e w r i t t e n a s ?(v) = ($",f) =
(8)
T h i s i s a n expansion of For v =
2
?
<X(v)Gv
- c2 u
p-v
,f>
(u
E
N;u#+
1).
= Ff i n t e r m s of f and t h e e i g e n f u n c t i o n s o f B = A-lT.
vo, X(v) = h ( v ) = 0 .
Conversely, one may e x p r e s s f i n
? by
c o n s i d e r i n g t h e o p e r a t o r FTF-l
in
L2(N,o); t h i s o p e r a t o r i s d i a g o n a l i z e d by F-l and s e r v e s as t h e analogue of A-lT. A s i n 151 one f i n d s
(9)
f(u) =
(h(u)6,,
+
$ A,?),
(?
E
Lip"),
-1
= b ' o r ; e q u i v a l e n t l y , S c S * . A s e l f a d j o i n t s u b s p a c e H i s one which s a t i s f i e s H = H*. We have t h e f o l l o w i n g r e s u l t (cf. [21, I 7 1 ) . Theorem 2 . 1 . K S
b e a symmetric s u b s p a c e i n H 2 and l e t
MS(L) = { { h , k } E S* Then ( i ) dim M S ( L )
$'
=
( i i ) S*
=
i s constant f o r L E
E $ (Im S
G
M S (9.)
L
i MS
I
E
4.
4-, where
4'
and f o r L E
4'
( d i r e c t sum),
01,
(a),
L E
( i i i ) there e x i s t selfadjoint extensions H H & H2 which s a t i s f y S c H = H* c S* L E
, .P
k = Lh}
of S & H 2 ,
i . e . , subspaces i f and only i f dim M S ( L ) = dim MS(L)
4+,
( i v ) t h e r e a l w a y s e x i s t H i l b e r t s p a c e s R c o n t a i n i n g H a s a s u b s p a c e and RL s u c h t h a t S c H . s e l f a d j o i n t subspaces H For any s u b s p a c e T i n H2 we may w r i t e T = Ts d T_, where T_ = { { f , g } E T
I
f=o
1,
T,
=
T
e
T_
The s u b s p a c e T , i s c a l l e d t h e o p e r a t o r a r t o f T. It is a c l o s e d o p e r a t o r i n dense i n T * ( E ) i and R(Ts) c T ( 0 ) l . H with D ( T ~ = ) D(T) The f o l l o w i n g theorem i s d u e t o R . Arens [ I ] .
INTEGRAL-ORDINARY DIFFERENTIAL-BOUNDARY SUBSPACES
Theorem 2.2. If H = H d Hm is a selfadjoint subspace in H 2 , H is a dens" defingd selfadjoint operator in H ( 0 ) l .
201
then
Let L be a system of n first (a,b) c IR:
3 . The basic linear ordinary differential operators.
order ordinary differential expressions on L = P D 1
where P. is an det P 0,x the edtries of Otherwise L is
(a)*
+
I =
Po , D = d/dx,
nxn matrix whose entries belong to CJ(i), j = O,I, and E 1.The system L is called regular if 7 = [a,b] is compact, P. belong to C J ( i ) , j = O , I , and det P,(x) 0 , x E 7. called singular. The Lagrange adjoint of L is defined by
*
+
L
-DPI* + P
=
0
*
= 0 D 1
+ Q
0'
where 0 = -PI*, Q = P * - (P I ) * . Thus if L is regular then so is L+ and L+ = IL. The sys?em Lois caljed formally symmetric if L = L+. +
Let H = L 2 (I), the Hilbertspace of nxl matrix-valued functions on with innerproduct b g*f , f,E E L2 ( 1 ) . (f,g) = a
I
I
In H 2 we define the minimal operators T
and T
+
associated with L and L+ by
To = {{f,Lf}l f E C~(I)]~, To+ = {{f+,L+f+ll f+
E C;(I)}',
where c denotes the closure in H2. It is easy t o see that = 0, i.e., that T and T are formally adjoint. Their adjoints T 2ndoT+ are called tge maxigal operators and are given by +
T
=
(T~+)* = {{f,Lf} If
T+ = (To)*
=
E m
AC
{{f+,L+f+} If+€
loc
Hn
(I), Lf E H},
ACloc (I), L+f+E H 1
By Green's formula the semi-bilinear form on H2 x H2 restricted to T deals with the behavior of the functions f E D ( T ) , f+ E D ( T ' ) at the end points of I: x+b + + + Pl(x) f(x)l x-ta ( 3 . 1 ) = (f+(x))* =
lim(f+(x))* x+b
~,(x) f(x)
-
lim (f+(x))* x+a
. x
T+
~,(x) f(x).
If L is regular then the functions f E D (T), f+ E O(T+) can be continuously extended to all of 7 and ( 3 . 1 ) reads (3.2)
= (f+(b))*
Pl(b) f(b)
- (f+(a))*
Suppose that the subspaces S and its adjoint S* (3.3)
T c S
and
T
+
Pl(a) f(a).
in H 2 satisfy
c S*.
Then S c T, S* c T+, S and S* are operators and it can be shown that there exist finite dimensional subspaces C and C+ of T and T+ such that
2n2
A .DIJKSMA
S = T n(C+)* and S* = T+
n c*.
It f o l l o w s from ( 3 . 1 ) t h a t S i s a r e s t r i c t i o n of T and S* is a r e s t r i c t i o n of T+ by means of a f i n i t e number of c o n d i t i o n s a t t h e two e n d p o i n t s of I . C o n v e r s e l y , i f S i s a r e s t r i c t i o n of T by a f i n i t e number of s u c h two p o i n t boundary c o n d i t i o n s , t h e n ( 3 . 3 ) h o l d s . F o r example, l e t L be r e g u l a r . I f S = T t h e n w e have u s i n g ( 3 . 2 )
S = {{f,Lf)ET and i f S = T
I
f(a)
=
f f b ) = 01,
t h e n we h a v e
+ +
+
S*= { I f .L f ) ET'
1
f + ( a ) = f + ( b ) = 0) = To+.
