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 <. 00 with e -OA = 1 and has the semi-group property: -tA -sA -(t+s)A e e =e for all s, t ~ O. The generation of such semi-groups has been discussed in detail in Professor Phillips' lectures. We reproduce here some of his results that we need, with several
addition~l
remarks.
Definition 1. 1. We say -A €: (Bo) if 1) A is densely defined and closed. 2) any
'A > 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<
B,
DC
e -i g
t >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
-<0-
T. This means that there are
such that
(1) This theorem is first proved by Tanabe (see [10, 11, 12
J ).
The last proposition regarding the analyticity is due to Komatsu (7]. The proof given here is somewhat different from theirs. See also Yosida [17]. 14
- 13 T. Kato
and
I
A(t) -1
ii) DA(t)
I~ M.
= D is independent of t. This implies that A(t)A(s) -1 E B
for any sand t. iii) A(t) A(O)
that
-1
II
is Holder continuous (in norm), This is equivalent to
IA(t) A(s) -1 - 1 I
.:: M(t-s) i
,
Then there exist a unique evolution operator U(t, s) with the following properties: 1) U(t, s) f B and strongly continuous for 0 ~ s !f t ~ T,
2) U(t, s) U(s, r) = U(t, r), U(t, t) = 1 , 3) U(t, s) is strongly continuously differentiable in t for t
IA(t) U(t, s) I ~
C"J
.::st U(t, s) = - A(t) U(t, s)
>s
, with
const. t-s
4) If u f D, then ';s U(t, s) u = U(t, s) A(s) u ,
s
~
t ,
If, in addition, A(t) -1 is analytic in t , then U(t, s) has an analytic continua-
I
I
tion for complex s, t such that arg (t-s) are sufficiently small. Remark 2.2. a) ii) and iii) express thatA(t) depends on t "smoothly", Note that
U)
of ii) is equal to the iN -
b) The assumption \A(t) -1
I (M
is
~
of Def. 1. 5, for some fixed
~ade
~
.
only for simplicity; if this is not the
case, then we may make a transformation u(t) _>
e-
~ t u(t) in (E) so
that the new equation has A(t) satisfying the above condition (this is allowed since we are interested only in a finite interval 0 :: t ::: T).
c) The assurop··
tions ii) and iii) will be weakened later. But it should be remarked that these
15
- 14 T. Kato
2 are satisfied if A(t) is an operator in X = L (.Il.) determined from an elliptic differential operator with smooth coefficients on a bounded region
.n..
with a smooth boundary, with the Dirichlet boundary condition. In this case DA(t) = H2m( il )
2.
n H:(.Q..)
and is independent of t.
Proof of Theorem 2. 1.
We construct the approximating operators A (t) by (4). A (t) is n n bounded by An(t) ~ (M+l)n since n(t) ~ M. But the important fact is
I
IJ
I
I
that -An(t) belongs to the class (HO) uniformly in t and n in the sense stated in i) of Theorem 2. 1'. We have namely ( 5)
(6)
A (t) n
-1
= A(t)
-1-1
+n
,
Also a straightforward computation leads to 2 (~+A (t))-1 = n 2 (n~ +A(t))-1 +~ n (n+ A) n+" n+t1
from which an elementary geometric consideration gives
I arg 1\1/ ~ 2'11 + oJ
(7) where M' is in general different from
M but may be taken independent of
n. Finally, we have A (t) A (s)-1 = A (t) (A(s)-1 + 1..) = J (t) A(t) Als)-1 + 1 - J (tl, n n n n n . n (7')
A (t) A (s)-1 - 1 = J (tl (A(t)A(s)-1 - 1) , n n n hence (8)
I An(t) An (s)-1 - 1 I ~ I
M jA(t)A(s)-1 - 1/
I
~
M2(t_s)
In particular A (t) - A (s) ~IA (t) A (s)-1 - 111 A (s) 2 en n n n n ~ M (M+l) n(t-s)- , so that A (t) is itself Hglder continuous.
n
16
co
I
~
- 15 T. Kato
Therefore, the evolution operator U (t, s) for A (t) can be construc-
n
n
ted as stated above. To prove that s-lim U (t, s) exists, however, we have to deduce other "1->00 n expressions and estimates for U (t, s). To this end we first note the identity
n
:
"s
U (t, s)e -(s-r)An(r) = U(t, s)(A (s) _ A (rne -(s-r)An(r) n n n n
Integrating on r (9)
~
s
~
U (b, r) = e n
t , one obtains
-(t-r)A (r) ) n +
t r
U (t, s) K (s, r)ds , n n
where (10)
K (s, r) = - [A (s) A (r) -1 - 1J A (r)e -(s-r)An(r) n n n n
K (s, r) has the estimate
n
(11)
, Kn(s, r)
I :;
C1~ 9
(s-r)
where C is a constant independent of n. This follows from (8) and -(s-r)A (r) -1 An(r)e n ~ C(s-r) ,which is in turn a consequence of
I
I
-A (r) E- (HO) (Theorem 1. 7. ). n
(9) may be considered an integral equation for U (t, r). This may be
n
written symoblically U n
(12)
= U(o) + U n
n
'*
K n
and can be solved by successive approximation: :.c
( 13)
Sn = "'--
U
k=o
U(k) = U(k-1) n n
*
K . n
The possibility of this successive approximation is guaranteed by the estima. te (11) 1) for Kn and the eshmate
IU(0) (t, r) I
~
C (we use the same symbol
(1) The important fact is that K are dominated by an integrable kernel inden pendent of n. 17
- 16 -
T. Kato
C to denote different constants). It follows that the series (13) is convergent uniformly, being majorized by a series independent of n • Now we can let n -> oc·· (14)
•
Then
U(o)(t s) = e -(t-s)A(s) '
U(o)(t, s) -,. n s
(14) is exactly the construction of the semi-group e - 1: A(s) as the limit of e-
1:' An(s)
(see Phillips' lectures).
Also (15 )
K (t, s)
n
by (7') and J (t) n
-;>
s
--7>
K(t, s) = -(A(t)A(s)
-1
-l)A(s)e
-(t-s)A(s)
1 (which is a basic fact proved in the generation of semi-
groups, see Phillips) and 't"')O,
(16)
which follows from (3) of
§
1. We note that the strong convergence (14) and
(15) are uniform in sand t (for t-s
~
\)(
>0
in (15) ).
It follows from 04) and (15) that U(l) (t, s) = uta) *" K (t, s) -> u(1)(t, s), (1) (2f, n n s then U (t, s) = U 't K (t, s) ---i> 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
<" .'(
<: 1,
= A'" AI:>
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(
<114m.
Outline of the proof. Here it is impossible to give a complete proof which is in principle the same as the proof of Theorem 2. 1. We shall indicate briefly the essential points in the proof.
(1) A similar theorem was stated by Sobolewski [9J in the special case where A(t) are a positive self -adjoint operator in a Hilbert space.
23
- 22 -
T. Kato
We again construct a sequence
tAn(t) f of bounded operators that appro-
ximate A(t). Here it is convenient to choose A (t) = A(t)J (t) , n n
(28)
J (t) = (1 + ~ A(t)h) -m , n n
that is, (28')
It is easy to show, as in the case h = 1, that A (t) satisfies the conditions i),
n E), iii) with constants M independent of n. The evolution operators U (t, s) n for A (t) can be cons+ructed as before, and we want to show that s-lim U (t, s)= n n
U(t, s) exists. To this end we deduce an expression, similar to (13), which is uniform in n. Again we use the identity (9), but the estimat e (11) is no longer true for m
> 1.
At this point we use the following identity, essentially due to So-
bolewski (see [9 (29)
J). m
A (s) - A (r) = \"""" A (s) Ln n n p=l
1- h h -h h P (A (s) A (r) - 1) A (r)p .
n
n
n
If we put (29) into (10), the identity (9) becomes an integral equation involving the quantities (30)
X (t, s) = U (t, s) A (s) nq n n
1-qh
q = 1, ... , m
,
To get a closed system of integral equations for X ,we multiply (q) from nq 1-qh _ right with A (r) , q - 1, ... , m. The result may be written symbolically n m (31)
X
nq
= X(o)
nq
+\
~1
X
np
'*
K
npq
where
24
q = 1, ... , m ,
- 23 -
T. Kato
X(o) (t, s) nq
e -It-slAnts) A (s)l-qh = A (s)l-qh e -It-slAnts) , n
n
(32)
The system (~l) of integral equations can be solved by successive approximation : c()
X
(33)
nq
=2-
k=o
m
x(k) = '\ x(k-1) np nq
x(k) nq
f=-1
*
K npq
The question is whether there is a uniform (independent of n ) majorizing series to (33). Such a series exists since we have the uniform estimates
I X(o) (t, s) \ ~ nq -
c (t_s)l-qh
(34)
\K npq (t,s)lz.,
C (t-s) l+ph-qh-9
(34) follows from the fact that -A (t) satisfies i), ii), iii). (note (27")). n The strongest singularities on the right of (34) occurs for q = 1 and
p = m. Then l-qh = 1-h and l+ph-qh-
9
= 2-h-
<. 1 by hypothesis. Thus
these singularities are not too strong, and the series (33) converges uniformly in n, as we wished to show. It follows, quite in the same way as in Theorem 2. 1. that s-lim
U (t-s) = U(t, s) exists and satisfies the conditions 1), 2) . n
To prove the differentiability of U(t, s) and the existence of A(t)U(t, s)
E- B , we need another expression for A (t)U (t, s) similar to n
n
(21). This can be obtained by deducing from (18) a system of integral equations satisfied by (35)
Y
nq
(t, s)
A (t)qh u (t, s) n
q
n
25
= 1, ... ,
m ,
- 24 T. Kato
and solving this system by successive approximation. For q
=
m ,
Y (t, s) = A (t)U (t, s) and this will give the desired differentiability of nq n n U(t, s) as in the proof of Theorem 2. 1. Here again we have a slight complication in constructing the first approximation y(1), similar to that encountered nq in (22) . But the main feature of the proof is the same as before, and the details may be omitted.
6.
Theorems 2.4. and 2. 5, generalize the earlier theorems, Since, howe-
ver, the fractional power A(t)h of a given operator is defined only in an abstract fashion and not given in a simple form, the question arises how one. can verify the assumptions of these theorems. There is a rather general case in which these assumptions are known to be satisfied. Let X and X' be two Hilbert spaces such that X' C X (algebraically and topologically), Let a(u, v) be a continuous sesquilinear form on X' such that Re a(u, u)
~ aI u 1,2,
J> 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, <.
t;
c- (HO) such that DA C X'
DA and v r:: X' (d. Lions'lectures). Now it
is independent of A (dependin,g only on X') for
1/2 (see[4J),
Consider next a family a(t X' such that Re a(1; u,u)
j
u, v) of continuous sesquilinear forms on
~ SI u 1,2.
Let A(t) be the operator associated
with a(t; u, v) in the way described above. Then DA(tt
o : ; . '" ( ponent
is constant if
1/2, Suppose, moreover,that a(t; u, v) is Holder continuous with ex-
e.v
for each fixed u, v e: X' . Then it can be proved that A(t) ", A(O)- )(
is H~ider continuous with exponent such a case with m = 3, h = 1/3 if
g .
Thus Theorem 2. 4. is applicable to
9 > 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_" <..
,..t
)
cr
1
ds 2 s
-=-
\~
j
"t'
~
o
(see (27")). But (set s = r
C
dS2,
s
V:o, )
r"r+'
dr
1
~
DJ 0
err>
~
. ;,
(t'
8
-
I)
\'>
(t _o-){'o )
~
Ilcr A +?> (q. e. d.)
By Lemma 2.7. we can estimate 11 :
J
t -1+9 '-1+9 -1+29 1 11 (t, r) \ ~ C r (t-s) (s-r) ds f C(t-r) ,
It is easy to estimate 12 :
\
12(t, r)
I ~ C(t-r) -1+~
13 is further divided into two integrals 't .. t
1 (t r) = I' + I" 3'
3
where
3
=
JT 2-
t
I~(t,r) « J t
:;: C
(\H(t,s)\ +\H(t,r')I) \A(r)e-(s-r)A(r) Ids
Ht i.
J [(t-s) -1+0 + (t-r) -1+[j] (s-r) -1 ds 'c.t! .<:
28
~ C(t-r)
-1+9
- 27 T. Kato
3
In estimating 1 (t, r), we use the identity: H(t,s) - H(t,r)
= A(t) [e-(t-S)A(t) _ e-(t-r)A(t~(A(t)A(s)-l - 1)
_ A(t) e-(t-r)A(t) A(t)A(s)-l(A(s)A(r)-l - 1) Using the second inequality of Leznma 2. 7 with
C -1- (J' 9fit IH(t, s)-H(t, r) I~ ,&(t-s) (s-r) (t-s)
(1 = 8
+C(t~)
-1
, we have 9
(s-r) ~ C(t-r) l'f
and hence
<. t+r s "-2-
-1
(s-r)
,
r+t
II3(t,r)l~ C \~
(t-r)
-1
Q
(s-r)
Collecting the above results, we obtain (22).
29
(s-r)
-1
ds
~
C(t-r)
-1+9
.
9
- 28 -
T. Kato
§3.
THE CASE IN WHICH -A(t) ARE GENERATORS OF ANALYTIC SEMI-GROUPS WITH VARIABLE DOMAIN.
1. In this section we prove a theorem, due to Tanabe 1). in which no assumption is made on the constancy of DA(t) or of DA(t)n . Theorem 3. 1. Assume that i) -A(t) E- (HO) uniformly in the sense of Theorem 2. 1. , i).
!
ii) A(t)-l is differentiable, with
A(t)-l HHlder continuous (in
norm). This implies that the derivative exists in norm. Also this implies that d
ill (A
+A(t))
-1
A(t)
=
A +A(t)
d
(ill A(t)
-1
)
A(t) ;\ + A(t)
(1)
A
d
= (1 - .\ + A(t) ) (ill A(t) exists and is analytic in
-1
A
)(1 - -A-':+""-A'-(-t) )
A.
iii)
M
for some constants M,
5'
such that
O~
) c:: 1 .
Then there exists a unique evolution operator U(t, s) with the properties stated in Theorem 2. 1. (including the case in which A(t) -1 is analytic). Remark 3. 2. If one is satisfied with construncting U(t, s) which is only weakly differentiable, one may omit the assumption of HHlder continuity of d -1 dt A(t)
(such U(t, s) is still unique).
(1) See (13] and [6].
The proof given below is slightly different from the
original.
31
- 29 -
T: Kato
Proof of Theorem 3. 1. Again we use the approximating sequence A (t) un
sed in the proof of Theorem 2. 1. and construct the approximating evolution operators U (t, s). n
To obtain a uniform expression for U (t, s), however, we make slight n
modifications in the method used in Theorem 2. 1.
§ 2,
Instead of (9) of
we ded\lce a different integral equation for Un'
starting from the identity ru-(t-s)An(s) . e U (s,r ) = K ( t-s,s) U (s,r),
~ . vS
(2)
n
n
n
where
f') J s
K (we . s) = _ _
(3)
n'
(
e
- 1:' An s)
.
Note that two other derivatives in (2) cancel each other since 'J . r'\ _ r::A (s) - 't'A (s) ;:::- U (s, r) = -A (s)U (s, r) and ~'J'""" e n = -e n A (s) • . vS n n n v n On integration with respect to s on (r, t), (2) gives the integral equation
u
(4)
n
= U(o)
n
+~
n
*,"U
n
where
t
~~Ol(t. sl ~ e -(t-sIAn(s)
(5)
" K (t, s) = K (t-s' s) n n J
(U(o) and K n
n
differ from the operators U(o) and K n
n
used in the proof of
Theorem 2. 1. ). Now it can be proved that 1\.
(6)
\
Kn(t, s)
C I ~ --~ (t-s)
where C is independent of n (for the proof see n. 2), Since
f <.
1 , (4) can
be solved by a successive approximation used in the proof of Theorem 2. I, ,
32
- 30 -
T. Kato
yielding a series expression of U (t, s) converging uniformly in n. It foln
lows by the same argument that s-lim U (t, s) = U(t, s) exists and has the 'Vi":'')oO
n
properties 1), 2) of Theorem 2. 1. To prove 3), we use another integral equation obtained from
ln U t(, s)e -(s-r)An(s) = U (t, s)K 'vS
n
n
n
(s-r,. s) .
After integration and multiplication from left by A (t) , this gives the fo11on
wing integral equation for Y (t, s) n
(7)
y(o) + Y
Y
n
i
where
n
n
=
A (t) U (t, s) : n
n
* ~n
y(o)(t, s) = A (t) e -(t-s)An(t) ,
(8)
n
n
A.
H (t, s)
n
=K
(t-s; t) .
n
Again we have the uniform estimate (see n. 2)
"
(9)
\ H (t, s) \ n
c9
~
(t-s)
I
. For Yn(0) ,however, we have only the estImate Yn(0) (t, s) I ::; C(t-s) -1 of which the right member is not integrable. Thus the estimate of y(1)(t, s) n
is not quite simple. If one uses the Hl:llder continuity of :t A(t) -1 , however, we can carry out an estimation of y(l) , obtaining n
I y~l)(t'S)1 ~
'(10)
(t-s)
where C is independent of nand tyof
d
.dt An(t)
-1
d
= dt A(t)
-1
c
c 9 +
9
(t-s)
1-9
is the exponent of the Hl:llder continui~
(see (5) of '.) 2).
