Luigi Amerio ( E d.)
Equazioni differenziali astratte Lectures given at the Centro Internazionale Matematico Estivo (C.I.M.E.), held in Varenna (Como), Italy, May 3 0 - J une 8 , 1 9 6 3
C.I.M.E. Foundation c/o Dipartimento di Matematica “U. Dini” Viale Morgagni n. 67/a 50134 Firenze Italy
[email protected] ISBN 978-3-642-11003-0 e-ISBN: 978-3-642-11005-4 DOI:10.1007/978-3-642-11005-4 Springer Heidelberg Dordrecht London New York
©Springer-Verlag Berlin Heidelberg 2011 st Reprint of the 1 ed. C.I.M.E., Ed. Cremonese, Roma, 1963 With kind permission of C.I.M.E.
Printed on acid-free paper
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CENTRO INTERNATIONALE MATEMATICO ESTIVO (C.I.M.E)
Reprint of the 1st ed.- Varenna, Italy, May 30-June 8, 1963
EQUAZIONI DIFFERENZIALI ASTRATTE
T. Kato:
Semi-groups and temporally inhomogenous evolution equations ................................................................ 1
J. L. Lions:
Équations différentielles opérationelles dans les espaces de Hilbert .................................................... 45
L. Nirenberg:
Equazioni differenziali ordinarie negli spazi di Banach .............................................................................. 123
R. S. Phillips:
Semi-groups of contraction operators .................................. 171
L. Amerio:
Almost-periodic equations in Hilbert Spaces ....................... 223
G. C. Rota:
A limit theorem for the time-dependent evolution equation ............................................................... 241
S. Zaidman:
Existence and almost-periodicity for some differential equations in Hilbert Spaces................................ 259
CENTRO INTERNAZIONALE MA TEMATIC 0 ESTIVO (C, L M. E. )
TOSIO KAT 0
SEMI-GROUPS AND TEMPORALLY INHOMOGENOUS EVOL UTION EQUATIONS
ROMA - Istituto Matematico dell'UniversitA
1
SEMI-GROUPS AND TEMPORALLY INHOMOGENOUS EVOL UTION EQUATIONS by T. KATO
INTRODUCTION
These lectures are concerned with the Cauchy problem for the timeindependent evolution equation du dt
(E)
u(o) = u • o
+ A(t)u = fIt) ,
The unknown u = u(t) and the given function fIt) take values in a Banach space X ; A(t) is a (in general unbounded) linear operator in X depending on t. It will suffice to mention here only a few examples of (E).
Ex. 1. A parabolic differential equation
2
n
~ - r:
a (x t) ~ j',k=1 jk ' UXjoxk
() t
n
()
- £. a,(x, t) ~ - a(x, t)u = fIx, t) j=l
J
()Xj
is in the form (E) with an obvious definition of A(t). The boundary conditions, which may depend on t , are included in the definition of A(t).
" Ex. 2. The Schrodinger equatiop -1
i
uUUt
+
A L.l
_
U -
x
V(x, t)u - 0 ,
also has the form (E). Here A(t) = i( ~
= IJ.
u
3
3
R
- VI. , tl) is i times a self.a-
djoint operator (at least formally) in X = L 2(R 3). Ex. 3. The wave equation
~
- 2T. Kato
in x E. R
3
may be reduced to the form (E) by writing
o
u
u
.:L
J
()t
uXl
.iL
o
0'
o
Q
o
o
o
()x 2
uX 3
o
where vI' v 2' v 3 are auxiliary functions. This has the form (E), where u is replaced by the 4-(' Jmponent vector function (u, vI' v 2' v 3)' In what follows I want to deduce several sufficient conditions on A(t) and f(t) in order that (E) has a unique. solution.
4
- 3T. Kato
§ 1.
GENERATION OF DIFFERENT TYPES OF SEMI-GROUPS
1. Let us consider (E) first in the special case when A(t) = A is independent of t : du dt
-+ Au
(E )
o
= f(t)
u(o) = u •
o
''
The solution is formally given by (So)
=e
u(t)
-tA
Uo
'+
ft
0
e
-(t-s)A
f(s)ds.
The problem is, therE:iore, essentially that of constructing the exponential function e -tAo This is exactly the problem of generating a semi-group . -tA) . e from a given operator A .
t
f
In what follows we consider only strongly continuous semi-groups on
[0,
00 ).
l e -tA}
Thus
is a semi-group if and only if e -tA is strongly con-
tinuous for 0 !; t 0
\ ( A +A) -n
I~
belongs to the resolvent set p( -A) of .A, with
~
,
n = 1, 2, 3, ....
where M is a constant independent of "
or n.
Theorem 1. 2. Let -A e: (Bo). Then there exists a unique semi-group tA . \I. e- tA II with e \ {M such that
I
(D)
d dt
-
e
-tA
u = -Ae
-tA
u=- e
5
-tA
Au
~
4T. Kato
for u E- 0 A' e
-tA
commutes with (
1.-1 + A) .
1\
Proof. See Phillips I lectures Remark 1. 3. If -A f (Bo), it follows that all complex belong to
f
A with
A> 0
Re
(-A), with n
= 1, 2,.•. , •
This is seen by considering the Laplace transforms of t
n-l
• e
-tA
(see
Phillips). Definition 1. 4. If M - 1 above,
t e -tA f
is called a contraction semi-group.
The subset of (Bo) determined by M = 1 will be denoted by (Co). (Note that M ~ 1 in general).
2. We now introduce another subset of (Bo) which is important in our problems.
Definition 1. 5. We say '-A
t (Ho) if
1) A is densely defined and closed
2) The spectrum
Iarg AI ~f "'
tI'(A) of A ·is a subset of a sector
w , w >0, and
I
\ arg A £
for Remark 1. 6. (Ho)
~+
W
-
E
c (Bo). This is not obvious from the definition, but fol-
lows from Theorem 1.7. below (again take the Laplace transforms of n-l -tA t
e
).
Theorem 1. 7. Let -A f (Ho). Then there exists a unique semi-group \ -tA I l e j such that for t 0
>
(0 1)
(I) X.
-e
-tA
A ~ -
d -tA e = -A e -tA dt
E
B 1)
We denote by B the set of all bounded linear operator in X with domain 6
- 5T, Kato
i 1
I
can be continued analyticallY to the sector arg t I..::: c0 , t r 0, -tA -tA: with (D') preserved. Furthermore, e and tAe are uniformly bounded e -tA
in any smaner sector :
Ie -tA
~
1\
f.ll
\ til Ae -tA
,
£. M ~
I
Iarg t I f
LV -
~
tA and e- _
1)
1 strongly when t~" 0 in this smaller sector -tA We can define e by the Dunford integral:
Proof.
j
e- tA = _,1_. 2lf 1
(1)
e A\A+A)-l dA
r-,
9
0,
'
C
. .m · a curve, runmng h C 1S were
where ~ .: :
f
€
(A) f rom
t0
,;.0
ei
9
~ + w . Thus the integral is absolutely convergent and
defines an operator of
8
The semi-group property follows from the standard argument with Dunford integrals. We have namely (2)
e
-sA = _1_
2-'
J
IS
C,e
A (\, 1\
+ A)-l d )..
I
" 1
where C' is obtained from C by a slight shift to right. Multiplying (1), (2) and using the resolved equation, we have
e-tAe- sA = (_1_. 211" 1
)2J (
e t\t+A's_l_i( A+A)-l_(A '+A)-lJdAdA',
N-A
cJ c ,
L
where the order of integration is arbitrary. Now
1
eAt _d_A-
=0
AS 2 _. 11 1 e
and
C X-\ since C lies to left of C', Hence e-tAe-SA=_21. if 1
1 C
e)"(t,,+S)(A+A)-l dA = e -(tts)A , ;
(1) For analytic semi-groups, see [15
Jas well as the book
lips.
7
by Hille-Phil-
- 6T. Kato
l e -tA} .
proving the semi-group property of
That e -tA has an analytic continuation is obvious from (1). In fact the integral of (1) converges for any t with
/L
ny t with ! arg t d dt
(3)
e
-tA
1
= 211" i
j
C ~e
At \
-1
1« B-
t
S suitably.
,by taking
to
1 arg
, hence for
a-
Moreover,
\
( fI tA) d II f B,
Jc
A(A
~
t
r0•
tA)-l = 1 - A(A tA)-l and AeAt dA = 0, \ -1 1-1 (D') follows from (3) (note that A(I\ tA) AC A(/\ tA) ).
Since
To prove the uniform boundednessof e -tA, we change the integration variable from
A'
A to
=
At in
(1). The corresponding integration path
tC can be deformed to a path C', independent of t , which runs from -i 9 ' ti 9 \ 00 e to fX) e with = ~ + t , t >0 being very small.
r;;,
The resulting expression
e -tA = -12lf i is true for any t
r0
\ (f+A)-l/:;
with
1 C'
e A' (A'-
t A)
t
I arg t \ ~ w-c.
-1 -d A' t
Since
M/lfl=Mlt!/IA'I,
it follows that
I .-tA! ~ 2~ ~,I.A'I II~~'II •
(3)'
M'
~
I
In the same way one proves Ae -tA { Milt lit ,. -tA To prove e -> 1 , t -> 0 , we note that
I
.1
A -liudA
(e- tA _l)M-= -12 e At [(AtA)-1 lfl C if u
c
DA' Hence for t ~ 0 (e -tA - l)u
1
-?> - - - , 2lr 1
.-
J\
(II tA) -1 Au -d).
(the integrand is O( A-2) for A-) 00
A
=
2~' r·eAt(AtA)-IAU ~A Ill) 1\
=0
Re /\ ). 0).
,
Since e -tA is uniformly bounded for I arg t 1ft;; - C
8
as proved a-
- 7T. Kato
bove and since DAis dense, this proves that e
-tA
strongly, q, e, d,
-7
1)
•
Remark 1. 8, Theorem 1. 7, implies that, if -M(HO), e -tA sends X into -tA DA and e u is always differentiable for any u EX, if t f O. This is a great difference from the case -Af(BO), where e -tAu expected only for u Remark 1. 9.
e
E
DAis in general
DA'
There are many examples of operators of (HO). Generally spea-
king, any strongly elliptic partial differential operator with "ordinary" boundary conditions belongs to (HO), Furthermore if X is a Hilbert space, there is a rather general sufficient condition for -A E- (HO), Suppose that the
l
numerical range NA = {(Au, u) II u I = 1, u E DA of A is a subset of a sector
I arg,.\\ ~ ~ - w,
w>O, If, in addition, there is at least one point II
f (A),
exterior to NA that belongs to
then -A E (HO)2).
3. We now consider the solution of the inhomogeneous equation (E ). o Definition I, 10, By a solution of (E) we mean a function u(t) with the fol-
lowing properties, 1) u(t) is (strongly) continuous for 0 ;:; t ::; T , u(o)
2) u(t) is (strongly) differentiable for 0 < t 3) u(t) E DA(t) for 0 4) (E) is true for 0
L..
< t
~
= Uo '
T,
t ~ T so that A(t)u(t) makes sense. ~
T.
The same definition applies to (E ) when A(t) = A is constant. o Theorem 1, 11. Let -A ~ (BO), Then any solution of (E ) is given by (S )
-
0
0
if f(t) is continuous for 0 ~ t ~ T, Conversely, u(t) given by (S ) is a soluo tion of (E ) if u 6 DA and f(t) is continuously differentiable. In this cao 0 (1) The uniqueness of ~ e -tAl with the properties stated follows from rem 1,2.
I
I
Theo~
(2) This is due to the fact that (A +A)u ~ dAI u I for any u fDA where dA is the distance of A. from N A' It follows, under the condition stated, that
( \ +A) -11
:; 1I dA~ Mil AI ' -
9
-8T. Kato
se Au(t) and du(t) / dt are continuous 1). Proof.
Let u(t) be a solution of (E ). Then o
() -(t-s)AA us+e () -(t-s)A( -us+s A ( ) f( )) = -de -(t-s)A us=e ds = e -(t-s)f(s) since u(s) E DA and e
-(t-s)A
E
B. Integration on s then gives (So) im-
mediately. Conversely, suppose
.:::: DA and f(t) is continuously differentiable.
Uo
Since e -tAu
satisfies the homogeneous equation and the initial condition by o Theorem 1. 2, we need only to consider the second term of (S ). In other
o
O'Js
words, we may assume U o = Noting that f(s) = f(o) + u(t) =
f
f'(r)dr , we have then
tOt e-(t-s)Af(o)dS
+j j
t
0
r
dr
o
e -(t-s)Af'(r)ds .
But (see Lemma 1. 12 below) A
j
t
e
-(t-s)A
ds = A
o
t
Je 0
-sA
ds = 1 - e
-tA
1- e
, -(t-r)A
Hence Au(t) exists and Au(t) = (1 - e
-tA
t
J
- t-r A (1 - e ( ) )f'(r)dr
)f(o) +
= f(t) - e -tAf(o) -
J °t
0
e -sAf'(t-s)ds .
On the other hand
J.
(1) This theorem is due to Phillips [8
10
.
- 9T. Kato d ill Hence
d u(t) = ill
Jt
e
-sA
f(t-s)ds = e
-tA
J
flo) +
o
!
t
e
-sA
fl(t-s)ds
0
u(t) = - Au(t) + f(t). as we wished to show. d By the way, the continuity of illu(t) and of Au(t) are obvious from
the above expressions. Lemma 1. 12. Let -A A
Je t
-sA
E (Bo). Then
ds
=e
-rA
-e
-sA
,
r
Proof. If u E DA, we have Ae
-sA
d u =- e -sA u ds
= e -sAAu
(wh'lC h is . cont'l~
nuou8 in t). Hence A
J
t
e
-sA
u ds =
)t
r
Ae
-sA
u ds = (e
-rA
- e
-tA
)u .
r
(The first equality is a direct consequence of the closure of A). For any t -sA -sA v ~ X , let u fDA' U -> v. Then e u ds -> e v ds and t n n r n r -sA -rA -tA -rA -tA A( e u ds) = (e - e ) u -> (e - e ) v. It follows, again by n n r t -sA -rA -tA the closure of A, that A e v ds exists and equals (e - e ) v. r q. e. d.
It
J
J
j
Theorem 1. 13. If -A ~ (HO), the continuous differentiability of f(t) in the se
" cond part of Theorem 1.11 can be replaced by a Holder continuity. Furthermore, u(t) of (So) is analytic if f(t) is analytic on [0, T] . Proof. Again we may assuIX\e u = O. Then
o
u{t) • ; : .(t-s)A{f{s) _ f{t))ds
J:
+
e-{t-s)A1{t)dt .
Therefore (see Lemma 1. 12) Au(t)
=) t
0
A e -(t-s)A(f(s) - f(t))ds + (1 - e -tAl f(t) .
11
- 10 T. Kato
.
I
and f{s) - fIt)
I~
-{t-s)A
.
I
const{t-s)&
,9 > 0 •
Note that the mtegral eXIsts because ,Ae
I~
const t:s-
(see Theorem 1. 7)
This shows that Au{t) exists and (closure of A{t)!) A(t)u = A
t
Joe -(t-s)A(f{s) - fIt)) ds + (I - e -tA)f{t) .
On the other hand, the construction of ()u{t)/u t requires a little detour. We define
u It) =
e - (t-s )A f{s) ds
0
t
Obviously u ~ It) d Cit
t-i:
J
-';> u(t)
u f (t) = e
for t -'> 0 , locally uniformly in t. Also
- tA
f(t- t ) -
Jt- £ A e -(t-s)A f{s) ds 0
a leX1S ' t ' Ae -(t-s)A.IS cont·muous for s . tegr th e 10 s smce easy to see that the limit for
~~0
~
t -
;
t.
But 1't IS .
of this integral exists and equals Au(t)
II
(use again the Holder continuity of fIt)), so that d Cit
u £ (t) -) fIt) - Au(t) .
Moreover, this convergence is locally uniform in t. Hence it follows that
d~
u(t) exists and equals fIt) - Au(t), by.a well known theorem in differential
calculus. If fIt) is analytic, u f It) is also analytic: dUE (t)/dt given above exi-
sts for complex t in some neighborhood of the interval [2 € , T] • But u i (t)
~
u(t) is true locally uniformly in t for these complex t. It follows
that u(t) is analytic. q. e. d.
12
- 11 T.Kato
§ 2.
THE CASE IN WHICH .A(t) ARE GENERATORS OF ANALYTIC SEMI·GROUPS WITH CONSTANT DO· MAIN FOR A(t)h.
1. First we note that the equation (E) is very simply dealt with if A(t) Eo B and strongly continuous in t. If we consider the homogeneous equation du/dt
+ A(t)u = 0, the Elolution can be constructed by a straightforward succes-
sive approximation: ClCJ
u(t)
=.L
uk(t) ,
u (t)
k=o
uk(t)
=·fo
o
=u0 k = 1,2,3, ...
A(s)uk _1(s) ds ,
This is equivalent to writing u(t) = U(t,O)u
o
and deterrmining U(t, 0) from the
differential equation dU(t,O)/dt = .A(t)U(t,O), U(O,O) = 1 , by successive approximation (the derivative is strong derivative). More generally, we can solve the differential equation
C> () t U(t, s)
(1)
=
U(s, s) = 1
.A(t)U(t, s),
by successive approximation. The family of operators U(t, s) constructed in this way will be called the evolution operator (or the Green function). The evolution operator has, in addition to (1), the following properties:
71
U(t, s)
= U(t, s)
A(s)
(2)
~
(3)
U(t, s) U(s, r) = U(t, r).
"' s
To prove this, it is convenient to consider another differentiable equation
?
~ V(t, s)
uS
= V(t, s) A(s),
V(t, t)
13
=1 •
- 12 T. Kato
This can again be solved by successive approximatiori. Then
V(t, s) U(s, r):: V (t. s)(A(s) - A(s)) U(s, r) = 0
'Vs
so that V(t, s) U(s, r) is independent of s. Putting s = t and s = r , we obtain U(t, r) = V(t, r) , and hence U(t, s) U(s, r) = U(t, r). q.
e. d.
With the use of the evolution operator, the solution of (E) can be expressed by (S)
u(t)
~ U(t, O)u
0
J:
+
U(t, ,) 1(,) ds ,
Now the above method does
~ot
work when A(t) is not bounded. The-
refore we want to construct the evolution operator for unbounded A(t) by a limiting procedure, by approximating A(t) by a sequence A (tl of bounded n operators (this is the way the semi-group e -tA was constructed in Phillips' lectures as the limit of e -tAn, A being bounded). We choose n (4)
A (t) = A(t)J (tl = n{1 - J (t)) , n
n
n
1 -1 A (t)) , n n
J (t) :: {1 + n
n = 1,2,3, .•
If A (t) (or J (t)) is strongly continuous in t , we can construct the evolution n n operator U (t, s) by the simple method described above. Then we want to n show that s-lim U (t, s) exists, which will be the evolution operator U(t, s) n
for the unbounded case. This method is seen to wo:rkunder certain conditions on A(t). We have namely Theorem 2. 1.
1)
Assume that
1) -A(t) f (HO) uniformly for 0 f
constants M
> 0,
t.V
>0
t
- U (t, s), and so on. In view of the fact n n n s that the series (13) is uniformly majorized, we conclude that (2)
(17)
U (t ' s) = L. \ U(k)(t , s) n
k
n
?
=
U(k) (t, s) _ U(t, s),.
I U(t, s) I~ C .
Since it is easily seen that the strong convergence (17) is uniform in s, t for s ::: t , U(t, s) is strongly continuous for s
~
t. Also 2) of Theorem 2. 1. fol-
lows from the corresponding relation for U (t, s). n
18
- 17 T. Kato 3.