4 . I n t e g r a l - o r d i n a r y d i f f e r e n t i a l - b o u n d a r y subspaces. I n o r d e r t o o b t a i n subspaces w i t h s i d e c o n d i t i o n s o t h e r t h a n two point-boundary c o n d i t i o n s w e make T and T + s m a l l e r . We i n t r o d u c e two f i n i t e d i m e n s i o n a l s u b s p a c e s B, B+ i n H2 agd d e f i n e t h e f o r m a l l y a d j o i n t o p e r a t o r s So = Ton (B+)*
So
and
+
= To
+
I l B*.
I t can be shown t h a t t h e a d j o i n t s of So , So+ a r e g i v e n by S
*
T+
=
4
B'and
(So+)* = T
G B.
Without l o s s of g e n e r a l i t y w e may and d o assume t h a t t h e s e a l g e b r a i c sums a r e d i r e c t . We want t o c h a r a c t e r i z e a l l s u b s p a c e s S and t h e i r a d j o i n t s S* t h a t satisfy So c S
, dim
S
I3 S o = d ,
I f ( 4 . 1 ) h o l d s t h e n i t c a n b e shown t h a t we must have t h a t d + d + = dim T 0 To
+ dim
and t h a t t h e r e e x i s t s u b s p a c e s C , C+ of T dim C+ = d+ s u c h t h a t S = (T
B)n(C+)*
4B
B + dim B+
and T+
and S* = (T+
4
,
B+ w i t h dim C = d ,
B+) nC*.
The s u b s p a c e s C and C+ r e p r e s e n t t h e s i d e c o n d i t i o n s which now a r e a m i x t u r e of two p o i n t boundary and i n t e g r a l c o n d i t i o n s . We want t o make t h e s e c o n d i t i o n s more e x p l i c i t . I n o r d e r t o do so we s h a l l make u s e of t h e f o l l o w i n g n o t a t i o n s . To i n d i c a t e t h a t a m a t r i x A h a s p rows and q columns ve w r i t e A ( p x q ) . Thus f E H i s o f t h e form f ( n x I ) . The p x q z e r o m a t r i x and t h e and I By p ( n x p) E H , p x p i d e n t i t y m a t r i x a r e d e n o t e d by 0 6 ( n X p) E D(T) i s l i n e a r l y i n d e p e n d e g t mod Dp(T ) o r 'm(n x p) i s a b a s i s f o r S(0) w e mean t h a t t h e p columns of t h e s e m a t r y c e s have t h e s e p r o p e r t i e s . For F(n x p ) , G(n x q) E H we d e f i n e t h e " m a t r i x i n n e r p r o d u c t " (F,G) t o be t h e 4 x p m a t r i x whose i , j - t h element i s b n
'
.
203
INTEGRAL-ORDINARY DIFFERENTIAL-BOUNDARY SUBSPACES
If C(p
r), D ( q
x
x
s ) are constant matrices, then
,
(FC,G) = (F,G)C If I S , L6} ( n
p) E T , ( 6 + , L+6+1 (n
x
+ + +
< { S , L ~ I, ( 6 ,L 6
(F,GD) = D* (F,G).
E T+ then + +
x
q)
=
(LS,~+)-(~,L6
)
is well defined. We denote by (F:G) the matrix whose columns are obtained by placing the columns of G next to those of 5 in the order indicated. We shall also make use of the following definitions:
and
s 1 = T~ n (B,+)* , s I += T ~ +n B]* It is not difficult to see that S
0
c S
1
BI =(So+)*
c(S + ) * = T 1
=
T iB
4
1
((So+)*),
(direct sums),
+.
and that a similar result-holds starting with S Yoreover, it is easy + are operators, Do ( S I ) is dense in H, to see that B , S I * = T+ dim ( S +)*(O)l= dim B2, e:cfllThe following result is a special case of a theo?em that will be stated and proved elsewhere. Theorem 4 . 1 . (4.2)
(4.3)
S , S* satisfy ( 4 . 1 ) and suppose that
I (hp :
1
(n
2to)
2tn+)
: ' 0 (
(n
x x
( s l + s2))
+
+
(sl + s2 ))
is a basis for
S(O),
is a basis for
S*(O),
i s a basis for
(So+)*(0),
is a basis for
So*(0).
Then there exist u
= (y, ~y
I E T ,a
v (4.5)
=
V+ =
( 6 , L6
+ + +
( 6 ,L 6
,8
1 E T
1 E T+ , 6'
and a constant matrix E ( s 2 (4.6)
E B]
= { l a , 1,
}
= {'a,
} E
,
(n
x
s2+),
, (n
x
(d-sl
+ + +
(4.4)
x
2~
= {20+,2T+]
Bl
E B','
,
+
s 2 ) such that
,
6
+ 2u
is linearly independent mod D ( S 1 ) ,
I
6'
+
is linearly independent mod D (S,'),
20+
(n
+
(d
-
- s,+)) , s1
+
-
s2)),
204
A . DIJKSMA
and if (4.8)
(4.9)
then (4.10)
h E D (T),
S = {{h + a, Lh + T + ID]
(h + a, (h + a,j and hence and B 1 = B, B I = B+. Let S and S* satisfy ( 4 . 1 ) with d+d+ = dim T 0 T + 2n. Then S and S* are operators and by Theorem 4 . 1 can be described as follgws.
205
INTEGRAL-ORDINARY DIFFERENTIAL-BOUNDARY SUBSPACES
S = {{f,Lf)
I
(6+(x))*
f E H fl ACloc (I\{c)), P,(x) f(x)
S*= {{f+,L+f+l
I
f+ E H
P,(c) f(c-0)
n ACloc (idc}), L+f+
x+b + 6*(c)P1*(c) Pl*(x) f+ (x) /x+a
6*(x)
-
f(c+o) = 0'+1, d
PI(c) + (D+)*]
[(S+(c))*
Lf E H , and
x+b /x+a +(s+(c))*
E H and
f+(c-o)-[(6*(c))Pl*(c)-D*lf+(c+o)
= 0;).
+ + +
Here {6,L6} E T, 16 , L 6 ) E T+ and the constant matrices D(n are such that {6,L6)+{01,~llD 20'
,
+ + +
x
d), D+(n
{6 , L 6 ) + {o+1,,+1}D+satisfy(4.6)with
X
d+)
20=01D,
D+ and
=
which is (4.7) of Theorem 4.1. Selfadjoint integral-ordinary differential-boundary subspaces. In this section we assume that L is formally symmetric and B = B+. Then So = So+ is a symmetric operator and we want to characterize the selfadjoint extensions H of S in H 2 . As to the existence of such extensions we remark that it can be pro8ed that (k) =
dim M,
(P.)