Since the right member of (10) is integrable, there is no further difficulty in proceeding with higher approximations y(k) , k n
=
2,3, .... The remai-
ning arguments are exactly the same as in the proof of Theorem 2. 1. and 33
- 31 T. Kato
may be omitted, q. e. d.
2.
Proof of (6) and (9). It follows from (6) of
U
"'ut
\
(1\
+An(t))
§2
-1
=
that
n2 (n+ A)
2
l
'J t
(~+ A(t))-l n+A
Hence by iii)
c where C is independent of n. The last inequality has the same form as iii) for A(t). Therefore, we II
1\
may prove, instead of (6) and (9), the same estimates for K and H. omitting the subscript n . From (1) of ~ 1 we have
'J e -t' A(s) =- 1 .
J
I!-~ 2~"
\ IAI1-~
K( 1:' . s) :: -
21T'1
'I\)s
Hence
I K(l' ;s)
as in the proof of (3') of
§ 1.
)C \e At' \ /\
C
e
h'J. -1 ( A+A(s)) d A '1)8
~AI ..,--L Irl~ II
Since K(t, s) = K(t-8;8) and H(t, s) = K(t-s;t),
(6) and (9) follow immediately.
34
- 32'-
T. Kato
~ 4. THE CASE IN WHICH -A(t) ARE CONTRACTION SEMI-GROUPS.
1. So far we have been considering only the cases -A(t) E- (HO). If we assume only that -A(t) E (BO), there arise several difficulties. In this case we can still construct the approximating sequence
LAn(t) t
and the associated
evolution operators Un(t, s), but the uniform expression such as (13) of
§2
is difficult to obtain, for it is essentially tied up with the property that -A(t) f (HO). It is easy to prove that U (t, r) - U (t, r) = ( n m )
t r
U (t, s)(A (s)-A (s)'U (s, r)ds, m m n~n
but it appears to be very difficult to prove the strong convergence of Un from '. this identity, although this is the very identity used in the construction of U(t, r) = e -(t-r)A when A(t) = A is independent of t (see Phillips) -(s-r)A n, a fact essential In that case A - A commutes with U (s, r) = e m n n in that construction. Here we will state a theorem in which, among others, we assume that -A(t) E- (CO) (generator of a contraction semi-group). No satisfactory theorem is known in the general case of (BO). Theorem 4. 1. Assume that i) -A(t)
e (CO),
that is,
(for convenience) that A(t) -1 ii)
DA(t)
I (A +A(t))-I I~.; ~
, A>0.
Also we assume
B, \ A(t) -1\ ~ M •
= D = const (again this implies that A(t)A(s)
-1
6: B).
iii) B(t) = A(t)A(o)-I IE- B is strongly continuously differentiable. (This implies that B(t) is continuous in norm, so that, since -1 -1 B(t) = A(o)A(t) If. B by ii),
I B(t)
I ,;
M ,
35
~
33
~
T. Kato
Then there exists a unique evolution operator U(t, s) such that 1) U(t, s) E B , strongly continuous, for 0
~
s
~
t
6-
T.
2) U(t, sluts, r) = U(t, r) , U(t, t) = 1 . 3) A(t)U(t, s)A(s)-1 ;:: W(t, s) t:- B , strongly contiIluouS in s, t . 4) U(t, s)A(sr l is strongly continuously differentiable in t for t
? s, with U -1 -1 J t U(t, slAts) = A(t)U(t, slAts) ~ B .
Remark 4. 2. 3) implies that U(t, s) D cD, and 4) implies that
J
-;;-t U(t, s)u = A(t)U(t, s)u , v
u E- D
Theorem 4. 3. Under the assumptions of Theorem 4. 1., the results of Theorem 1. 11. are true (with (E ) 'and (S ) replaced by (E) and (S), respectively). o 0 2. It is not possible to give here a complete proof of these theorems. We shall give a sketch of the proof of Theorem 4. 1. , referring for details to [1] . The proof of Theorem 4. 2. is similar to that of Theorem 1.11. The idea of the proof is to use the approximation of A(t) by step
func~
tions. This is equivalent to approximate U(T, 0) by a product (1)
where A. = A(t.) and 11 denotes the partition 0 = t .(. tl .( ... '- t = T J J o n of the interval [0, TJ . For simplicity we introduce the following notations
(2)
\ X = e ~(trtj_l)Aj
B.
l
W. = A,U'.JA JJ h 0
U;k" Xj Xp ooX k '
J
= B(t.) J
-1
Since -A j E- (CO), we have \ Xj \ ~1 and \U jk 1~1. It can also be shown that W. are uniformly bounded. To see this we use the recurrence formuJ 36
- 34 T. Kato
la (3)
which can be verified in a straightforward way if one notes the commutativityof Ak and X k : AkXk :? XkAk • (3) is a recurrence equation of Volterra type, and can be solved by successive approximation (actually in a finite number of steps). The solution may be written W.
J
=
(4)
oc
L
W(o) = U j jl
yjp)
p=o
j
,
J:.
(p+l) _ ) -1 W(p) W. -_ U'k (Bk - Bk _1)B k _1 k-l ' J k=l J Since \ B~~ 1 \ f M by iii) and (5)
{\Bk -Bk _1 \ k=l
~ JT\dB(t)\~JT\B'(t)!dt 0
::N,
0
it is easy to see that the series (4) for Wj converges uniformly (with respect to the partition (j ,if one supposes that the sequence {Wj
1has been exten-
ded to a step function defined for all t), In particulat Wj are
un~formly
boun-
ded:
I Wj
(6)
1 L. C
= const.
3. We now show that U Lt (T, 0) has a strong limit for
\
~ \ =-
mrx \ t j - t j -1
\ -;:, 0 ,
To this end we consider a partition
IJ
of
of
!J.'
IJ
t,
Let
!J.
11'
of [0, TJ and a sub partition
be 0 = to <. t 1 .( .. ' <. tn = T. The partition points
form a subset of {to' t 1, .. , , tn ~ . By using the semi-group pro-
37
- 35 -
T. Kato
perty of
~ e - 1: Aj ~
,however we can write
D L\ = DA (0, T) = Xn, ... ,Xl'
Xj = e
D
X! J
= D ,(0, T) = X' , ... ,Xl' , &1 L\ n
-(t.-t. I)A.
J J-
J,
I)X, = e -(t.-t· J JJ,
with the same number n of factors. Here A. = A(t.) , A! = A.(t!) , where t! is equal to sopte tk such J J J JJ J thatltj-\
I ~ I~'I
We have then
n
(DLl.,-D. )A- 1 0
u.
=L j =1
D' . l(X!-X.lD. I 1A-1 n, J+ J J J-, 0 DI . 1 (X! n, J+ J
X.)A~ll J J-
W. 1 J-
where Dol = 1, D'
n,n+ I = 1 by convention. Hence
because
I D'n, J+"" . 1 I<
and \W. 1 J-
I
I~ c .
It follows from Lemma 4. 4. (to be proved below) tha.t
I <.) (X! - X.)A~ll I A.A~11 I I(X!J - X.)A~ll J JJ J J J JL
-
M (t. - t. 1)
J
J-
.( M (t. - t. 1) .... J Jt:..
-
M2(t. - t. 1) J J-
-1 I(NJ - A.)A. I J J
I(B!J - B.)B~ll J J I B~J - B.J I .
Summing this over subintervals in
6. 1 gives
a quantity
~ M2( f
dB(t))
• l'
38
/J.
of a fixed subinterval II..y of
I
(t. - t._ 1) J J
~
M2/1J II
r .J
I'
dB(t).
-;
- 36 -
T. Kato
Summing the results over all
= M2N \ 4
'I.
I'~
we get a quantity '" M2
I_ 'I J:dBltl ~
Hence by (7) (8)
[0,
J. If
!l
their common subpartition, we have (8) and a similar inequality with
f:..
Now let
replaced by
Ll' tl. ".
11 "
and
be arbitrary partitions of
T
is J
Hence
(9)
This shows clearly that lim Ut.
A~ 1 exists in norm when
I
I~ I -7 O.
is uniformly bounded ( Ut.I ~ 1) and DAis dense, it follows
Since U to
o
that U = U(T,O) = s - lim U!J. (T,O) exists. j!J.I->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
~
s .( t k , -
is a given partition of [ 0, TJ . Then it is easy to show that
39
- 37T. Kato
I
(10)
[U(t, s; d
) - U(t, s)]
I
A~1 ~ c
IIII
so that (11)
I
!1 ) - U(t,S)JA~1
[U(t,s;
lill->o.
\ -"> 0,
Thus U(t, s)A -1 is the uniform (in s, t) limit in norm of U(t, s;
o
d )A- 1 0
for 1111 -> 0 , so that U(t, s) is the strong limit of U(t, s; L1 ) (uniformly in s, t). Since U(t, s;
!J )
is strongly continuous
with U(t, s), with U(t, t) = 1 (since U(t, t;
4.
in s, t , the same is true
L1 ) = 1).
To prove 3), we return to the expression (4) for W.. After extending
tWj } and tw~p) 1as step functions for all values of t (~s mentioned after W~p)
(5)), we can prove by ridtictfon that each term function W(p)(t) for respect to W II = W Ll
il ,
In view of
unifor~ convergence of (4) with
it follows that W. -?>W(t) =
r..
W(p)(t). In particular p=o -1 (T, 0) = Wn = A(T)U A (T ,0 )A 0 -'> W = W(T, 0). Hence
J
-1 U(T,O)A o = lim that
\1\ -) O.
tends strongly to a
-1. -1 U Ll (T,O)A o = lim A(T) Wt,
-L = A(T) -W(T,O) so
Ll -'>0
A(T)U(T, O)A -1 = WiT, 0) If B
o
This is a special case of 3). The general case and the continuity of Wit, s) can be dealt with in the same way. To prove the differentiability 4), we note that U(t+ C s) - U(t, s) = (U(t+ J
(U(t+ t, s) - U(t, s))A(s) where Wit, s) = A(t)U(t, s)A(s)
-1
-1
e
1 -1 / (U(t+t,s) -1) A(t) c
~
,t) - I)U(t, s),
= (U(t+E
J
t)-l)A(t)
-1
Wit, s) ,
B. To prove 4) it suffices to prove that
-> 40
-1
J
I(;
J0
.
- 38 -
T. Kato Now we have (apply (ga) to the trivial partition t = to
Ct,
t+~J)
I (U(t+ ~,t)
()
- e- tAt )A(t)-1
I~
C t
j
t+t
<
t1 = t+
~
of
/BI(s)idS
t
Hence :
) (U(t+ L t) - e -
~ A(t))A(t) -1 j-?:o 0 ,
cJ 0 .
But \ :
I ->
(e- EA(t) _ 1)A(t)-1 + 1
0
by Theorem 1. 2.. Hence
I ! (U(t+ ( ,t) - 1 ) A(t) -1 + 1 I -»
~~0
0 ,
as we wished to show. Actually this proves only that ~
''Jt)
+
U(t, slAts)
-1
= W(t, s)
J
+ denotes the right derivative. But since W(t, s) is known to be '""J -1 strongly continuous, it follows that 1t U(t, slAts) exists and equals
when
(~)
W(t, s).
5. Remarks. a) It is not easy to extend for \ Xj
I~ 1
Theor~m
4.1. to the case -A(t) f (BO),
is used essentially. Also it is not easy to weaken the assum-
ption that DA(t) is constant. However, these generalizations are possible under certain circumstances. Suppose that there is an operator othly together with Q(t)
-1
f
Q (t)
~
B, such that -Q(t)
B , depending on t smo-1
A(t)Q(t) E- (CO). Suppose
furthermore, that there is another operator R(t) E B, again depending 41
- 39 -
T. Kato
smoothly on t together with R(t) -1 E B , such that R(t) -1 A(t)R(t) has a constant domain and a continuity property similar to iii) required of A(t) there. Then Theorems 4. 1. and 4. 2. are seen to hold (se~ [2
J).
b) In the beginning of this section, we stated that it is difficult to prove the existence of lim U (t, s). Itt -;:.00
Once the existence of U(t, s) has been esta-
n
blished as in Theorem 2. 1., however, it is not difficult to prove that U (t, s) -> U(t, s) strongly.
n
c) The method used in the proof of Theorem 4. 1. is analogous to the difference approximation to the differential equation, but not exactly the same. The true difference method would be to use the scheme u(t.) - u(t. 1) J J+ A(t.)u(t.) = 0 t.-t. 1 J J J J-
(12)
(we set f
= 0 for simplicity)' choosing the backward difference. This gives
u(t.) = (1 + (t.-t. I)A.) J J JJ
-1
u(t. 1) , J-
A. = A(t.) J J
and so u(t.)
J
= (1 + (t. - t. l)A.) J
J-
-1
J
.......
It is expected, then, that (13)
lim (1 + (t - t
\.(\ 1_> 0
n
l)A) n-
-1
....... (1 + (t
n
r
t )A 1) 0
-1
= U(T,O)
.
-(t.-t. l)A. This means that we should be able to replace X. = e J JJ used in the J above proof by (1 + (t. - t. I)A,}-l , which is an operator of more elemenJ JJ tary character than X .. But the direct proof of the existence of the limit J (13) is not known, although it can be proved after one has proved the exi-
stence of U(t, s).
42
- 40 T. Kato
BIBLIOGRAPHY
[1] T. KATO, J. Math. Soc. Japan
E. (1953),
208-234.
[2] T. KATO, Comm. Pure Appl. Math. ~ (1956), 479-486. [3] T. KATO,
Proc. Japan Acad. 36 (1960), 94-96.
[4] T. KATO, J. Math. Soc. Japan
11 (1961),
246-274.
[5 J T. KATO, Nagoya J. Math. ~ (1961), 93-125. [6J T. KATO and H. TANABE, Osaka Math. J .
.!i (1962),
107-133.
[7J H. KOMATSU, J. Fac. Sci. Univ. Tokyo, Sec. I, Vol. 9, Part 1 (1961), I-II. [8J R. S. PHILLIPS, Trans. Amer. Math. Soc. 74 (1954), 199-221. [9] P. E. SOBOLEWSKI, Doklady Akad. Nauk U. S. S. R. ~ (1958). 984-987. ~O] H. TANABE, Osaka Math. J .
[l1J H. TANABE, Osaka Math. J. [12] H. TANABE, Osaka Math, J.
.!l. (1959), 11 (1960), 11 (1960),
121-145. 145-166. 363-376.
(13] H. TANABE, Froc. Japan Acad. 37 (1961), 610-613. [14J H.TANABE, Technical Report, NSF, G-22982, Dept. Math. , Univ.California, 1963.
-
f151 K. YOSIDA, Proc. Japan Acad. 34 (1958), 337-340.
[16J K. YOSIDA, Proc. Japan Acad, 36 (1960), 86-89. ~
[17] K. YOSIDA, Berkeley Sympsium, 1960.
43
CENTRO INTERNAZIONALE MATE MATICO ESTIVO (C.I,M.E.)
J. L.LIONS
EQUATIONS DIFFERENTIELLES OPERATIONELLES DANS LES ESPACES DE HILBERT
ROMA - Istituto Matematico delllUniversita
45
,
,
,
EQUATIONS DIFFERENTIELLES OPERATIONELLES DANS LES ESPACES DE HILBERT par
J. L. LIONS
Soit H un espace de Hilbert, t un parametre valle (0, T); et
A(t)
On appelle
une famille
d'op~rateurs
non
r~el,
~
~quation diff~rentielle op~rationelle
dans un interdans
H.
dans H une
~quation
de la forme:
A(t) u (t) +
(1)
2 (ou + d ~ ) dt
d~t(t)
= f(t),
f
donn~
avec u(o)
et
u(t)
~tant
donn~
(ou u(o) et duro) dt
dans Ie domaine de
A(t)
donn~s)
pour que
(1) ait un sens.
Le chapitre I donne les outils permettant de definir des non
born~s
commodes pour notre objet et qui couvrent un tres grand nom-
bre d lop~rateurs
diff~rentielles
avec des conditions aux limites
(pour des exemples, nous renvoyons tielles
op~rateurs
op~rationelles
a notre
vari~es
,
ouvrage: Equations
diff~ren-
et problemes aux limites, Springer, Collection Jau-
ne, t.111(1961)). Le chapitre II donne quelques r~sultats pour les ~quations diff~rentieler. du les du 1 ordre (1. e. contenant seulement ---cit; A(t) peut etre un op~· A
rateur - ou un systeme -
diff~rentiel
d I ordre quelconque) et contient une
introduction tres breve aux problemes non homogenes (Ies equations d I~VO lution non homogenes sont
~tudi~es
par E. Magenes et l'A. dans une note
auxC.R. Acad.Sc., Paris, t.251, 2118- 2120,(1960), etdansuntravail aux Rend. dei Lincei, 1963).