Proof of Theorem 2. 1., continued, We uae another identity -dd e -(t-a)An(t) U (a, r) = e -(t-a)An(t)(A (t) - A (a)) U (a, r) , a n n n n
whence we obtain
U (t,r) ;::.e-(t-s)An(t) n
+J t e-(t-s)An(t)(An(t)An(s)-1_ I )An(s)Un(s,r)ds. r
Multiply this equation with A It) from'left and write Y (t, s) = A It) U (t, s); n n n n then (IS)
Y (t,r) = n
y{o~{t,r) +Jt n
H (t,a) Y (s,r)ds
r
n
n
where
(IS), may be written symbolically as (20)
Y = y(o) + H n n n
*
Y . n
We want to solve (20) again by successive approximation: IX)
(21)
L
=
Y
n
k=o
y(k) = H ¥ y{k-1) • n n n
y(k) n
Here, however, we have a slight difficulty that did not exist in (13), for y(o) has the uniform (independent of n) estimate
ly~O){t,S) I~ C{t:-s)-1
n
where
(t_s)-l is not integrable. Thus the uniform estimate of y(1) is not quite sim-
--
n
pIe, although its existence is obvious (An (t) E B!). Here we give only the result : (22)
s) I I y(1)(t, n
~
C 1- e (t-s)
19
- 18 T. Kato
of which the proof will be given in n. 7. Once (22) is established, the further successive approximation proceeds smoothly, for the right member of (22) is integrable as well as that of
I Hn(t, s)
(23)
\:::
Cl _ 9
(t-s)
which is proved as in (11). By an argument similar to that given in n. 2, it (k) has a strong n
follows that (21) is uniformly majorized and that each term Y limit y(k) for n -> 00. Hence A (t) U (t, s) = Y (t, s) _.l> Y(t; sl, n n n s
(24)
(24) gives (see also (5))
U (t, s) n Since U (t, s)
n
=A
n
~U(t,
s
(t)
-1
Y (t, s) n
~
s
A(t)
-1
Y(t, s) .
s) by (17), we must have U(t, s) = A(t)
means that A(t)U(t, s) exists and equals Y(t, s)
E.-
B if t
-1
>s
Y(t, s). This
. Thus we ha-
ve proved (25)
I A(t)
U(t, s)
I !:
C t-s
The differentiability of U(t, s) is proved in the following way. Since JU (t, s)u/u t = - A (t)U (t, s)u and U Itt, s)u ~U(t, s)u , A (t)U (t, s)u-) n n n n n n -~Y(t, s)u = A(t)U(t, s)u uniformly for t ~ s+a , it follows that ()U(t, s)u/2 t exists and is equal to -A(t)U(t, s)u. That is, U(t,s) is strongly differentiable in t for t ) s , with the strong derivative -A(t)U(.t, s) E B. Similarly, we have ()
;:;- U (t, s)u = U (t, s)A (s)u (IS n n n If u E D = DA(s)' we have An(s)u -l>'A(s)u, n-.> x , uniformly in s (see
"
Phillips) so that ,; s Un(t, s)u
----7
U(t, s)A(s)u. The same argument as above
20
- 19 T. Kato
U then proves that -;::- U(t, s)u ~'s
= U(t, s)A(s)u .
If A(t) -1 is analytic in t in a neighborhood !J of 0 f. t =- T , the -1 -1-1 same is true with A (t) = (A(t) + n ) . Therefore U (t, s) can be conti-
n
n
nued analytically to t ~ /j , s ~ ~ . Now the expression of U (t, s) by the sen
ries (13) holds true when the variaples t, s, r are supposed to lie on a straight line in
d
having a small angle
with respect to
9
9
with the positive real axis, uniformly
,and each term U(k) is seen to converge for n n
to U(k) uniformly (on the line as well as in
8 ).
-'> 00
Thus U (t, s) converges
n
strongly and aocally)un~formly to a U(t, s) as long as J arg(t-s)
I are sufficien-
tly small. It follows that U(t,5) is strongly analytic in such a region of t and s. But since strongly analyticity is equivalent to analyticity (in norm), U(t, s) is analytic. This completes the proof 0f Theorem 2. 1.
4.
We now consider the inhomogeneous equation (E).
Theorem 2. 3. Let the assumptions of Theorem 2. 1. be satisfied. Then the conclusions of Theorem 1. 13. are true (with (Eo) and (8 0 ) replaced by (E) and (8), respectively). Proof. Almost the same as for Theorem 1. 13. The only modification required is to note that A(t)U(t, s)
= A(t)e -(t-s)A(t)+ Y'(t, s) 00
Y'(t, s) =
r
IY'(t, s) I ~
y(k}(t, s)
k=l
c
(t_s)1-9
see (20), (22), (24). Hence A(t)
Jt r
U(t, s)ds
J
J
r
r
= t A(t}U(t, s)ds :: 1 - e -(t-r)A(t) + t Y'(t, s)ds
by Lemma 1. 12 (Y'(t, s) is absolutely integrable).
21
- 20 T. Kato
5.
Generalizations, To improve Theorems 2. 1. and 2. 3. , we need the fractional powers
A(t) 0\
of A(t).
e
When -A
(BO), the fractional powers A C\ can be defined in a na-
tural way 1). Here we assume, for simplicity, that A-I E-
B in addition.
Then we can first construct the Dunford integral (26)
A
J
1
_Ii(
= - -2-' 11 1
where L is a curve from -
_01
z
L
to -
rYQ
(z - A)
Do
-1
dz
E B ,
passing between z
0
0 •
= 0 and
(j
(A).
The integral converges absolutely since
I(z_A)-1 I { M/Im(-z). Since this is a Dunford integral it is easy to see that A o{+:'l,
If 0
L may be taken as the
double ray (0, - xc ), yielding
A
(27)
sin 1fC{
-0(
IT
Then we define A" since A
as the inverse of A-' ; note that A-X. is invertible -n -(n- x) -'x -n u = 0 implies A u = A A u = A u = 0 , u = 0 , where
n is a positive integer larger than
~
.
We need also the following expression, which if3 valid for -A t (HO). (27')
A
i.,
e- LA = _1_. 2111
1(-A)~ C
eAt' (A+A)-l d A .
This can be proved by verifying that (27') gives e _t A when multiplied by (26) (cf. the proof of Theorem 1. 7). It follows from (27') that (27")
lAx'
e-lX,IL_C_,
- 1(;\
'I
(1) These fractional powers are considered by many authors; see, for exampIe, [3] , [4] , [16] and the references given ther~. 22
- 21 T. Kato
We can now state generalization of Theorems 2. 1. and 2. 3. Theorem 2. 4. 1) Assume that i) - A(t) f (HO) uniformly (as in i), Theorem 2. 1. ). ii) DA(t)h
= Dh = const. for some h = 11m with a positive integer m.
This implies that A(t)h A(s) -h iii) A(t)h A(e ) -h is so that \ A(t) h A(s) -h - 1
I
c
H~lder ~
B for any sand t. continuous with an exponent
~. > 1 -
h ,
M(t-s) 9 .
Then the conclusions of Theorem 2. 1. are true (with D replaced by Dh in 4)). In the last statement of Theorem 2. 1. (analyticity), the analyticity of A(t)
-h
Theorem 2.5.
should be assumed. Under the assumptions of'Theorem2. 4., the conclusions of
Theorem 2. 3. are true. Remark 2. 6, Theorem 2. 1. is a special case of Theorem 2. 4. for m = 1. The assumption that DA(t)h = const. is supposed to be weaker than that DA(t) = const., but there is no general proof valid for Banach spaces X. In any case this is true for accretive operators A(t) in a Hilbert space. In other words, DA = DB implies DA ex = DB:.x if X is a Hilbert space and -A, -B ( (BO) are such that Re(Au, u)
:?
0 , Re(Bu, u) ~ 0 (u E: DA = DB)'
Furthermore, it has been proved by Lions that, when A is an operator in X = L 2( il) determined from a strongly elliptic differential operator of order 2m on a domain Sl.. with a smooth boundary, A 0(, has a domain independent of the coefficients or of boundary conditions for
D(
0 (lui'
is the norm in X'), for all
u f X' , Then there exists a linear operator -A and a(u, v) = (Au, v) for u can be shown that DA"
o ~ x, 2/3 (see r 4l). ~
,.;
Also it is very likely that the assumptions of Theorem 2.4. are satisfied when A(t) is a family of strongly elliptic differential operator, if the re26
- 25 -
T. Kato
gion
.:.1. ,
the boundary conditions and the coefficients are sufficiently regu-
lar in X as well as in t (see Remark 2. 6. ).
7. Here we shall show that y(1)(t, s) has the estimate (22). We note, once
n
for all, that there is no question about the existence of y(I)(t,s), for all funn
ctions such as yO(t, s) and H (t, s) are smooth at t = s because A (t) n n n The only question is to obtain a uniform estimate such as (22).
c-
B.
In this nO we shall omit the subscripts n and write A(t), Y(t), H(t), ... in place of A (t), Y (t), H (t), ••• (so that A(t) E B in the following). n n n By definition we have (36)
where
y(1) = H ~ y(o) = I + I + I 1 2 3'
t II (t, r) = r H(1;, s) [ A(s)e -(s-r)A(s) - A(r)e -(s-r)A(r)] ds ,
f
(37)
J:
I 2(t, r) = =
I 3(t,r) =
H(t, r) A(r)e -(s-r)A(r) ds
H(t, r)(1 - e -(t-r)A(r))
(see Lemma 1.12),
t
J [H(t,s) - H(t,r)] A(r)e-(s-r)A(r)dS r
We now need Lemma 2.7. Under the assumptions of Theorem 2. 1. we have \ A(t)e -tA(t) _ A(s)e -1:'A(s)
I~
~
(t-S)0 ,
11:1
c rr 1+\',.
27
,
- 26 -
T. Kato f:.,
Proof. From the expression (3) of ::s-1 for -Ae A(t)e-l"A(t)_ A(s)e-L'A(s) =
2~i
II (1\ +A(s))-II~ 1 + M
we have
J/'" (1\ +A(t)rl(A(t)-A(s~A +A(s))-I.~dA
Since A(t) - A(s) = (A(t)A(s) -1 -1) A(s) =\ 1 -
-7::A
and A(s)( 1\\ +A(s))rl =
, we have i) and by iii)
'\ A(t)e - L'A(t) _ A(s)e - tAts) \
J\ I~
~ C(t-st
dA
eAt
"c
IA\
(t-st
It',
Again (in the proof of the second inequality we may write A(t) = A)
I JA2e -sA ds t'
I A(e -'(A - e -
0-
A, ) I=
1_" o In the same way we can construct U(t, s) = s - lim UA (t, s) by consi\A 1->0 de ring partitions 1 of [ s, t ] . The relation U(t, sluts, r) = U(t, r) then follows immediately from U
I
A
(t, s)U
11
" (s, r) = U
11
are partitions of [s, t ], [r, s [r, t
Jobtained by joining
/l"
(t, r) by going to the limit, when L1',
J, respectively, and
4'
and
~
is the partition of
4. ".
Incidentally we note
I (U Ll
(9a)
-
U)A~l I ~
which follows from (9) by letting
T 2 M 11l' 0 B'(t), dt
JI
I11 " , --> 0,
t1, = !J
To prove the continuity of U(t, s), it is convenient to introduce the notation , U(t, s,
-(t'_1-t'_2)A'_1 -(tk-s)Ak Ii ) _- e -(t-t'_l)A, J Je J J J , , ••• ' e if
where
!J.
t , 1
L
1\
donne:
(-li p/ D2p u
Ifl~rf\
est un isomorphisme de
m H (0). SUr o
(faible) du probleme de Dirichlet pour
H
-m
(0). C'est une formulation
l'op~rateur
Ifll~
Autre exemple. Si 1'on prend butions sur
3. Dne
0
V
2:.
(-1) IPI D 2p
m
= Hm (0) ,alors V' n'est pas un espace de distri-
.
propriet~
3.1. Le triplet
du triplet
fv,
tV, H, V']
possede la
H, V') .
propri~t~
suivante:
Theoreme 3.1. Si fT est dans l'espace £(V';V') et a la propriete d 'appliquer contin~ment
i> (V;V),
•
de nor me 1/
rrll ),
alors
11
(de norme
V dans lui m~me (i. e. et sa nor me
E .B(H;H)
I frl
IlfYl/')
fT E
dans cet
espace verifie:
I rr I .$ max ((I rr 1/ ,II rr II') . NOllS ne demontrerons pas ici ce tMoreme (cf. commentaires) mais nous en demontrerons un cas particulier, du
a p. D.
Lax.
Enon9ons d'abord ce cas particulier: TMoreme 3.2. Soit
(3.1.)
11
E £(V;V)
,de norme (/
(11 u, v) =(u, 1'1" v)
Alors
IT
V~rifions
E
~ (H;H)
et
d'abord que ce
(11/
,tel que
pour tout
u, vE V.
I rrl~ II rr II th~oreme
me 3.1..
55
est un cas particulier du TMore-
- 9-
A
fI
En effet, si
E '0 (V;V)
(3.1) signifie que
fr#
rllme 3.1, avee II
nil' = IIrrll
J. L. Lions
~
,alors son adjoint (I E
rr . Done rr
prolonge
~ (V';V')
~
et
a les propri~t~s du TMo-
d'ou Ie TMorllme.
I
3.2. Donnons maintenent une
d~mostration
direete du TMoreme 3.2,
dl1e aLax. Soit
Iu I = 1.
u E V ,avee
So = lu /2 = 1. On a
Posons:
(gr~ee a (3.1)):
rr
"fI
s = (11 u, n
'11
u) = ( ('j
n+l
S
n
u,
=(
"1\
rr u,
1/
r\
u),
n=I,2, .... ,
rr n-l u)
done 2 n -
s
l~mes
Probl~mes
V . Pour
aux limites "elliptiques", nous renvoyons
aux limites en tMorie des distributions, Acta
Math. t. 94 (1955), p.13-153. Les resultats du N.5 sont des cas particuliers de Lions, Espaces intermediaires entre espaces hilbertiens et applications, Bull. Math. R. P. R. Bucarest, 2(1958), p.419-432.
61
- 15 -
J. L. Lions
Chapitre II EQUATIONS LINEAIRES DU 1er ORDRE
1. Notations. Exemples
1. 1. On
d~signe
par
bles. On suppose que et
K dense dans
K et
H deux espaces de Hilbert,
K C H ,I 'injection de
H. Si
f, g E H, (f, g)
K dans
H
s~para
~tant
continue
est leur produit scalaire dans
H et
I
(f = (f, f)I/2 Si
u, v E K, ((u, v))
est leur produit scalaire dans K et
Ilull
= ((u, u))
1/2
On considere une variable t - Ie temps - dans l'intervalle [0, TJ . Pour chaque ferm~
dans
t E (0, T)
K ,V(t)
Le couple
~tant
{V(t), HI
On supposera que suivant :
d~signons
K sur
V(t).
par
V(t) Pit)
,on se donne un sous espace vectoriel V(t) dense dans
H.
jouera Ie role du couple {V, "d~pend
HI
du Chap. 1.
mesurablement" de t ,au sens
l'op~rateur
de projection orthogonale dans
On supposera alors (1.1 )
rpour tout
I [0,
TJ
k Eo K ,la fonction
a valeurs
dans
t
-~P(t)k
est mesurable sur
K .
1. 2. Exemples. On prend
H = L 2 (n)
, n ouvert de 63
n
R
- 16 -
J. L. Lions
Muni de la norme
K est un espace de Hilbert. Llespace
V(t)
est defini par des conditions aux limites, de finis par
des relations lineaires (8. coefficients dependants de t ) entre les traces sur la frontiere
r
de 0 ( rest supposee assez regumre) de
et des derivees dlordre
~
u E Hm(O)
m-l .
Sous des hypotheses raisonnables de regularite, en t sur les coefficients, on montre alors que les
1\
P(t)
dependent mesurablement et meme
"regulierement II de t. 1. 3. Formes
a(t;u, v).
On se donne, pour chaque tinue sur
I
E;
(0, T)
,une forme sesquilineaire con-
K ,avec la propriete:
pour tout
(1.2)
t
et
u, v E K ,la fonction
~
/a(t;u, v)i
M Ilull
On designe maintenant par
t ->a(t;u, v)
Ii vii
est mesurable
M = constante, t E [0,
L2 (0, T;V(t))
l'espace des (classes de)
fonctions u telles que : 2
i)
u E L (0, T;K)
ii)
u(t) E V(t) p.p.
On constate facilement que lIon definit ainsi un sous espace vectoriel ferme de
L 2(0, T;K)
;donc
L 2(0, T;V(t))
Nous aurons besoin du
64
est un espace de Hilbert.
J
'£1
- 17 J, L. Lions
2
Lemme 1.1. Si (1. 3)
t
.
u, vEL (0, T;V(t)) ~
,la fonetion
a(t;u(t), vet))
1 L (0, T) .
est dans
Demonstration, Puisque
Ia(t;u(t), vet)) I ~ M
II v(t)/I
Ilu(t)1I
,la seule chose
a
montrer est la mesurabilite de la fonetion (1. 3). Comme
a(t;u, v) a(t;u, ,-)
est continue sur K ,on peut eerire
et de m~me,. ,a(t;u, v) a(t;u, v) Alors
= ((
fi.
etant continue sur (t)u, v)),
FUt)p(t)
t
a(t;u(t), vet))
->fat)u(t)
pour tout
zi(K;K).
V(t):
u, v E Vet),
(( tPJ(t) P(t)u, P(t)v))
On a:
~(t) E
= ((6) (t)u, v))
R(t)
E
~ (V(t);V(T)).
= (( fi (t)P(t) u, pet) v)) d'ot!
= pet) (jj(t)P(t).
= (( F (t)u(t), vet))) et il suffit de montrer que
est mesurable dans K ,done (K etant separable) que
k E K,
(( }L (t)u(t), k)) = ((
rt (t)u(t), P(t)k)) = ((u(t), Rit(t) P(t)k))
est mesurable - Done il suffit de montrer que surable done que
t -~((k1'
J1
;If
(t)P(t)k))
((k 1, pet) (J){t)P(t)k)) = ((P(t)k 1, Done il suffit de montrer que
t
t --rJt(t)P(t)k
est mesurable. Or vaut
~(t)P(t)k)).
-~(t)P(t)k 65
est me-
est mesurable, done
- 18 J. L. Lions
que
t -;) (( cf3(t)P(t)k, k 1))
= ((P(t)k,
tf) (t)k 1))
©(t)k 1
fin de eomptes, il suffit de verifier que (( (b(t)k 1, k))
que
est mesurable, done
l'est-
a(t;k 1, k)
Or vaut
est mesurable. Done en
qui est mesurable par hypotehese - e. q. f. d.
2. Le probleme. Theoreme d'existenee 2 u E L (0, T;V(t))
2.1. Nous eherehons
telle que T
T
)0 [a(t;u(t), 'fit II - (u(t), cP ItIIJ dt· 10 Ii (t), cp (tlld!+(u0' 0/ (0))
(2.1)
pour toute fonetion
cp
(2.2)
cp
telle que
2 E L (0, T;V(t)),
Dans (2.1), Remarque:,
2 cp I = ~ dt E L (0, T;H),
f est- donne dans
CP(T) = o.
(L 2(0, T;V(t)))' et u
Gr~ee au Lemme 1. 1,
o
dans H.
T
)0 a(t;u(t), 0/ (t))dt
a un sens.
2.2. Formellement, la relation (2.1) signifie que
2
(2. 3)
u E L (0, T;V(t)), du { A(t)u(t) + dt = f u.( 0)
dans (0, T)
=uo •
En fait Ie probleme 2. 1 est une formulation faible (on generalisee) du_ systeme (2.3). On preeisera eela eompletement dans Ie eas OU dant de t.