+
dim B, P. E
$'
0
Hence as a consequence of Theorem 2.1 (iii) we have the following result (cf. t 5 1 ) . Theorem 5.1. There exist selfadjoint extensions H & H2 of S = T CI B* H where T is the symmetric minimal operator associated with Lo= &"'L i f d %ly if To has selfadjoint extensions in H'. We now also assume that dim M
(P.)=
dim M
(e)= d, say, k E $+.
Using Theorem 4.1 we obtain the following characterization of the selfadjoint extension H of S in H2 which is Theorem ( 3 . 3 ) of [ 5 ] applied to differential operators.
H satisfy
Theorem 5.2.
(5.1)
S cH
=
H* c S o * ,
and suppose that
(5.2) (5.3)
('CO
"0
(n
x
s,)
:2cp)
(n
x
(s,
is a basis for H ( O ) ,
+ s2))
is a basis for S *(O).
Then there exist (5.4)
u = {Y,LY} E T,
a = {'o,~T) E B , ,
(n
x
s2),
(5.5)
v = {6,L6]
8 = { * o , * T } E B 1 , (n
x
(d-sI-s2)),
E T,
206
A. DIJKSMA
and a constant matrix E = E*
(s2 x s )
2
such that
(5.6)
6 + * u is linearly independent mod D ( S ) ,
(5.7)
],
S
1 is arbitrary).
Moreover, if the bases in ( 5 . 2 ) , ( 5 . 3 ) are orthonormal then the operator part H of H is given by s (5.11) H (h + u ) = Lh + T - 'tO(Lh + T , '(0) + S
+
'C?[(h
+ o,Y)
-
.
Converse1 , if ( 5 . 3 ) and the constant matrix E = E*(s x s ) are iven, the 2 elements fn ( 5 . 4 ) , ( 5 . 5 ) exist satisf in ( 5 . 6 ) , ( 5 . 7 f and $,r; afe defined by ( 5 . 8 ) , ( 5 . 9 ) , then H defined by ( 5 Y I 0 7 satisfies ( 5 . 1 ) e ( 5 . 2 ) . If the basis in ( 5 . 3 ) is orthonormal then H is given by ( 5 . 1 1 . ) . Proof. Let H satisfy ( 5 . 1 ) , '-to+
= lo,
b+= %,
sl+ =
( 5 . 2 ) and let ( 5 . 3 ) hold. By Theorem 4.1 with s I , s2+ = s 2 , d+ = d, and S = H = H* = S*,
there
exist elements as in ( 4 . 4 ) , ( 4 . 5 ) satisfying ( 4 . 6 ) , ( 4 . 7 ) and a constant matrix E ( s 2 x s 2 ) such that if Y,Y+,t,,t,+ are defined by ( 4 . 8 ) , ( 4 . 9 ) then H = H* is described by ( 4 . 1 0 ) as well as by ( 4 . 1 1 ) . Thus H satisfies ( 5 . 1 0 ) . We shall show that v+5 satisfies ( 5 . 7 ) , T i n ( 4 . 8 ) can be written as in ( 5 . 8 ) for some hermitian matrix E, and 5 in ( 4 . 9 ) can be written as in ( 5 . 9 ) . The operator S, i s densely defined (and symmetric) in H and therefore there exist a w(n x ( s I + s 2 ) ) E s 1 such that
207
INTEGRAL-ORDINARY DIFFERENTIAL-BOUNDARY SUBSPACES
Then by ( 4 . 1 0 ) , ( 4 . 1 1 ) of Theorem 4.1 and the facts that S = S*, S c S * and 1 1 u+a, u++a+, v+B E S l * , we have that for all {h + a, L h + r + co} E S = H
2
'
= Osl,
2
and hence (5.12)
v+B + WA + { O , C j
(5.13)
u +a
+ + -
u
E H,
- a + wB + {O,Y+} -
{O,Y} E H .
From ( 5 . 1 2 ) it follows that d-s = which is (5.7). From ( 5 . 1 3 ) and the description of Y+ - Y
+ +
and from ( 4 . 8 ) it follows that
I+ - T
= 2cn
Since the elements of (5.14)
E
-1
[E*
-
E
-4
to
d-s
-S
1
2
'
in (5.10) we deduce that
+ +
+ w~,u+a>= -2m < u +a -u-a,u+a>,
< u +a -u-a
= -2m
-S
= 0
+ +
< u +a , u + +~A
+ +
I.
are linearly independent it follows that
'Q
+ +
=
+A
E*
=[E
+ +
< u +a -u-a,
u+a>
+ +
-1
]*.
=
'tfl[F,-i ] ,
Now, Y in ( 4 . 8 ) can be written as Y = 'Y,
[E
-4
+ +
I
+ +
where F = E - 4 c u + a , u +a -u-a> = F* by ( 5 . 1 4 ) . This proves ( 5 . 8 ) where w$ have written E instead of F. From ( 5 . 1 2 ) and ( 5 . 1 3 ) it follows that = , which shows that 5 in ( 4 . 9 ) can be written as in ( 5 . 9 ) . The orthonormality of the bases in ( 5 . 2 ) and ( 5 . 3 ) and (5.10) clearly imply ( 5 . 1 1 ) . The second part of Theorem 5.2 i s a trivial consequence of the second part of Theorem 4 . 1 . This completes the proof. Example 5 . 3 . Consider Example 4 . 3 , where we suppose that L = L+. Then P I = P * and therefore we may and do assume that B = B+. 1
Suppose also that T has selfadjoint extensions. Then all selfadjoint extensions H in H2 of S = T O n B* are operators described by 0
0
H = {If, Lf}
I
f E
H
ll
ACLoc (I\{c}),
Lf E H and
where d = 4 dim T 0 T0 +. n and { A , L6}(n x d) E T,D (n x d) are such that 6 + 0' D i s linearly independent mod D (S 1 ) = D (S ) and
A . DIJKSMA
208
x+b (x+a -D*S(c)
6*(x)Pl(x)6(x)
t
D + D* P -'(c)
6*(c)
D = Od
d '
1
Example 5 . 4 . Let L = L+ be regular. We define the symmetric operator S by
D(S )
=
b
I
I
I f E 17 ( T o )
a
dp*. f = 0, j = I , '
...,
=
If E D ( T o )
1
( I )=
n, L E
,4:
0
T.