47
- 2-
J. L. Lions
Le chapitre III cont'.ent, apres quelques indications problemes non
lin~aires,
g~n~rales
sur les
unexpose,dans un cas particulier assez significa-
tif, dlun travail de I. M. Visik, Mat. Sbornik, t.59(101), p.289-325 (1962). Le chapitre IV revient aux problemes
lin~aires,
contenant cette fois
des d~riv~es du 2eme ordre en t. Nous avons insiste sur la regularite en t
de la solution, ainsi que
sur la dependance en les coefficients. Les chapitres I, II, IV ont evidemment des points communs avec notre ouvrage, loco cit. Mais nous avons essaye de presenter ici soit des resultats
a ceux de
nouveaux par rapport
notre livre, soit avec des methodes nouvel-
les des resultats deja etablis dans notre livre. (Des indications bibliographiques et des remarques sur les methodes ou les problemes non resolus sont donnees dans les commentaires places
a la fin de chaque
TABLE DES MATIEBES
Chapitre I. - Couples d lespaces hilbertiens 1. Hypotheses. Exemples. 2. L'espace VI. Exemples. 3. Vne propriete du triplet
tV, H, VI}
4. Operateur defini par a(u, v). 5. Vne propriete de certains espaces fonctionnels. c.ommentaires sur Ie Chap. I.
Chapitre II. Equations lineaires du 1er ordre. 1. Notations. Exemples.
2. Le probleme. 3.
D~monstration
4. Le cas V(t)
=
Th~oreme
d'existence.
du theoreme d I existence 2. 1. V.
48
chapitre.)
- 3J. L.Lions
5. Une application de la
m~thode
de transposition.
Commentaires sur Ie Chap. II.
Chapitre III. Introduction
a certains problemes non lin~aires.
1. Position du probleme.
G~n~ralit~s.
2. Les estimations
a priori fondamentales.
3. Le resultat final.
Th~oreme
Applications.
d'existence et
d'unicit~.
4. Cas des problemes mixtes.
Commentaires sur Ie Chap. III. ' IV . E' quatlOns , 1'mt:alres ... ' du 2~me or dreo Ch apltre 1. Position du
probl~me.
2.
des solutions
Propri~te's
(~ventuelles)
de (1. 7).
3. Theoreme d'existence. 4.
Propri~t~s suppl~mentaires
de la solution.
5. Propri~t~ de dependance continue de la solution en art; u, v)et B(t).
Commentaires sur Ie Chap. IV.
49
- 4J. L. Lions
Chapitre I COUPLES D'ESPACES HILBERTIENS
1. Hypotheses. Exemples 1,1 - On
d~signe
par
Si f, g E H, (f, g) Si
V et
H deux espaces de Hilbert.
VC H ,1J injection de
~ • J.:
u, v~ V, ((u, v)) est leur produit scalaire dans
On suppose q~.e
I f I = (f, f)
est leur produit scalaire dans H,
V, 1/ ull =((u, u)):t..
V dans
H ~tant continue.
Donc il existe une costante c telle que (1 • 1)
pour tout
lul:scl/ul/ On suppose enfin que
Vest dense dans
uE V. H.
1.2 - Exemples.
Si
0
fonctions de
On
est un ouvert de carr~
d~signe
~quivalant
Rn
sommable sur
par
m
H (0)
,on prend 0
H=L
2
(0)
pour la mesure de Lebesgue
1'espace de Sobolev d 'ordre
m
m:" u E H (0)"
a pour tout
p
avec
lci
et
(classes de
DP u est calcul~e au sens des distributions sur Si lIon pose:
51
0 .
I pi
~ m"
- 5J.L. Lions
Y9. dx)
(1.2)
on definit sur On
Hm(O)
d~signe par
tiables sur
0
une structure d'espace de Hilbert.
iJ (0)
l'espace des fonctions
et a support compact dans
0
ind~finiment
differen-
(muni, lorsqu'il y a lieu,
de la topologie de Schwartz). On Si Hrg- (0)
d~signe
0
par
m I'i'i Ho (0) l'adMrence de ~(O)
est un ouvert de frontiere
r
consiste en Ie sous espace de
(Nous nlentendons pas etudier ici Si
Co
ra verifier que
a quel
est "tres petit", alors Hm(Rn)
= H~
(Rn)
m
H (0).
suffisamment reguliere,
Hm (0)
form~ des u telles que:
r
pour
sur
(1. 3)
dans
I p/.:S
m-1.
sens (1. 3) a lieu). Hm(O)
= H~
(0). Par ex, on pour-
(r~gularisation et troncature).
On peut alors prendre: (1. 4)
(avec
/I u I =
Iu 1m) ,
ferm~
quelconque de
ou bien (1. 5)
ou, plus generalement, un sous espace
m
H (0), con-
Hm (0) . On verifie sans peine que dans tous ces cas, les hypotMo ses du point 1. 1 on lieu.
tenant
Autre exemple. Voici un autre exemple (utile) et OU bolev.
52
V nlest pas un espace de So-
- 6-
J. L. Lions
On designe par
V 1'espace des (classes de) fonctions u telle que
2 u <;;; L (0)
(1. 6)
IJ,
ou
I
a'(t + ....... + t~
=
~x~
vXc
c 'est un espace de Hilbert, et les hypotheses du point 1.1 ont lieu. On verifiera facilement, en prenant par ex. pour
~ u =0
Ie plan et
~ o x.
premieres
I
0
un disque dans
que (1.6) n 'entraine pas en general que les
soient dans
L2(0)
deriv~es
; V n'est pas un espacede So-
1
boley.
2. L'espace V'. Exemples. 2.1. On
d~signe
lin~aires
Si
continues sur V fE H,
done definit un voque de
a
par V' l'anti-dual de V ,i. e. l'espace des formes anti-
v -; (f, v) ~l~ment
H -..,. V'
(car
est une forme
Lf de
V'
anti-lin~aire
;1 'applications
Vest dense dans
continue sur
V,
f ....... Lfest biuni-
H ). On identifie alors Lf
f .Donc:
ve He V'.
(2. 1)
Si f E V' , v E V , (f, v) l'anti-dualit~.
L'espace
V'
designera leur produit scalaire - dans
est un Hilbert pour la norme
. I (f,v) I II f 1/11 V' -_sup 1/ v /I ,v E V.
53
- 7J. L.Lions
Naturellement, si definit un
~l~ment
u E V, v .... ((u, v))
1\ u de
est continue sur
V ,donc
V'
((u, v)) = (A u, v).
(2.2) On dMinit ainsi
1\
avec:
1\
(2.3)
(ou, de
fa~on g~n~rale,
continues de
X dans
E
.8 (X;Y) Y );
J8 (V;V')
d~signe l'espace
1\
des applications
est tin isomorphisme de
V sur
lin~aires V'
(isomorphisme canonique). Notons que
Si
u, vE V,
on
a ~videmment j(u, v)/
(2.5)
:::
(Cauchy-Schwarz)
Iu/·/vl
On a aussi (2.6)
2.2-Exemples. V = Hm (0), alors o est un espace de distributions sur 0 (grace au fait que 9) (0) estSi nous prenons les exemples du point 1.2, avec
V'
par dMinition - dense dans H~ (0) espace de ~'(O)
,on peut identifier
V'
a un sous
= distributions sur 0 ). On posera :
(2.7) est
compos~
des distributions
T sur 0 de la forme f
(2.8)
54
e
P
L 2 (0),
J. L. Lions
la d~composition (2.8) ri'etant pas unique. On
v~rifiera
que dans ce cas 1'isomorphisme
u->
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 <
S
S
n+l n-l
done S1 s2 sl = S ~ s~ o 1
sn ......... ~ -sn-l
.:s ........
dloll
n s. ~ s
(3.2)
1
n
sn=lr;nuI2~
Mais
e2
/1n- nu l/ 2
(par (1.1)), done
2/ ,2n 2 s n ~ e 117" I · 1/ u II
(3.3)
De (3.2) et (3.3) I 'on tire:
s 1 ::; e
2/n
// u
/I
2/n
/ITI/I
2
dlou, en faisant augmenter indMiniment '" sl ~
Irr u/ ~
II rrll
2
/I IT /I·lu I
I rru/ ~ 1111"11
, i. e.
pour tout
56
u E V; d lo1l1e
si
lui
th~orllme.
=1, i.e.
- 10 -
J.L.Lions 4. Operateur dMini par 4.1. Soit
a(u, v)
a(u, v)
une forme sesquilineaire continue sur
V ,i; e.
lineaire en u ,anti-lineaire en v ,et telle que a(u, v) ~ M /I ul/1/v II Si
u E V, v -t a(u, v) a(u, v)
(4.1)
M
= constante.
est donc anti-lineaire continue sur
= (Au, v)
V, donc
AuE VI ,
et AE/!,(V;VI) .
(4.2)
Reciproquement, si lion se donne
A E .s(V, VI) ,alors,
definie par (4.1), est sesquilineaire continue sur 4.2-0n designe par
D(A)
l'espace des
a(u, v)
V.
u E V tels
que
AUEH.
On verifiera sans peine que la condition necessaire et suffisante pour qulun element v
-?
a(u, v) D(A)
u
de V soit dans
soit continue sur
D(A) est que la forme anti-lineaire
V pour la topologie induite par
est Ie domaine de l'operateur (non borne)
A dans
H. H.
4.3. - Dans la suite, nous considererons une famille d'operateurs A(t)
dependant du parametre reel t (Ie temps), definis dans H a partir
de V (ou V(t))
et dlune famille de formes
a(t;u, v).
5. Dne propriete de certains espaces fonctionnels 5.1- De ra par
fa~on
generale, si X est un espace de Banach, on designe-
LP(0(; ~ ;X) l'espace de Banach des (classes de) fonctions f sur
( ~, ~ ) mesurables (fortement) a valeurs dans X ,telles que
1llf(t)II~dtr/P<~. 15p<~. 57
- 11 -
J. L. Lions
5.2- On designe par
W(01,
llespace des (classes de) fonctions
~)
u telles que (5. 1) (5.2)
( Le sens de (5.2) est Ie suivant: u definit une distribution sur ] 0( I
~[ a valeurs
dans V ;alors
~~
est une distribution
a valeurs dans
V ,donc dans VI et (5.2) a un sens).
est un espace de Hilbert pour la norme /I ull W( ()( , ~ ) :;
W (C( , ~)
2
(IJullC-(O('~;V)+
11'1/2 1/2 u C(Il/,~;VI)) .
r
Nous voulons demontrer Ie TMoreme 5.1. Toute fonction u de une fonction continue de Soit
C(
[01, ~ ] ; H)
[0(, ~ ] -) H
[Ol, ~J -)
est p.p. egale
W( oi, ~)
a
H.
l'espace des fonctions continues sur
,munI de la topologie de Ia convergence uniforme. Alors
(5.3) avec injection continue. Pour demontrer Ie theoreme 5.1, Lemme 5.1. Soit ction
a [c( ,~J ~
commen~ons
0<'< 0« ~<~'.~
U E: W (
i: ?')
par Ie
u E W (IX,
~)
est restri-
,telle que U soit nulle dans un voi-
sinage de o{l et ~' . Demostration Supposon (ce qui ne restreint pas la generalM ~) que ~:; 0 . De finissons v dans v(t)
= u(-t)
(- ~ ,~)
par: si
t
< 0,
u(t) 58
si
t
> 0.
- 12 J. L. Lions
On v~rifie que
v E W (- ~,~ ) • Multipliant v par une fonction
differentiable, nulle au voisinage de (0, ~), on obtient
et
w=u
p.p. sur
E W( - ~
qui
w=(Jv
J3
-
(O,r).
dans
n existe
U
m
[o(,l,P] a valeurs dans
V
1 au voisinage de
,p) ,nulle au voisinage de -0,
ProcMe analogue par symetrie aut~ur de Lemme 5. 2.
a
et egale
r.
Le resultat suit.
, fonctions indefiniment differentiables ,nulles au voisinage de r::J.' ~
r'
les que U
m
-) U
dans
W (0<
I
'r(l,' )
(U ayant les proprietes enoncees au Lemme 5. 1.). Demonstration Par regularisation en t Mais pour
U
m
,on a:
I :0 t
)U m (t)) 2 =
(Um(O'), Um (6)) d(t=
i
t
=
J ;,('
[(U'm (t5), Um (6)) + (U m (6),
u'm (()))]
d~
,
t
<2 \
(J
II Um (C5) I Ilu~(())llv'
dcr
(j.
d'ot
(5.5)
, t E
59
(c/,P')'
,tel-
- 13 . J.L. Lions De (5. 5) U
dans
r~su1te
(avec (5.4)) que
H ,de sorte que
converge
U m
uniform~ment
vers
U ~ C ([/, J3}H) (apres modification ~ven-
tuelle sur un ensemble de mesure nulle), dloll
r~su1te
Ie tMoreme 5.1
(avec Ie Lemme 5.1).
5. 3.
Compl~ments
Du Lemme 5. 2 r~sulte aussi que l'espace des fonctions indMiniment
diff~rentiables
dans [~I
rJ ->
Par prolengement par
V
est ~ dans
continuit~,
en utilisant Ie
W( ri. , ~ ). 5.1 , on
th~oreme
obtient la formule de Green:
p
j
(5.6)
L{U'(t), v(t»
+ (u(t), VI(t))J dt = (u(~ ), v(P)) - (u(OI ), v(ex»
tI,
pour u, v, E W( rJ. , ~). (Dans cette formule, la valeur en
0(
u( d), v(ci), ••.. d~signe
de Ia fonction continue dans H p.p.
~gale
a u, v).
Commentaires sur Ie Chapitre I N! 1. La condition n~cessaire et suffisante pour que est que
en = F
o
soit m-polaire, i.e. qu'il n'esiste pas de distribution
T E H-m(R n) a support dans F qui soit La
Hm(O)=Hm(n)
propri~t~
d'interpolation du
f
D.
r~sultat
du N. 3 donn~e dans Lions,
Bull. Math. R. P. R., Bucarest, 2(1958), p. 419-432, est un cas particulier du
r~sultat
suivant (pour lequel nous renvoyons
Espaces de moyennes,a paraitre); soit
B un espace de Banach, BeH,
H =Hilbert, B dense dans H ,et soit BI Be He BI . Alors si IT aussi un
~l~ment
de
est un
a J. L. Lions - J. Peetre,
~I~ment de
llantidual de B. Donc t(B';B')
ni (B;B)
z£ (H;H) • Cette propriete est inexacte
pas un Banach - Par exemple si lIon prend
,clest
si
B n lest
'f
= espa-
H = L 2(R), B =
ce de Frechet des fonctions indefiniment differentiables a decroissance 60
- 14 J. L. Lions
rapide (d. L. Schwartz, TMorie des distributions, Paris, Hermann, t. 2), alors
B'
=J
I:::
espace des distributions temperees et
SC
L2
les operateurs de derivation sont bien lineaires et continus de lui
m~me
et de -
fo
I
dans lui
m~me
C
31 ;
8
dans
mais n 'appliquent pas L 2 dans lui
m~me.
La demonstration du texte du
TMor~me
"
3.2 est due
a p. Lax,
On
symetrisable transformations, Comm. Pure Applied Maths. Le N. 4 intorduit part Ie couple
brievement les operateurs non bornes definis
V, H et une forme sesquilineaire continue sur
des applications aux
a J. L. Lions,
tr~s
proJ:>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(
cE
U
W
unique tel que b~ (u e
(3.4)
En outre, si
(3.5)
W ,il exi-
I:
II u,
I
,'f) = L('f)
L(cp)1
~
(t) II 2 dt +
pour tout
IILjlllrllw
t
I:
,on a:
lUI (t) /2 dt +
68
Cf E W.
h
(0)1 2 (
con,tac.t,.
- 21 J. L.Lions
Nous prendrons en particulier :
f
T
(3.6)
L(tf)
=
(f(t),
r
(t)) dt
+ (u o '
f (0)).
o D'apres (3.5) on peut extraire de
U
c
uc"
une suite
I
C -~ 0
,tel-
Ie que:
2 . uc' ~ w dans L (0, T;V(t)) falble
(3.7)
r;\ dt d
(3.8)
~ l.
Alors
Uti'
X
d
r[
f )->
bE' (ut"
2 L (0, T;H)
ans
.(t;w(t),
r
faible.
(til - (w(t),
r'
(t))] dt
o
(car
I
T
l
I
du I (_t dt
,11-) dt -,)
0)
o
Par consequent w satisfait
~
:
2 wE L (0, T;V(t)),
Tl.a(t;w(t), r(t)) - (w(t), f (t)) ]
r
I
dt
=
o
fT
(f(t),
r
(t)) dt
+ (u o '
f (0))
0
et on peut done prendre
u
=w
,ce qui acheve la demonstration du Theo-
reme. 3.3. Remarque. Le probleme "perturbe II (3.4), avec Ie choix (3. 6) de respond formellement
a:
A(t)
ut (t) + ul (t) -cu'!
l
CUt (0) = u 0
u (0) -
(t) = f,
'
ue,.(T)=O.
69
L(
r)
,cor-
- 22 -
J. L.Lions
4. Le cas 4. 1. Si fonctions
f
(4.1 )
{
V(t) = V.
V(t) = V
ind~pendant
de t ,on peut dans (2. 1) prendre les
de la fason suivante :
f (t) = ~ (t)v, v E V,
Alors (2.1)
lIT
0/ E
s'~crit
2 L (0, T),
'r'IE L 2(0, T) , If (T) = 0.