66
V(t)
= Vest indepen-
- 19 -
J. L. Lions
2.3. Dans la suite nous
d~montrerons
Ie tMoreme d'existence:
TMoreme 2. 1. On suppose que (1. 1), (1. 2) ont lieu, et qu 'n existe une constahte ~ et une constahte d.. (2.4)
AIv I 2 ?
Re a(t;v, v) +
> 0 telle que
rX.I/vi/2,
pour tout
v E V(t).
Alors il existe une fonction u solution du probleme 2.1. ~
Le probleme de
l'unicit~,
sans hypotheses
suppl~mentaires,
est
ouvert. eF't
Le
th~oreme
3.
D~monstration
3.1.
d~montr~
au N. suivant.
du tMoreme d'existence 2. L
R~duction pr~liminaire
:
par changement de u en ektu, k convenable supposer que (2.4) a lieu avec 3.2. On va utiliser une ram~tre
On
E :> 0
d~signe
destin~
par
,on peut toujours
A =0 .
m~thode
de perturbation. On introduit un pa-
a tendre vers
O.
W l'espace des (classes de) functions u telles que
2
u E L (0, T;V(t)) ,
(3.1. )
C~ ~)
u' E L 2(0, T;H),
u(T)
= O.
Muni de la nor me T
/Iu /I w ~ (~
(iU(t)// 2 + IU'(t)/2) dt) 1/2
o
crest un espace de Hilbert. Pour (3.2)
u, \
fEW
(u.
~~ I
r[
,on pose : a(t;u(t).
o
'f (tl - (u(tl. r\tl + E(u '(tl. rp '(til J dt. 67
- 20 -
J.L. Lions
La forme
u,
f->
bE (u,
cp)
est sesquilineaire continue sur W ;
verifions Ie : Lemme 3.1. On a
(3.3)
~
Re b, (u, u)
Gi jT
II u(t)/1 2dt + d T Iu'(t) /2dt + )0
o
+/ 0) u(
/2.
Demonstration
2 Re bE (u, u)"
\: 2 Re a(t;u, u)dt + 2
£
J: /
U'(td 2 dt -
-r[
(u, u') +(u', uJ dt
et Ie dernier terme vaut
puis que
o
J:
-
! I
I
u(t) / 2 dt" IU(O) 2 - /U(Tl/ 2" IU( 0
f
u(T) = O.
D'ot! (3.3) (En utilisant (2.4) avec
A= 0)
Consequence: Si ste
c.p -> L( 0 .Il existe alors une fonction umsati--
sfaisant a (2.2), (2. 3), (2.4). Nous allons alors
d~montrer
Ia
Proposition 2. 1. Hypothese du Lemme 2. 1 sur f . Il existe une fonction u telle que (2.5)
(2.6)
00 2 n u E L (0, T;L (R )) ,
~ ~.
E L 2k (Rnx(O, T))
,
1
(2.7)
(2.8)
1\ u =OT du u(x, 0)
n
-6 '0"dXi i =1
= O. 81
Cl u U·l (h.) 1
= f dans t > 0 ,
- 34 J. L. Lions
Remarque 2.1. La condition (2.8) a un sens. En effet, posons : 2K
d u ~ r· t
Alors (--r;)
=P
l.~ n E L {.l.K-W (R x(O, T))
=LP
I
n (R X (0, T))
1
OU
....!.. + ~ = 1. Donc, en s~parant les variables: p
p
'd u ~/(-J. (0 x.)
(2.9)
E
L
pi
pi n (0, T;L (R )).
1
Soit
et soit
-1 pi n I p n W ' (R) = dual de W ' (R ) .Alol's
~
ox.
est un operateur
1
lin~aire
continu de
pi n L (R)
~
w- 1, pl(Rn)
et donc (2,9) entraine :
(2.10)
De (2.7), (2.10) lion
d~duit
n )) ~ o t ~ LPI(O ' T'W-1,pl(R , •
(2.11)
Mais (2.11) implique en particulier que u est (p.p. ~gale a une fon~ ] -1 pi n ction) continue de [0, T ~ W ' (R) (on peut pr~ciser : de (2.5) et (2.11) r~sulte que u est continue de
[0, TJ a valeurs
dans un espaee
interm~diaire entre L 2(Rn) et W- 1, pl(Rn)) . Done (2.8) a un sens. Avant de
d~montrer
la Proposition 2.1
in~galit~s
a priori.
~tabilissons
les
inegalit~s
a priori: 2.2. Les
Nous multiplions (2.3) par
gim(t)
82
et sommons en
.11 vient, en
- 35 J. L.Lions
int~grant
en
t sur (0, T) :
(B f, u (t)) dt , m
(2.12)
ott
J
2
=
r r
(- fj f\
u + Au, u ) dt . m m m
0
Mais
J
= 1
(T-t)
d
dUm
(IT AUm' dt)
dt
0
f
T
=
(T-t)
(Ul~(t), u~(t)) dt +
JT
(T-t) (
o
~t; n
f
1
rt
dU tKo' (~)
Rn
0
1
---..,....J(i) 1
Sait
J
1,0
Ie premier terme et
J 1(i)
83
les autres, de sorte que
- 36 -
J. L. Lions
J
1
=J
n
+
1,0
22 . 1
1=
J (i) 1 .
Le premier vaut
J
(2.13)
rI
1 2
U'mlt)I' dt .
=-
1,0
o
~crire J 1(i)
On peut
J/
i)
sous 1a forme:
= (2k-1) fT IT-t) dt
J
Rn
o Introduisons 1a fonction
f
(2.14)
A10rs (comme
d U m~I(·t
(~)
\l
I
~ ) ~ ( ~K ISign, A ) si
dUm = 0
-f'l-
o x.1
a\m (a
si k impair
Xi~t)
2
k pair.
1a ou elle change de signe) :
1
=17
r
d Ai dUm, 1 2 L at.): (~~
Donc:
(2. 15)
J. (i) 1
2k-1
=7
Ca1cu1ons main,tenant J 2 ; posonS :
((u, v))
=
J
Rn
n
(uv+
L i =1
84
d u dv -0-)
() x.
1
x.
1
dx.
- 37 J.L. Lions
Alors
f
T
J =
2
((u'rn(t), urn(t) )) dt
°
+
fT [
d Urn
11
2k-1
dans L
2k/2 (k-1) (Rnx(O, T))faible.
1
Pas sons maintenant ~ la D~monstration
de la Proposition 2.1
~ jest une
Si
fonction scalaire, par exemple dans
r
on multiplie (2. 3) (~crite pour m= \I ) par
j
Si 1'on pose: r.p . (t) w. IJ J
il vient :
(B d lOU en
1\
um(t),
int~grant
~ (t)) = (B f, ~ (t)) ,
sur
J
T
o (u'ro, B
(m = V'
)
(0, T) puis par parties:
ce que 1'on peut encore
~crire
n
Y(t)) dt +[ ;
i
j f T
dum2k-1
0
n
(~) 1
R
88
0, T [ ),
et 1'on somme en j,
j ~ mo ~ m.
(2.25 )
{j) (]
d
~x. B 1
- 41 -
J. L. Lions et gr~ce
a (2.20),
f
T
a la limite; il vient
(2.24) on peut passer
(Wl(t), B
n
'r' (t)) dt +~
o
i
r ('0d
JT
=1
).
:i)
:
2k-1
(
'J a xi B 'i' )dx dt
Rn
0
(2.26)
r
Cette relatior a lieu puor toute
de la forme (2.25).
fj (t)
Enfait tout va bien aussi si lion prend
, (2.27)
( Lf j (
r
satisfaisant
a:
est deux fois contintment diffe'rentiable dans [0, T [ ,
~ j (0)
= 0,
/t)
born~e lorsque
t
~
T.
Admettons un istant Ie Lemme 2.2. On peut choisir une base
r
w. telle que les fonctions J
j satisfaisant
soient telles que les
B If
forment un
LP(O,T;W 1,P(R n )) nL 2(R nx(O,T)) Alors (2. 26~ qui peut
sl~crire
syst~me
a (2. 27)
dense dans
p=2k.
, :
0) telle que
(3.6)
Alors on sait
(N.2)
qu'il existe
92
u
m
satisfaisant aux conditions
- 45 -
J. L. Lions
analogues
a (3.2),
(3.3), (3.4), avec
f
m
au lieu de f.
M. Visik montre (Visik, loco cit., p. 315 et suivantes) que, lorsque m ---+
00 ,
les solutions
u
m
convergent vers la solution du probleme.
4. Cas des problemes mixtes. Si lIon considere maintenant Ie probleme mixte :
x En, t > 0 ,
(4.1)
ot
n
est un ouvert born~ de u(x, 0)
(4.2)
= uo(x)
Rn
,de frontiere assez r~guilere, avec
donn~,
et si
u(x, t) = 0
(4.3)
XEr,t>O
(r = frontiere de n),
on a des reaultats analogues. La m~thode aussi est analogue, avec la diff~ rence suivante : sur un ouvert parant" ties en prendra
(4.4)
B x
si lIon prend encore
1lop~rateur "S~_
sous la forme (2.1), il appara1t, dans les int~grations par par,des
B
n,
int~grales
de surface; pour supprimer ces integrales, on
sous la forme:
Bu =
-0/ Ll u + u - i,(T - t)
ot test une fonction r~guliere dans
1i
93
* ,telle que
- 46 -
J. L. Lions
(x) > 0
si
xE
n,
=0
si
xE
r,
xE
r.
(x)
~X) On
consid~re
alors, au lieu de (2.3); Ie
* "U
(B
,
>0
m
(t), w.) J
syst~me
*' f, w.)J
= (B
j
Nous renvoyons au-travail de Visik pour les
= 1, ...•. ,
m
d~tails.
Commentaires sur Ie Chap. III. Tous les r~sultats de ce Chapitre sont d~s
a 1. M. Visik,
Mat. Sbornik,
t.59 (101), 1962, p.289-325. On trouvera dans cet article des tre ayant pur seul but d'introduire Le N. 1 montre que Ie Green - Galerkin,
utilis~es
r~sultats
plus
g~n~raux,
a ce travail.
m~thodes
dans les
"usuelles" -
m~thode
de Faedo -
non
lin~aires
par E. Hopf,
probl~mes
Math. Nachr. 4 (1951), p.213-231 - ne conduisent pas ici tif. Les
m~thodes utilis~es
ce chapi-
dans les
~quations
a un r~sultat posi-
de Navier-Stokes et
~quations
similaires (cf. enparticulier J:L:Lions, C.R.Acad. Sc., t.252 (1961), p. 657-659) donnent des estimations
a priori sur
-8+ .
l'exemple que no us choisissons) besoin d'estimations 'J.
C) u
8
2. u
Mais on a ici (dans
a priori sur
d x. 'U t '0 x.'C)xJ. . 1
1
C'est l'objet du N. 2, qui contient les
id~es
Visik.
94
essentielles introduites par 1.1'1:.
- 47 J. L. Lions
Le N.3 donne un th~or~me d'existence et d 'unicit~ et Ie N.4 indique bri~vement
comment
~tendre
Ia
m~thode
95
aux
probl~mes
mixtes.
- 48 J. L. Lions
Chapitre IV
EQUATIONS LINEAIRES DU DEUXIEME ORDRE
l. Position du Probleme. 1.l. On considere V et H comme au Chapitre I (et au Chapitre II,
N.4); V et H sont
s~parables.
On donne une famille de formes ~ires
t
a(t;u, v)
,t
continues sur V ;on suppose que, pour tout
~
a(t;u, v)
E (0, T), sesquilin~..
u, v e V ,la fonction
est mesurable et que t E (0, T).
(1.1) On nonne
~galement une
famille
d'op~rateurs
B(t) E ,&(H, H) ,tels
que
(1.2) ment
pour tout
f,
gE H
diff~rentiable dans
(B(t)f, g)
est une fois continu-
[ 0, TJ
'UJ"
On d~signera par
t ~
,
l'espace des -(classes de) fonctions u telles
que 2
u E L (0, T;V)
(1. 3) (1. 4)
du
dt
2
E L (0, T;H)
(Pour Ie sens de (1. 4) , cf. Chap. I, N.5). Muni de Ia nor me
1/2
97
- 49 J. L.Lions
uP est un espace de Hilbert. Notation: pour u, v
vJ , on pose:
e
J. [ T
(1.5)
E(u, v),
a(t;u(t), v(t)) - (u'(t), v'(t))
Comme on Ie
v~rifie
n~aire continue sur
W.
Naturellement, si Ie
sans peine,
E(u, v)
+ ((B(t)u(t))' ,V(t))j
dt.
est une forme sesquili-
uJ. alors en particulier u est (p.p. ~ga [0, TJ--; H . On pourra donc parler de
uE
a une fonction) cvntinue de
u(O), u(T).
1. 2 Le Probleme. On cherche
u
E;
(1.6)
u(O)
= uo
(1.7)
E(u,
f)'
tel que sont
cp..T) = 0
donn~s
(1.8)
'
uJ ,satisfaisant a. donn~
Uo
r
dans V
(/(t), 'f(t)) dt
+ (U 1'
rp(O))
pour tout
ou dans Ie deuxieme membre de (1. 7),
fEiJfl
f et u l
avec : f
e
2
L (0, T;H)
•
(1. 9)
Naturellement. sans hypotheses a(t;u, v)
,Ie probleme
pr~c~dent
suppl~mentaires,
notamment sur
n'admet pas de solution.
Nous allons dans la suite donner des conditions suffisantes permettant d 'affirmer l'existence et
l'unicit~
d 'une solution du probleme
98
pr~c~dent.
- 50 J. L.Lions
1. 3. Interpretation for melle du
probl~me
Utilisant les operateurs
A(t) E
1. 2.
210 (V;VI)
(cf. Chap. I et II) et inte-
grant formellement par parties dans (1. 5), il vient : (1.10)
A(t) u(t)
+ u"(t) + (B(t) U(t))1 = f('t) ;
les conditions initiales
s~nt,
u(O)
=uo
ul(O)
=u
(1. 6)
dlabord (1,6) :
puis (1. 7)
1 Nous a.llons justifier cela au N. suivant.
2. Proprietes des solutions (eventuelles) de (1. 7).
Theor~me
2. lSi
u E West une solution du probleme 1. 2, alors
elle ales proprietes suivantes :
(2. 1)
u" E L 2(0, T;V')
(2.2)
u l (0)
=u 1
.
(Noter que (2.1), Joint au fait que continue dans
[0, TJ ----? VI
ulE L2(O, T;H)
,implique que nlest
,de sorte que (2,2) a un sens).
Demonstration. On peut prendre dans (1,7)
~(t) = y(t)v
(2, 3)
Alors (1. 7) se reduit
a
+ ((B(t)u(t))'
,
YE3)(JO,T[).
T
): [a(t;u(t), v)
=
rT
J0
v~ cP (,)dt - J -'-
(f(t), v) cp.t)dt
99
o
(u I(t), v)
~I(t)dt =
• 51 "
J. L. Lions
Jo,
d'ol'l, au sens des distributions sur a(t;u(t), v)
+ ((B(t)U(t))'
, v)
d2
+ -2 dt
T[ :
(u(t), v)
= (f(t),
v),
pour tout
ve V,
ou encore A(t)u(t)
(2.4)
+ (B(t)U{t))' + ull(t) = f(t)
(au sens des distributions sur JO, T[ - ) VI).
A(t) u E L 2 (0, T;VI)
Mais on sait (cf. Chap. II) que (B(t)U(t))' E L 2(0, T;H)
;par ailleurs
de sorte que (2,4) implique (2, 1).
Mais alors, (cf. Chap. I, N.5, 5. 3), si
T \
.
f uJ , Iii!
T
(Ull{t), f(t)) dt = (u'(T), f(T)) - (ul(O),
f (0)) -J
o
(ul(t), f'(t)) dt ; 0
si done 1'on prend Ie produit scalaire de (2,4) avec
CP(T) = 0
I: [
1
f
(t),
r
E
u),
et
alors :
a(t;u(t).
cP (t)) + ((B(t) u (t))'. fIt)) - (u 'ttl.
f'(t))] dt - (u '(0).
flO)) •
• \: (f(t). f(t)) dt dlol'l
E(u.
~ ).
I:
(f(t). 'fit)) dt
+ (u'(O). 140))
ce qui, en comparant avec (1. 7) donne stration du
ul{O)
theor~me.
100
=u1
et acheve la demon-
- 52 -
J. L. Lions
3. TMoreme d'existence. 3.1. TMoreme 3,1. On fait les hypotheses suivantes : t
~(t;u, v)
est une rois continCment
u, v E V ; a(t;u, v)
(3.1)
= a(t;v, u) et
diff~relltiable dans
i1 existe
A et d. ~o
vEV
[
(3.2)
B(t)
[0, TJ '
tels que
;
est hermitien dans H ,pour tout t, et t --?(B(t)f, g) est
une flois contintment diff~rentiable dans
[0, .~
pour tout f, g E H.
Alors, 11 existe une solution u et une seule du Probleme 1. 2ayant. en autre les
propri~t~s
suivantes :
(3.3)
00 uEL (O,T;V)
(3,4)
u' E L 00(0, T;H)
(et, naturellement, les 3.2.
propri~t~s donn~es
D~monstration
au TMoreme 3.1).
de l'existence. kt
Notens qu'un changement de u en e u change a(t;u, v) en a(t;u,v)
.
2
et
+ k(B(t)u, v) + k (u, v)
B(t)
en
B(t) + 2kI
(I = identit~ dans H ). On peut donctoujours se ramener au cas on dans (3.1) f),=0. Soit approch~e
W 1 ••...
wm •.•. une base de V . On dMinit
d'ordre m ,par m u (t) = \ ' m L-
(3.5)
i=1
on les
um(t) , solution
g. (t) 1m
sont
d~finis
g.
1m
(t)w;
par Ie systeme
101
diff~rential (lin~aire)
.; 53 .;
J. L. Lions
(3.6)
: (f(t) , w.)
j : 1, 2 ••••.. , m
J
avec les conditions initiales
g. (0):
(3.7)
1m
ou les ~. fl.. 1m '''' 1m
eX.1m
sont choisis de facon que (
m
I"
,..j.
. IJ\ 1m
(3.8)
w. - ) u 1 0
dans V lorsque
Etablissons maintenant des majorations
m -~
a priori pour
00,
u (t).
m
Nous posons : (3.9)
u' (t)/ 2 t a(t;u (t), u (t)), m m m
(3.1'0)
Multiplions (3.6) par (u ff z
m
(t), u'
m
gi. (t)
Jm
(t))ta(t;u (t),u' (t))t((B(t)u (t))',u' (t)): m m m m
(f(t), u' (t)). m
Prenons la relation complexe (3.11)
et sommons en j ;nous obtenons :
d tit
a(t;u, v)
conjugu~e
= a'(t;u, v)
102
et ajoutons ; si nous posons :
- 54 -
J. L. Lions
nous obtenons :
- a'(t;u (t), u (t)) m m Utilisant (3.9) et
- 'fm(tl·
int~grant,
r l[
= 2Re (f(t), u',m (t)).
il vient :
t
'm
a '(15', um(o I um("'II - 2 Re( (B(
de
a une fonct!0n, not~e u.,., continue de
,V 1/ 2 H1/ 2 ,cf. commentaires), de
~,TJ->VI
deriv~e u~
[ai, TJ-7> H continue
(etnteme_>H 1/ 2 (V,)1/2).
En autre: pour tout
(4.1) et (4.2)
t _ ) ((~(t), v))
(4.3)
dt
d
~(t) E H
est continue dans
[0, TJ
pour tout
et (4.4)
dl.W(t) t ----7 (-d-t-
pour tout v E V ;
,g)
est continue dans
rLO, T,1J
pour tout g ~ H;
enfin quels que soient v E V, g E H, (4.5)
{ uo' u l '
f} --,>
est continue de (ou
{ ((
l'application
~(t), v)), (d~
l.lJr.(t), g)
J
VxHxL 2(O, T;H)-'> C(O, T)xC(O, T)
C(O, T) = fonctions continues dans
[0, TJ
gence uniforme).