It can be
(Tof,a)=Okll.
Thus, if B is spanned by {o,Ol dim HT
c T
kl,
where i~ are n x 1 matrix-valued functions of bounded variation on shown '(cf. [ 5 ] ) that there exist i o , O l (n x k) E H such that O(So)
0
then So = T
il
B*. Since dim If
selfadjoint extensions H of S
in H 2
(L) = TO
exist and are
0
characterized by Theorem 5.2. 6. S mmetric subspaces and generalized spectral families. This section i s a
czntinuation of section 2. Let S be a symmetric subspace in H 2 and let H = H B Ha be a selfadjoint extension o f S in R2, where R i s a Hilbertspace contasning H as a subspace (cf. Theorem Z.I(iv)). By Theorem 2.2 and the Spectral Theorem for selfadjoint operators we have that m
A d Es(X),
Hs = -m
where E = { E ( A ) I X E El is the unique suitably normalized s ectral family of przjectigns in H ( 0 ) l = R 0 H(0) for Hs. The resolvent R 0: Hs can be written as
For X E IR let the linear operators E ( h ) on R be defined by
E(h) f
=I
f E H(O)l,
Es(X)f
f E H(0).
0
Then E = [ E ( h ) I X E I R 1 is called the spectral family of projections in P for H. The selfadjoint subspace extension H of S in RT i s called minimal if the set {E(X)f I f E ff} L' H is fundamental in R . The resolvent R of H defined by H RH(L) where I For L E
is
4-
=
(H - L I ) - '
,
L E
4' ,
the identity operator on R , is an operator valued function on RH(L) is bounded on R and can be expressed in terms of E :
.4:
209
INTEGRAL-ORDINARY DIFFERENTIAL-BOUNDARY SUBSPACES
Let P be the orthogonal projection of R onto H and let R (L)f = P%(L)f
,
F (X)f
, f E H,
=
PE(X)f
f E H. L E C?, h E IR
Then R is called the generalized resolvent and F the generalized spectral family for S corresponding to H. We remark that all generalized spectral families for S can always be constructed using the above method by starting with a minimal selfadjoint extension of S. For f E H it f o l l o w s from (6.1) that
and an inversion of this equality yields for f E H
- F(u))f,f)
((“(A)
=
limn
IX
ESO
I
!J
Im(R (v+iE)f,f)dv,
where A, p are continuity points of F. For more details we refer to [ 4 ] , [5] and [7]. In the next section we shall give a method of calculating all generalized spectral families for S = T n B* as considered in section 5. 0
0
7. Eigenfunction expansions. As in section 5 we assume that L = L+ and B = B+ and hence that S o = So+ is symmetric in H 2 . We do not assume that dim M ( a ) = dim M
( i ) ,P.
E
4
.
Let H = H 8 H_ be a fixed minimal selfadjoint extension of S
in R2
and F = i F ( h ) I X E IR 1 the corresponding generalized spectral family as described in the previous section. IJe shall indicate how one can obtain a suitable expression for F which immediately leads to eigen function expansions. Let c E I be fixed. Let p = dim B and { u , T ~ (n x p) a basis for B. Let s1 (x,i) (n x n) and u(x,L) (n x p) be the unique solutions of
where k E
4.
(L-L)
S’(L)
(L-L)
U
=
(k) =
0, &O
S’(C,k) = I
n’
-
T,
U
(C,k)
=
onp.
Let s 2 ( L ) = u(k) + u and s(L) = ( s ’ ( L ) : s 2 ( L ) ) .
One can easily verify that for all h E {PRs(i)h,k
PRs(k)h
H(0)
+ P h l E So*
= =
T
0 H ( 0 ) and I! E
:4
B.
Hence there exist a unique element. {f,Lf} E’T and a unique vector a(Pxl) E such that .{PRs(L)h, We define
LPRs(P.)h
+ Ph} = {f.Lf}
+
{U,T}
a.
4‘
210
A . DI JKSMA
It can be proved that the linear map hl+ r(PR (L)h) from (H(0))i into #n+p is continuous. The Riesz Representatioh Theorgm implies that there exist Gn+,(L)), Gi(L) E ( H ( O ) l , j = l , . . , n+p, such that G(L) = (G,(L)
,...,
T(PRs(I1)
,
h E (H(0))’
h) = (h,G(L)),
¶.
E $‘
,
We define
-
t(i) = (H - to)(Es(X) - Es(0)) G(Lo),X where
E
¶.
4%
E IR,
is fixed, and put P(X)
=
T(Pt(h))
=
((Hs - Lo)t(h),
A E IR
G(Lo)),
I
Theorem 7 . 1 . The (n+p) x (n+p) matrix-valued function p 0” IR is hermitian, non decreasing and continuous from the right, and for a11 h E H, a,B E IR
(7.1)
(F(8)
-
x
8
I
F(a)) h =
s(X)dx
(h,
a
If h E C
(1)
then
(7.1)
(FfB) - F(a))
s(v)dp
(11)).
0
can be written as B
.f
h =
i,
s(v) dp ( v )
(v),
a where -
b {(u)
=
(h, s ( v ) ) =
S*
(x,v) h(x) dx.
a The matrix p is called the spectral matrix (properly normalized p is unique). It can be shown that if H has a pure point spectrum then p consists of stepfunctions o n l y . Let
f^,
be (n+p)
1 matrix-valued functions on
x
IR and define
-m ^
^
Since p is non decreasing we have (f,f))Oand Let H be the Hilbertspace defined by
i
= L*
(P)
=
t
;I 1 1
we can define
i: 1 1
0. Then for all
s E 'I
(R:
x
Cr,a)
we have the estimate
----Proof.
Suppose x =
Cm(Py,Cr) and let
s E
(X',Xn),
x'
E
IRn-l, x
IR,
E
Then we have
Using the inequality of Cauchy-Schwarz we see that the last expression is not greater than
Since for fixed a > 0 both sides of inequality functions of
s E
V (1R:
x
Cr,a)
we have established inequality
(4.1) in
case s
E
(4.1) are continuous as
is dense in Vo($+
and the set Cz(lP,:,(?)
V (IR" o
+
x
x
Cr,a)
rr,a), 'Phis completes
the proof of lemma 1.