:
a(',;u(t), v)
f
T
r (t) dt -
o
o
(4.2)
(f(t), v)
-
'I' (t) dt
+ (uo' v)
(u(t), v) ~'(t) dt =
tt-
(0).
lei f E L 2(0, T;V') (=(L 2(0, T;V))'). Si en particulier nous prenons
r e fJ (] pact dans
r
0, T[) (fonction indMiniment
(0, T) ) ,alors (4,2) se rMuit ~
(a(';u('), v) + :, (u(,), v))
0/ (') dt·
o
pour tout
f
T (1('), v)
~ (t) dt
0
r
E
~ (]
0, T[ ),
butions dans ] 0, T [ (4.3)
diff~rentiable ~ support com-
ou
:t (u (t), v)
• Donc : d
a(t;u(t), v) + Cit (u(t), v) = (f(t), v)
(JJ'( ] 0, T[ ) = ~
est prise au sens des distri-
(J 0, T[ )'
~I
= distributions sur] 0, T[ ).
Mais(cf. Chap. I) :
70
-
au sens de ~ ( ] 0, TL )
- 23 J. L. Lions
a(t;u(t), v)
(4.4)
= (A(t)
u (t), v) ,
A(t) u (t) E VI.
Notons Ie : 2 A(t) u (t) E L (0, T;VI).
Lemme 4.1.
D~monstration.
done que, pour t
*
--+
n suffit de v~rifier la mesurabilit~
v E V, t --+ (A(t) u (t), v)
(u(t), A (t)v)
t --? (w, A't)v)
llest, done que
est mesurable. Done que
•
t -+ A(t)v
Pest; done que
de eette fonetion;
est mesurable, done que
t ---t (A(t)w, v)
= a(t;w, v) l'est, ee qui
est vrai par hypothese. C. q. f. d. Alors (4.3) dOLle:
(4.5)
A(t) u (t)
et par
+ d~~t) = f(t)
eons~quent
d~~t) = f(t)
_ A(t) u (t) E L 2(O, T;VI).
Done: Th~oreme
4.1. Si
V(t) = V
ind~pendant
de t ,et si u est solu-
tion du Probleme 3.1, alors du L2 TtE (O,T; VI ).
(4.6)
Corollaire 4.1. Sous les hypotheses du
~gale
a une fonetion)
continue de
[0, TJ
Th~oreme
--+
H .!!..
4.1, u est (p.p. u(O)
= uo '
D~monstration.
La 1ere partie du tMoreme est En outre, on peut
-
f
T (u(t), f'(t)) dt
o
r
~erire
=
eons~quenee
du Chap. let (4.6).
maintenant (ef. Chap. I) que
(u'(t),
r
(t)) dt
o
71
+ (u(O),
r
(0))
- 24 -
J. L. Lions
(puisque
(
r
(T):; 0). Done:
~ (t)) + (u'(t). rIt))J
[a(t;u(t),
dt
+ (u(O),
f (0)) "S T (f(t) , 'f(t))dt+(U
0'
o
0
et
d'apr~s
(4.5):
rL T_
r
a(t;u(t),
(t))
+ (u'(t),
r
T
(t)0 dt:;
o
done
r
(f(t),
r
(t)) dt
0
(u(O), ~ (0))
f (0))
(UO)
done
u(O):; u 0' ee qui ach~ve Ia demonstra-
tion du eorollaire. 4.2.
Unicit~
TMoreme 4.2. Hypotheses du Ie
TMor~me
2.1
~
V(t):; V. Alors
2.1 admet une sulution unique.
probl~me
D~monstration.
D'apres Ie point 4.1, il faut
2
(4.7)
u E L (0, T;V) ,
(4.8)
A(t) u + -
(4.9)
u(O) :; 0,
alors
(4.10)
)
T
r~sulte
a(t;u(t),u(t)) dt
o
eeci : si u verifie
du 2 dt ~ L (0; T;V'),
du :; 0 dt '
u :: O. Or de (4. 8)
r
d~montrer
:
+ fT (du(t) d t ' u(t)) . dt-- 0, 0
Mais (d. Chap. I)
72
r
(0))
- 25 -
J. L. Lions
f [(d~\t), T
d~\t)
u(t)) + (u(t),
)]
I
= I u(T) I 2 _I u(o) 2 = IU(T) /2
dt
o donc prenant la partie
f
T
r~elle
de (4.10), il vient
1 Re a (t;u(t), u(t)) dt +"2
I u(T) I 2 = 0
o
et comme
Re a(t;u(t), u(t))
~ 0{
II
1/2 , 0(
u(t)
> 0 ,on en
d~duit
que
u :: O. c.q.f.d.
5. Une application de la
m~thode
5.1. On peut
r~sultat
~noncer
Ie
de transposition final du N.4 (en prenant u = 0) o
sous la forme suivante : u
~
A(t)u +
est un isomorphisme de l1espace
~~ u," L 2(O, T;V)
X(des
2
tels que
2
u l E L (0, T;VI), u(O) = 0) sur llespace
L (0, T;VI).
En Ilinversant Ie sens du temps II et
changean~
A(t)
;(
en
A (t)
,
alors v ~
~
\'
dv
L. v = A (t) v - ill 2
est un isomorphisme de llespace X (des vEL (0, T;V) 2 + 2 VI," L (0, T;VI), v(T) = 0) sur llespace L (0, T;VI). 5.2. On transpose maintenant ce un isomorphisme de
(5.1)
f
L 2 (O, T;V)
etant donn~e X+, il existe
sur
v ---7 L(v)
r~sultat: L:*(adjoint (X,~)'. Donc:
tels que
de
L )est
,forme anti-lin~aire continue sur
u E L 2 (O, T;V)
unique tel que
(u'L v) = L(v) pour tout v E X+ . 73
- 26 J.L. Lions
(u, Z; v)
2
d~signe ici Ie produit scalaire entre
L (0, T;V)
et
2
L (0, T;V')-
5.3. Cas particulier. Soit
(5.2. )
f
€;
L(v)·
r
112
L (0, T;H)
(L
(f(t). v(t)) dt
et non L
+ (u0
!)
donn~e
et soit
• v(O)).
o
Camme (ef. Chap. I) 0, T --+
H),
X+ C C(O, T;V) (fonctions continues de
dMinie par (5.2.) est bien un ~lement de
L(v)
(N. B. :on pourrait m~me prendre pour f une mesure
a valeurs
(X)'
dans
+
H).
Done:
(5.3)
il existe
2 u E L (0, T;V)
j
I
T (u(t),
(t)v(t) _ v'(t)) dt.
o ou
(5.4)
d~duit
J
T (f(t), v(t)) dt
+ (uo. viOl)
0
f
e
1
L (0, T;H), u E H ,pour tout o
v' E L 2(0, T;V'), v(T) On
unique, telle que
2
vEL (0, T;V) tel que
= 0.
de (5. 3) que
A(t) u(t)
+ u'(t) = fit)
dans] 0, T[
Done, cette fois : (5.5) On peut montrer que dans ces conditions u est encore (p. p. egale
a) une fonction continue a valeurs dans H. Alors (5.5)
ufO)
=. u0
.
74
- 27 J. L. Lions
Commentaires sur Ie Chap. II Le probleme de
dans Ie
l'unicit~
Par ex. prenons;
=L
H
2
L
tiere assez r~guliere, et soit
K
(0),
2.1 n 'est pas
Th~oreme
=H
1
0 ouvert
(0),
une partie de
(t)
r
r~solu.
born~
de fron-
= frontiere de
0
d~pendant de t ,par ex. contin~ment. Prenons : V(t)
=
=H
u.E K
1 (0) ,
f od
=In
a(t;u, v) :; a(u, v)
i=1
= 0 sur
u
av
u x.
'0 x.
1
0
1
l..
(t)}
dx.
Le probleme correspondant au Probleme 2.1 n'est pas
l
notre connaissance (sauf, evidemmen~ si
(t)
r~solu,
a
ne depend pas de t !).
Il correspond, formellement, au probleme suivant :
A
- D
u(x, 0) u(x, t)
~u
u(x, t)
x
0 t
0 x ] 0, T[ ,
donn~
=0
si
~u On (x, t) = 0 Si lion
dans
+ -r:- = f
~mpose
xE
si
L. (t),
xE
tE(O,T)
r -6
des hypotheses
(t), t E (0, T).
suppl~mentaires,
alors des
r~sultats
d 'unicit~ sont connus : 1) Si lIon suppose que , et que P(t) est
d~montr~e
d~pend
a(t;u, v)
d~pend
de fayon
diff~rentiable
de
aussi de faeon diff~rentRble de t ,alors l'unicM
dans
T. Kato - H. Tanabe, On the abstract evolution equation, Osaka Math. J. 14 (1962), p. 107-133. 2) L' hypothese de
diff~rentiabilite
sur
75
a(t;u, v) (mais pas sur P(t))
- 28 -
J. L. Lions
a
~t~ supprim~e
dans
J. L. Lions, Remarques sur les
~quations diff~tentiel1es op~rationelles,
a
para1tre. On trouvera une Mmonstration du donn~e
2. 1 diff~rente de celle
TMor~me
au N. 3 dans
J.L.Lions, Springer Collection Jaune, t.ll1,1961. Le N. 5 est plus important pour la te
m~thode
Ie dans les
de transposition, probl~mes
inaugur~e
aux limites non
m~thode
compl~ments
r~sultat;
cet-
par Visik et Sobolev, est essentiel-
homog~nes ~tudi~s
Lions - et aussi Schechter - Pour Ie cas quelques
que pour Ie
pr~sent,
par Magenes et
oh pourra consulter pour
:
J. L. Lions, Quelques remarques sur les nelles, Rend. di Padova, 1963.
76
~quations diff~rentielles op~ratio
- 29 J.L.Lions
Chapitre III ,
/
INTRODUCTION A CERTAINS PROBLEMES NON LINEAIRES
1. Position du probleme.
Nous voulons
r~soudre
Ie probleme de Cauchy pour l' ~quation
dun
Au =""'-:-t -~.
(1. I)
() ];
k entier
~
G~n~ralit~s.
-4 1='.
d
;) u 1K-i
'0 x.1
(r.r-) = f (jJ
x.
1
'
2.
Donc on cherche une"solution"(dans un sens ou dans un autre! )de (1. I)
d~finie pour t > 0 et x E Rn ,telle que u(x,O)
(1.2)
u(x, t) ce
~tant
a l'infini
= uo(x),
assujettie, pour chaque
U o donn~e,
t
~
0
, a une condition de croissan-
(en x).
Toutes les fonctions
consid~r~es,
dans ce chapitre sont
a valeurs
r~eHes.
Pour les problemes non lin~aires d '~volution (et m~me pour les problemes
, lin~aires
m~thode
- comme on verra -) une
m~thode
tres puissante est la
des solutions approcMes.
Expliquons cette
m~thode
en l'adaptant au cas du probleme ci dessus;
on va d 'ailleurs voir que dans ce cas elle est insuffisante ..... On prend une suit de fonctions
WI' w 2'·····, w m ,······ telles que
77
-' 30 J. L, Lions
w. E ~ (Rn)
pour tout i ,
1-
quel que soit m ,les
Wl' . , , . Wm
dentes, les combinaisons
lin~aires
sont
lin~airement ind~pen-
L
5 Wi' tiE R i
,sont
finie (R n ),
denses dans
$
On peut remplacer r~gulieres
a d~croissance
(Rn)
par n'importe quel espace de fonctions
suffisamment rapide
On cherche alors une ! 'solution
a l'infini.
approch~e"
u
m
sous la forme
m
g. (t) w. 1m 1
um(t) =\'
(1. 3)
L-.. i =1
m
(ou encore:
ou les
g. (t) 1m
sont
E g. (t) w.(x)), i=1 1m 1
=
determin~es
de facon >
a satisfaire a :
d
U m Zl( l dum(t) ~ dWI (dt ,w.)+~ ((-'0-) ,~.)= 0, j =·l""m J i=1 \ rj x
(1. 4)
ou de
u (x, t) m
o
fa~on g~n~rale
(f, g) =
J
f(x)g(x) dx;
Rn on ajoute
a (1.4) les conditions
g. (0) = 0(. ,
(1. 5)
les
initiales :
1m
I)( im
1m
~tant choisis de fa~on que
m
220< im Wi i=1
vergence ayant lieu dans un espace convenable (on
78
~
U
o
,la con-
pr~ciseracela
dans la
- 31 J. L. Lions
suite !.). Le systeme (1. 4) (1. 5) est un systeme d I~quations ordinaires non
diff~rentielles
lin~aires.
Montrons que (1. 4) (1.5) dMinissent globalement (i. e. pour tout t
~
0) les
gim (t) . Pour cela, multiplions (1. 4) par gim (t)
et sommons
en j; nous obtenons :
ou encore
d
1 d 2 dt
urn lK
( ~)
dx = 0
(I I =norme dans L
2
1
et done
Supposons alors que dans
2 n
E L (R·)
m
et que
L.. o( i=j.
1m
w. 1
-7 u o
L 2(R n ). De (1. 6)
(1. 7)
Uo
t
r~sulte
alors que, pour
o::::t~T,
T fini quelconque
I urn (t) I ~ constante dUm
'0 \
born~
de tout ceci T~sulte que
dans
urn (t)
L
2k
n (R x(O, T))
lorsque m varie (et
est dMini pour tout
Le probleme est maintenant de passer
t
~ 0).
a la limite,
en extrayant (si
possible!) de urn une suite u y telle que u v converge vers une solu-
79
(Rn))
- 32 J. L. Lions
tion du probleme initial. Or, dlapres (1. 8) u y
~
~ '0 xi
(1. 9)
On
(1.7), on peut extraire une suite u y telle que
w dans
----7
v~rifie
Xi
L 00(0, T;L 2(R n )) dans
L
2k
faible,
n
(R x(O, T))
faible.
sans peine que
(1,10)
d
A
tK-!
La difficult~ principale (la seule meme!) est de montrer que (..!..2})
o X.1
converge,m~me dans un sens tres faible, vers
Par exemple, puisque dlapres (1. 7),
d urn
(~)
u·!
est
born~
dans
1
2.1<
L (h.-t) (Rnx(O, T))
on peut supposer que
mais, a cause de la convergence faible, on ne peut en conclure que W. I 1
~ w ~('I
= (~) U X.
•
1
a priori fournies par (1. 6) ne sont pas suffisantes; Ie probleme est d lobtenir des estimations a priori portant sur ()~ u '01 urn Autrement dit, les
'gx''bt ' 1
in~galit~s
~x.qx . .
J
1
2) Les estimations
a priori fondamentales.
Applications.
2. 1. On va utiliser 11 op~rateur auxiliaire B
80
donn~
par
• 33 • J. L. Lions II
(2.1a)
Bu = - L.l u
+U
-
d (T - t) IT ~u . 'ft
On simplifie un peu en prenant
w. J
~tant
Uo
= 0 dans Ie N.1 - et, la base
choisiecomme au N.1, on cherchea
d~terminer
u (t)
m
par:
m
um(t) = ) ' g.
(2.2)
4-1 i
1m
=1
(t) w. , 1
avec: (2.3)
(B
1\ u
(ce qui suppose f assez (2.4)
= (B f, w.),
(t), wJ.)
m
um(O)
= 0,
j=l, ...• , m,
J
r~guliere)
u'm(O~
avec les conditions
= 0,
um(t)
born~
lorsque t
~
T.
Nous admettons ici Ie Iemme suivant (pour Ie quel nous renvoyons a Visik, M. Visik utilise ici la
m~thode
de Leray-Schauder)
Lemme 2.1. On suppose que f est indefiniment diffl!rentiable a support compact dans l'ouvert ,
t > 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<
(~)
dx dt
+L.
I ,'---o__S1 _ _ _ _ _---<1
J
n
i:: i
JI 2,0
2,0
ob
Mais
et par une trasforrnation analogue
(2. 16)
J(i)
2
= 2k-l kt
Par ailleurs
(2.17)
J 2,o •
f,
IT
J =1.
0
~
r :,
+
a eelle effeetu~e pour
J l' Rn
II
d
0Xj
1)?""Oiiu~)] .1'
Um(td/ 2 dt'
o
Done:
85
J 1(i) :
+II
2 dx dt.
uro(T) 1/ 2
J 2(i)
- 38 -
J. L. Lions
n
J
1,0
+ \' (J (i) + J (i) ) + JI 1
L.,
2
2,0
+ J.. 2
II um(T) II
2
=
i =:1
"r
T (B f, um (t))dt, o
(2.18)
J1
,0
donn~
par (2.13),
J(i) (resp. J(i) 1 2 )
JI 2,0
=
donn~
par (2.15) (re.sp. (2.16)),
fT Jr o
(2
On dMuit de la que :
J 1,0 +
(2.19)
n
L i
(i) (i) I (J 1 + J 2 ) + J 2, 0
est born~ lorsque m ~ +00
=1
2.3. - Utilisation des
in~galit~s
a priori -
D~monstration
de la Pro-
position 2. 1. De (2.19) 1Ion
d~duit
que 1Ion peut extraire de urn une suite
telle que:
86
U
v
- 39 -
J. L. Lions
Q J) ~
w dans
, d
Uy uy ="OT
~
00 2 n L (0, T;L (R))
dW
IT
2 2 n L (0, T;L (R )) faible ,
dans
(2.20)
dans L
2k n (R x (0, T))
dans L
dUm
(TXt)
2k/(2k-l)
n (R x (0, T)) faible,
~tre d~montr~: d'apres l'expression de J~i) + J~i) ,
Seul (2.21) doit
~
.
falble,
converge p.p.