107
,topologie de la conyer,.
.. 59 " J. L, Lions
4.2. -
D~monstration
Pour
uE
VI.
de (4.1)!!. (4.3).
v E V ,nous poserons :
d2
~
"t (u(t), v) = a(t;u(t), v) + ((B(t)u(t) )', v) + - 2 (u(t), v)
(4.6)
dt
ce qui est une distribution sur ]0, T [ • En prenant toutes les au sens des distriQutions, (4.6) a un sens pour On a, si u est solution du (4.7)
mU(t), v)
= (f(t), v)
m
(4.7bis)
(u.. (t), v)
Soit maintena.nt
Probl~me
d~riv~es
u E L2(O, T;V).
1.2:
pour tout
v E V, ou
= (f(t), v).
t" un nombre fix~,
0
< 1:'< T .
Designons par: Y't1
(,
0(1/)
la fonction nulle pour t < 1:"" ,= 1 pour t > L;
= masse + 1 au point t';
~(~)
=
:t 1t)
Alors
fll(~U~(t), v) = (Yt'f(t), v) + (~('t), v) S(I~t 1 + [(B('t")~('C')'V)+(U,\..(t;.)'V)J('t)
(4.8)
D1un autre
cot~,
revenant ala
d~monstration
du
TMor~me
voit, en utilisant (3.13), que lion peut supposer que la suite propri~t~s
(4.9)
(3.14)
v~rifie
{
en autre
uy
('t')
-> X. dans
v("d~ X"
U
et
108
V faible
dans H faible
3.1, on
u y ayantles
- 60 •
J. L. Lions
Mais
em, (Y'l;"ur(t),w j ) = (Y'trf(t), wj ) + (uy('t'), Wj)~(tt I
+ et en passant
0
B (1;')Uy ('I:), wj ) + (u~ (1"), Wj0
a la limite
selon
Y ,
SIr)
on obtient (en utilisant (4.9) ):
flQ,( Y'Cu(t), wj ) = (Y,/:,f(t), wj ) + (}.o' Wj)$;'t) + + [(B(t),\, wj ) + CXL' wj )
Srt)
Cette realtion a alors lieu pour toute eombinaison
lin~aire
finie des
w. et done J (4.11)
Evidemmentf{fb (Yt"u(t), v) =~ (Y't'~(t), v) pliquent
et eomme
Xo E
~(t') =Ao' u!,.(~) =:Xi V, Xd.,E H, on a (4.1) et (4.2). En 0utre,
I ,\(tI1l 2 +lu;" (tl
(4.121 4.3.
D~monstration
Soit
'Ii,.,
Dlapres (4.12),
et done (4.8) (4.11) im-
r"
's ( liuo 112+
h 12 +
(4.10) donne:
n
f((J'1 2d{J1esp(,S TI·
o
de (4.2) (4.4),
,telleqUet"I'f\-,)~' O<J./i(t)f(t)
Par
(J 2 V((t) ~ (t) dans L (0, T;V) fort,
-+
par parties, llexpression
au lieu de
- (B
~O)Uo'
~
r:
((B~(t)U~(t))I,
r
(t))dt
I
)vaut : T
f (0)) -
(BF(t)u?(t), '('(t)) dt =
)
o
• - IB
~IO)Uo' 'f (0)) -
r
Iu Pit),
B~lt) ~'It)) dt
o
et eeci converge vers
- (B(O)u o '
~ (0)) -
r
T
(w(t), B(t)
rI(t)) dt
(utiliser
o (5.5)).
DIOll (5.18). 2)
D~monstration
Chaque
de (5.11).
u ~ satisfait
a
A~(t)u\3'(t) +..i... (B ~(t)U~(t)) + L u ~(t) = fit) dt dt2 On a done (5.11) si lIon
v~rifie
que
(5.20)
115
dans
JO,
Tl
.. 67 ... J. L. Lions
d~
(5.21)
(B ~ (t)u ~(t))
~
d~
(B(t)u(t))
dans
L 2(0, T;V') faible,
Pour (5.20)·onnoteque, si 'fEL2(O,T;V) ,ona:
et Ie
r~sultat
suit comme dans la
d~monstration
de (5. 17).
Pour (5.21) ,on note que
et on en deduit (5.24) en utilisant (5.5). 3)
D~monstration
de (5.12), (5.13), (5.14), (5.15).
On introduit maintenant (cf. N.4) :
1YYl~ (u(t), v) = a ~(t;U(t), v) + d~ (B~(t)U(t), v) + d: (u(t), v) "l dt
(5. 22)
ce qui a un sens pour (5. 23)
u E L2(0, T;V)
et on note (cf. (4.8)) que:
~(Y~Ur(t), v) = (~f(t), v) + (~(O"), v)~;+ (B'(C')U!(V')+ U{(Ci), v) ~0'
~ fix~ dans JO, T[ .D'apres (5.16), on peut extraire une suite de ~ , que nous d~signerons encore par ~ pour simplifier, Supposons
telle que
(5.24)
?
uf (~)
d
~
dans V faible,
---,) ~o
ill u~ (G';)
....
X1.
dans H faible
Admettons pour un instant Ie
116
- 68 -
J. L. Lions ~
~ ~ co, ffr!.~Ya-u ~(t), v) ...., 11(, (Yo-u(t), v) (distribution sur J 0, T[ ), .uniformement pour (j ~ [0, TJ.
Lemme 5.1 Lorsque dans $'(0, T)
D'apres (5.23), il resulte du Lemme 5,1 que (5,25 ) converge vers (5.26) dans
9:/(0, T)
,uniformement pour (f
Eo
[0, TJ
.
Donc: (u! (/J'), v)
->
?
(B? (())u (a') + u ~
(u. (OJ, v)
uniformement en (J, v E V,
,! (G'"), v) --? (B( O")u (~) +u' ~
~
~
((1"), v) unifor-
mement en CS, ce qui montre (5.13) (5.15). Par ailleurs, d'apres (5.24) l'expression (5.25) converge (pour 0" = 0",,) vers
d'ou en comparant avec l'expression (5.26), u,((j'o)= Donc
u!
(~) ~
u.(Cfo)
dans V faible,
Xo' u~( ((B(t)Yo-U(t))I,v)
(5. 29)
2 -d2 (Y(j u ~ (t), v)
~
dt
r
D~monstration
Soit
2 -d2 dt
au meme sens que dans(5.27);
'"
(~u(t),v) au me me sens que dans (5. 27).
de (5.27).
e; 1) (0, T) . n faut montrer que
)Ta It;Y~ u~ It), YIt)v)dt ... fT alt;Y~ ult), 'l'lt)v)dt o en
uniform~ment
0 (j •
Le premier membre vaut
Jr
T
((u
~(t), Y(j ~ P(t) ~ (t)v)) dt =
o
+
r
lIu
~t), v9 ~)'QiIt)v)) dt.
o
T
La derniere expression converge vers
J
o
uniform~ment
en 0- pour que, Iorsque (J varie,
((U(t),~ (t)Ya- If (t)v))dt
II. (t)Y(j r (t)v
re dans un compact de L2(O, T;V). On a done Ie
f
r~sultat
T Ilu PIt). I
si
R~lt) • fi It)) Y(J" f It)v)) dt
o
uniform~ment en
(j. Or
118
-7
0
demeu-
• 70 •
J. L. Lions
d'oll Ie
r~sultat.
Commentaires sur Ie Chapitre IV. Les n~s
r~sultats
de ce Chapitre complNent queIque peu les
dans Lions, Equations
diff~rentielles op~rationelles,
ction Jaune, t.1l1, 1961. Les versit~
de
Montr~al,
r~sultats.
r~sultats
don-
Springer, Colle-
des N. 3 et 40nt N~
a l'Uni-
donn~s
Ecole d 'Et~, Juillet 1962.
II serait int~ressant d'affaiblir les hypotheses de diffe'rentiabilit~ faites sur
a(t;u, v) ;a(t;u, v)
continue et {-a(t;u, v)
mesurable et bor
n~e suffit (m~me d~monstration que celle du texte !); mais il serait par ex. int~ressant
a(t;u, v)
de savoir s'il est suffisant, pour I 'existence et
v~rifie
Le
famil~e
r~sultat
que
une condition de Lipschitz en t .
Nous n 'avons que des V par une
l'unicit~,
r~sultats
d'espace
de
V(t)
d~pendance
fragmentaires si nous rempla90ns
comme au Chap. II.
continue en .les "coefficients"
donn~
au
N.5 peut etre compl~t~ par un r~sultat de dependance continue en les espaces, mais cela n'est pas donne ici. Les
r~sultats
des N.4 et 5 apportent
des reponses (peut ~tre insuffisantes .... ) a des questions posees par M. M. L. Amerio et S. Zaidman. On peut donner une
d~monstration
de l'existence d'une solution (ef.
Th. 3.1.) par un procede de perturbation singuliere en
consid~rant
Ie pro-
bleme comme limite de problemes "elliptiques" (dans un certain sens ... ) (pous avons utilise un proc~de de ce genre pour Ia demonstration du tMoreme
d 'existence dans Ie cas des equations differentielles operationelles du 1er ordre). En voici Ie principe. On se place sur la demidroite (0, +00); a(t;u, v), B(t)
sont supposes
donn~s
sur (0,00) avec:
119
J. L. Lions
/a(t;u, V)/
+ /a'(t;u, V)I ~ M Ilu/l'
+ IB'(t)1 ~
IB(tli
II vII '
t
~ 0,
t ~O,
Cl ,
a(t;v, v) ~ 0( 1/ v liZ,
0( > 0,
t ~ O.
Notons que, dans ces conditions:
(1)
roo
2 Re
((B(t)
~ (t))',
r
e-2 t '/' 'It)) dt "C2 JOO
o
(/1
I I),
ou
'1"(2)dt
0
" (on peut meme remplacer dans Ie 2~me memi;Jre la norme me
{PerttIe ~rt
(> J
Alors on ~
II II par la nor-
quelconque, C2' ind~pendant de (
y de fa 10n que
(Xi> 0, v E V, t
~
0
(2)
On (3)
d~signe
e-(tu~ L: (V), Pour
(4)
par
u, v E
~ (u, v) =
+
+
~
roo
l
W( OO
Wr
2
1'espace des u avec
(L+ (X)
=L
e-(t'-1.' EL:{V), e-(tu " E L! (H),u(O)
2
(O,oo;X)):
= O.
,on pose :
a{t;u{t),. e
-2(t
v'(t)) dt
(00
+ t)
o
((e
-!t
u'{t), e
_p v'(t) ))clt +
0
((B{t)u{t))', e -2(t V '(t)dt _ fOO (u', (e -2(t v')') dt
0
0
J(00 (u", (e-
E
211' t Q
v')') dt,
o
120
t
> O.
+
- 72 ..
J. L. Lions
On v~rifie que, pour
E. ~ 1/4y , on a :
Re~(V,V);~
r
(5)
o
(11:l'vr +(tv"12) dt,
l
+
J00 dJe-rtvI12+le·rtv'12)dt+~lv'(o)12+
o
Alors, il existe
(6)
1ft,(Uf ,V)
uEE Wy unique, tel que
Jroo (e -rt f,e .alit V' )dt+(u 1,V'(O)),pourtoutvEW(,
=
o u1
donn~
dans H, f
donn~
avec
e
·at
2 f E Lt(H).
E.. ---+ 0, u c. converge au sens
On montre ensuite que lorsque suivant : • ., t
e
\I
u
l.
• "t
e , u'
a
ou u satisfait
!
~
~
e - (tu
dans
L 2 (V) faible, T
- t tdans 2 L t (H) faible,
eu'
(7)
"J(00 0
pour tout
r
E
On
r f) dt + (u1, frO))
2 t
W( et
u(O)
(8)
(t(t), e -
v~rifie
=0 comme dans notre livre, Chap. VIII, p.152, 153, que la
121
~
73 " J. L. Lions
restriction de u
a l'intervaUe
(OJ T) est solution du
probl~me
1.2 avec
u(O) = O. On a donc obtenu 1'existence d'l,me solution comme limite des problemea" elliptiques II (6).
122
CENTRO INTERNAZIONALE MATEMATICO ESTIVO ( C. I. M. E. )
L.
N IRE N B ERG
EQUAZIONI DIFFERENZIALI ORDINARIE NEGLI SPAZI DI BANACH
ROMA - Istituto Matematico dell'Universita
123
EQUAZIONI DIFFERENZIALl ORDINARIE NEGLI SPAZI DI BANACH di L. NIRENBERG
Capitolo I INTRODU ZIONE
1..1. Ci proponiamo di descrivere i risultati di un recente lavoro in collaborazione con Agmon
[2] . Questo lavoro riguarda 10 studio delle equazio-
ni della forma (1.l)
Lu
1 du
= T ill - Au = f
e in particolare il comportamento delle soluzioni quando
t -. + 00. Le fun-
zioni assumono i lora valori in uno spazio di Banach.Noi nontratteremoilproblema dei valori iniziali: per la classe di equazioni considerata questa problema non
~
ben posto. Infatti noi tratteremo equazioni che provengono da
equazioni differenziali a derivate parziali in un cilindro che ha l'asse t come generatrice, per esempio equazioni ellittiche. (L'operatore A rappresenta un operatore differenziale a derivate parziaU nelle altre variabili). Quindi noi consideremo proprieta delle soluzioni, non l'esistenza di esse, Parecchie delle questioni qui considerate sono state suggerite da ricerche dovute a Lax
[8], [9] ,
[10J .
In questa capitolo noi descriveremo i problemi e mostl'eremo co-
125
- 2-
L. Nirenberg
me Ie condizioni richieste siano verificate sia per equazioni ellittiche che per altre piu generali. Dopo di cia noi ci limiteremo principalmente alle equazioni astratte (1. 1) con pochi ulteriori riferimenti alle equazioni differenziali alle derivate parzialL Noi considereremo i seguenti problemi: (i) Sviluppi asintotici per grandi valori di t delle soluzioni di Lu = 0
(1.. 2)
t >0
come somma delle "soluzioni esponenziali ". Completezza di queste soluzioni esponenziali. (ii) Regolarita delle soluzioni di
Lu
= f.
(iii) Unicita del problema di Cauchy includendo il caso del problema di Cauchy all'infinito cioe : la soluzione nulla e la sola soluzione che tende a zero rapidamente all'infinito. Considereremo qui piu generalmente funzioni u che soddisfano disuguaglianze del tipo
0. Una tale soluzione valori
Ao
tali che
e
O(e
Im A0 >0.
-at
)
dove
a
= min 1m A\
fra gli autoo Si puo arrche dimostrare la tendenza a zero
in modo esponenziale delle soluzioni di quadrato integrabile di alcuni sistemi con coefficienti variabili:
1
du - A(t)u dt
supposto che la mat rice dei coefficienti a una matrice limite
A quando
t -+
=0 J A(t)
0()
•
tenda con sufficiente rapidita
Allora noi scriveremo 1'equazio:.
ne sotto la forma 1
0,
du - Au dt
= (A(t) - A)u
piu generalmente, sotto la forma(1. 3) ,dove
I?(t)
tende a zero oppor-
tunamente pet' t .. 00 • Senza alcuna condizione suI modo di tendere a z~rodi A(t) - A,
0
di
N,
(1. 22)
Poiche sostituendo Dt
nell1operatore con e i
8nt , dove
f} e un numero
piccolo, Poperatore rimane ellittico, otteniamo la maggiorazione (1. 22) per arg
A=.±. & . Allora
segue la maggiorazione desiderata (1. 11) per RS(
e analitica
nel settore (1.10). Siccome RS( AJ
in
A nel settore (1
1
A)
10), se-
gue facilmente la (1.12). Inoltre segue dalla teoriadelle equazioni ellittiche che R(),.)
e
meromorfa nell1intero piano. Consideriamo altri due semplici esempi connessi, i quali sebbene riguardano problemi non ellittici possono essere trattati per mezzo del teorema 1.1 applicato ad un operatore opportunamente modificato, Esempio 3. - Di nuovo nel cilindro, consideriamo I1operatore D u - A(x, D )u t x
dove A
e un operatore di ordine
2m, insieme con, per semplicita, Ie con-
dizioni al contorno di Dirichlet (1. 15). Assumiamo inoltre che, se AI(X, Dx)
e la parte principale di· A,
in ogni punto x, si abbia
(1. 24)
't_AI(X,~)
= 0,
per t:
t=
reali solo se
,~
0,
~
= O.
2m Allora in particolare D + A(x, D ) sono operatori ellittici. In t
x
-
134
- 11 -
L. Nirenberg
conseguenza) se 10 spazio di Banach e L2 ,troviamo di nuovo che la risolvente
R(A)
ratori
Questo si dimostra corne prima appHcando il teorema 1. 1 agli ope2m D + A(x,D ). t
soddisfa la (1.11) e la (1. 12) nel doppio settore (1.19).
x
-
Esempio 4. - Nel solito cilindro consideriamo l'operatore
D2 u _ ~ 2 u t
x
il quale agisca sulle funzioni che si annullaho con Ie derivate prime sulla superficie laterale del cilindro. Scrivendo l'operatore corne un sistema del primo ordine per due funzioni e usando il fatto che
D; +b.
~
e ellittico) cos)
che il Teorema 1.1 pub essere applicato,noi troviamo di nuovo che la (1. 11) e la (1. 12) sono soddisfatte per
RS( A), cioe per R~) ristretto ai vettori
(0, f).
l. 6. Noi faremo spesso uso di condizioni corne (1. 11), (1. 12); corne abbiamo
visto queste possono non valere in pratica per
R(),)
cioe per RCA) ristretta ad un opportuno sottospazio di
rna solo per Y
RS(A))
. Molti dei risul-
tati possono essere estesi ai casi nei quali Ie (1. 11). (1.12) valgono soltanto per
RS(
A)
,:r;na per semplicita noinonprenderemo in considerazione que-
sto raffinamento (cf.
[2)).
Osserviamo che il problema (iv) sulla tendenza a zero di tipo esponenziale pub essere usato per dimostrare che 10 spazio delle soluzioni di quadrato sommabile di alcuni problemi al contorno ellittici omogenei)i cui coefficienti tendono rapidamente a valori limiti quando
t -+ + 00
I
ha dimen-
sione finita. Si potrebbe sperare che questo sia il caso generale per un problema ellittico uniforme rna e possibile dare un semplice contro esempio per mezzo di un esempio di Plis [12}. PHs ha costruito un operatore ellit-
135
- 12 -
L. Nirenberg
tieo line are ne non banale
T
con coefficienti principali reali per il quale vi e una soluziov
con supporto nella sfera unitaria. Sia ora
sopra la sfera e sia
L
un operatore eHittico in
dici (con periodo 271') nella direzione che
L
tero
j
r
r
il cilindro
con coefficienti perio-
xn+1 della generatrice in modo tale
= T nella sfera unit aria col centro nell'origine. AHora per ogni inponiamo
Vj
= v(X1 , .... , xn' xn+1 -21Tj) dove vela soluzione
costruita da Plis. Le soluzioni
v.
J
hanno dati di Cauchy nulli e sono linear-
mente indipendenti. Da ora in poi. noi ci limiteremo principalmente aHa teoria astratta rna converra tener presente questi esempi. Noi daremo dimostrazioni quasi complete perche Ie tecniche usate, che forse hanno interesse maggiore che i risultati stessi, non sono molto complicate - sebbene qualche volta un poco artificiose. Noi faremo costante uso della trasformata di Fourier, della teoria; elementare delle funzioni di variabile complessa, in particolare del teorema di
Phragm~n-LindelOf
e del teorema di Paley- Wiener.