-------
Lemma 2. For each and all y
Proof. -----
> 0
E
E
there exists a 6 > 0 such that for all
s E
1
W (R"
x
Cr)
IR"with IyI < 6.
Let E > 0 and suppose that s theorem that
E
Cm(IRn,Cr).Then it follows from Plancherel's
220
R. MARTINI
where 5 d e n o t e s t h e F o u r i e r t r a n s f o r m of s d e f i n e d by
E v i d e n t l y , t h e r i g h t hand s i d e of ( 4 . 3 ) e q u a l s
and it can e a s i l y be e s t a b l i s h e d t h a t t h e r e e x i s t s a such t h a t for a l l
Thus f o r all y
6
5
E
IR" and a l l y
E
F
> 0 , depending
only on
E,
B n w i t h ( y ( < fi
En w i t h ( y l < 6 it f o l l o w s t h a t
For f i x e d y f u n c t i o n s of s
6
(4.4) i s
dominated by
R n b o t h s i d e s of i n e q u a l i t y ( 4 . 2 ) a r e c o n t i n u o u s a s
W'($
Cr) and C:(l#,f)
lhr'(18x
is dense i n
IT1(#
X
Cr). Fence f o r a l l
Cr)
i n e q u a l i t y ( 4 . 2 ) i s v a l i d . T h i s com-
uniformly on t h e c l o s e d u n i t b a l l 1: o f Vo(R:
Cr,a); i'.e., u n i f o r m l y on t h e s e t
y
E
Finwith lyl < 6 and a l l s
6
p l e t e s t h e proof of lemma 2 .
K c o n s i s t i n g of a l l s
-----
Proof.
6
Vo(B:
Cr,a) such t h a t
We may r e s t r i c t o u r s e l v e s t o t h e c a s e where t h e l i m i t i s t a k e n over a l l y = (y,,
..., y n )
iRn w i t h y
2 7,
Now f o r any a > 0 we o b t a i n
0.
In
t h i s c a s e we have
221
DEGENERATED DIFFERENTIAL OPERATORS
Let
for a l l y
> 0 b e Riven. By lemma 1 we may f i x a n a > 0 and a 6 1 such t h a t
E
I R n with y
E
Thus for a l l y
E
t 0 and IyI < 6 ,
R n w i t h yn 2 0 and
yI < 6 1 t h e r i g h t hand s i d e of ( 4 . 5 ) is n o t
greater than 2dx.
Next i t follows t h a t
1
Using t h e f a c t t h a t a-' {x = ( x , , with 0
8~. 1 ~ 1 5
< 6 , such t h a t f o r a l l y
1+6 E
IF?
1
>
t h e r e e x i s t s a c o n s t a n t c and a 6 w i t h IyI < 6 2 and y
e x p r e s s i o n i s dominated by
E v i d e n t l y t h i s e x p r e s s i o n is n o t g r e a t e r t h a n m
m
t 0 the last
2
222
R . MARTINI
Vow by lemma 2 t h e r e e x i s t s a 6 y
E
IRn
with y
Thus f o r a l l y t i m a t e d by
E.
Bn w i t h y
E
3
with 0 < 6
0 and A ' > 0 such t h a t u
p(y)
S
Ayn and yn
5
A'F(y) f o r a l l y = (y,,
...( y n )
- L
6
S. This f a c t can be proved
-
as follows. Let I be t h e canonical m a p p i n e f t h e tangent space ):?I( r e i o n t o t h e tangent space ?(HI:) x
E
IRy l e t A
tinuous posit:ve
E
f o r any
(I?"+).Then A i s a conIRy and by t h e compactness of % it follbws t h a t
be t h e norm of t h e s s t r i c y i o n Ix of I t o f u n c t i o n of x
with Riemannian s t r u c t u -
with Euclidean s t r u c t u r e e.JMoreover, T
R. MARTINI
226
(A Ix 51 i s f i n i t e . Denote by L ( c ) , L ' ( c ) t h e l e n g t h s of a C ' c u r v e i n -AD =w isup th r e s p e c t t o t h e s t r u c t u r e s and e r e s p e c t i v e l y . PVidently we have L ( c ) s E
for a l l y
5 R e ' ( c ) . Hence
y,.
5
E
5,
t a k i n g l e a s t upper hounds, it f o l l o w s t h a t p ( y )
s
Changing g and e i n t h e x a s o n i n g made ahove we s e e t h a t t h e r e e x i s t s a l s o
a c o n s t a n t A ' > 0 such t h a t for a l l y
y,
E
E
Al;(y).
Now it follows t h a t
and
YE for a l l y
Vo(B:
x
E
2 IA'F(y)lP = ( A ' ) C i ( y )
5.
y = (yl,
Thus Vo(B: and F(B:
Cr,;)
..., y n )
bedding of Vo(B: a
I
x
and H ( B y x C r , z . )
Cr,ii)
f,;)
a r e isomorphic t o
r e s p e c t i v e l y , where a ( y ) = y:
f o r each
RY. Hence what remains t o b e proved i s t h e f a c t t h a t t h e em-
E
x
x
i n t o F(By x Cr,a) i s compact. However, by a s s u m p t i o n
?,;)
dYn Y n B
0 such t h a t ( 5 . 5 ) i s dominated by
CI
228
R . MARTINI
(t).
Using matrix n o t a t i o n we have.for a l l s , t
/
(5.7)
< Os,t > dp =
U'
/
< u
-1
(%s)b
,v-'G
E
r
(M,F) t h e e v a l u a t i o n s
> dp =
U'
here t h e prime denotes m a t r i x t r a n s p o s i t i o n and 3 i s a c e r t a i n Wemitian matrix function. If t h e l o c a l expression of 0 with r e s p e c t t o t h e c h a r t c and t h e t r i -
v i a l i z a t i o n (n,v) i s given by
..
.
where a l J , b J , c
E
C"(+(U),L(Cr,Cr))(i
= 1, 2,
..., n ) then
( 5 . 7 ) equals
By i n t e g r a t i o n by p a r t s we s e e t h a t t h e l a s t expression equals
+ I: j p
1
F'(hbj
( u.' .) .
-
1
XD.(haij))D.sdx +
i 1
Now h , a l J ,bJc a r e Cp) en
+ (U) and
;'
Q (U') i s continuous on Q (U), i n a d d i t i o n U ' c c U .