(2.21)
.I.
faible,
demeure dans un born~ de
1 n
H (R x(O, T-l))
conque; on peut donc extraire une suite u'J telle que
,
ge fortement dans L
2
, ~ > 0 quel-
'1 llv) J. ca ~ X.
conver-
1
sur tout compact de
n
R x(O, T)
et donc, parune
nouvelle extraction, on peut supposer que (2.21) a lieu. Mais (cf. (2.14)) la fonction
~ (~) est monotone, de sorte que
(2.21) impUque :
(2.22)
dU y
-0- converge p.p. xi
Mais de (2.22) et du fait que r6sulte en T que, si des distributions; donc
e.
1
dUv ~ o xi
est
born~
dans
a
u V dlA est la limite p.p. de ~, --1..9. () Xi
aXi.
1
au sens
et donc (2.22) peut se pr6ciser: p.p.
(2.23) 87
- 40 -
J. L. Lions
De 180 on
d~duit
que (en utUisant (2.20)) :
dW
(2.24)
("F'X.)
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.
, :
f
,
By> = 0
i =1.
(dualit~ entre L2 et lui m~me et LPI (W- 1,pl) et LP(W1,P) -cf. Remarque 2. 1), entraine :
89
- 42 J. L. Lions
n
-l=
~7
i
1
dw
~
On peut alors prendre sous
r~serve
2k-l
l""7.1 """Ox:""1 (
= f.
)
= w et la Proposition 2.1 est
l.l
de la
D~monstration
Soient
du Lemme 2.2.
fJk(t)
d
les fonctions propres de
d
-Tt(T-t) Tt
k=I, .... ,
fJk(t)
Soit ensuite
V km (x)
lorsque
born~e
t
~
t\ >0 T.
v~rifiant
-/)'1) km +(1+f')11 k
=€
m
km.5 m
€ m' m = 1, ...• , formant une base de Q)
les
demontr~e,
= 1, ..•.
n (R)
, avec, par ex. ,
2 n
V' km E L (R ). Alors, les
B
(l..
~
ff km
~tant des nombres r~els quelconques :
=);
km vkm(x) fJk(t))
k,m
t
km
5m
(x) fJk(t)
k,m
(k, m variant dans en ensemble fini ). En prenant les w. -
tTk
m
mais,a
1
~gaux
' on obtient Ie r~sultat d~sir~. Les w. ne sont pas dans 1
d~croissance
,
a l'infini suffisante pour valider tout
90
ce qui
aux
5tJ (Rn) pr~cede.
- 43 -
J. L. Lion's
3. Le
3,1.
final.
r~sultat
Enonc~
On pose:
du
d'unicit~.
r~sultat.
2k = p;
Th~or~me
d'existence et
TMor~me
p'
est dMini par
lip
+ lip'
=1
3.1. On donne f avec
(3.1.)
11 existe une fonction u et une seule telle que
{
(3.2)
uE L
co2 n (0, T; L (R ))
u E L P (0, T; Wl,P(R n ))
dU _ ~
(3.3)
ot
~
i::: 1
d du
h(h) 1
2k-1
dans Rn x] 0, T [ ,
=f
1
u(x,O) = O.
(3.4)
Remarque 3. 1 Comme ala Remarque 2; 1, il
r~sulte
des hypotheses faites que
de sorte que (3.4) a un sens. 3. 2.
D~monstration
de 1'unicit~.
Soient u et w deux solutions du probleme. Alors w = u-v tisfait aux conditions analogues a (3.2), (3.4) et a
(3.5)
dW
l
d
du
'dT - i =1 '0 \ ( "t' (d
91
\ '
() v
'& w
- 0
a x,) c:rx: )-
sa-
- 44 -
J.L.Lions
Ilr r
oi'l
Comme
(a, b)
=
a
2k -1
- b
2k-1
a - b
w(t) E LP (W 1,P(R n ))
et
W'(t)
='Bt
E LPI(W-1,pl(Rn )),
a un sens p. p. (dualit~ entre
l'expression
(w' (t), w (t))
W- 1,P'(R n ))
et dMinit une fonction sommabIe. De m~me
*tl~ut : ~)~
- (
~
x.
1
x.
'~
1
x.
X.
1
WI, p(Rn)
et
. un sens p.p. en t,. a;; t_ 11
, w(t))
1
d 'ailleurs, cette quantitE! vaut
quantitE!
~
0 • Donc (W'(t), w(t)) , 0
i. e.
Jw{t)/ 2
dE!croissante. Comme
L'unicitE! est
w(O)
= 0 ,on en dE!duit que W:: 0 .
dE!montr~e.
3. 3. Principe de Ia dE!monstration de l' existence. On considE!re une suite
fm
de
V
(dans I'ouvert t
> 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(<5" Ium(I)' II', u (1)'11
m(d} +
o
L t
+ 2Re
(f(l)'l, u
'm (
(les c. d~signent des constantes). J De l'expression de rm(O) et de (3.8) on dMuit fm(O)
< c 2( Iluoll
2 + lu112)
et comme
(3. 12)
103
~.
.. 55 J. L. Lions
On en
d~duit,
par un lemme classique de Gronwall :
(3.13)
:
Cons~quence
u
(resp. u' ) demeure dans un ensemble
m
m
born~
de
00
L (0, T;V)
00
(resp. de L (0, T;H)). On peut donc extraire une suite
u~
= uy telle que
00
dans L (0, T; V) faible, (topologie faible de dual de L1 (0, T;V))
(3. 14) 00
.
dans L (0, T;H) falble. Montrons que w est solution du probleme. Pour cela, soit une fonction une fois
e.(T) = ° J ej(t)
et
;utilisons
i.nt~grons,
e.(t)
J
contin~ment diff~rentiable dans [0, TJ ,avec (3.6) avec m = V et j fix~ < V ; multiplions par
posant
(3.15) il vient, apres
(3.lB)
int~gration
E('V,
~
Mais grace relation,
x.) ~
et ceci pour toute
j:
((f(t)'
a (3.14),
(u~(o), jJO))
(3.17)
par parties:
A
E(ty,.x) - ) E(w,J) et grace
--7
E(w.x)
X(t)) dt + ("I (0), X(0)). a (3.8),2 erne
(u 1' X(O)), Donc :
~
j:
X de la forme
(f(t), X(t)) dt + (U 1.;( (0))
(3.15) - donc aussi pour toute combimti-
104
J. L. Lions
son
lin~aire
Xe 1)[ telle que
f.,(T)
= O.
Par ailleurs de (3.14) suit que
u\i(o) ---;Pw(o)
(par ex) et eomme d'apres (3.8), 1ere relation, que
w(o) = Uo
aeheve done la pri~t~s
dans H faible
l),(o) - , ) Uo ,on voit
ee qui montre que w est solution du probleme 1. 2 et de l'existenee d 'une solution ayant les pro-
d~monstration
(3.3), (3.4).
3. 3.
D~monstration
Nous allons
=0
de
d~montrer
E(U'i) = 0
(3.18)
alors u
a la limite, pour toute
finie de telles fonetions et par passage
l'unieit~.
que si
pour tout
LAf: satisfait
uE
r
U;, tel que
E
r
u(o)
= 0 et
(T) = 0
,
•
Pour eela, nous prenons s avec
< s < T et nous dMinissons
0
~ (t) par
-Jt
S
si t
u(O") d6'""
<s ,
(3.19)
o On a bien 2
~
E
L 2 (0, T;V)
si t > s et eomme
'P' = u
done E L (0, T;H). Enfin 1(T)
(3.20)
j
s [a(t;
= O. Done on peut prendre eette
'r"
r
La derni're in"grale vaut
E L 2 (0, T;V)
dans (3.18); il vient:
f" f) -(U',U)]- dt + IS ((B(t)u(t))',
o
.
r
(t)) dt = o.
0
-
J:
(B(t)u(t), u(t) ) dt
105
;prenant deux
- 57 -
J. L. Lions
fois la partie
r~elle
de (3.20), il vient :
d'ou
11 en resulte (iue
Mais si lIon pose
t t
(3.22) alors
"It) f(t) = v(t) - v(s)
,
u(O')
O~ 8 , 8
o
80
et (3. 21)
t <s
~(') 1/ 2 + /U(.)/2 {; '6
Soit
d,,-,
I:
sl~crit
:
I I')
(IIV(t) - vis) 1/ 2 + u(t)
tel que, par exemple,
II 2+ ~(s) 12 ~ c7
~
1 - 2 c 6 So = 1/2. Alors, pour
,on a:
Ilv(8)
dt
j: (1IV(t)/l2 106
+
/U(t)/2 ) dt
a
58 ..
J.,L, Lions
donc
u
=
° dans
(0, so), En recommencant, on trouve que
u
=
° dans
(so' 28 0 ) etc. Ceci acheve la
4.
d~mQnstration
du tMoreme 3.1.
de la solution.
Propri~t~s suppl~mentaires
4.1. Th~oreme 4.1. Hypotheses du TMoreme 3.1. Soit u la solu· Hon du
Probl~me
1. 2. D'apres les TMoremes 2.1 et 3.1 nous savons
que u est p.p. ~gale (et
m~e _>
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
une suite de
(O,T)
109
• 61 .. J. L. Lions
u*(()n) - )
t
u~ ( 0:) -)
?/
{
(4.13)
Nous allons
d~montrer
dans V faible, dans H faible.
que
(4.14) Alors (4.2) et (4.4) en
r~sultent.
Pour montrer (4.14), notons d 'abord que
2
L (0, T;V); done Puis
B(t)~
a(t;Ya;, u(t), v)
u --.;:. B{t)"1 u
((B(t)\ u)', v) - ) '" De meme,
a(t;Y~
--:I>
u(t), v)
2 L (0, T;H)
dans
Y~
u
~ Y~
u
dans
2
dans
L (0, T).
done
]0, T[ .
((B(t)~ u)', \1') au sens des distributions sur
d2 - 2 (Y.., u(t), v) - - ; ) ~ dt
2
d Cit(lfo (u(t), v)
dans
ILl' rY/ (0, T).
Done
'TTl (\ u(t), v) -.> m, (~ u(t), v)
(4.15 ) Mais
t11(~1\ u(t), v) = (~f(t), v) + (u'1'(O""n)' v) +
+(
g; '(0, T).
, S(u. . )+
·OB( ~)u1f( er) v) + (u*(()n)' v]S{~)
et d'apres (4.13), ceci converge vers (ipf(t), v)
dans
t 'V)~;~) +
(B(
f )t + f4 ,v) S(~) .
Utilisant (4.15), on en tire (4.16)
'Yl1. (Y~u(t), v) = (~f(t), v) + (10 ' V)S/~) + (B( f) 110
t + I4'
v)
~(~j.
- 62 -
J. L. Lions
Comparant avec (4.8) (pour 1; =0 ), il vient (4.14), dlou (4.2)
r f.
(4.4). La
d~monstration pr~cedente
suppose
fit de faire la modofication suivante : on de u par 0 pour gement de r)')
t <0
a(t;u, v), B(t) rv
'V
et on
'\I
= 0, il suf-
par u Ie prolongement rv "v par a (t;u, v) , B (t) Ie prolon-
< O. Si lion pose:
par 0 pour t
d2
+ ( (B(t) u (t) )1, v) + 2 rv
r
rv
d~signe
d~signe
~galement
1(., (u (t), v) = a (t;u(t), v)
O. Pour
'"
dt
'V
(u (t), v)
alors
rn, (0t), v) = (0t), v) + (ull'(O), v) ~;) + (B(O)u)t(O) + u. (0)) v) ~O) On introduit
rn
(J'
n
comme
(Ylr ~(t), v) ~
""
pr~c~demment
~ (;(t), v) dans
distributions sur
J -00, T[
dans V faible,
ul~((Jn) ~ ~i
1'& (Y(7"'II 'G"(t) , v) ~ (t(;), v) + ( ~ dlou
u,t(O)=fo, 4.4.
R
0'
lJ
v~rifie
1(-00, T) = espace des
to
faible, et donc
~~) + (B(O) fo + €1 ' v) ~O)
v)
U~(O)=f:1 ,dlouencoreler~sultatd~sir~.
D~monstration
de (4.5)
11 suffit de montrer que Ie graphe de llnpplication est donc une suite
que
pr~c~demment, u~( O"n) ~
. Mais, comme dans
et lIon
fu(m) () , u 1(m) ' f(m)] -+
0
ferm~.
Soit
dans VxRxL2(0 . , ToR) , ,
telle que
f( (u ~m) (t), v)), Comme
(
um
)
-t
0
(d~ u~m) (t), V)} dans
00
L (0, T;V),
111
converge dans du(m)
Cit
~
0
C(O, T) x e(O, T). 00
dans L (0, T:H),
- 63 -
J. L. Lions
i1 en
r~sulte que la limite de
f oj
cessairement
0,
[(
(U~) (t), v)), (d~ u~m) (t), v)}
,d'oll (4,5), ce qui acheve la
est ne-
d~monstration du theo-
r~me.
4.5. Remarque. De (4.12)
~
[u o, u 1,f} dans
U~(t)l
[uj((t),
pour tout t
r~sulte,
estcontinuede
fix~;
l'application
VxHxL 2(O,T;V)
V x H fort.
5. Propriete de d~pendance continue de la solution en
a(t;u, v) et
B(t).
a~(t;u, v)
5.1. On se donne une famille de formes
ayant les
0(
0(
(5.2)
(5.3)
(5.4)
est une fois continUment differen-
,pour tout
u, v E V, et il existe 1\
ind~pendants de ~ ) ;
( I. '(t;u. V)/ + I!
.~(t;u. V)/ $
M
l/u/l'I/ vII.
\ stante ind~pendente de t et de ~
(
(
et
> 0 tels que: a ~(t;v, v) + ~ /v/ 2 ~ eX/Iv 1/2 ,v E V (A et
a F(t;u, v)
~
u E V fixe,
pour v
uniform~ment
continument
t fixe,
I+ I
diff~rentiable :t
~ --,) 00,
d~' ill B (t)
f ~
t
dans
B~(t) I:s M; dans d ill
->
M ~tant une cone-
;
a(t;u, v) , pour tout
B~(t)e:~(H;H) ,hermitien; IB~(t)
(5.5)
, a?(t;u, v)
Co, TJ
tiable dans
(5.1)
B~(t),
propri~t~s suivantes :
~
pour chaque
et d'operateurs
t ~
E [0, T] born~
fixe,
~~
00,
de V;
(B~(t)f, g)
est une fois
[0, TJ pour tout
f, g E H;
B?(t)f ---iI B(t)f dans H fort,
[0, TJ B(t) f.
112
' f fixe dans H; au
m~mesel!s,
- 64 J. L. Lions
On a Ie
r~sultat suivant ; en d~signant par
I:
E~IU~.I') ~
15 • 6)
r
ou
Iflt). fit)) dt+ IUj' flO)) •
r)~ [a~lt;Ult). ~, r lU, r =
15.7) E flu.
u ?la solution de
rlt)Hu'lt). f'lt)) +
((B(t)u~))' •!fIt)] dt,
o
u
(5.8)
E
f.
(T)
U~(O)=u o . B. On pourrait aussi prendre une famille
~..,
(5.9)
u!, J. uf
00, on a :
~ ~
u
~
d
d~
(5.10)
f~,
5.1. On suppose que (5.1) ..... (5. 5) ont lieu. Alors lor-
Th~oreme
sque
et avec
0,
u dans, L 00(0, T;V) faible (dual faible de L 1(O, T;V», du
~ ill dans L
d2u~
d 21l
(5.11)
----". dt~
(5.12)
u!(t) --1
~
00
faible,
2 L(O, T;V')
dans
dt 2
(0, T;H)
faible,
dans V faible,lorsque t est
u*(t)
fix~,
et
~
(5.13)
(u~
v
(B
(5.15)
~
dtu~
(
(u.(t), v)
~
uniform~ment
en t., pour tout
dans V,
fix~
d
(5. 14)
(t), v)
(t) ~
~(t)u! (t) + !
pour v
d
Ttu.(t)
dans H faible, lorsque t
u!(t), v) --> (B(t)
fix~, uniform~ment
113
en t.
u~(t) + d~
u.(t), v)
est fix~,
- 65 J.. L. Lions
D~monstration.
?
1) D'aprlls (4.12) et les hypotMses faites sur
(oilles
C.
1
d~signent
C. sont en particulier 1
dt
diff~rentes
ind~pendants
~
u
w dans L(lI,(O,T;V)
~~
dw dt
dans
La premillre chose sera
r
j
~
L 00(0, T;H)
). de
T (u' (t),
r~
telle que
r
I (t)) dt
-j
L 1(0,T;V)),
faible.
a montrer est que
o
~(O) = uo
~
~ ,on a:
de celles du N.4; les
faible(dualfaiblede
d~montr~.
Comme
u
de
Done on peut extraire une suite
u~~ ...2....
des constantes,
a ,B
r
w
= u; alors (5.4) (5.10)
(w'(t), 1'(t ll dt, et oomme
0
,il nous reste
a montrer que
(5. 17)
et que
J
T
(5,18)
~
,
'(B (t).I;t II',
o D~monstration
On peut
r
T·
(til dt -.
S
((B(t)w(t))',
r
(til dt.
0
de (5. 17)
~crire: a ~(t;u, v) = (( Ji~t)U, v)) : ((u, ~~t)V)) (car a~(t;u,., v)= 114
.. 66 ..