136
- 13 -
L. Nirenberg Capitolo 2 Sviluppi in serie di soluzioni esponenziali.
2.1. In questo capitolo noi considereremo alcuni risultati abbastanza semplici per llequazione
(2; 1)
Lu
0 Tt - A)u = 0
1
= (i
>0
t
i quali illustreranno l'uso della trasformata di Fourier e della teoria delle funzioni di variabile cC'mplessa. Noi supporremo che bile nell'intervallo
0 < t < 00
•
Se noi poniamo
sideriamo la trasformata di Fourier di
u(t)
I u(t) I
sia integra-
= 0 per t < 0 e con-
u
allora troveremo, prendendo la trasformata di Fourier in (2. 1) \
(2.2)
1\
(1\ -A) u (A)
1
= '1m.;;;- u (0) Iv2Tr
o
A
(2.3)
t
A
per i quali
1
A
V 2Tr
< 0 la funzione u( A)
litica nel semipiano
e regolare reale A.
1
= • ,In'::
per quei valori reali di nulla per
\
u (J\)
1m
A< 0
"), R ( ,/\ ) u (0)
R( A)
esiste. Poiche
u(t)
si an-
pub essere estesa in una funzione ana-
la quale
e continua
in 1m A < 0 . Se
R('\)
in qualche regione del semipiano superiore la quale tocchi l'ass, (come (1.10)) allora la (~. 3) pub essere usata per estendere ana-
137
- 14-
L. Nirenberg
~(A)
liticamente
in questa regione.
Noi cominciamo con un semplice risultato per il problema di Cauchy finito, essenzialmente dovuto a Lyubi~
(11]
. Per uniformita noi formuliamo questo risultato assumendo proviamo che
t r:J...
n risultato e molto preciso in quanta sariamente per
~
il quale vada all'infi-
~
nito e rimanga in un angolo chiuso nel semipiano aperto
~
u(T) = 0 e
u(t)
non si annulla neces,..
come possiamo vedere nell'esempio 1 della sezio-
ne 1 quando cj.. e reale e positivo. Dim. : Estendendo
u(t)
col valore ze-
ro per t fuori dall'intervallo
0:;; t S T
rier, come prima, troviamo che
;?().. ) e una funzione vettoriale intera che
soddisfa la (2.2). Poiche sulla curva ~
S}lK .
D'altra parte sull'asse reale
e prendendo la trasformata di Fou ..
la (2.3) e verificata, si ha
I G(.~ ) I e limitata.
Per il teorema di 'Phragmen-Lindelof concludiamo che
nell'intero semipiano superiore. Segue dal teorema di Paley-Wiener che u(t)=O
per
t>o{.
2.2. Consideriamo ora alcuni risultati connessi con i problemi (i), (ii), (iii). Formuliamo Ie seguenti ipotesi
138
- 15 L. Nirenberg
(a)
e regolare nelle
R(~.l
due regioni angolari
O$arg(,A-N)$
(2.4)
e,
con qualche costante positiva
0.::: 1T -arg().+N).::: N per
1T e 0 tutte Ie condizioni. f teorema sono soddisfatte; infatti non vi sono soluzioni esponenziali, e di pit
139
,1
- 16 L. Nirenberg
per t < eX. la soluzione non
e necessariamente analitica e la
(2.6) non vale.
La dimostrazione del teorema segue la via diretta. La relazione (2.3) serve ad estendere
in una funzione analitica nella regione an-
(t( A)
golare (2.4). Noi possiamo scrivere
ult)
~ 2~i
[
e iAt RIA)uIO)d).
+
2~ i
r
e iAt RIA)u(O)d)
c
c +_1_ (
1{2ff)
eiAt U(.A)dA
-c
1 primi due integrali possono essere calcolati lungo i lati obbliqui degli angoli definiti da (2.4) (questo puo essere giustificato facilmente con l'uso di un argomento che fa uso del teorema di Phragmen-Lindelof); questi integrali convengono assolutamente per t nella regione (2.5) (per esempio PinteN+e i8x grale ) N ei,}· t R(A ) u (0) d A converge assolutamente per t nel semipiano
0
< arg(t- 0( ) + 8 to)
1/21(>
e-
fJt
se (}
< rr/2
(2.7) \u(t)1 ~ lu(o)1 e
JAo
- f!,tlogt
se (} = 'if/2
Noi daremo un'idea della dimostrazi{)ne nel caso (} = ~ • Sia -
se Consideriamo la striscia M + C( < 0 < T + ex. + )0
--
U. J"o
ttl
)
0
•
nella quale la
(2.6 1) certamente vale. Possiamo anche asserire che sullato destro della striscia
I u(t) I .:: costante I u(T) I eN 11:'1 considerando T al posta delliorigine. Siamo aHora in condizione di applicare il "teorema delle tre rette" (in una forma opportuna) alIa funzione analitica
u(t)
in una striscia ,.
e la disuguaglianza
(2.8) ne risulta.
2.4. Prendiamo ora in considerazione la questione della completezza delle soluzioni esponenziali. Noi supporremo che Ie precedenti ipotesi (a), (b) valgano in modo che,per il teorema 2. 2)ogni soluzione u di positiv~
Lu = 0
sulliasse
possiede uno sviluppo asintotico come somma di soluzioni esponenzia-
li r.u .(t) . Noi diremo che Ie soluzioni esponenziali sommabil~ sono comJ '
plete fra Ie soluzioni integrabili per
t
~
a
se si verifica quanto segue: sia
u una soluzi~ne delliequazione Lu = 0 integrabile per
e [. u .(t) J sia il suo sviluppo asintotico; assegnate due costanti positive E., C esiste una combinazione lineare finita
If (t)
Uj(t + costante) tale che
142
delle
uj(t)
t >0
e delle lora traslate
- 19 -
L. Nirenberg
Iu(t) - 'I' (t) I ~ E. e -Ct
(2.9)
se
> a.
t
Per provare la completezza noi faremo uso della nozione di "Ordine ":
R( A) ~ di ordine finito
w~ 0
1m,,{ ~ 0
per
se per ogni
C. > 0 esiste una successione di curve di Jordan differenziabili nute nel semipiano
In
conte-
J
1m)... > 0 eccettuati i punti terminali che appa:dengono
all'asse reale da parte opposta rispetto all'origine (con la distanza di dall'origine tendenie all'infinito),tali che
i) R(A)
esiste su
In
In
e soddi-
sfa la disuguaglianza /R(A) /
ii) w
~
~e
IJ.. ItJ+E
il pili piccolo numero non negativo soddisfacente questa proprieta .
Per un operatore ellittico in un cilindro come nell'esempio 2 Agmon [ 1] ha dimostrato che la risolvente corrispondente a tale operatore to forma di sistema del primo ordine ha "ordine"
~
n ; n
A sot-
essendo la di-
mensione della base del cilindro. Anche per l'operatore dell'esempio 3 egli ha dim'ostrato che
n
w ~
a
eche
l' ...... , ,
~
k
uscenti da un punta dell'asse reale e d'altra parte appartenenti al semipiano 1m.A. > 0 tali che ognuna delle )
.
k + 1 regioni nelle quali il semipiano
visa da questi archi sia contenuta in un angolo con apertura R( A)
0 ,si abbia Di pili
0
quando
143
e di-
I
gran-
- 20 -
L. Nirenberg
Teorema 2.3. : ~ R().)
soddisfa Ie condizioni (a), (b), (c) allora Ie so-
luzioni esponenziali integrabili sono complete nel senso precisato sopra, sult ~ (j. + ~+
l'intervallo
8
8>0 •
per ogni
Come dimostra l'esempio 1 Ie soluzioni esponenziali possono non essere complete sull'intero interv111 ne di
Lu = 0
t > 0 . Dim. : (1) Sia u una soluzio-
sommabile sull'asse positivo e
po asintotico. Dapprima dimostriamo che dato zione lineare finita
u( ~ )
cioe
'f (t)
~ppartiene
delle
uj
L Uj e >0
ne sia il suo svilupesiste una combina-
e delle lora traslate tale che
alIa chiusura della varieta gene rata dalle
Uj
e dal-
Ie loro traslate. Per provare cia e sufficiente most rare che se htl e un fun,. zionale lineare continuo definito su
Y il quale sia zero su tutte Ie - u/t)
-It
per tutti i valori di t ,aHora h (u( ~ ))= 0 . Questo si dimostra di nuovo con
~;
l'ausilio della trasformata di Fourier piano poli di
u()...), data dalla (2. 3) nel semi-
Im.A. ~ 0 ,e meromorfa .nell'intero piano con poli in ). j ,gli stessi '\
R( J\). La funzione
-III
...
h (u (A))
e allora una funzione scalare ana-
litica nelPintero piano eccettuato' al pili nei poli ti delle potenze negative di relativo al punto J.... j da
(.\ -
Aj
. Comunque i coefficien-
.A j) nellosviluppo di Laurent di ~().;)
sono vettori che appartengono alla varieta' gene rata
u. e dalle sue traslate valutate n~l punto t = 0 (cfr. [4] capitolo VII). J 'l' Di conseguenza h *' si annulla su questi coefficienti e h*(u (,..))
e una funzione intera. La funzione h* (~(.A)) e limitata nel semipiano tale e
\G(.A) I • Di pili
Im.A:: 0 percM
noi possiamo applicare il teorema di Phragm~n-Lin
delCif in ciascuno degli angoli nei quali Ie curve superiore, e possiamo coricludere che
h"'(il(A))
144
tj
dividono il semipiano
= 0(efo 1mA ) per Im.A>O.
- 21 L. Nirenberg
Poiche
It
1\
h (u ().. ))
e la trasformata di Fourier di
°
h*'(u (t)) =
rema di Paley-Wiener che
per
u(t)
segue usando il teo-
t~ ~.
(2) Per completare la dimostrazione del teorema supponiamo 1m 1. > C per J
j >m
e consideriamo la funzione m
v(t) = u(t) -
L
1
u. (t). J
Ovviamente 10 sviluppo asintotico di il risultato ottenuto in II) alIa funzione
v(t)
esiste una combinazione ]neare finita I{' (t) te,
j > m , tale che
alla funziorie
v(t) -
I v( ~) \fJ (t)
- r..f
(~)
v(t)
t=.
u. (t) . Applicando 1>'111 J troviamo che, dato un 6' > 0,
delle
e
u.
J
e delle lora trasla-
I .5. e..' . Se noi ora applichiamo la
(2.6)
troviamo; e
-Ct pert> - ,~+O(+~.
Combinando insieme questa disuguaglianza con la precedente otteniamo il risultato desiderato. Q.E.D. Come illustrazione dell'uso del teorema 2.3 consideriamo }foperatore differenziale parabolico
nel solito cilindro con base in uno spazio n- dimensionale, applicato aHe funzioni che si an nulla no sUlla superficie laterale. Per il risultato di Agmon precedentemente citato la risolvente corrispondente all'operatore ha "ord> ne".5. n/2. D'altra parte per ogni numero complesso non sia pur.amente immaginario 1'operatore
145
a
di modulo unoche
L. Nirenberg
soddisfa la condizione dell'esempio 3 e ne segue che su ogni raggio
'iT '-231T 8 f"2
la risolvente
de e soddisfa Ie condizioni di
\. R( A) eSlste per
(c)
con
~
=0
Quindi Ie soluzioni esponenziali di t~
in ogni intervallo
>O.
S >0
semiasse
t
sull'asse
t ~ 0 ) anche se
I .A. i I
arg).. =8,
sufficientemente. gran-
. Lu
=0
,sommabili sono dense
nell'insieme di tutte la solU'Zioni sommabili suI
(lnfatti si puo dimostrare che tali soluzioni sono complete
D. x e sostituito da un qualunque
operatore forte-
mente ellittico il quale agisca su funzioni che hanno dati di Dirichlet nulli sulla superficie laterale del cilindro).
2.5. Terminiamo questo capitolo con un risultato relativo alle soluzioni di Lu = 0
definite per tutti i valori di t, il qua1e risultato noi chiameremo un
principio astrf).tto di Weinstein. Per semplicita noi non 10 presenteremo nella sua forma pili generale. Quando applicato a certe equazioni differenziali a dertvate parziali in un cilindro comp1eto -oooo. Su ciascuna retta della successione
esista fuo~i di un segmento di lunghezza s e abbia norma limi-
R(~)
tata da M. Nell'ipotesi
(H)
la poSizione del segmento di lunghezza
S puo
variare per ogni retta della successione. Teorema 3.3. - Sia
u una soluzione della disequazione
ILul sull'intervallo
05 t $ T
Esiste una costante
S
tp(t)
con
lui u(T) = 0 e supponiamo che valga la (H).
c tale che se
'P (t) < c
allora
u .. O.
La dimostrazione segue da vicino quella del teorema 3.2; con ant u(t) = 0 per t > T consideriamo v(t) = e ; (t) u (t), (1.+ i an) v = f (t) e otteniamo la (3.3) come prima. La disuguaglianza (3.4) allora vale pertutti i valori reali di
A
nel complemento
I di un intervallo di lunghezza s
per modo che
!I~ I
00
(.A) I
2 d A ::; M2
) -QC)
152
If(AlI 2d A.
- 29 -
L. Nirenberg Poiche v(t) mostrare che ~( .A) in
L2
di
I~(.A )I
ha il supporto nell'intervallo
0 $ t $ T possiamo
ha la proprieta che su ogni retta 1m ~ = a la norma e limitata da
e Ia I T moltiplicato per la norma in L2
"
calcolata sull'asse reale. Poiche v().)
e una funzione intera non e diffici-
Ie dimostrare (per esempio, per assurdo) che c'e una costante 00
f
I~(.A II 2 dA $ k
..
f I~ (A l/
2 dA
k tale che
.
I
Conseguentemente abb:amo 00
00
J 1~(A1I2dA~kM2f 1~12d). .00
_~
dopo di che si procede come per il teorema 3.2. Nella stessa situazione del teorema 3.2 sarebbe desiderabile ottenere anche limitazioni inferiori per Ie soluzioni della (3.2). In condizioni piii onerose possiamo ottenere limitazioni inferiori per
per ogni ~ > 0 invece che per
I u(t} I .
Teorema 3.4. Assumiamo che su ogni retta
ImA R( A)
= costante
ci sia un intervallo di lunghezza
s
mitata in norma da una costante
M fissa. Esiste una costante e
~ t9(t)
fuori del quale
esiste e sia litale che
::; c allora ogni soluzione della (3.2) soddisfa la disuguaglianza
153
~ 0
• 30 -
L. Nirenberg
HP
LI dove
Ko' K1 ,
f
u(,,)
I
sono costanti fisse e
e una
~
costante dipendente dal-
la soluzione.
e basata
La dimostrazione
su un argomento di convessifa ed
e trop-
po complicata per essere data qui. Osserviamo che il metodo della dimostrazione del teorema 3.1 puo essere anche trasportc.to alllequazione con coefficienti variabili (ancora in uno spazio di Hilbert con prodotto scalare (
I
»)
du
ill - B (t) u = 0 •
(3.5)
Ci limitiamo ad enunciare il risultato. Per ogni t J B (t) so definito in un insieme dense dello spazio ed nio di
B{t)
che a quello di
u (t)
sia un operatore chiu-
appartenga sia al domi-
B. Assumiamo inoltre che esso dipenda rego-
larmente da t e che sia quasi autoaggiunto. Noi esprimiamo queste condizioni sotto la forma: esistono due costanti u (t)
di classe
tali che per ogni soluzione
C2 valga la seguente relazione
Re :t
(3.5 1)
k, c
(B (t) u (t), u (t))
~
t
I: (B+B~) u I 2 + + c Re ((B-k)u, ul.
Sotto queste condizioni se u log Ie -kt u(t) I strazione
e una soluzione
di (3.5) di classe
e una funzione convessa della variabile
e analoga
C2 allara
1:' = e ct . La dimo-
a quella del teorema 3. 1 in quanto si dimostra che la de-
154
- 31 L. Nirenberg
rivata seconda di
I
I
log e -kt u (t)
rispetto a"C
e non negativa.
3.3. Ci occupiamo ora di un teorema di convessita nello spazio di Banach Y
per Ie soluzioni della (3.2) sotto opportune condizioni sull'operatore A.
Il risultato che noi presentiamo
e una generalizzazione
di un risultato recen-
te di Cohen e Lees [3] . Faremo uso del "teorema delle tre rette" di Hadamard per funzioni analitiche in una striscia. L'operatore
Asia della for-
rna
(3.6)
e un operatore line are chiuso ed e un generatore infinitesimale o di un gruppo forte mente continuo T(t) di operatori. Assumeremo che gli dove
iA
operatori
T(t)
(3.7)
siano uniformemente limitatit IT(t)1
::;;K
sebbene analoghi risultati valgimo anche nel caso che
\T(t) I .$ K e wit I,
(Nel caso che
Y sia uno spazio di Hilbert, un operatore autoaggiunto A o soddisfa certamente la.(3; 7)) Dapprima noi daremo una semplice estensione del teorema 3.1. Questa
e ottenuta estendendo analiticamente la soluzione nel
campo comples-
so. Consideriamo questa estensione anche per l'.equazione inomogenea
(3.8)
du
"""'dt - i Arj.. u = if. Nel caso rX.= 0, iAo
essendo generatore di
mula (1. 6)
155
T(t) , abbiamo la fu-
- 32 L. Nirenberg r
\0
u(s+r):;T(r)u(s)+i
ri.f
Per
0 c Ie una formula di rappresentazione simile nel caso che
sia olomorfa per valori cornplessi di che
T(;t)f(t-A)dA.
t:; 1: + i 0' in una striscia
e data da t\ltti valori complessi di
-ooO. Per esempio si pub dimostrare il seguente Teorema 3.7.
~ ~
(3.17) soddisfa, per
dove
.f
(t) = H(l+t)
-k
,k, H ;? 0
;allora una soluzione u della
t 2 1 ,la disuguaglianza
lu(t)l ~ /u(oH ~
t
lu(t)/ ;; /u(o)/ ~
t
e una costante fissa e
e
-u. t r'
e-~(t+1)
~
se K> 1 2
se K
=0
dipende dalla soluzione.
In pratica si potrebbe applicare tale risultato ad una equazione differenziale a derivate parziali in un cilindro tale che diventi iperbolica quando si sostituisca
it
con
e i4
ft-
J
cioe a queUe equazioni per Ie quali il
problema iniziale e con condizioni al contorno eben posto sia per valori positivi che per valori negativi del tempo.
162
- 39-
L. Nirenberg
Capitolo 4 Stabilita all 'infinito ..
4.1. In questa capitolo noi presentiamo un risultato relativo al problema (iv); supponiamo che Y sia uno spazio di Hilbert. Nel caso di dimensioni finite il risuHato risale a Dunkel [5] . Nelle applicazioni di questa risuHato ad ope-
ratori differenziali in cilindri come negli esempi 2,3,4 si considerano opera,.. tori differem;iali i cui coefficienti possono dipendere da t rna tali che Ie differenze di essi coefficienti dai loro valori limiti (dipendenti solo da x) sono limitate da una costante per
t- k per qualche
k > O. Allora si conclude che
Ie soluzioni che sono di quadrato integrabile sull 'asse positivo tendono a zero esponenzialmente. 11 numero k dev'essere preso almena uguale all 'ordine massimo dei poli reali della risolvente R( A). Noi consideriamo soluzioni della disuguaglianza
(4.1)
ILul ::;
c
(1+t)
k
I uI
che sono di quadrato sommabile per t>o ,e assumeremo che
R(,A.)
sia re-
golare sull'intero asse reale eccettuato al piu per un numero finito di poli rea-
I R(A) 1= 0(1)
Ii AI, •.••• , .Am' e che
Teorema 4.1. Se ciascuno dei poli
e sufficientemente piccolo allora
I.AI _ sull'asse reale. AI' ...... , Am e di ordine ::; h ~ c quando
00
esistono due costanti positive
che per ogni soluzione della (4.1) di quadrato integrabile si ha:
(4.2)
r
1
Ieat u I 2 dt < C r0
o
163
Iu I 2 dt
a, C tali
- 40 -
L. Nirenberg
Noi abbiamo visto nella sezione 1 che l'ipotesi su c non puo essere eliminata. La dimostrazione del teorema
e basata sulla seguente disuguaglian-
za a priori. R( A) ---soddisfi Ie condizioni precedenti. Sia
Lemma 4. 1. Supponiamo che v(t)
una funzione tale che
v( 0) = 0
quadrato sommabile sull'asse
JOOo
(4.3)
dove la costante
C1
~ I v(t) I
positiv~.
n teorema
siano di
Allora si ha
2
I v I dt < C1
(00
)
o
dipende solo dall'operatore A.