JIence t h e r e e x i s t constants k6 > 0 , k
7
(5.10)
T'hcs.
> 0 and k8 > 0 such t h a t
DEGENERATED D I F F E R E N T I A L OPERATORS
Similarly, we obtain for all s,t
E
229
To(M,E) the evaluations
If the local expression of the differential operator Y' with respect to
the chart c and the trivialization ( n , v ) is given by (5.12)
w
ys =
-
i zd D.s + es,
i where dl,e
E
1
C"@ (U), L(Cr,Cr)) ( i = 1 , 2 ,
..., n) then the last expression equals
230
R.MARTIN1 .
1
Thus by t h e c o n t i n u i t y o f h,dl,e,%' t h a t t h e r e e x i s t c o n s t a n t s kq > 0 and k
10
on
0 (U) and
s i n c e U'ccU it f o l l o w s
> 0 such t h a t
E s t i m a t i n g t h e l a s t t e r m of t h e r i g h t hand s i d e of
( 5 . 4 ) we o b t a i n f o r
some c o n s t a n t k l l > 0
*
where 5 t =
$ and f
E
C"(+(U),
Now choose an a t l a s t h a t for any i = 1 , 2,
{U!} ( i = 1 , 2 ,
L(C
a=
..., m E
..., m) b e
r
,C
r
1).
{ c . = (Ui,pi)}
(i = 1 , 2,
..., m) f o r M
such
a d m i t s a t r i v i a l i z a t i o n ( n , v i ) over U. and l e t
a c o v e r i n g of M such t h a t IJ!ccUi.
Then it f o l l o w s from
231
DEGENERATED DIFFERENTIAL OPERATORS
t h e e s t i m a t e s made above t h a t t h e r e e x i s t c o n s t a n t s {ITi}
( i = 1 , 2,
..., m )
such
t h a t f o r any complex number
5
max KiCPc. , v . ,u!( s ) i=1,2,. ..,m 1 1 1 1
fPCi,". ,U! ( t ) .
*
1
1
A
Thus BX i s a c o n t i n u o u s s e s q u i - l i n e a r form on ; ( F , u )
T'(P,@).
T i s completes t h e
proof o f lemma 5. A
Lemma 5 i m p l i e s t h a t f o r any complex number A . nuous e x t e n s i o n B
X
t o V (E,o) O
Bx h a s a unique c o n t i -
V (8,~).
x
The next p r o p e r t y we want t o p r o v e about t h e f a m i l y (B ) ( A A
E
C) i s the
following Lemma 6. -------
Let
1 1 ILv
f i x e d norm f o r Vo(F,o).
be a
d e f i n e d on Vo(E,a)
x
Then t h e s e s o u i - l i n e a r form R X V o ( E , a ) i s c o e r c i v e f o r a l l complex numbers X w i t h
r e a l p a r t Re X s u f f i c i e n t l y l a r g e ; i . e . ,
t h e r e e x i s t c o n s t a n t s Ic > 0 and X 1 > o
such t h a t (5.17)
f o r a l l X w i t h Re X 2 P r o o f . Let
-----
KI ( s I ;1
Re BX ( s , s ) 5.
,x=
X, and a l l
{ck = (U ,$
k k
f o r any k = 1 , 2 ,
s E Vo(E,o).
) }(k = 1 , 2,
..., m
U
k
..., m ) h e a f i x e d a t l a s f a r M such t h a t
a d m i t s a t r i v i a l i z a t i o n of F o v e r Uk. I n a d d i -
t i o n , l e t b e g i v e n a m o d i f i e d Cw p a r t i t i o n o f u n i t y w i t h r e s p e c t t o t h e a t l a s
b
such t h a t Ew2(x) = 1 f o r each x E 11. Denote by ok t h e d e n s i t y of t h e measure $ ( p ) k k k w i t h r e s p e c t t o t h e Lebesgue measure on 4 (U). Then d r o p p i n g f a r a moment t h e
k
index k we have t h e f o l l o w i n g e q u a l i t i e s and e s t i m a t e s f o r an s
E
ro(M,E)
232
R.MARTIN1
S i n c e @ i s a s t r o n g l y e A & i p t i c d i f f e r e n t i d o p e r a t o r o f t h e second o r d e r we may a p p l y G k d i n g ' s i n e q u a l i t y , f o r m u l a t e d f o r s y s t e m s , and t h e n w e obt a i n t h e f o l l o w i n g . There e x i s t c o n s t a n t s k,2 > 0 and k
13
such t h a t
DEGENERATED DIFFERENTIAL OPERATORS
233
Using t h e i n e q u a l i t y (5.20)
21x1 Iyl
1
5 ~ 1 x 1 ' + -$y12.
which i s v a l i d for any
E
> 0 , we s e e t h a t t h e e x p r e s s i o n ( 5 . 1 9 ) i s n o t smaller
than
.. . Now h , a l J ,bJ
,;
a r e Cm on $ (U) and
li s
s u p p o r t e d i n 4 (U). Thus t h e r e
e x i s t s a c o n s t a n t k,4 such t h a t
Using t h e i n e q u a l i t y (5.20) a g a i n we s e e t h a t for e v e r y
E
> 0 the last
e x p r e s s i o n i s e s t i m a t e d by
.. S i n c e a l J ,h,c,w a r e Cm on 0 (U) and e x i s t s a constant k
15
i s supported i n 9 (IT) t h e r e
such t h a t
Using s i m i l a r arguments we s e e t h a t t h e r e e x i s t s a c o n s t a n t k16 such t h a t t h e exp r e s s i o n (5.21) i s g r e a t e r t h a n
R.MARTINI
234
Combining (5.18) - ( 5 . 2 5 ) , ( 5 . 2 3 ) w i t h a s u i t a h l y chosen i s bounded o n $ ( I T ) we s e e t h a t t h e r e e x i s t c o n s t a n t s 1.. bering t h a t Ly
t,
17
and remem-
> 0 and k , 8
such t h a t
For t h e p a r t o f t h e s e s q u i - l i n e a r
form B A which c o n t a i n s t h e f i r s t or-
d e r d i f f e r e n t i a l o p e r a t o r '? we o b t a i n
a
S i n c e '3,h,e a r e C
-
on Q (U) and a i s hounded on 4 (LJ) t h e r e e x i s t s a con-
s t a n t k l g such t h a t
Using t h e i n e q u a l i t y ( 5 . 2 0 ) a g a i n we s e e t h a t f o r any E > 0 t h e l a s t e x p r e s s i o n is e s t i m a t e d by
w,h,e a r e Cm on 4 ( U ) and such t h a t
- i s bound-ed o n 8 (U). Thus t h e r e e x i s t s a c o n s t a n t 0 .
k.20
DEGENERATED DIFFERENTIAL OPERATORS
(5.30)
i1w2
< Ys,s >
41
0 ( k = 1 , 2,
..., m )
and F ( k = 1 , 2 , k
..., m )
such t h a t
"ow l e t C be a number such t h a t t h e s e t s {U') (k = 1 , 2,
k
..., m )
d e f i n e d by
Uk = {x form a c o v e r i n g of
8' such t h a t
E
Mluk(x) > C ) ?I.