J. L. Lions
= a f(t;v, u)) et done (5.17) est vrai si lIon a :
11~·
(5. 19)
\I
Or
r (t)f(t)
IIIl~t) ~ (t)~ ~ M II
D~monstration
(on
de (5.18).
int~gration
~erit ~
et dlapres (5. 3),
f EL 2(0, T;V).
Jf~t) ~ (t) ->
dans V fort ,p.p.en t ,d IOll(5.19).
->./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~( <1;) =X1. UI! (Cio) --+
f (~)
u
dans H faible, pour une suite extraite; mais la limite etant independante de la suite extraite, on a Ie resultat (5.12) (5.14). Le theoreme est donc completement demontre sous reserve de la Verification du Lemme. 5. 1. II Y a trois choses (5. 27)
a demontrer
a~(t;YClU~(t)' v) ~ ment en
:
a(t;Y(!"u(t), v) dans
(J*;
117
lJ'(O, T) faible, uniforme-
.. 69 • J. L. Liol1fl
(5.28)
((B~(t)YfU~(t))I,V) -> ((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
<
(1. 3)
dove
I
rappresentano la norma nella spazio di Banach. Noi otterremo an-
che limitazioni inferiori per Ie soluzioni in vari casi mostrando che Ie soluzioni non possono tendere a zero troppo rapidamente; queste limitazioni sono ottenute con considerazioni di convessita . (iv) stabilita all'infinito; cioe Ie soluzioni di quadrato integrabile della (1.3) tendono a zero in modo esponenziale. 1. 2. -
Per vedere quali condizioni sull 'operatore
re, cuminciamo col richiamare quale
A e opportuno imp'lr-
e la situazione in uno spazio di dimen-
sioni finite; cioe per un sistema di un numero finito di equazioni differenziali
126
- 3L. Nirenberg
ordinarie. Ogni operatore
i A
e aHora limitato ed e il generatore
di un
semigruppo (nel easo attuale di un gruppo) limitato di operatori. (1. 4) essendo
T(t) T(s+t)
= e itA
= T(slT\(~}, T(o) = I dT dt
(1. 5) e la soluzione di (1.1) con
(1. 6)
u
= A T(t)
assegnata, per fissare le:idee, per
ult) "Tlt-s)uls)+i
J:
Tlt-'t")1(
~)d t
Le soluzioni deH'equazione omogenea
Lu
=
t =s
.
°sono combinazioni
lineari di polinomi esponenziali ehe chiameremo semplicemente soluzioni iA t esponenziali X(t) = e 0 p(t) I dove Ao e un autovalore di A e p un polinomio in
t
(1. 7)
= l?
p(t)
,
e
i cui coeffieienti sono vettori dello spazio:
m
+it
II)
Tm-
(it)m-l , m-l).
1 + ..... + (
IPI
Qui (1. 8)
m
(A-A) \D 1= 0, (A o 1
e l'indiee
A0 ) It) . = ~. l' TJ J-
j
= 2, ...•.. , m;
della soluzione esponenziale. Ogni autovalore
della risolvente
R( A)
= R( A, A) = (AI
fa. Il massimo indiee ehe una soluzione
Ao e un polo
e una funzione i.A t esponenziale e 0 p(t)
-A) -1
che
meromorpuo avel'e
e semplicemente l'ordine del polo Ao . (Anehe nel caso generale noi eonsideriamo soluzioni esponenziali,
127
- 4L. Nirenberg
e automaticamente verificata,
definite nella stesso modo, aHora la (1. 8)
esse provengono dai poli dell 'operatore risolvente
R(.x ,A)
Osserviamo che la norma dell 'operatore tivi di
e limitata da una costante
t
I T(t) I
moltiplicata per
Lu =0
= (A I-A) -I). per valori posidove
-w
e il
A" o•
minima delle parti immaginarie degli autovalori
E' chiaro che Ie soluzioni di
e
wt
ed
che sono di quadrato integra.-
bile sull 'asse positivo tendono a zero esponenzialmente per
t ...
00
,
infatti
queste sono ovviamente combinazioni lineari di soluzioni esponenziali con
ItnA o >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
<.p (t) , non
e vero in generale
che soluzioni di quadrato in-
tegrabile tendono a zero esponenzialmente. Consideriamo il seguente semplice esempio: un'equazione in una sola variabile scalare (1. 9)
dv dt
-c =1+t
v,
128
t
>0
- 5L. Nirenberg
dove ovviamente, nenzialmente ma
A
= 0 . Una soluzione v = (1+t) -c non tende a zero espo-
e di quadrato
La stessa funzione
integrabile se v = (1 +t) -c
c >
+.
e anche
soluzione dell 'equazione
di ordine superiore, C k v = -(-1~+-t)ir/{- v, C = (-1) c(c+1) ..... (c+k-l).
(1. 9~
Questa equazione puo essere scritta come un sistema del primo ordine per
u
= (u o' ...... , uk_I) con uo = v du, J dt
- u =0 j+1
(1.9") du
k-l dt
C
u
o
Allora noi vediamo ehe Ie soluzioni di quadrato sommabile non tendono a zero esponenzialmente se
1
c >2' Infatti in questo caso la corrispon-
dente risolvente ha un polo di ordine k nell'origine; per poter concluderela tendenza a zero in modo esponenziale, occorre che
~
(t)
(in questa caso
C(1 +t) -k) sia limitata da cost. (1+t) -k e la costante sia sufficientemente piccola. Noi vedremo (capitolo 4) che la situazione
e analoga per un operatore
generale in uno spazio di Banach.
1. 3. - Per trattare i casi di dimensione infinita'noi assumeremo generalmente che
A sia un operatore chiuso e che la sua ,risolvente 1 R().) = R(A, A) = (.AI - A) sis: una funzione meromorfa in una regione del
129
- 6L. Nirenberg
A.
piano complesso
seguente proprieta
In molti problemi collegati con equazioni ellittiche la
e soddisfatta:
R( A)
e regolare
Iarg ±.A I ~ ~ ,
(1.10)
in un "doppio settore II
I.AI ~
N
e soddisfa la condizione c ,.
(1.11)
/A I.... 00
inoltre sull'asse reale, per
,si ha:
(1.12)
Consideriamo alcuni esempi di equazioni differenziali aIle derivate parziali in un cilindro e mostriamo come si possano dedtirre Ie proprieta richieste per la risolvente. Osserviamo dapprima che
e spes so molto
utile du considerare u(t) appartenente ad uno spazio di Banach X mentre dt e Au appartengono ad un altro spazio di Banach Y; comunque, per semplicita, noi lavoreremo con un unico spazio di Banach
Y la cui norma
II
indicheremo con
1..4 ..-Nel primo esempio die considereremo non ci sono soluzioni esponenziaIi diverse da zero. Esempio 1. - Lo spazio di Banach vallo
0 <:: x
funzioni continue neIl'inter-
< 1 ;A e l'operatore ~~, dove 01.. e una costante; A
agisce sulle funzioni
e
e queUo delle
'\ uguale a (A -
i
I()
C1 , che si annullano per x
di -1
Ci '() x )
= 0 . Allora
ed esiste per tutti i valori domplessi di
R( A) f
= i ri..
fe
-i),o(.(X-y)f(y)dy .
130
'\
J\
R(A)
- 7L. Nirenberg
Si verifica facilmente che, per una opportuna costante
C ,si ha
1mA~
< c ~1m--:-A-<X.e se 1mAC\. >0 ,
\
R(.A)/ .:. c (1 - 1mJ\(X. )
-1
Consideriamo ora I' equazione per
o~ t.::,. T, 0.::,. x.::,. 1. Se
ha la sola soluzione
iL
<
0
u(x, t)
tlu
u
1mol=rO
l'equazione(1.13)e ellitticae
= 0 . Se 1m eX. = 0 supponiamo cl. > 0 ;in modo che
risulta essere una derivata direzionale nel piano (x, t); aHora la sd:
luzione si annulla per tra parte
i
1m Ad.
Lu =i- -dt - oL - =0 ()x
(1.13) in
du
I
se
u =0
t > x t)..
rna non necessariamente per
e la sola soluzione che si annulla per
t
t <x
nella spaziQ Euclideo (poniamo
$ 2m-j
n-dimensionale come base:
1,
CflJ (x; D ,D ) u x
A2
'
m-J
2m
come generatrice e un dominic limitato /J)
:?x)' n
DX = (..!. i !L 0x1' ...... , .1i
(1.14)
dove gli
t
• D'al-
= O.
1. 5. Esempio 2. - Consideriamo un'equazione ellittica di or dine in un cilindro avente I 'asse
0(
t
.(J
* Dt2m
Dt = -1: l0t ) ,'
2n-l Dj u + "\ A u =f i2m-j t o
sono operatori differenziali neUe variabili
con coefficienti indipendenti da
condi.zioni altcontorno di ordine minore di
t. Supponiamo che 2m
di ordine
u
soddisfi
"coercitive"sulla superfi-
cie laterale del cilindro con coefficienti indipendenti da
131
x
t
della forma
- 8-
L. Nirenberg
B.(x,D )u J x
(1.15)
=0
j=l, .... ,m.
Introducendo come nuove incognite Ie derivate rispetto a
t
j=O, ..•... , 2m-I) possiamo scrivere la (1. 14) come un sistema del primo ordine in
t
j=0, ...... ,2m-2 (1.16)
Le proprieta (1. 11), (1: 12) per la risolvente dell' operatore che ne risulta si ottengono dalla seguente nota proprieta per Ie soluzioni del problema (1.14), (1.15): (A) L'integrale della somma dei quadrati di tutte Ie derivate di
I'ordine
2m
l'integrale di t2
su una sezione
t1 < t < t2
u
e maggiorato da una
fino al-
costante per
(I f 12 + lui 2) in una sezione piu larga tl < t < t 2 , t1
> +2' Da cib segue:
Teorema1. 1. - Sia
v
una funzione definita in
1>
che soddisfi Ie condi-
zioni al contorno (1.15), esistono allora tre costanti positive
5, c, N
che
dipendono soltanto dal problema di val~ri al contorno ellittico,tali che per nel "doppio settore" (1.10) : 'arg:!:
AI 2: S, I.AI
~ N,
A
si ha
2m
(1.17)
L.
j=o Qui
~
II
rappresenta la norma di
presenta la somma delle norme in no all'ordine
L2
di
j.
132
v
L2
in [)
e Ivll. rapJ e di tutte Ie sue derivate fi-
- 9-
L. Nirenberg
A
Per valori di
reali nel "doppio settore" Ia disuguaglianza iAt (1. 17) si ottiene applicandola (A) ana funzione u(x. t) = v(x)e con tl = O. tl = 1. t2 = 2. t2 =3 . Consideriamo ora 10 spazio di Banach U = (u •.•...••• u2 1) o m-
Y dei vettori
completato rispetto alla norma
IUI
(1. 18)
2m-l
= Z. lI u .11
. 1 J 2m-J-
j=o
e prendiamo come dominio dell 'operatore, A i vettori)n
Coo) U dove
uo soddisfa Ie condizioni.al contorno (1.15). Per ottenere informazioni sulla risolvente occorre invertire il sistema. dove
.. , .... ., ..
(1.19)
F = (f •.....• f2
o
m-
1)
j=O ••...• 2m-2.
m- 1+ tA2 m-J.u.J = f2 m- 1
AU 2
Supponiamo che sia per
j
fJ' = 0 per
j
< 2m-I. Allora eliminando Ie u
j
> 0 con I 'osservazione che
(1. 20)
U.
J
=
troviamo (1. 21) Dalla (1. 18) e (1. 20) segue che per valori reali di
A .1).\
~ N,
\).U\':C\F\ Allora se noi indichiamo con S 1'insieme dei vettori F con 133
- 10 L. Nirenberg
fj
= 0, j < 2m-I, e con
1\
reale,
la restrizione di R(A) a S abbiamo per
RS(.~)
\A\ > 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
u(t) = 0 per
Teorema 2.1 : Sia
u(t)
una soluzione di (2.1) per
Supponiamo che ci sia un arco di Jordan semplice
0~t
R( A)
e definito e soddisfa
0<. ~ 0 . ~ T > 0( allora
R( A) = 0 (e o<.lm A)
u(t) = 0
per
t < 0(.
T
~
u(T)= O.
Im A > 0 I nel quaper qualche costante
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 <2
'
e soddisfa la condizione
IR(A)I= 0 (eol.senSIAI)
(2.4') (b)R(A)
e una funzione
mo con
AI'
e
per 1fi. I ......
meromorfa nel semipiano chiuso
00
ImA? O. Indichia-
}.2' ......•. i poli nell'insieme 1m A> 0, numerati in ordine
crescente delle parti iumaginarie. Teorema 2.2 : Nella ipotesi (a) u(t) pUO essere esteso in una funzione analitica di
t =(5' +i 1:
(con
val~ri
(2.5)
in Y) nella regione angolare
larg(t-o(.)l<e. Assumendo che anche l'ipotesi (b) valga e indicando con
uJ" = e i A} t p. (t) . J
il residuo di
/M R(.A. ) u (0)
rappresenta uno sviluppo asintotico per no i poli,contenuti nella striscia per qualche
€ > 0 ,e se
f
u(t). Inoltre se
J
Al' ..... , -\n' so-
0 < 1m A< a .. in modo che
= (5 - eX + t cotg e aHora
m
(2.6)
.A.. , aHora
nel polo
Iu(t) - 2: Uk (t) I < costante Iu (0)1
e
1m
Am
N 11:1
1
(nel settore (2.5)). Nell 'esempio 1 della sezione 1 con 01.. > 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
mentre l'ultimo
Cost il primo asserto del teorema
e una funzione
intera di t .
e provato.
Nel caso che l'ipotesi (b) valga noi possiamo usare la (2.3) per estendere come
u( A )
in una funzione meromorfa nel semipiano
I G().. ) I e limitato
1m .A ~ 0 e sic-
sull'asse reale non vi sono ivi polio Allora possia-
mo sostituire il terzo integrale con l'integrale di so a una curva che congiunge
-N
Ie in modo da evitare i poli reali di
con
2~i
e iA t R( A) u (0) este-
N e spostarla di poco dall'asse rea-
R(A). I1 contorno che ne risulta, costi-
tuito dai lati degli angoli (2.4) e della curva che congiunge
·N con
puo essere spostatb verso I 'alto in modo che I'integrale dia i residui la disuguaglianza (2.6)
e facilmente
dimostrata.
140
N
u .(t) J
e
- 17 L. Nirenberg Un simile aI'gomento, inte,grando lungo i lati obliqui relativi alla trasformata di Fourier di u ,dimostra che se
R(.A.)
e regolare nella regione
e soddisfa la relazione
/R()d I = 0 (e cx.lImAI ), allora ogni soluzione di
Lu
=f
litica sull'intervallo I t I~ T versamente se Ie soluzioni di
eX> 0
appartenente alla classe
,e
analitica nell'intervallo
Lu
=f
1 C con f ana-
I t I ~ T - eX
• In-
con f analitica sono analitiche aHo-
ra si ottengono, dal teorema del grafico chiuso, condizioni necessarie che sono molto simili aIle condizioni sufficientL
2.3. E' interessante osservare che dalla (2.6) si possono dedurre sia limitazioni inferiori che limitazioni superiori per to gia usato da Krein e Prozorovskaya si (a) e che '
RCA)
ro finito di polL Sia
[7].
I u(t) I
usando un argomen-
Supponiamo che valga l'ipote- .
sia regolare sull'asse reale eccetto al piu per un nume-
Cf'
un numero
positiv~
< (} . Allora se to > ex c'e
un numero ~) dipendente dalla soluzione e da to) tale che)per t
Iu(t)\ ~ I u(o)!
> 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 <:: t o- 0( - fto un nUmero positiv~ < to - 0( e poniamo 0 = T per t ~ ttl'
La disuguaglianza (2.1) segue dalla disuguaglianza di convessita:
141
- 18 -
L. Nirenberg
(2.8)
fissando
to
e ponendo
T=t. Per ipotesi la disuguaglianza (2.6) vale per
m = 0 nella forma (qui t = G' + i 1:')
I u(t) I .:: costante I u(O) I e NI'Z:'J
(2.6 1)
6'-cx.>-
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<- 2m
Noi consideriamo operatori che soddisfano la seguente condizione: (c) Assumiamo che
R(A)
abbia "ordine" w in
esistano archi semplici differenziabili non sovrapponentisi
1mA-> ~
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)
<~ .
esista su ciascun V. per valori di I.A. . 0J di e inoltre per qualche costante > 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 -oo
a
u(t)
una soluzione di
> 0,
e -a I t I lu(t)
I
Lu = 0
sjg. sommabile. Sup.poniamo che
sia una funzione meromorfa nella striscia disfi la disuguaglianza qualche
k
>0
Z : I 1m )..1 ~ a
IR(.A ) I= Ore k 1)..1)
. AHora
u(t)
sull'intero asse t tale
dove essa sod-.
per valori grandi di
e la somma di un numero finito
esponenziali corrispondenti ai poli di
R(
A)
~(A.)
p. I
e per
di soluzioni
nella striscia aperta
11m A/
Se ciascun polo nella striscia aperta ha molteplicita finita allora l'insieme . delle soluzioni u per cui
e -al t I lu(t)
I e sommabile ha dimensione.finita.
146
- 23 L. Nirenberg
[4) , § VII.