Assumiamo dapprima il lemma e il teorema.
I (1+t k )v(t) I
e
v~diamo
come da esso si deduca
segue con un ben noto ragionamento daIla disuguaglian-
za: T
f
(4.4)
I u(t) I 2dt
T-l
dove
T> 1; la costante
C2
e indipendente da
T. Nel seguito
C3, C4 , ••.
sono costanti indipendenti da u e- da T. Per dimostrare la (4.4.) consideriamo una funzione crescente indefinitamente derivabile ~ (t) 2: 0 la quaI
Ie sia uguale a zero per
t
< 0 ed uguale a uno per t 2:
v(t) = E; (t) u (t + T - 1) con
v(t).= 0 per
1, e poniamo
t,2:0,
t< O. Dallemma abbiamo
164
- 41 L. Nirenberg
l
CO lu/2dt$ (CO Ivl 2 dt::;;C 1 fCO o
T
\(1+tk)
Lvi
2dt
) 0
I~
.$ C1
1(l+tk)
Lu(t+T-l)
12 dt +
1
1
+C 3
(
(1+tk)2
)0 con ,qualche costante
C3
tI
LU(t+T.1)1 2 + IU(t+T-1)1 2 } dt
dipendente da ~
; quindi
(1 +tk)2ILU(t)l2dt+C4
f
T
2
2
(/Lul + lui )dt.
T-l U sando la (4. 1) abbiamo
(~
T
lu(t)/ 2dt + C5
T la quale da la (4.4) se
C1 c
2
~
f
/u(t)/ 2dt
T·l 1 2" .
4.2. °Dimostrazione del Lemma 4.1 : c 1' c 2' •..... indicheranno costanti dipendenti soltanto da A. Se matadi Fourier
(A -A) ~ = f
Lv
=f
noi abbiamo come al solito per la trasfor-
,0
165
~
42
~
L. Nirenberg
dovunque
R( A)
componiamo
sia regolare. Siano i poli
R().)
Al
< ,\ 2
to' Denoting this state by S(t, tolY ,it is clear that the solution will be continuable only if we require that We can now c .impute
(t l'
~2
S(t, to)Y
S(to + II + C' 2' to)Y
in turn belong to Dt . either directly
> 0) or indirectly, using SltO + C;' l' to)Y as initial data and
obtaining the solution at a time C' 2 later as
The uniqueness condition (i) then implies the semi-group property
As was pointed out by Hadamard, this semi-group property is reflected in certain addition theorems, Stabllity requires that y
.~
n
limn S(t, to)Yn
= S(t, to)Y whenever
y ,assuming of course that all of the data involved belong to Dt ' 0
In other words
S(t, to)
is a continuous operator on DtO' Moreover to
say that y is the initial value for a solution means that lim
t ....t o
S(t, t ) Y = y, In the usual terminology this means that operate '."S 0
S(t, to) converge strongly to the identity as
176
t
~
to+ •
- 4R. S. Phillips
In addition to the above mentioned Hadamard conditions, we now assume that the problem is basically time-invariant. Physically this means that the underlying mechanism does not depend on time or, equivalently, that the corresponding differential equation including the boundary conditions are time invariant. In terms of the above notation this amounts to iii) Dt is independent of t , iv). S(t 2, \) depends only on t2 - t 1. We shall call the common set of initial data D and set S( "t)
=
= S(t + 't', t). The semi-group property then becomes 1)
In some physical problems the initial data determines the entire past as well as the future of the system. For such mechanisms the restriction t l' t2 > 0
need not be imposed and (1) will hold for all real t l' t2 . The
resulting family of operators defines a group of operators. We shall also assume that v). S(t) is linear, Condition (v) will satisfied if the associated differential operator is linear and the boundary conditions determining D are homogeneous. Thus existence, uniqueness, stability, time-invariance, and linearity constitute our principal assumptions. It is clear that a large class of initial value problems from mathematical physics satisfy tl\ese requirements and consequently have solutions which can be described by strongly continuous semi-groups of linear operators. Implicit in the above discussion is a topology on D. We shall hence-
177
·5R. S. Phillips
forth assume that D lies in a Banach space X and without loss of generality we may assume that D is dense in X (since otherwise we can take X tu be tbe closure of D ). If for fixed t ,the operator S(t) is linear and continuous on D ,then by the usual argument it can be shown that S(t) has a unique linear bounded extension on all of X. We denote the so-extended operator again by S(t) and it is readily verified that the resulting operators again have the semi-group property. The condition limt -+0 S(t)y = y,
y
~
D,
is not sufficient for the development of a simple theory. There are many ways to supplement this hypothesis, the simplest being (vi)
limt ----t 0 S(t)y = y,
y f X.
Equivalently we could assume taht S(t) is bounded in norme near the origin (by the uniform boundedness theorem). This condition is satisfied by pratically all of the applications of the theory.
178
- 6R. S. Phillips
2. Strongly continuous semi-groups of operators. We shall now derive some of the elementary properties of strongly continuous semi-groups of operators. These are one-parameter families of bounded operators subject to the following conditions: S(t 1 + t 2) = S(t 1)S(t 2),
i).
=I ;
S(O) ii). lim ,S(t)y
t-.O+
= y,
y
eX.
It follows from the uniform boundedness theorem that
for some positive constants M and
= n cl + 1:',
cl .Since any positive
< t < cl, we see that
ten as
t
where
w = (log M)I ~ • Actually more is true. Lemma l.
0
I I) It = lim
wO;; inf (log S(t) t>o
Proof. Setting f(t) f(tl + t 2)
t-+co
(log fS(t)/ )/t.
= 10g\S(t)i ,we see that
= log IS(t 1 +t 2) k
so that f(t)
t can be writ-
log IS(t1)11 S(t 2)1
= f(t 1) + f(t 2),
is subadditive. Moreover, as we have just remarked
f(t) ~ log M + wt. Now mer case, given
fo = inft>O
C> 0
f(t)/t
there is an
is either finite or-co. In the for-
'1. > 0 such that fo
Hence for (n-l)'l < t< n'1' we have
179
~ f( "1 )/'YI ~E + E .
- 7-
R. S. Phillips
It follows that lim sup t~oo
lim t-+oo
fo .The case ~
f(t)/t =
f(t)/t ( ~ + E ,and hence that
= -00 is treated in a similar way.
The parameter Wo plays an important role in the theory of semigroups of operators. It is called the type
of
[S(t)].
For one thing exp (wO t) is the spectral radius of the operator S(t) ; in fact
I,S(tt /1/
n
= exp
[
I
~t
log S(nt)1
]
~
exp wot.
Also it is clear that for each wI > Wo there exists an M > 0 such that
IS(t)1 ~ M exp (wl't) .. Lemma 2. The semi-group Proof. For arbitrary
y
€
[S(t); t
~ 0]
is strongly continuous.
X and t2 > tl ) 0 ,we have
and the right member tends to zero as t2 - tl
~
O.
The initial-value problem is usually presented in the form of a differential equation of the type
d S(t)y/ dt equal to a spatial operator whe-
re y belongs to the set of initial data D . In the process of abstracLw; the defining conditions (i) and (ii) we lost track of
D . On the other hand
we cannot expect an arbitrary y E X to serve as initial data for the class
180
- 8R. S. Phillips
of problems we wish to consider. Fortunately it is possible to retrieve D from the semi-group of Qperators. This is accomplished by means of the infinitesimal generator A ,defined as follows: Setting
= S(?J ) - I
A
1.
't we define
Ay = lim A.. y
,--to+'(..
whenever this limit exists. The set of elements y for which the limit exists is the domain of A, denoted by DA • nis clear that DA is a linear subspace and that A is a linear operator. It turns out that DA corresponds to the above mentioned set of initial data D. We now show that DAis dense in X and that A is a closed operator. Lemma 3.
DA is dense in X.
Proof. We set
y.
>
r
8(t)y dt.
o
It is clear that
ments
l yo(J
0( -1
YI1\~ 0( as
0( - t
0+
and hence that the ele-
are dense in X. On the other hand
,
-1
= ~-1
1'\8(t
o
Jr~+1
+1. )y - 8(t)y] dt
S(t)y dt -
,.-1
l'
S(t)y dt
0
~
181
- 9R, S, Phillips
so that lim A'll yrX, :: S( o()y - y. Hence the Yo( belong to DA' " .... 0 + L Lemma 4. If Y €. DA ,then so does S(t)y and d S(t)y/ dt = AS(t)y = S(t)Ay ,
(2)
t
> O.
Proof. For ,. > 0 ,we have
Assuming y
€ DA ,we see that S(t)y € DA and that
AS(t)y
= d+S(t)y/dt :: S(t)Ay.
't.- 1 [S(t)y - Sit - ~)y]
Moreover
:: S(t
-l)A~7
l-t 0 +
is strongly continuous, the limit as
,and since 8(t)
exists and
d-S(t)y/ dt = S(t)Ay . Corollary. For y E. DA '
(3)
S(t)y _ Y =
It
S( 'L)Ay d 1:'::
o
Jo(ot
S( 't')Y d 1:;
Proof. The first two equalities follow from integrating the relation (2) and the third from the proof. of Lemma 3,
Lemma 5. A is a closed linear operator. Proof. Suppose the sequence Ayn ~ z. Then
l
yn
S('t')Ay n --" S('t")z
jeDA
,that yn --'" y
uniformly on
[0, "l1
ce by (3)
S('t)z d 't
182
~
z
as
and that and hen-
- 10 R. S. Phillips As a consequence y £ DA
and
Ay = Z ,showing that A is closed.
We have now returned to our starting point, namely, the initial value problem. Setting D
= DA ,we have shown that D is dense in X ,that
S(t)y provides a solution to the initial value problem d.S(t)y/ dt
= AS(t)y
S(O)y = y for y in D which possess all of the desidered features. Moreover this gives the only soluCon to this problem. For if y(t) is a strongly continuously differentiable function on (0,00) to X such that
t > 0,
d y (t)/ dt = Ay(t),
(4)
lim y(t) = Yo ' t -+0+
then y(t)
= S(t)yo . In fact for 0 < 1:< t ,the function S(t - "t')y('t") is
strongly continuously differentiable in 't". with derivative d S(t -,?:)y(~)/d"t'= S(t-t')dY(1;')/d,?; - S(t - c)Ay('t') = O. The desired result follows on integrating from 0 to t. The classical procedure in solving the initial-value problem (4) is to take the Laplace transform. Heuristically y(t)
= S(t)y =[exp
(tA)] y
and
J(0 exp (-At)S(t) dt = 1000
exp(-~I - A)t dt = (AI-A)- 1= R).,(A).
We now derive this result rigorously. Theorem 1. Let [S(t) ]
be a semi-group of type
simal generator A . Then for all
A with 183
r'e
A> Wo
'
wo and infinite-
- 11 -
R. S. Phillips
Rl.(A)y
In
0
00
A with re It
Proof. For a
As a consequence,
Ry
0
> wI > WO'
r~
Next we show that
As
l
e A"'I. -1
OOo
exp (- -\. t) S(t)y dt
Ry
e
-A.t
I R I~ M/ (6' - WI)'
€ DA . In fact for 'Yl, > 0
eA."!.
S(t)y dt - - -
'1.
l
"l - A.t e
S(t)y dt .
o
,...., 0+,
e)"!' -1
~ A
and
1
'1
't Conseguently (5)
A= G" + i Y ,
exp( -At) S(t)y is integrable and
defines a linear bounded operator with norm
=
y E x.
exp (- ). t) S(t)y dt ,
Ry
J't 0
E. DA and
AR = A.R - Y y y
184
e -A.t S(t)y dt
~
y.
- 12 -
R. S. Phillips
On the other hand, for
y
€
DA ,
A exp(- ilt) S(t)y = exp(-.t t) S(t)Ay
is strongly continuous and majorized by M IAy1 exp (both
exp( -
A. t)
S(t)y
and
A exp( - A.. t) S(t)y
(e- - WI]
t). Thus
are integrable and since
A is closed we may conclude that
In other words
ARy
= RAy. Combining this with (5), we see that
O.I - A)Ry = y,
y €. X,
R( AI - A)y
y
=Y ,
E. DA '
from which it follows that R = R.t(A) , the resolvent of A The previous theorem shows that re [6'"(A)
J~ Wo
; here we use
6"(A) to denote the spectrum of A • In particular, then, the resolvent set of any infinitesimal generator contains a right half plane. The problem of when a closed linear operator is the infinitesimal generator of a semi-group of operators is basic in the applications of the theory. As we have seen, the resolvent
R).. (A) is the Laplace transform
of S(t) when A is its generator. It is natural to ask for conditions on the resolvent of an operator which suffice to make this operator a Laplace transform. This problem has its classical counterpart in the numerical case and the criteria which have been developed for generators have much the same form as the classical criteria. However the method of proof is quite different since the usual compactness arguments are not available; this deficiency is more than balanced by the special properties peculiar to the resolvent of an operator.
185
- 13 -
R. S. Phillips
The first and in many ways the most useful generation theorem was obtained independently by E . Hille and K. Yosida in 1948. We shall derive their result as a corollary of. Theorem 2. A necessary and sufficient condition for a closed linear operator U with dense domain to generate a semi-group there exist real constants
[S(t) ]
is that
M > 0 and w such that
(6)
A> wand
for all real
n = 1, 2, . . •. In this case t
(7)
> O.
Proof. Suppose first that S(t) is a semigroup with generator A bounded as in (7). According to the previous theorem,
R ).. (A)y =
J(OCJ o
e
-.l. t
A,>w.
S(t)y dt ,
It is easy to see that one can interchange the order of integration and dif-
ferentiation and so abtain
1
(n) (A) (l)n OCJ n -At S() RA y= 0 t e ty
d't.
Consequently
I A I(Jo R
(n)
(A)y
(OCJ
n - At wt t e M e dt
= n!
M/ ( A - w)
On the other hand it is known fOIl resolvent operators that
186
n+l
.
- 14 -
R.S.Phillips
and combining this with the previous inequality gives (6). The converse argument which we present is modelled after K. Yosi.. da's proof of the Hille- Yosida theorem. We shall divide the proof into a ; number of steps. a). Setting B). = lim
A2 R;.. (U) - AI
,we show that
BA, y = Uy,
y
A~oo
BA. Y = A(). R;.. (U)y - y) =AR;.. (U)Uy
Now
AR). (U)x - ; x for all x
-->
as )..
00
e. Du·
so that it suffices to show that
EX. Again if x €
DU
then
;here we have used the inequality (6) for the case n = 1.
Approximating an arbitrary x in X by a sequence in fact that the operators
[
AR). (U)
Du
and usingthe
J are uniformly bounded in norm for
).. sufficiently large, the result now follows by the double limit theorem. b). For each
fl> 1
there is a ~fo such that
In fact
n=o and making use of (6) we get
\etB ,\ l ~ e - At L o 00
t n A~'11 M A.. -n-!- (A _w)n = M exp (t w -;x-:w)'
187
- 15 -
R. S. Phillips
1\.]3
It sufficies to choose
c). lim
so that
exp(tB)..)y
~
.A().- W)-l(
exists for each y
,\,~OO
for
,A >
Aj3.
E. X uniformly with re-
lO, 00 1 .To prove this we defi-
speet to t in each finite subinterval of ne the auxiliqry function.
Both factors are continuously differentiable with respect to (: in the uniform operator topology and
since
R,t(U)
and
Rp-(U)
commute. Integrating from 0 to t we get
I
t
exp(tB).A.-) - exp(tB),)
=
ViZ) (B;.v -B)..) d
c: .
o
If we now make use of the estimate obtained in (b) we have
Du
If
YE
in
t in each finite subinterval of
,then according to step (a) this converges to zero uniformly [0,00) . Finally an arbitrary y in X
can be approximated by a sequence in rators
L
exp(tBA.)
1
Du
and using the fact that the ope-
are uniformly bounded in norm for
sufficientlylar-
ge, the result follows by the double limit theorem. d). Setting
S(t)y = lim
exp(tB)..)y, we now show that
A~OO
188
- 16 -
R. S. Phillips
[S(t); t
>01
is a semi-group of operators. It is obvious that the approxi-
mating operators, namely [exp(tB),) ]
are semi-groups. Hence
since for strong limits the limit of a product is equal to the product of the limits. Further since the limit is uniform in t on subintervals, it follows that S(t)y is continuous in
t;, 0 ;in particular
S(t)y ~ y
as t -+0+.
Finally we note that the inequality (7) is an immediate consequence of (b). e). U is the infinitesimal generator of [S(t)] . It is clear that
For y €
Du
we have lim
B). y = Uy
so that the integrand converges
)..~oo
uniformly on [0, t
J
and we get
S(t)y - y"
i
t S( 't)Uy d't .
Consequently
At = "1.. -1
J"I.0 S('t)Uy
as '1.->0+. Denoting the generator of DA ::J DU
and
Ay
de- ~ Uy
I S(t) ]
by A, it follows that
= Uy on DU . On the other hand for A > w, R;L (U)
exists by hypothesis and R A(A)
exists by
Theorem 1. Hence
189
- 17 -
R. S. Phillips
OJ - U)% = f A1- A)D A and this shows that
DU = DA
and therefore that
A = U . This concludes
the proof of Theorem 2. An operator of norm less than or equal to one is called a contraction operator. Semi-groups of contraction operators will constitute the main theme of these lectures. For such semi-groups the generation theorem takes on a particularly simple form. Corollary (Hille- Yosida) A necessary and sufficient condition for a closed linear operator U with dense domain to generate a
semi~group
of
contraction operators is that
).. > O.
(8)
Proof. If M = 1 ,it is readilyseenthat (8) impUes (6) with w=O. The assertion now follows directly from Theorem 2. An example will serve to illustrate this theorem, For
x
= Co(-00, 00),
the space of continuous complex-valued functions which tend to zero at infinity, consider the following initial-value problem: u,(O, x)
= f(x),
-00< x 0 , then for f = Ay - Ly we have (9)
Proof. Clearly 2 A(y, y)
~ 2 A(y, y) -
[(LY, y) + (y, Ly) ] = (f, y) + (y, f) ~ 2 \y
I ,f I
from which (9) follows. Remark 1. One consequence of (9) is that (f.. I - L)-l is a bounded operator on the range of If the operator
CA I
(AI - L)-l
- L) which is a closed subspace if and only
is closed. Thus
rator if and only if the range of
Remark 2. The map: range of I - L
(,H - L)
y -; f
=Y -
(,~.r -' L)
is a closed ope-
is a closed subspace.
Ly
is one-to-one so that if the
is all of H ,then L is necessariely maximal dissipa-
tive. As we shaIT see the converse is also true for dissipative operators with dense domains.