Then from ( 7 . 1 8 ) it f o l l o w s t h a t t h e r e e x i s t s a c o n s t a n t
R.MARTIN1
244
f o r a l l U c D and s i n c e elements o f D can b e approximated with r e s p e c t t o t h e topology of V (F,a)by f u n c t i o n s belonging t o r o ( V , F ) we s e e by t a k i n g l i m i t s a t both s i d e s of ( 7 . 1 9 ) t h a t (7.20)
Re ( L X s , s )2 ( A
for a l l U
E
8.
-
f i ' ) l l s l I o2
D and a l l X 2 0. This completes t h e proof of theorem
9.
A method f o r t h e c o n s t r u c t i o n of semi-groups.
We r e c a l l t h a t i n s e c t i o n Vo(E,a) into H(E,a).
4 we
have i n v e s t i g a t e d t h e embedding of
The q u e s t i o n whether t h i s embedding i s compact was our main
i n t e r e s t and a p o s i t i v e r e s u l t concerning t h i s p o i n t was s t a t e d i n theorems 2 and
4. In o r d e r t o g i v e a method f o r t h e c o n s t r u c t i o n of semi-groups we suupose t h a t t h i s indeed i s t h e c a s e . Thus we suppose t h e embedding J of V ( E , a ) i n t o H ( E , a ) t o be compact. I n s e c t i o n
5 we proved t h a t t h e o p e r a t o r LX = 0 X de-
f i n e s a b i j e c t i o n of t h e s e t D = { s
V (E,a)lOs
Re
E
s u f f i c i e n t l y l a r g e , l e t say f o r Re X t X 1 .
nuous i n v e r s e of
L f o r Qe X A
E
H ( E , a ) } onto H ( E , a ) f o r
By GX we have denoted t h e c o n t i -
2 A,.
Now consider t h e f o l l o w i n s diagram
It follows t h a t t h e o p e r a t o r G i = J o GX, G i : H(E,(r) + H ( E , a ) ,
i s compact a s a
composition of a compact l i n e a r and a continuous l i n e a r o p e r a t o r . I n a d d i t i o n i f t h e o p e r a t o r @ i s symmetric on D i n H ( E , a ) with r e s p e c t t o some norm
I I I IH;
i.e.,
(8.1)
( o s , t ) H= (s,Ot&
for a l l s , t
E
D , it follows t h a t f o r any r e a l number X w i t h X t X 1 t h e o p e r a t o r
G i i s s e l f a d j o i n t . Thus i n t h i s c a s e
it i s p o s s i b l e t o apply t h e s p e c t r a l t h e o r y
f o r compact s e l f a d j o i n t o p e r a t o r s t o G;. Then G i has t h e r e p r e s e n t a t i o n
245
DEGENERATED D I F F E R E N T I A L OPERATORS
Tor any s
H(E,a), where
E
{p
x1 k
i s t h e s e t of e i g e n v a l u e s o f G{ and where {Skk}
( E c I ) i s a n orthonormal set of e i g e n v e c t o r s b e l o n g i n g t o t h e e i g e n v a l u e p x
k
k'
Now G' c o i n c i d e s w i t h t h e r e s o l v e n t R ( X ; L ) and between t h e semi-group h
{ T ( t ) } ( t t 0) c o r r e s p o n d i n g t o t h e i n f i n i t e s i m a l g e n e r a t o r L and t h e r e s o l v e n t
R ( A ; L ) t h e r e e x i s t s t h e r e l a t i o n ( s e e ( 2 . 5 ) i n c h a p t e r I)
m
(8.3)
R(h;L)f = Je-ItT(t)f 0
dt.
Hence
m
(8.4)
G X f = /e
-At
T ( t ) f dt.
0 So by i n v e r s i o n o f
(t
9.
2
(8.4) we o b t a i n a r e p r e s e n t a t i o n f o r t h e semi-Rroup { T ( t ) l
0 ) c o r r e s p o n d i n g t o t h e i n f i n i t e s i m a l g e n e r a t o r L.
Examples.
I n t h i s s e c t i o n we s h a l l g i v e some examples t o i l l u s t r a t e t h e r e s u l t s of t h e p r e c e d i n g s e c t i o n s .
-
Example -------- 1 . Let M b e t h e u n i t - h a l l Bn.i n t h e n-dimensional E u c l i d e a n s p a c e lRn l e t E be t h e t r i v i a l complex bundle Bn x C'. operator 0 : rO(M,E)
+
and
Define t h e d i f f e r e n t i a l
r o ( M , E ) by
(Os)(x) = (x,(1
-
IX~~)~AS(X))
if s ( x ) = ( x , Z ( x ) ) and where A d e n o t e s t h e Laplacean ID: and where [ x I 2 = Cx?. i i 1 Then 0 s a t i s f i e s t h e assumptions of theorem a(x) = ( 1
-
8 i n case p
> 0 . Moreover, t a k e
1xI2)' and l e t p ( x ) be t h e d i s t a n c e o f a p o i n t x
o f M. Then s i n c e f o r each x
E
E I4
t o t h e boundary
M we have
it f o l l o w s from theorem 4 t h a t t h e embedding o f V o ( E , a ) i n t o l I ( E , a ) i s compact p r o v i d e d t h a t p < 2.
Also u s i n g G r e e n ' s f o r m u l a it can e a s i l y be s e e n t h a t t h e e x t e n s i o n o f 0 d e f i n e d i n theorem 6 i s symmetric on D . Thus i n c a s e 0 < p < 2 t h e c o n s t r u c t i o n method of s e c t i o n 8 c a n b e a p p l i e d .