Ricordiamo (cfr.
3) che se
Ao e un polo
di ordine 1:' aHora la dimensione dello spazio degli zeri di finita,
e chiamata la
di
R(A) C:"
(). 1 - A) ,se
molteplicita di )..0'
r
Dim.: Poniamo
~±
(1)
=
v;n
+00
e- iAt u(t)dt
o
1:1+ ( )..)
Per l'ipotesi formulata su u,
e analitica nel semipiano
1m
A < a ed e limitatct e continua nella chiusura del detto semipiano, mentre
~j
A) e analitica nel
semipiano
1m).
> -a ed e limitata e continua nel-
la sua chiusura. Di pili nel semipiano corrispondente abbiamo
o
A,
u t (J\) =
dovunque
R()...)
e definito.
1
i'V'21f
R o..)u(o)
R( A)
Sic come
e meromorfa nella striscia
11m).. I < a
questa formula fornisce la comune estensione analitica di
e
in questa striscia come funzioni meromorfe
COSl
sono estensioni analitiche l'una dell'altra. Indichiamo con
che
w(
A) la funzio-
ne meromorfa nell'intero piano definita da queste. Gli unici poli di sono i poli
.A l' ....... ,.Am
di
R( )..)
SuI bordo della striscia per \ A I~
00
formulate su
nella stris cia aperta
w( A)
e limitata,
infatti
~+()..)
w( )-.)
I 1m AI < a. w( J...)
= 0 (1)
per il teorema di Riemann-Lebesgue. A causa delle ipotesl R( A)
per dedurre che
possiamo applicare un teorema di PhragmEm-Linde16£
I W(A) I =0(1)
per
1).1 ~ 147
00
nella striscia. Se
Rj(,A)
- 24 -
L. Nirenberg
e la parte singolare dello sviluppo di Laurent. di
R(.A)
relativo al polo
.Aj
si vede che la funzione
w(A) -
.~ 211'
1
r.J RJ'(A) u(O)
e una funzione intera limitata tendente a zero per \). \ ....
00
sull 'asse reale.
Per il teorema di Liouville questa funzione e zero, per modo che
Prendendo la trasformata inversa di Fourier si trova formalmente che e uguale ana somma dei residui di
e i At R( A)u (0)
Ie a una somma di soluzioni esponenziali.
148
nei punti
A., J
u(t)
cioe ugua-
- 25 -
L. Nirenberg Capitolo 3 Unicita per il problema di Cauchy e proprieta di convessita.
3.1. In questa capitolo noi studieremo il problema (iii) per Ie soluzioni delle disuguaglianze (1. 3) - del resto parecchi risultati essendo gia stati ottenuti
n nostro scopo qui e quello d'illustrare l'uso della
nella precedente sezione.
convessita per ottenere limiti inferiori per la soluzione quando t -
+ 0()
•
Come semplice esempio abbiamo il seguente nota risultato dove 10 spazio Y e uno spazio di Hilbert. Teorema 3.1. Sia
u(t) Lu
(3. 1)
du
1
=-i -dt - Au = U
= ~ B,. '! essendo una costante e B un operatore simmetrico. Al-
dove
A
lora
log
I u(t),
e una funzione convessa di t.
Allora per
0,
una soluzione di
0 $. t
o
~
t . troviamo
un limite inferiore
.
/u(t) I> /u(o)J (
Iu(to) I
tit 0
Iu( 0) I )
quindi di qui la tendenza a zero per
I u(t) leal pili esponenziale.
Per dimostrare il teorema basta osservare, derivando, che la derivata seconda di
log lu(t)/
2
enon negativa.
E' utile nella pratica avere tali risultati non soltanto per Ie soluzioni deU'equazione
Lu
= 0 rna anche per Ie soluzioni della disequazione (1. 3) :
ILu I $. f (t) I u I
. A questo proposito in
[8J Lax ha dato una dimostrazione
delrisultato noto che ogm soluzione dell equazione a derivate parziali ;,Au +
'La.
1
*u x.
+ au = 0
Ii. quale 8i annulli pili rapidamen-
1
149
- 26 -
L. Nirenberg
Je che ogni potenza nell' origine
e
ideriticamentezero . Con un in-
gegnoso ragionamento egli riduce la dimostrazione al seguente teorema elementare (dove Y
e uno spazio di Hilbert).
Teorema 3.2. Sia
una soluzione della disuguaglianza
I Lu I .$ tf (t) I u I
(3.2) dove
u(t)
u(t) .!..
du at
sono di quadrato integrabile. Assumiamo che vi sia una
successione di linee nel piano complesso }.. an --+ 00
con te
M.
allora
sulle '{uali
parallele all'asse reale; 1m>. =a n
IR(). ) \ e uniformemente limitata da una costan-
~ lP (t) .$ c < M- 1 ~ I u(t) I
= 0 (eat)
per ogni valore reale a,
u:. O. Il teorema si dimostra facilmente con uso delle trasformate di Fou-
rier e del teorema di ParsevaI. Sia di
COO
niamo
la quahe fi annulli per v(t) = e n ~(t) u (t)
~ (t)
,;;. 0 una funzione monotona di t
,con v (t) = 0 per
t < 0 , e posto
prendendo Ie trasformate di Fourier, a,bbiamoJper)..
'i
t ~ "t . Po-
t.$ 0 e sia uguale ad uno per
;..
reale,
A
(.)I. + 1 an - A) v = f.
(3.3)
AHora per l'ipotesi su
R()..)
abbiamo ancora per
A
reale
(3.4)
e dal teorema di Parseval segue che, per un1opportuna costante dente da
n
:
150
C indipecl-
- 27 L. Nirenberg
~
00
00
I:v(t)
~ I f(t) I 2 dt
I 2dt $ M2
o
o
t
) Ie
ant
Lu
I
2
dt
?:
JOO
2 2a n"t 2 2 $MCe +cM
e
2a n t
2
lut dt
't
a causa dell'ipotesi fatte.
dove
Limitando l'integrazione a primo membro all'intervallo: a t v = e n u ,troviamo,poiche eM < 1 2 2 ( (1 - c M ) J
00
e
2a nt
IuI
2
t
> 1:'
2 2a n 't dt $ M C e •
"r
Questa disuguaglianza vale per tutti gli u(t} = 0 per Lax eM
ne e
t > 1:'
[8]
,0
U;
0
poiche
n
r
,e ne segue di conseguenza che e arbitrario.
ha anche dato un esempio per mostrare che la condizio-
< 1 e essenziale; in questa esempio i A e autoaggiunto, M= 1
u
+'
e una soluzione non banale della disuguaglianza I Lu I $ ("2 + e.) I u I ~ 2 -at ( per un dato 0 ) la quale tende a zero come e per qualche costante
a .
3.2. I teoremi precedenti sono molto particolari; per esempio nel secondo ) ) se l'operatore iA e autoaggiuntojallora la condizione del teorema richiede che il complemento della spettro all'asse reale non sia: limitato superior-
151
- 28 -
L. Nirenb'erg
mente su questo. E' naturalmente interessante ottenere estensiorii di questi risultati che possono essere utili in pratica, e i risultati successivi di questa sezione possono essere considerati come generalizzazioni di essi. In questa sezione noi descriveremo soltanto due semplici generalizzazioni. Consideriamo dapprima il teorema 3.2, noi cercheremo di avere un risultato analogo che sia valida per ogni operatore
1m A = costante
aggiunto. In questa caso ogni segmento so
A
puo contenere punti dello spettro di
immaginario. Assumendo ancora che
A con
i A auto-
nel piano comples-
Ache appartengono all'asse
Y sia uno spazio di Hilbert noi otterre-
mo un teorema di unicitaper il problema di Cauchy finito sotto Ie seg-uenti ipotesi: (H) Consideriamo una successione di linee ImA=a n , an->oo. 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
-oo
(s, r)
t
u(t):;u(s+re
t:;s+re 1
,a< s < b ,
come coordinate oblique nella striscia. La
seguente formula fornisce una estensione olomorfa di
(3.9)
Z:; Z (a, b; o()
'ex.
della forma
f
, iQ(. (
io(
):;T(r)u(s)+le
r
)0
u
"'\
in
T(.I')f(t-Ae
iQ(
2:: )dA
Questo si verifica con un facile calcolo. Nel caso
I u(t) I $ K I u(b) I
f:;O
noi vediamo dalla (3.7) che
I u(t) I ::; K I uta) I ,
valgono rispettivamente nellato sinistro e destro della
striscia. Applicando il "teorema delle tre rette' I di' Hadamard otteniamo il seguente Teorema 3.1' : Se -vale la (3.7), ogni soluzione di
Lu:; 0
~(a,
b) soddisfa
la disuguaglianza di convessita
\u(t)\:;:Klu(a)1
b-t b-a
.lu(b)1
Supponiamo ora che
i Ao
t-a b-a
a
invece di essere il generatore infinite-
simale di un gruppo sia il generatore infinitesimale di un semigruppo per
t
~
T(t)
0 fortemente continuo. La formula (3.9) da ancora un estensione
analitica di una soluzione di
Lu:; 0
su (a, b),
156
~
33L. Nirenberg
u(stre
(3.10) in una semistriscia
id.
= T(r) u(s)
)
r >0 . Allora vale il seguente teorema sulla continua-
zione unica all'infinito (assumendo 0/. Teorema 3.5. Se
iAo
~
> 0).
un generatore infinitesimale di un semigruppo
fortemente continuo e se una soluzione u ,di Lu:: 0 per t > 0 -Eo t 1T!c)/. -soddisfa la disuguaglianza I u(t) I = 0 ( e ) per qualche f > 0, aHora T(t)
U
=O. Dim. - Sic come la norma di
te noi possiamo sostituire ;
temente
u
con
ue
-~t
T( t)
cresce al pili esponenzialmen-
con T(t)e -ct ,dove -iot ce ).
T(t) (
~=
c
>0
e conseguen-
Questa soluzione soddisfa ancora aIle stesse ipotesi cosl che possiamo ridurci al caso che ca di
I T(t) I ~ K • La (3.10) fornisce
u, uniformemente limitata, nell'angolo
allora un'estensione analiti0
< arg t < oL
,la quale
~
fortemente continua nella chiusura dell'angolo e soddisfa la condizione _E.tiT/r:J. lu(t) I = Ore ) sull'asse reale positivo. n cambiamento di variabile z = tTr/rX trasforma l'angolo nel semipiano superiore del pianoz ;ponendo norma di
v(z)
Ie positivo si ha
nel semipiano
I v(z) I = Ore
soddisfa la condizione
Iv(zJl
Imz
-!z
2: 0
u(z ol/T!') = v(z)
limitata mentre suIl'asfe reaTz ). Allora la funzione analitica e v(z) E
~
= Ore - T 1z I)
suIl'asse reale. Per un classi-
co teorema di Carlson (cfr. [13J p.185) concludiamo che anche
u
~
vediamo che la
v
=0
e quindi
O.
3.4. Le limitazioni inferiori per Ie soluzioni della (3.2) sono basate su .disuguaglianze per una soluzione dell'equazione non omogenea
(3. 11)
du 'A =1'f ...."..-1 UL 0(. ,
157
a
I
- 34 -
L. Nirenberg
per mezzo dei suoi valori agli estremi dell'intervallo (f e u sono supposte fortemente continue nell'intervallo chiuso
[a, b]
).Noi continueremo a sup-
porre valida la (3.7). Teorema 3.6. Esiste una costante fissa tale che ogni soluzione della (3.11) soddisfa la disuguaglianza b
(3.12)
I u(t) I $C K {l u(a)l .
+ lu(b) I + max
astSb
If(x) 1 (i+log
)
I~~~ I )dx }.
a
Prima di dimostrare il teorema osserviamo che se 10 spazio considerato
Y
e uno
spazio di Hilbert e se
e autoaggiunto allora vale
iA
una disuguaglianza piu forte (e pub essere provata in modo piu semplice): per
a$; t $ b b
IU(t)
I 2 $ 2 I ural I 2 + 2 I u(b) I 2 + 4
J I ~~ - i Au /2 dt
[ (
]
2,
a Questa conduce conseguentemente ad un migliore risultato sulla convessita. Dim. del Teorema 3.6. Noi possiamo assumere che tima si realiz'Za sostituendo t con
a
= 0. e b = 1 I l'ul-
t/b e cib non cambia Ie condizioni del
teorema. Per' semplificare la discussione ancora di piu noi considereremo solo il caso Ol
= ~ •Estendiamo u nella striscia Z;: 0 $ Re t
$. 1 ponen-
do (3. 13)
u(t)
= u(s + ir) = T(r)u(s).
Applicando l'operatore di Cauchy-Riemann
158
(()
1
0
.Ic)
0f =2" ('(;s + 1 Tr)
- 35 -
L. Nirenberg
troviamo
%ft )
(3.14)
=
+
T(r) f(s).
Sui lati della striscia abbiamo
lu(t) I ~ K( 1 u(o) 1 + lu(l) I
)a
causa del-
la (3.7). Poniamoci ora in condizioni di usare il "teorema delle tre rette t, di Hadamard. A questo scopo sottraiamo da u una soluzione particolare della (3.14) in
Z . La
e allora olomorfa ed il teorema e applicabile. scegliamo (z = x+iy, e ~ indica ~~)
dJfferenza
Come particolare soluzione
(_1_ +
z-t
11 nucleo dell'integrale
e una
z+l+t
)dxdy.
soluzione fondamentale per l'operatore di Cau-
chy-Riemann e si verifica subito che l'integrale
e assolutamente
convergen-
te e che
AHora
h(t)
= u(t) - w(t) e una funzione olomorfa in
~
.
Prima di procedere oltre osserviamo che vale la seguente disuguaglianza, facile da verificarsi, con qualche costante
J
00
-00
\_1_ +~ I z-t
z+1+t
dy < C (1 + log
~_1--:-Ix-si
Segue di qui e dalla (3.14), avendo posto
159
C~ 1 :
se x f s.
- 36L. Nirenberg
1
I fix) I (1 + log
\
max
(3.15)
O~s':::l
I x-s 1 f
,)dx
= F,
o
che (3.16)
Iw(t)1
c :5"2 K F.
Consideriamo ora la funzione olomorfa h. Dalla (3.16) e dalla limitazione trovata per
u(t)
L. ) vediamo che sugli stessi lati
sui lati di
Ih(t)1 SKi lu(o)1
+ lu(I)1 + 2C F).
Per il IIteorema delle tre rette II la stessa disuguaglianza vale nell 'interno di
:E , e segue che per
t reale
lu(t)1 ::; Ih(t)1 + Iw(t)1 che
e proprio la
:s K ( I u(o) 1+
lu(l)1 + C F)
Q.E.D.
(3.12) cercata.
Possiamo ora facilmente dedurre un risultato di convessita per Ie soluzioni della (3.2). (3.17)
\Lu\::;
\P(t)
lui.
Applicando la (3.12) abbiamo max \u(t)1 ast$b
:s C K (
lu(a)
I + I u(b) 1+ B
dove
160
max a$t$b
I u(t)1
)
- 37 L. Nirenberg
(b \ ~ (x)( l+log
B = sup a$t::;b
b-a
Ix _ t I ) dx •
a
e abbastanza piccolo in modo che
Se l'intervallo (a, b)
B K C ~ 1/2
allora
si ha (3.18)
max a$t$b
I u(t)l S 2
C K( I uta) I
+ I u(b)! ) •
Ora per ogm arbitrario numero reale
v(t) = e
-d.. t
~
poniamo
u(t),
v soddisfa la disuguaglianza dv , I T1 ili+ (1\-
Chiaramente
"\ -ict Ar£A: e (Ao -
simale del gruppo
I T(t) I ~ K
A(/) vi ;:; c.p(t) I v(t)
~)
=e
T(t) = e -i ~ t T(t)
-iO(.-,... A, e i A
IS =~sen
0(
e un generatore
infinite-
il quale soddisfa anche la condizione
. Quindi la disuguaglianza (3.18)
v(t) • Allora ponendo
I
e anche valida per la
funzione
abbiamo per ogni numero reale 0"
Tenendo T fisso e scegliendo la costante 6
in modo che i due termini nel
secondo membro siano ugualiltteniamo il seguente risultato di convessita : Cor: Sia u una soluzione della (3.17). Se l'intervallo (a,b) lo che
161
e
COS!
picco-
- 38 -
L. Nirenberg
l
b
b-a L{J(x)(l+log \x'-tl
1
)dx$2CK
/
a
allora per
a
< t < b ,si ha b-t
lu(t)l~4CKlu(a)1
(3.19)
t-a
o:a. o:a ·lu(b)1
Applicando questo risultato ripetutamente a intervalli consecutivi possiamo ottenere limitazioniinferior'iper Ie soluzioni della (3. 17) per t >O. 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
\(1+tk)
Lvi
2dt
) 0
I~
.$ C1
1(l+tk)
Lu(t+T-l)
12 dt +
1
1
+C 3
(
(1+tk)2
)0 con ,qualche costante
C3
tI
LU(t+T.1)1 2 + IU(t+T-1)1 2 } dt
dipendente da ~
; quindi
(1 +tk)2ILU(t)l2dt+C4
f
T
2
2
(/Lul + lui )dt.
T-l U sando la (4. 1) abbiamo
(~
T
lu(t)/ 2dt + C5
T la quale da la (4.4) se
C1 c
2
~
f
/u(t)/ 2dt
T·l 1 2" .