193
- 21 R. S. Phillips
Next we develop a Cayley transform theory which has Theorem 3 as one of its consequences.lt will also give us some insight into the construction of the dissipative extension of a given dissipative operator. We define J
= (I +L) (I _ L)-1
DJ= range of I - L , and show that J is a contraction operator. In fact for
u
E DJ
u = y -Ly
( 10)
Ju = y + Ly
for some
y E DL . Hence
(Ju,Ju) = (y,y) + (Ly,Ly) + [(Ly,y) + (y,LY)] (u, u) = (y, y) + (Ly, Ly) - [(LY, y) + (y; Ly) ] so that
We can recover L from J by means of y = 1/2(Ju + u) (11)·
Ly
= 1/2(Ju -u)
from which it follows that I + J must be one-to-one and that
1l = range
(I + J) . Note that J is closed if and only if range (I - L)
is closed and hence (by Remark 1) if and only if L is closed. Conversely, suppose that J is a contraction operator with I + .J one-to-one. Then (11) defines a dissipative operator L with
194
- 22 R. S. Phillips
DL
= range
(I
+ J) ;in fact
(y, Ly) + (Ly,y) = 1/4 [(JU - u, Ju + u) + (Ju + u, Ju -
[I Ju 12 - \ u /2 J ~ O.
= 1/2 Finally we show that I + J of
(I + J)
is automatically one-to-one if the range
is dense. For suppose there is a Uo
and set y = Jv + v for arbitrary
U)]
v E. DJ
f0
such that Ju o + Uo=0
. Then
and expanding the extreme elements of this inequality gives
Since this holds for arbitrary
0(
we conclude that
ce y ranges over a dense set this implies that
(u, y) o
= 0 and sin-
Uo = 0 ,wnich is impos-
sible.
Remark 3. It is always possible to extend a contraction operator Jo
to be a contraction operator with domain H . To accomplish this,fir&t
close up the operator and then set
Ju
.L
= 0 for all u t DJ . o
We summarize the above in Theorem 4. If Lo
is dissipative then J 0 defined as in (10) is a
contraction and they are closed together. If J 0 is a contraction with range
(1:t J o ) dense then Lo defined as in (11) is dissipative with dense
domain and convers ely. The relations (10) and (11) establish a one -to- one inclusion preserving correspondence between dissipative extensions of Lo
195
- 23 R. S. Phillips
(ll
o
dense) and contraction extensions of J o . In particular, the maximal
dissipative extensions of
Lo (DL
o
dense) correspond to contraction ex-
tehsions of J 0 with domain H Corollary. If ~ > 0 and L is dissipative with dense domain, then L is maximal dissipative if and only if range ().. I - L) = H. Proof. Land ther. But
;r 1 L
are dissipative and maximal dissipative toge-
).. -1 L is maximal dissipative if and only if range
0.1 -
= range (I -
L)
Proof of Theorem 3. If
[S(t)]
\ -1
I\.
L)
= H.
is a semi-group of contraction ope-
rators then as we have seen above its generator A is dissipative with den-
A>0
se domain. Moreover by Theorem 2, for A and hence range
(AI -A)
belongs to the resolvent set
= H . Thus A is maximal dissipative.
Conversely, if L is maximal dissipative with dense domain then for
A. > 0 6,
,range
(). I - L)
1('\.1 - L)-ll~
A- 1
= H by the Corollary to Theorem 4 and by Lemma . Thus
exists and is of norm
R). (L)
~
A-i.
The Hille- Yosida Theorem applies and asserts that L generates a semigroup of contraction operators. Lemma 7. If L is maximal dissipative with dense domain, then
Be
is LX ,the adjoint of L . Proof. Let J be the Cayley-transform of L . Then J is a contraction,
D. J
traction with 1+ J
= H and range (I + J) is dense. Obviously DJlC = H . Suppose range
is a con-
were not dense, then
would have a non-trivial zero which as we have seen in the proof of
Theorem 4 is contrary to range I +J IC
(I + Jl 0 such that
or H . Further for any x. H tbere is a unique decom-
We now define
and in view of the above inequality we get 1/mlx/2< /xl:<mlxI2 Moreover if we set
Wx 1
=
x for x
I:. H W x =-x for x c. H. we see '-+' 1 lit-
201
- 29 R. S, Phillips
that
Q(x, y) = (W 1 x, y)1 . Since
W~
=I
this is the desidered normali-
zation. F.rom now on we drop the subscript 1. Let E ,E
+
-
be the orthogonal projections into H+ and H ,re-
spectively. For a given dual pair of subspaces x
[N,
p]
with
= x+ + x _ E ~ and y = y+ + y _ € P we set
(12 )
Jx
-
=x +
and extend J linearly on
Jy
E _N + E+P
+
=Y-
. In the first place J is a well
defined contraction operator. In fact, the inequalities Q(x,x)
1x./2 (0
= /x+12 _
I
I
Q(y, y) = y+ \2 _ y -' 2
IJ(x _ + y+) 12 = 1x+ + y./ 2 :: Secondly,
for x E Nand
~0
for y
£
Ix+ 12 + y _12 ~ Ix _ \2 +1 y+ \2
I
J is symmetric. To see this we note that for
x
Pimply
= \ x_ + y+ 12.
E N, yEP
the duality of Nand P gives
Hence for arbitrary u, v €. D Y (Ju, v)
= (Ju -
+ Ju+, v +v ) -+
= (Ju -,v ) + (Ju ,v ) + +-
Thirdly, ,J anticommutes with W ,that is JW since
DJ
= E _N + E+P. Further
202
= -WJ
. In fact,WD/D,r
- 30 -
R. S. Phillips
JWx=J(x -x )=Jx -Jx =-W(Jx +Jx )=-WJx. + +. + Conversely given J a symmetric contraction which anti-commutes
wi~h
W we !tee that WDJ = DJ
Setting
D± = Et DJ
we have
so that the same is true of D± C DJ ' DJ
=D+ + D_
E1 =2- 1(J±W),
' and we may defi-
ne the subspaces ( 13)
N = [x+ Jx; xED.1 P
= [x+
Jx ; x
~ D+
1.
We shall show that Nand P form a dual pair of subspaces, . In the first place
JD + C H_ and
Wy. = -y _
Hence
JD. C H+' In fact if
xeD
and
y _E H_ then
so that
Jx_l..H_, that is Jx.EH+, Similarly Jx+EH_ foranyx+6·D+,
It now follows from J being a contraction that N is a negative subspace
and P is a positive subspace, To prove duality let u E N, ve. P ,Then u ,v '- DJ and u = u - + •
+ Ju , v = v + Jv . Hence + +
Q(u,v) = (Ju.,v+) - (u.' Jv+) = 0 by the simmetry of J. Consequently [N, P ]
form a dual pair,
It is clear that by starting with the dual pair
[N,
p]
and using (12)
to define J ,that we then return to the original dual pair by means of (13). Further an extending dual pair [N l' PI
J defines via (12) an extending
operator J 1 • Now N is a maximal negative subspace if and only if
203
- 31 -
R. S. Phillips E N = Hand P is a maximal positive subspace if and only if E+P = H+' Hence if DJ
= H then the corresponding subspaces
[N, P ]
are dual ma-
ximal subspaces. We have therefore proved Lemma 9. There is a one-to-one inclusion preserving correspondence given by (12) and (13) between dual pairs of subspaces and symmetric contractions which anti-commute with W . The dual pairs of maximal subspaces correspond in ,this way to symmetric contractions defined on all of H which anti-commute with W . It remains to ahow that any symmetric contraction which anti-com-
mutes with W can be extended to an operator with domain H having these same properties.M. Krein (Recucil Math. 20 (1947) pp 431-495) has shown how to extend a symmetric contraction Jb to a symmetric contraction B defined in all H . Since B:J J,o x€. DJ
o
WJ
o
= -J 0 W we see for
that WBW x
Hence
and since
= WJ 0 Wx = -W 2J 0 x = -J 0 x = -Bx
- W B W also extends J 0 as does
Thus so defined J is a symmetric contraction extending J 0 which anti-commutes with W_ . Consequently J defines via (13) a dual pair of maximal subspaces which extends the dual pair defining J . o Remark 1. If we set
204
- 32 -
R. S. Phillips
then the previous extension can be made to a maximal dissipative operator L with corresponding
¥:;
~o.
Remark 2. The above theorem can be applied to the study of conjugate symmetric dissipative operators and their extensions. To see what this means we recall that a conjugation C has the propertiG!s : (i) (ii)
C(ax
+ by) :; aCx + b Cy
C2:; I
(iii) (Cx, Cy; :; (y, x) An operator L is called conjugate symmetric if for all x, y £ DL ,that is if
(Lx, Cy) :: (Ly, Cx)
L C C LX C. An operator L is called
self-conjugate symmetric if
L:; C L l( C. Using Theorem. 6 it is not dif-
ficult to prove the following Theorem due to 1. M. Glssman : A dissipative conjugate symmetric operator with dense domain has a dissipative self-conjugate extension. We can dispense with the dense domain if
to
f(t) L 2(y) - continuous in J , 0
j
f(t)Y -a. p. _S2 ¢::> f(t)
L 2(Y)_a. p. (according Bochner's definition).
o
Moreover, we shall say that f(t) is Y -w. a. p. _S2 if f(t) is L 2(y) .. w.. a. p. , that
o
~s
if,
'r/
(f(t), g) 2' Lo(Y)
geL 2(y), the scalar product 0
=
is a Bohr a. p. function. In what follows, we shall write ·f(t) ·to indicate f(t), and we shall add the indication of the space where f(t) has to be considered: thus the notations 2
2 or f(t) L (Y)-a. p. , o 2 f(t) Y-w. a. p. -S or f(t) L o2(Y)-w. .a. p. , f(t) Y-a.
p. -S
are equivalent.
231
-8L. Amerio
b) Let Y and V
s;;
Y be two Hilbert spaces, V dense in Y : assu-
me, moreover, that the immersion of V in Y is continuous
k
(lxll Y ~
k I\x, V'
> 0). Let us consider the second order (22) linear functional equation: Q(x;h)
=
J {(x'(t),
h'(t))y - (A(t)x(t), h(t))V
+
h(t~}
dt
J
(3. 1)
+ (B(\) x'(t), h(t))V + (C(t)x(t),
=
J (f(t), h(t))ydt , J
where x(t) (unknown function), h(t) (test function), f(t) (known term) satisfy,
V
compact
6 , the conditions : x(t), ~t) €
(3. 2)
2
L (
!::. ,
x'(t), h'(t) €
V);
2
L (
A,
Y) ;
h(t) has a compact. support; f(t) E L 2(
(3. 3)
A,
Y).
The derivatives are taken in the sense of distributions. By (3. 2) and (3. 3), x (t), h(.t) are L 2(V) - continuous and x'(t), h'(t), o f(t} are L 2(y) ~ continuou!>. ,
.
0
Let us consider the following Hilbert space
with the scalar product
1 2)
(w 1 ,w 2 )W=(w 1 ,w 2) 2 + (w ,w 2 L (V) L (Y) o 0 Hence, by (3. 2), x(t) = {x(H, ); 'l~ Ao} W, and it is
Ilx(t)~ W =
U 1:10
/lx(H
~ )0 ~d'1
+
Therefore x(t), h(t) are W- continuous.
232
is a function from J to
• 9-
L. Amerio
In equation (3. 1) the operators A(t), B(t), Cit) are bounded,
Vt e
J; precisely:
A(t) ~
J. (V, V),
B(t) E
.t (Y, V),
Cit) E
.t (V, Y)
and we aSS'olme, moreover, that A(t), B(t), Cit) are continuous functions of t , in the uniform topologies of their spaces. In particular, A(t), B(t), Cit) can be supposed a. p. functions of t. In this case we shall call re/iUlar any real sequence
A = {A~l
such
that (3.4)
lim A(t+ An) = AX (t) , lim B(t+ An) = BA (t) , n-t 00
n... oo
lim C (t+A n)
= C). (t)
n......
uniformly on J. Hence A). (t),
B~
(t), C" (t) are a. p. and, by fundamen·
tal Bochner's criterium, any real sequence contains a regular subsequence. Moreover, we shall consider (according to Favard's theory for ordina-
'tf
ry systems),
(3.5)
regular
A.
,the homogeneous equation: Q (u;h) = Q(u;h) , o
Q)" (u;h) = 0 ,
that is the equation
J{(u'(t), h'(t))y - (A).. (t)u(t), h(t))V + (B ),(t)u'(t). h(t))V + J
+ (C>. (t) u(t) , h(t))y } dt
=
0.
Let zit) be a W-bounded function. Put
1> (z; 't" ) =
Sup I/z. (t+ 1" ) - zit) ~W ' tEJ
let us call "z the set of all W-bounded functions x(t) such that f(x;"t' ) ~
f
(z;"t' )
233
- 10 -
L. Amerio
Let
A
x(t) E
1\ z
Q f· be the set of all sohttions x(t) of (3. 1) such that z, , and let A Q be the set of all eigensolutions u(t) of the hoz,
mogeneous equation Q(u; h) = 0
such that u(t) = x 2(t) - xl (t), \(t) E.
1\
Z,
Q, f'
One can prove the following theorems I (Minimax theorem) - Assume that: Of.') there exists a W -bounded solution, x(tj, of (3. 1);
(j ')
V
u(t) E
1\ _ Q x,
it results
II u(t) II )
Inf t
eJ
Then (3. 1) possesses, in
O.
1\ _ Q f' one and only one minimal solux, ,
tion, Jt(t). Precisely, put f-(x) = Sup t 6 J
II x(tll\ W
'
N
)A = Inf )A (x)
"x, Q, f there exists,
J!!.. 1\-x, Q , f'
one and only one solution, }t(t), such that N
},,(x)
=r
....
.
Let us observe, moreover, that}f A(t), B(t), C(t), f(t) are periodic (with the same period) then i(t) is periodic. Let us assume now that: A(t), B(t), C(t) are a. p. operators, f(t)
~
L 2(y) _ w. a. p., A = f A ~ is a regular sequence. Then there exists a o -n suitable sequence (which we shall denote by {A n such that lim
* f(t + f\\ n) 2
W\-.oo
fA (t) Lo(Y)
1)
2 (fA (t) Lo(Y) - w. a. p.)
234
- 11 -
L. Amerio
uniformly on J. Moreover, if condition
* x (t +).. n). w=
lim Wi +01>
where ,M (i A' (t)) 6
,. (x; h)
Hence the set
=
J
(fA (t), h(t))y dt . J is not empty.
1\ _ Q
f x, >-' "II (Weak almost periodicity theorem). Assume that:
0(.11)
0. ") I""
there exists a W-bO.unded solution, i(tl, of (3. 1);
'f/ regular A
and
'" u(t) ,
-
1\ _x, QA
it results
>0 .
Inf Ilu(t)~
t, J
Then the minimal solution, 'itt) , is W-w.. a. p. In other words, the scalar product (x(tl, g)W
=
J
{itt + ~ ), g( 1 ))v + (X'I(t + '1
[1.
is a Bohr a. p. function,
'V
g
),
gl( 'I ))y}
d'l
e W.
d) In order to prove that the minimal solution is W-a. p., it is sufficient to prove that its range is W-relatively compact. To that purpose, we shalt restrict, first, the set where the minimal solution has to be defined. Let us assume that there exists a W-bounded and W-uniformly continuous solution itt). Because of the inequality
it follows lim ~~O
cp (x; 't"
)
~ lim Cf (x; 1." ) = 0 . ~?o
235
- 12 L. Amerio
Hence the minimal solution "i(t) is W-uniformly continuous. We enunciate now the conclusive statement. III - (Almost-periodicity theorem). - Assume that: 0(.
III) there exists a W-bounded and W-uniformly continuous solution, itt) ,
of (3. 1); ~ III)
'r/ regular
A and
"i
u(t)
e
A_ Q x, A
it results
lnf II u(t) II > 0 ; eJ 't III) the immersion of V in Y is completel,)' continuous; t
oIII) the hypothesis (of ellipticity) is satisfied: (A(t)x, xlV ~
2 V 1\ x~ V
(y
>
0).
Then the .minimal solution, ~(t), is W-~ Let us observe, at last, that, for the problem of vibrating membrane, considered at
§
2, conditions
~
III),
'0 III), 0 III)
are satisfied (23)
(1) Cfr. J. FAVARD, Lecons sur les fonctions presque-periodiques, GauthierVillars, Paris, 1933. (2) S. BOCHNER, Abstvakte fastperiodische Funktionen, Acta Math., 61(1933). (3) L. AMERIO, Sull'integrazione delle funzioni quasi-periodiche astratte, Ann. di Mat., 53( 1961); Sull'integrazione delle funzioni quasi-periodiche a valori in uno spazio hilbertiano, Rend. Acc. Naz. dei Lincei, 28(1960). See also: M. L. RICCI, P. RIZZONELLI, Sulle funzioni 11 -quasi-periodiche, Rend. 1st. Lombardo, 95 (1961); L. AMERIO, Sull'integrazione delle
236
- 13 -
L. Amerio funzioni IP {Xn\ -quasi-periodiche, con
1~ p
£ + QO
,
Ricerche di Mat.,
12 (1963).
(4) L. AMERIO, Problema misto e quasi-periodicita per l'equazione delle onnon omogenea, Ann. di Mat., 49 (1960). (5) C. F. MUCKENHAUPT, Almost-periodic functions and vibrating systems, Journ. of Math. and Phys. ,MIT, 8(1929). (6) S. BOCHNER, Fast-periodische Lgsungen der Wellen-Gleichung, Acta Math., 62 (1934). (7) S. BOCHNER, J. VON NEUMANN, On compact solutions of operational differential equatioJ!S, Ann. of Math., 36 (1935). (8) S.SOBOLEV, Sur la presque periodicite des solutions de l'equations des ondes, I, II, III, Compt. rerrl. Ac. Sc. U. R. S. S. (1945). (9) 0, A. LADYZHENSKAYA, Mixed problems for hyperbolic equations, Mo .. scov-Leningrad, 1953 (10) S. ZAIDMAN, Sur la presque-periodicite des solutions de l'equation des
ondes non homogene, Journ. of Math. and Mech., 8 (1959). (11) L. AMERIO, Quasi-periodicita degli integrali e.d energia limitata dell'equazione delle onde coptermitle noto quasi-perioaico, 1, II, III, Rend. Acc .. Naz. dei Lincei, 28 (1960). (12) Cfr. (11), II. This theorem has been generalized to the almost automor-
phic functions: S. BOCHNER, Uniform convergence of monotone sequences of functions,
Pr~
Nat. Acad. Sci., U. S. A., 47 (1961).
(13) L. AMERIO, Sull'equazione delle onde con termine noto quasi-periodico,
Rend. di Mat., 19 (1960).
237
- 14 L. Amerio
(14) J. KOPEC, On linear differential equations in Banach spaces, Zeszyty Nauk Univ. Mickiewicza. Mat. Chem. ,1(1957); L. AMERIO, Funzioni debolmente quasi-periodiche. Rend. Sem. Mat. Univ. di Padova, 30 (1960). (15) S. BOCHNER, Almost-periodic solutions of the inhomogeneous wave equation, Proc. Nat. Ac. Sc. ,46(1960). (16) S. ZAIDMAN, Solutions presque-periodiques des equations hyperboliques, Ann. Scient. Ec. Norm. Sup., 79(1962). (17) G. PROUSE, Analisi di alcuni classici problemi di propagazione, Rend. Sem. Mat. Univ. di Padova, 32 (1962). (18) V. A. IL'IN, On solvability of mixed problem for hyperbolic and parabolic equations, Uspehi Mat. Nauk, 15 (1960). (19) C. VAGHI, Soluzioni C-quasi-periodiche dell'equazione non omogenea delle onde, Ricerche di Mat., 12(1963). (20) L. AMERIO, Sulle equazioni lineari quasi-per iodiche negli spazi hilbertiani, I, II, Rend. Acc. Naz. dei Lincei, 31 (1961); Soluzioni quasi-periodiche delle equazioni lineari iperboliche quasi-periodiche, Rend. Acc. Naz. dei Lincei, 33 (1962); Soluzioni q1.j8si-periodiche di equazioni quasi-periodiche negli spazi hilbertiani, Ann. di Mat., 61(1963); §u un teorema di minimax per Ie equazioni differenziali astratte, Rend. Acc. Naz. dei Lincei, 1963; See also: L. AMERIO, Sulle equazioni differenziali quasi periodiche astratte, Ricerche di Mat., 9(1960), 10(1961); S. ZAIDMAN, Solutions presque periodiques dans Ie probleme de Cauchy, pour l'equation non
homoge~
ne des ondes, I, II, Rend. Ace.. Naz. dei Lincei, 30(1961). (21) Cfr. J. L. LIONS, Equations differentielles-operationelles et problemes
,],{
limites, Springer, Berlin, 1961;'Equations differentielles-operationelles dans les espaces de Hilbert, course C. 1. M. E.