246
R. MARTINI
Example --_-_-_-2. Let M be the cylinder R 1
x
I embedded canonically in the 3-dimensional is the 1-dimensional unit-spli~reand
Euclidean space I R 3 . Here 5'
I = r0,ll. Let E be the trivial complex vector bundle over
'1.
Define the linear differential operator 0 : r o ( M ,F ) + r o ( M , E ) by (Os)(x) = (x,a(x)A;(x)) if s(x) = (x,$(x)) and where in cylinder-coordinates a(x) = xp(l - x )' and where 3 3 A is the Laplacean given in the same coordinates by
Then0 satisfies the conditions of theorem and the conditions of theorem 8 in case p
2
7 in case p
> 0 and q > 0
2 and q 2 2. Fvidently, we can define
a similar differential operator on the cylinder Pn
X
I embedded in the (n+2)-
Of course, +* Sndenotes . the n-dimensional unitdimensional Euclidean space I!? sphere. Example -_- ------3. Let M be an oriented n-dimensional Riemannian compact manifold with boundary 3E.I and let
be the vector bundle of complex-valued differential forms on M. As usual, d : r(M,F)
Let
*
+
r ( M , E ) denotes exterior derivation.
be the real linear automorphism of E such that if e l , e2,
is an oriented orthonormal basis for T*(M) and 1 , 2,
...,
A
e2
*1 = e
if
0
A
A
... e2
A
A
e
...
= 1; A
e n ''
is even and
*eO(l) U
..., e
a permutation of the integers
n it follows that *el
if
0
A
eo(2)
is odd. Obviously,
A
..*
A
eo(p) = -eo(p+l)
* maps AP(T*(M))C
the automorphism of E which on AP(T*(M)), It follows that the operation
A
*.*
into An-P(T*(EI))C
and
**
=
w, where
is multiplication by (-l)p(n-p).
* induces a conjugate linear real automorphism of
r(M,E), which also will be denoted by
*.
is
247
DEGENERATED DIFFERENTIAL OPERATORS
The Riemannian s t r u c t u r e f o r
M
d e f i n e s a s t r i c t l y p o s i t i v e smooth
If t h e n o t a t i o n for t h e Riemannian s t r u c t u r e i s (
measure p on h4.
measure has w i t h r e s p e c t t o t h e c h a r t c = (U,$) a l o c a l d e n s i t y
0
,
)
M , then t h i s
which i s g i v e n
by
where g . . ( x ) = (dxOi, d x + j ) h n .
LI
I t a l s o i n d u c e s a Hermitian s t r u c t u r e E f o r E . T h i s we s h a l l d e s c r i b e now. Let e
,,
e2,
...,
s t r u c t u r e
E
{ei 1
he. l2
...
he. ) ( I s p s n , t s i , < i 2 < P
A . . .
< i s n ) P
form a n orthonormal b a s i s f o r A ( T * ( M ) ) . Let To(M,F) b e p r o v i d e d w i t h t h e i n n e r p r o d u c t , g i v e n by
Then, if w i t h r e s p e c t t o t h i s i n n e r p r o d u c t t h e f o r m a l a d j o i n t o f t h e e x t e r i o r d e r i v a t i o n d i s d e n o t e d by t d , it f o l l o w s t h a t
td=;*d
*w,
where w is t h e automorphism of E which is m u l t i p l i c a t i o n by ( - 1 ) ’
on AP(T*(M))
C
( s e e PALAIS C41, chp. IV, p . 76-77). Now l e t a b e a c o n t i n u o u s r e a l - v a l u e d f u n c t i o n d e f i n e d on M such t h a t m
t h e r e s t r i c t i o n of a ( x ) = 0 when x
E
t o t h e i n t e r i o r E4 \ aM i s C
ad : r o ( M , E ) + r0(M,E)
it f o l l o w s t h a t i t s f o r m a l a d j o i n t s a t i s f i e s t
(ad) = a ;
*
d
*
Hence some c a l c u l a t i o n s y i e l d
and
and p o s i t i v e and such t h a t
aM. Then f o r t h e d i f f e r e n t i a l o p e r a t o r
w +
*
(&)
A
*
w.
R.MARTINI
248
So i f t h e Laplacean A i s d e f i n e d by
A =
-
{d(td)
+ (td)d}
it f o l l o w s from t h e e q u a l i t y
t h a t t h e d i f f e r e n t i a l o p e r a t o r 0 : To(M,E)
-
0
t
{a3 ( a d )
-f
r
(M,F) d e f i n e d by
+ P(a3))ndj
equals 0
= a2A + aY,
where Y i s t h e d i f f e r e n t i a l o p e r a t o r o f t h e first o r d e r g i v e n by Y =
-
{(a")
A
w
*
d
*
W
+ d;
*
(da)
A
*
w
+
2 ;
*
(dn)
A
*
wd}
S i n c e A i s s t r o n g l y e l l i p t i c t h e d i s c u s s i o n above shows t h a t t h e t h e o r y developed i n t h e p r e c e d i n g s e c t i o n can b e a p p l i e d t o t h e d i f f e r e n t i a l o p e r a t o r 0 j u s t defined.
RCFmENCES
'11
DDNFORD, N . and J.".
c21
IIARTINI, R . , D i f f e r e n t i a l O p e r a t o r s d e g e n e r a t i n g a t t h e Qoundary a s I n f i n i t e s i m a l G e n e r a t o r s o f Semi-groups, m e s i s , D e l f t ( 1 0 7 5 ) .
r31
PiARASIMHAN,
C41
PALAIS, R., Seminar on t h e Atiyah-Singer Index Theorem, Ann. o f Math.,
141
YOSIDA, K . ,
Sf'Fl'ARTZ, L i n e a r O p e r a t o r s , Vol. 1 , I n t e r s c i e n c e P u b l i s h e r s ( 1958).
R., k n a l y s i s on Real and Complex M a n i f o l d s , North-Holland Publ. Comp. ( 1968). Study 57, P r i n c e t o n ( 1 9 6 5 ) . F u n c t i o n a l A n a l y s i s , Grundlehren d . Vath.
Iliss.,
SpringerV e r l a g ( 1968).