238
- 15 -
L. Amerio (22) First order a. p. equation x'(t) = Sx(t)+f(t) (S selfadjoint unbounded operator, f(t) a. p. ) has been recently studied by ZAIDMAN (Teoremi di quasi-periodicita per alcune equazioni differenziali operazionali, Rend. Sem. Mat. e Fis. di Milano, 33 (1963)), Zaidman proves, first, that u'(t) is a. p., by using spectral integral representation of the operator S; afterwards, by virtue of theorem on integration (loc. cit. at (3)) it follows that x(t) (bounded by hypothesis) is a. p.. Let us mention, moreover, on NavierStokes equation (first order, non linear, equation): C. FOJAS, Essais dans l'etude des solutions des equations de Navier-Stokes dans l'espace. L'unicite et la.presgue-periodicite des solutions "petites", Rend. Sem. Mat. Univ. di Padova, 32 (l962); G. PROUSE, Soluzioni quasi-periodiche dell'equazione di Navier-Stokes, Rend. Sem. Mat. Univ. di Padova (l963). (23) Let us assume only that the immersion of V in Y is continuous: then (strong) almost-periodicity of ~(t) can be proved if equation (3,1) admits a suitable theorem of continuous dependence (see L. AMERIO and S. ZAIDMAN, loco cit. at
(20)
).
239
CENTRO INTERNAZIONALE MATEMATICO ESTIVO ( C. I. M. E. )
GIAN .CARLO ROTA
A LIMIT THEOREM FOR THE TIME-DEPENDENT EVOLUTION EQUATION
ROMA· Istituto Matematico dell'Universita
241
A LIMIT THEOREM FOR THE TIME-DEPENDENT EVOLUTION EQUATION by GIAN-CARLO ROTA
1. INTRODUCTION.
The topic of the present lecture is slightly at variance with those treated in the lecture series of this course. The main theme has been the abstract differential equation du = A',t )u ill
(1)
t ~ 0
u (:- X ,
,
where X is a Banach space, and where A(t) is a family of unbounded linear operators. The main problem has been the solution of (1) under very weak conditions on A(t). Under suitable conditions, which we shall specify below, the solution u(t) of (1) which for t=O satisfies the initial condition u (0) = f, where f (2)
f:
X , can be expressed in the form u(t) = P(t,O)f
,
where P(t, s) is a family of bounded linear operators satisfying the evolution equation P(t,s)P(s,r) = P(t,r), t,s,r
(3)
P(t, t) =1.
~
0
Thus, we shall understand the expression "solving the initial value problem for the equation (1)" as meaning that a family of bounded linear operators P can be determined which satisfies (3) and which gives a solution of (1) by formula (2). Our present point of view will be to take the existence question of 243
- 2G. C. Rota (1) for granted, and to
inve~tigate
instead another problem related to the e-
volution equation, which for many investigations is as important, if not actually move, as the e 4 istence-uniqueness question. This is the limiting behavior of solutions u(t) as t ->00. To get an idea of the kind of question we have in mind, let us consider first briefly the case where A(t)
=
A is inde-
pendent of t. This case has been thoroughly studied: it is the theory of oneparameter semigroup of Hille and Phillips. The evolution operators (3) become in this case P(t, s) = P
t-s
,
where the right side is given by a one-parameter semigroup pt whose infinitesimal generator is the operator A. In this case it is well-known that the solution u(t) of du dt
-
(4)
= Au
u(O) = f
,
t
~
0
t
is given by u(t) = P f. The behavior at infinity of u(t) is described by the classical ergodic theorem. Under suitable assumptions on the semigroup pt - which can be equivalently stated as conditions on the infinitesimal generator A the limit ( 5)
lim T->>c
~) T 0
ptf dt
= lim T1
T~c>o.
fT u(t)dt 0
exists in the norm of the given Banachspace X. (Cfr. Dunford-Schwartz, Linear Operators, Vol. I, Ch. VIII). The limiting relation {5) is of fundamental importance in many investigations, and expresses a property of the semigroup which is analogous to a "boundary behavior" of sorts. In fact, to quote Norbert Wiener, the godic theorem is a kind of fundamental theorem of calculus at infinity. We shall be concerned below with the following two 244
questions~
Er-
- 3-
G. C. Rota
(a) When can the Cesaro limit in (5) be replaced by an ordinary limit? (b) Is there a "natural" analog of the ergodic theorem for the general evolution equation, corresponding to the
time~dependent
differential
equation (I)? These questions of course have innumerable answers, which depend largely on the kind of convergence that is required (norm convergence, weak convergence etc.). The type of convergence we shall require below is dominated convergence in Lp(S,
2:. , f' ) = X ,
P
> 1 . Specifically, we ta-
ke u(t) and f to belong to some Banach space Lp(S,
I '/)
where
Jv. (S) = 1; the operators A(t) and PIt, s) will operate on these spaces only I
and will be subjected to the further conditions specified below. We say that a sequence f natedly to f (in symbols, fn (I) f (s) n
->
ct>
in L (S, n p f) when
I
'f ) converges domi-
f(s) for almost every s in S, that is, f
n
conver-
ges to f pointwise; and
I
(II) the function f"'"(s) = s~p fn(s)
I beiongs to
Lp(S,
~ , j'- ).
We hasten to add that from this point on all equalities will be understood in the "almost everywhere" sense. Thus, a function is defined almost everywhere, etc. Our main concern will be to give an answer to questions (al and (b) when convergence is understood in the dominated sense. Questions related to dominated convergence appear in many contexts in both analysis and probability theory, and are usually considerably more difficult than the analogous questions for norm convergence; we need only recall, beside the G. D. Birkhoff ergodic theorem, the Riesz-Calderon-Zygmund theory of Hilbert transforms and singular integrals, the martingale theorem (ef. below), the results relating to the almost-everywhere convergence of Fourier series 245
-4G. C. Rota and oimore general orthogonal expansions (such as the Rademacher-Menchoff theorem), the various strong laws of large numbers, etc. Dominated convergence is the most satisfactory type of convergence that can be hoped for in problems of both analysis and probability; it is the nearest to actual pointwise convergence. It is very desirable to have a unified theory relating to this kind of convergimce, in much the same way as the theory of Banach spaces gives a unified approach to questions of convergence in the mean; unfQrtunately, no such theory exists at present, and we have to rely on various methods of varying degree of complication. We can now proceed to state our main result. First, we consider the discrete analog of equation (1). This is the difference equation (for integer n) (6)
IJ Un = (Pn - I)Un
,
n ~ 1.
The solution of this difference equation which satisfies the initial condition u 1 = f is easily seen to exist uniquely when the Pn form a sequence of bounded operators in the Banach space X. Indeed, since
L1 un = un+ 1 (7)
u we easily get n
Un+ 1 = P nUn = P nP n- 1Un- 1 = ... = P nP n-l ' " P 1f
The discrete analqg, of the evolution operators P(t, s) of (3) are the operators (8)
P(n, k) = P P l ' " Pk n n-
n
>k
P(n, n) =1. We now consider an adjoint equation to (6), (for fixed n), that is the equation (9)
k ':.1, 246
- 5G. B. Rota
where P: is the adjoint operator of Pk' The solution of this equation that at k = 1 takes a given value g is given by the following expression
* *"
(10)
*
Vk=Pn- k P n- k1,,·Pg + n
We shall call (9) the adjoint equation of (6) • The continuous analog of equa_ tion (9), related to (1), is the equation (for fixed t) (11) where
dw 'II: = A (t-r)w dr '
-
t! (r)
r
Ef. n.
Roughly speaking, a conditional expectation has much the same properties as an integral, for example, the Lebesgue bounded convergence theorem holds.
( t)
Martingales. An increasing (decreasing) martingale En
is a sequence of conditional expectations such that E E
n m
(n
~
= E n for n:::: m
m). There is an analogous notion of martingales with a conditions para-
meter. The theory of martingales has been developed largely by Doob. The main result is the martingale theorem, which states that if E gale, then E f n
---j>
d
f for f in L , p p
n
is a martin-
> 1 (actually the general theorem is
more far-reaching). The notion of a martingale is one of the most fundamental in analysis; there is no telling how vast their applications will be in future years, once analysts begin to realize their power. So far the notion has been used largely by probabilists. ( ~) Abstract L-spaces. An abstract L-space is a Banach lattice X such that if f, g ~ 0 in X, then II f+g /1
=
Ilfll
+ II gil. For the theo-
ry of these spaces, see Day, Normed Linear Spaces. The main result on abstract L-spaces we shall need is the following. Let X be a Banach lattice such that /I
111 = 1 , where 1 is a lattice
identity. (Cf. Birkhoff, Lattice theory). Then there exists a structurally unique probability space (S,
L ,fA )
such that X is latticially isomorphic 249
- 8 -
G. B. Rota
to Ll (S,
l ' f1 ).
This theorem is due to Kakutani.
( ~) Infinitesimal doubly stochastic operators. In keeping with the spirit of this Symposium, we consider the conditions to be satisfied by the operator function R(t) in (1) in order that the evolution operators (3) be doubly stochastic. These conditions were essentially determined by Phillips (Czech. Math. Journal, 1962), and we shall limit ourselves to stating them without proof, since we shall have no occasion of using them. They are; (1) A(t)1
= 0; (2) Af. (t)1 = 0; (3) for f in the domain of A(t), and f in
L () L ; (p p
-1
q
+q
-1
+
= 1) we have (A(t)f, f )
~
+ 0, where f = max(f,O).
3. MAIN THEOREM. After all these preliminaries we can now proceed to state our main result, in both discrete and continuous forms. Theorem. (1) Let PI' P 2' ... , P n" .. be an infinite sequence of doubly stochastic operators in L p (S, L. (S, ~, (15)
f ) is a probability space.
,f
),
where p
>1
Then for f in Lp(S, ~ , f1
lim P P ... P P "*.... P 1'" n n nn ... ~ 1 2
and where
)
PI( f
exists in the sense of dominated almost everywhere convergence. (2) Let P(t, s) be a family of doubly stochastic evolution operators in L p(S,
2:
'}J. ), where again (S, ~ , J-I
) is a probability space
(that is, operators satisfy (3)). Then, for f in Lp(S, (16)
lim P d(t, O)P(O, t)f = lim a t-l>\)Q
t~60
1.,fI ) the limit
p* (0, t)P(O, t)f
exists in the sense of dominated almost everywhere convergence. Proof. The main device in the proof con8ists"in reducing the convergence result to an application of the martingale theorem. For simplicity 250
-9G. B. Rota
we shall only prove the discrete case. The continuous case can be proved either by reducing it to the discrete case, or else by a similar construction to the one we shall use below for the discrete case. At any rate, the difficulties involved in passing from the discrete to the continuous case are purely technical and involve no new ideas. We begin by constructing what we shall call the path space of the sequence PI' P 2' ... To this end, we consider an infinite product of replicas (S
,Z n'
n
f.J I
n
)
of (S,
r.. , /.
I.J
)
for n = 0, 1, 2,,,. namely
0.,
(S'
)", LI , .u ' r
') ::
fT
n=o
(S
n'
L
IJ)
n'''· n
It is well known (cf. Halmos, Measure Theory) that this infinite product spa-
ce is a well-defined probability space. We shall now consider an algebra of real valued functions on (S',
~
"
r' ') defined as follows. A function F
L,', t' ') is a function of infinitely many variables F(s 0' sl' s2" .. ),
on (S',
s. €- S.. We say that F belongs to 1
U
1
cQ.
if F is a sum of finite products of
the form
where f. f L ~ (S., 1
1
Clearly
Z 1., J.A 1.j, 0\', is an algebra.
We define a linear functional L on
&as follows. We first define L on functions F of the form (17) by the formula L(F)::
S
foP1 [ fl2 [f2 ".
[phfhJ]
".]
df
S and then extend L by linearity. It is easy to see, using the fact that PI:: 1
n
for all n that L is well-defined. We shall now verify that L has a very important property; it
251
- 10 -
G. B. Rota
is a positive linear functional. In other words, we shall prove that if F is in
& and takes only non-negative values,
then L(F) ~ O. This appears at
first sight not to be a trivial statement, because a function F in
~ is a
linear combination of functions of the form (17), and each of the summands may take negative values, even though the sum is always non-negative. To prove this statement, we notice that if
for all so' sl"'"
sk' then, changing the variable
sk to s and remem-
bering that P k applies only to functions of the variable s, we get
(18)
This expression is non-negative for every value of the variables so' sl'"'' sk_1' s. because P is a positive operator. Next, we change the variable sk_l to s, and then apply the operator P k-l' that is
(19)
and again this expression is positive, because P k-l is a positive operator. Proceeding in this way down to k = 1, we finally obtain that
for all s in S. Hence, integrating, we get L(F) ~ 0 as we wanted to show.
We now remark that the algebra r!l.has a very important property. 252
- 11 -
G. B. Rota If F belongs to
8, and
F + = max(F, 0), then F + belongs to
eft.
This is
clear if F is of the form (17) for then
Therefore, to establish the assertion, it suffices to consider the case when both F and G depend upon two coordinates only: for then an easyinductlon
-
+
-
will yield the general statement. Let F = - min(F, 0). Then F = F - F , and furthermore the functions F this, it is easy to see
~hat
+
and F
have disjoint supports. From
if F = P-Q, where P and Q are non-negative
functions with disjoint supports, then P = F
+ and Q = F-. Using this fact,
the statement we are to prove reduces to proving that F+G can be written as the difference of two non-negatjye functions with disjoint supports. Let
= P -Q, say. The functions P and Q are non-negative and it is an easy verification + that their supports are disjoint. Thus, P = F and Q = F as we wanted to show. At this point we can define a seminorm on (20)
L(l) = 1 .
253
0( as follows
- 12 -
G. B. Rota
Since L is a positive linear functional, the positivity and triangle inequality
& whose norm in 0 form a linear subspace N ; taking the quotient vector space &IN = B we obtain canonically
follow at once. Clearly, those G in
a norm on B. We now complete B, thereby obtaining a Banach space C. We now claim that C is an abstract L-space. This is intuitively clear, but for the sake of completeness let us verify the statement. First we claim that C is a Banach lattice.
Now~
a
is certainly lattice-ordered;
thus, to show that C is a Banach lattice, it suffices to show that B is a lattice, since the completion of a lattice-ordered vector space is again lattice-ordered, and is in fact a Banach lattice. Thus, it all boils down to showing that if. F ~ 0 and 0 ~ G ~ F , with L(F) = 0, then L(G) = o. But this is an obvious consequence of the positivity of L, which gives L(F)
?
L(G)
~
O.
The fact that ~ x+y/I = /1 x II + II y /1 for x and y ~ 0 in C is also an evident consefluence of (20). Thus, C is an L-space. We now apply Kakutani's theorem and represent C as L 1(S t><J'
2:. />4'
f,.. ), where (S;o ,~..a, fA ~ ) is a probability space.
This is the space in which we shall now define a martingale. We shall need the following well-known (Cf. e. g. Rota, Padova Rendiconti, 1959) lemmas. Lemma 1. Let (S ta , / ' M
(both over probability spaces) such that T1 = 1
)
for all f
~
O. Then there is a
ff ) is isomorphic to
such that L1 (S, ~
,
Lemma 3. Let
~
be a
Ll (S
J:.. c
tXJ ,
Q -subfield of
,r
the orthogonal projection of L 2(S.;:oo
(Xj
c8
IT -subfield
' f p;:;
ZIXl'
0
of.l t><J
).
and let E be
' f ~ ) onto L 2(S..c '
r..;foa ),
Then E is a conditional expectation. In particular, E can be extended to a bounded operator in L p (S
ptJ
£. ~ , f!">(l )
->
f(s ) ~ x , where x l- C. This mapping preserves the o norms of all positive functions. Hence, by Lemmas 2 and 1 (a), there exists a
(j -subfield of
subfield with
£
.2::
isomorphic to £
'IQ
: we shall identify such a
altogether, and identify L 1(S,
r
,f' ) with
L~ (Soo ,L: 'f-ee)'
a
We now come to the crucial point. Consider the algebra of all functions of
rr-
&
n
which are "independent II of the coordinates so' sl'"
.. , s , Under the canonical map into Band C successively the closure of n
0(. n in L1(S
L 1(S pO ,
j,>¢,
z:. co' ft
L"f\ ,~IIJ
L
(8 "",
for all h. We shall take this dense subset to be
n'
f
t)O
)
it will be true
& n (or rather its image in
L t>4 , but for simplicity we are identifying the two). Thus, it suffices to choose h(s ,sl' s2"" ) = f (s ) f l(s 1)'" f k(s k)' Then the right o n n n+ n+ n+ n+ side of (22) equals for g = pI P~ 1 ... PI' f(s ), n
J(J8
n-
n
(>i*:i(
-
P 1P 2 · .. Pnl (P n P n - 1 .,' PI f) fn Pn+d fn+1, ..]
Now, for a doubly stochastic operator,
f
Pqdf
=
f
]
(p' l)q d(1 =
Hence this last expression simplifies to
Jr8
(P n P n -1 ... PI f) f n P n+ 1
=f8
f P 1P 2 .. · P n [ fn P n+1 [fn+1'"
=1
~*
t
[
f n+ 1 . ..
]
d
r -_
JJ .. J d~
hfdfl4 ,
810
as we wanted to show. This shows that E f = g. n
Next - and last - we prove that if q = q(sn)' then
256
df'
=
f
q df '
- 15 G. B. Rota E q = P 1P 2 ... P q(s ). This is very easy. It suffices to verify that for o n 0 f = f(s ) we have
o
But the integral on the right is (by definition! )
f
S
fP 1P 2 · .. PnQdr
hence the proof is complete.
4.
APPLICATIONS. It is easy to get now a solution of our problem (a). We have that
p 2nf converges dominatedly if P is selfadjoint. This last fact was discovered independently by E. M. Stein. In the continuous case, we infer the dominated convergence of
->.~ if pt, in addition to being doubly stochastic, is also selfa-
ptf as t djoint.
The reader may perhaps wonder how the result of the main theorem (originally published in Bulletin of the AMS, 1962, p. 94) was arrived at. The answer is that doubly stochastic operators are related to conditional expectation operators in much the same way as contraction operators in Hilbert space are related to orthogonal projections. It is this analogy that has suggested the present result, although of course the proof has to be based upon entirely different principles. As an interesting unsolved problem we shall mention the convergence of
where P. are doubly stochastic operators or even conditional expectations. 1
257
CENTRO INTERNAZIONALE MATEMATICO ESTIVO ( C. I. M. E. )
. S. ZAIDMAN
EXISTENCE AND ALMOST-PERIODICITY FOR SOME DIFFERENTIAL EQUATIONS IN HILBERT SPACES
ROMA - Istituto Matematico delllUniversitA
259
EXISTENCE AND ALMOST-PERIODICITY FOR SOME DIFFERENTIAL EQUATIONS IN HILBERT SPACES by S. ZAIDMAN
§ 1. Let H be a Hilbert space; ( , ) is the scalar product and
II II
the norm in this space. Consider a linear closed operator A in H , with domain DA dense in H, and let Alit be the adjoint of A . Denote by J the real axis - 00" t < +110
,
and by KA(KA'it ),