4.2. °Dimostrazione del Lemma 4.1 : c 1' c 2' •..... indicheranno costanti dipendenti soltanto da A. Se matadi Fourier
(A -A) ~ = f
Lv
=f
noi abbiamo come al solito per la trasfor-
,0
165
~
42
~
L. Nirenberg
dovunque
R( A)
componiamo
sia regolare. Siano i poli
R().)
Al
< ,\ 2<
...... < Am; de-
in una somma finita per mezzo di una partizione fini-
ta dell1unita data da (m+2) funzioni non negative di Coo: (J.()..) , m+l J 0". (..A): 1 Ie quali abbiano la proprieta che j\. non appartiene al o J 1 supporto delle r:J. eccettuato (y. . 1 supporti di "1' ...... , t!J sono limiJ I m 1 si estendono a ~OO ed a +00 ritati mentre i supporti di (f 0 e r:; m+ spetti vamente. Poniamo
r:
R.(A) J
= d.(A)R(A) J
~=LwJ'(.A).
permodoche
Inunintornodi
A.:~=w.
cosl che Iw.! J J J e di quadrato integrabile. Poiche w.( A.), j = 1, .... , m, ha supporto compatJ to essa e la trasformata di Fourier di una funzione analitica v.( t) che e J in L2 . Noi dimostreremo per ogni j Ie disuguaglianze
l
00
I v.1
(4.3 1)
J
2 dt
< costante
-
o Nel punta
I (1+tk) f (t) \ 2dt •
0
A., R( A)
ha un polo di ordine :;; k e dunque puo esJ sere sviluppata in un intervallo contenente il supporto di cr. nel modo seguenJ
te: ()'_A..)-r P +P CA) J r 0 dove
P, r=l, ....... ,k r
sonooperatorilimitatifissatie
P (J\) 0
e unope-
rat ore olomorfo nel rettangolo:
1.
in un intervallo aperto contenente il supporto di CJ., e 11m.A 1< i J "' Per la formula integrale di Cauchy abbiamo che Ie derivate
166
- 43 L. Nirenberg d
r
\
( dA) Po(A)
J
con r::; k
J
sono limitate in norma nel supporto di 6'j .AI-
lora abbiamo \ k w.(I\) ) (),-/\.)
J
1
k '" =,-
1
(A - ~.) k-r cr. ().) P J
J
r
" '\ + 6.(A) (/I-A.) ') k " P (;\)f(A). J J 0
f(l\)
Differenziando questa uguaglianza e tenendo conto del fatto che Ie derivate di P o().)
e di
t5.( J
A)
sono limitate abbiamo
per n:;:;k.
Poiche la trasformata di Fourier inversa di
t
(~) 1
n
'\ k (D - .A.) v.(t) t J J
t n i A. t = (-.-) e J 1
(ft)\). - .A/w.( A) J J
A·
k -i t D (e J v.) t
JI
segue dal teorema di Parseval che
Applichiamo ora una ben nota disuguaglianza di Hardy (cfr. [6] teorema 330) secondo la quale per funzioni scalari
a(t)
si ha
(cio si prova facilmente con ripetute integrazioni per parti).
167
- 44 L. Nirenberg
U sando Ie due ultime disuguaglianze abbiamo
Vjl2dt
~c3
r
(1+tk)2If(t)/2dt
o
cioe Ia disuguaglianza (4.3') cercata. Finalmente consideriamo Ie funzioni e
m+1. Sic come
e uniformemente limitata Iontano dai poli noi
deduciamo come prima che woe wm+1 funzioni di quadrato integrabile
sono trasformate di Fourier di
v0 (t), vm+1 (t)
Ie quali soddisfano Ia disu-
guaglianza
j=O e (m+l)
Q.E.D.
qUindi in particolare anche Ia (4.3'). Osserviamo che se ne
I'"
d:).
I
= O( 1)
per
R( A)
/A I~
00
soddisfa inoltre Ia ulteriore condiziosull'asse reale) nel teorema e neIIem-
rna precedenti si pub sostituire Ia norma in
1 < P < 00.
168
L2
con quella in
L p ,per
- 45L. Nirenberg
BIBLIOGRAFIA
{lJ S.Agmon
On the eigenfunctions and on the eigenvalues of general elliptic boundary value problems, Comm. Pure Appl. Math. 15 (1962), pp.119-l47.
[2J S. Agmon, L. Nirenberg - Properties of solutions of ordinary differential equations in Banach space. Comm. Pure Appl. Math. 16 (1963) pp.121-239. [3] P. Cohen, M. Lees - Asymptotic decay of solutions of differentlal inequalities, Pacific J. Math. 11 (1961) pp.1235-1249. [4] N. Dunford, J. T. Schwartz - Linear Operators I, Interscience Publications, New York 1958. [5] O.Dunkel
Regular singular points of a system of homogeneous linear differential equations of the first order, Proc.Amer. Acad. Sci. 38 (1912-1913) pp. 341-370.
[6J G.H.Hardy,J.E.Littlewood, G.Polya - Inequalities. Cambridge Univ.Press, London 1951. [7] S. G. Krein, O. I. Prozorovskaya - Analytic semigroups and incorrect problems for evolutionary equations, Doklady Akad. Nauk. SSSR, N.S. 133 (1960) pp. 277-280. English translation, Soviet Mathematics Amer. Math. Soc. 1 (1960) pp. 841-844. [8J P.D. Lax
A stability theorem for solutions of abstract differential equations and its application to the study of local behavior of solutions of elliptic equations. Comm. Pure Appl. Math. 9 (1956) pp. 747-766.
[9J _ __
A Phragm~n -Lindelof principle in harmonic analysi:o with application to the separation of variables in the theory of elliptic equations. Lecture Series of Sympo-
169
- 46 L. Nirenberg
sium on Partial Differential equations, Berkeley, Calif:." summer 1955. Univ. of Kansas 1957.
[10J _______ A Phragm~n-Lindelof theorem in
harmonic analysis and
its application to some questions in the theory of elliptic equations. Comm. Pure Appl. Math. 10 (1957) pp.361-389.
[llJYU.I. J..yubi~.
Conditions for the uniqueness of the solution to Cauchy IS abstract problem, Doklady Akad. Nauk. SSSR N. S. 130 (1960) pp. 969-972. English translation, Soviet Mathelmatics Amer.Math.Soc. 1 (1960)pp.ll0-113.
[1~ A. Plis
A smooth linear elliptic differential equation without any solution in a sphere. Comm.pure Appl. Math. 14 (1961) pp.599-617.
[13J E. C. Titchmarsh
The theory of functions. The Clarendon Press, Oxford, 1939.
170
CENTRO INTERNAZIONALE MATEMA TICO ESTIVO (C.I.M.E.)
R. S. PHILLIPS
SEMI"GROUPS OF CONTRACTION OPERATORS
Roma - Istituto Matematicb delllUriiversita
171
- 1-
SEMI-GROUPS OF CONTRACTION OPERATORS by R. S. PHILLIPS
Resume: 1 - Introduction: The initial-value problem as motivation for the theory of semi-groups of operators. 2 - Strongly-continuous semi-groups of operators: a short course in the theory of semi-groups of operators. 3 - Dissipative oper ",tors, the class of generators of semi-groups of contraction operators. A Cayley-transform theory for disEjipative operators. 4 - The double extension construction for pairs of dissipative operators contained in each other 1s adjoints. 5 - Boundary space theory relative to a given null subspace. 6 - Operator theory: the application of boundary space theory to operators and in particular to symmetric system of differential operators.
173
~
2R. S. Phillips
Semi-Groups of Contraction Operators
1. Introduction. The theory of semi-groups of linear operators had its origin in Stone's theorem on groups of unitary operators acting in a Hilbert space (1932}. Stone's theorem was motivated by the time dependent solution of Schroedinger's wave equation for quantum mechanics. The study of semi-groups of operators, rather than groups of operators, was undertaken by Hille in 1936 who became interested in the semi-group properties of certain classical singular integrals. However it was not until 1948 that the applicability of the theory was fully appreciated. At that time K. Yosida applied semigroup methods to the diffusion equation. In the hands of Feller, Hille, Kendall, Reuter, and Yosida, the theory became an integral part of the theory of probability. It has also been applied with profit to the Cauchy problem for the wave equation; here the contributions of Friedrichs, Lax, and Phillips should be mentioned. We shall use the initial value problems of mathematical physics to motivate the theory of semi-groups. A suitably abstract formulation can be obtained by the following considerations. It will be recalled that Hadamard called an initial value problem
well set if i) There is a unique solution to the problem for some given class of initial data, ii) The solution varies continuously with the initial data. These two requirements are eminently reasonable on physical grounds. The existence and uniqueness of the solution being an affirmation or the principle of scientific determinism, whereas the continuous dependen-
175
- 3-
R. S, Phillips
ce is an expression of the stability of the solution, Stability is essential in view of the impossibility of knowing, the precise initial state and the consequent desirability of approximating the true solution with approximate initial data, If
Dt
denotes the set of initial data for the problem at time t ).0,
then the state of the system with initial data y in DtO is determined for all t > 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 <00
We show that this has a semi-group solution with generator DU
= class of functions y(x) with y , ~ dx continuously
n
differentiable and y 'dx 2 It is clear that
DU
is dense in X • A solution of
()..I - U)y
2
d =AY --y =f
dx 2
190
in X.
18
~
~
R. S. Phillips
can be obtained by the method of variation of parameters in the form
One verifies that to show that
d2 dx 2
y E DU
IR~ (U) I~ )~ 1
and that Ay ~ ~
f~r A. >0
=f .
It remains only
. This estimate can be obtained
directly from the integral representation of R)..
3
~
Dissipative operators.
The theory of dissipative operators is intimately connected with semi-groupsof contraction operators. Such semi-groups appear as the solutions to initial-value problems where a basic quantity does not increase in time. For hyperbolic equations this quantity is the energy, for parabolic equations it is the mass density, and for Markov processes it is the probability density. The study of all of these phenomena can be subsumed under the theory of dissipative operators. Unfortunately, however, not too much is known about dissipative operators in a Banach space. Most of the results apply only to Hilbert spaces; the results in 11 are meager but promising and those for 1 , 1 < P < 00, are very sketchy. We shall limit p our discussion to dissipative operators on a Hilbert space and their applications to certain initial-value problems. Suppose [S(t); t ). 0] is a semi-group of contraction operators with generator A. Then for y
E DA
o,js(n)YI~-IYI2 "(S(~~-Y, -7 (Ay, y)
we have
S('[lY)+(Y'
+ (y, Ay) 191
S("')~Y)
- 19 -
R. S. Phillips
as
't.... 0 • Geometrically
Re(Ay, y) ~ 0 means that
origin side of the tangent plane to the sphere
Ay + Y lies on the
[z j Iz I =Iy I ] at the point
y ., This forces the path of
~ = Ay
Definition. A linear operator
y(t)
for
to stay within this sphere.
L will be called dissipative if
(y, Ly) + (Ly, y).~ 0 ,
y
E. DL '
and maximal dissipative if it has no proper dissipative extension.
The salient result here is Theorem 3.
A necessary and sufficient condition for an operator L with
dense domain to generate a semi-group of contraction operators is that L be maximal dissipative. This can be considered as an existence theorem. Thus given a dissipative operator Lo
with dense domain, the theorem together with Zorn's
lemma then asserts that there is a dissipative extension L of Lo wich generates a semi-group of contraction operators. For example, let H =1 2 D(L 0)
t !
= ultimately zero vectors y = '¥j,j
(LoY)· = L..a,,'YI. 1 lJ tJ where it is assumed that
L. Ia lJ.. \2 <
00
i
192
for all
and for all finite sets
- 20 -
R. S. Phillips
of integers
t
'rIri]
that the matrices (a
In this case L
o
n.n. 1 J
)+
(an.n. )~ 8. J
1
is dissipative with dense domain and hence by the theorem
there exists a dissipative extension which generates a semi-group of contraction operators. Before proving Theorem 3., we first prove a basic lemma. Lemma 6. If L is dissipatlve and ~ > 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<)
J)(
(I + J)
being dense. As a consequen<
is one-to-one and defines a maximal dissipative operator, say
196
?
• 24 R. S. Phillips
with dense domain. Now (Ju - u, J)(v + v) :: (Ju + u, J~ - v) so that (Ly, z) :: (y, Mz) , This shows that Hence if L~
M C L)(
. On the other hand range
were a proper extension of M ,then
(I - M) :: DJK :: H .
I - L)( would have
a non-trivial null vector, which is impossible when range
(I - L) :: H •
4- The double extension construction. The previous extension theory has the shortcoming that the maximal dissipative operator L extending the dissipative operator LQ not only has an unknown domain, but also has an unknown action on that part of
It
not in DL
. For most applications this is not good enough. Generally we o are given both a 'minimal' operator Lo and a rmaximal' operator LI and we wish to find a maximal dissipative operator L which is both an extension of L 0 and a restriction of LI . In this case the action of L is prescribed and only the domain of L is to be determined. In this direction we can prove Theorem 5. If Lo and Mo are dissipative with dense domains as well as restrictions of each others' adjoints, then there exist maximal dissipative extensions
L:l Lo and M:l Mo
which are each others' adjoints.
Corollary. Under the hypothesis of this Theorem there exists a ximal dissipat,ive operator L such that
197
ma~
- 25 -
R. S. Phillips
Example. Let
H =12
and suppose
the hypothesis:
(a ij ) is a matrix satisfying for all i
for all j
ii) each principal finite submatrix is dissipative.
Let
DL
o
=D
Mo
= class of all ultimately zero vectors and set
It is clear that Land M o 0
are dissipative with dense domains and that
they contain each others' adjoints. In fact
According to the Corollary to Theorem 3 there is a maximal dissipative operator L such that Lo C L C M~ and L serves as the generator for the initial-value problem
198
- 26 -
R. S. Phillips
We will develop a solution to Theorem 5 couched in the language of subspaces of a Hilbert space. To this end we consider a Hilbert space H with inner product (x, y) on which there is given a continuous Hermitian symmetric bilinear form Q(x, y) = (Wx, y) ,regular in the sense that W has a bounded
inve'r~e.
Definition. A subspace N (or P) will be called negative (positive) if Q(x, x)
~
0
(? 0)
for all x E N (x E P), and maximal negative (maximal positive) if N (or P ) is not properly contained in any otner negative (positive) subspace. The ordered pair of subspaces
[N, pJ
will be called dual
subsp~
ces if N is negative, P is positive, and Q(N, P) = 0 ;and they will be called dual maximal subspaces if, in addition,
N is maximal negative and
P is maximal positive. In therms of these concepts we shall prove Theorem 6 . Any pair of dual subspaces in.a pair of dual maximal subspaces
[N, p]
[No' Po : N :::l No'
J
are contained
P:::l Po. Moreo-
ver Nand P are complements of each other relative to Q. That this implies Theorem 5 is seen as follows: Let H = Ho x Ho withelements
x= lx ' ,X 2 ], y= [Y',Y2] ,andinnerproduct
2 2
(x,y) = (Xl,yl) + (x ,y ) If we set
then the graph of L
o
,G(L) ,is a negative subspace and G(-Mo) is apo0
199
- 27 -
R. S. Phillips
sitive subspace whenever Land M
are dissipative. Further if L
0 0 0
and Mo contain each others adjoints then
and hence [G(L 0), G( - Mo}
1
form a dual pair. Moreover Q is regular
in this case and WX = W. Assuming Theorem 6 ,there exists a maximal negative subspace N:::l G(Lol and a maximal positive subspace If Nand P
P:::l GC-M} such that Q(P, N) = O. o
are graphs, they are clearly graphs of maximal dissipati-
ve (respectively maximal accretive) operators. It suffices to consider the case of N . Definition • If N is a subspace of
H = Ho x Ho
' its domain DN
is defined as
Lemma 8. If N is a negative subspace with dense domain, then N is a graph. Proof. If N were not a graph, then there would exist an x
= r x', x2 ] EN such that xl = 0, x2
find a
y
1€
= [y.,y 2
N such that
+O.
(y"x2)
Since
rOo
~.
is dense we can
Then
However on N, -Q is a positive form and satisfies the Schwarz inequa.l i -
200
- 28 -
R. S. Phillips
ty, that is
I
\-Q(x, y) 2 since
~
t
-Q(x, x)
1
[-Q(y, y)
1=
0
Q(x, x) = 0 . Consequently N must be a graph. Defining Land M so that G(L)
= N , G( - M) = P it is clear that
Land M are each maximal dissipative operators. Since the graphs G(L) and
G(-M)
are dual it follows that
L C MJ and
MeL". By Lemma
7 we know that LX and MX are (maximal) dissipative and hence L
= M'l(
and M = LX . This concludes th~ proof of Theorem 5 from Theorem 6. Before getting into the proof of Theorem 6 ,it is convenient to re norm H so that W becomes unitary. The original W splits H into its positive and negative complementary subspaces H+ and H vely and since W is regular there is an m
for all
x € H+
respecti-
> 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 <
0 • This problem is non-trivial even in the finite dimensional
case. As a special case it solves the following: Given x , f such that o (xo, f) + (f, xo) ov :; -'--""'--"--'-'-'-...>1..1<0 o J x 12 + If/ 2 '
to find a dissipative matrix operator
L :; (a .. ) which is real symmetric 1J kes Xo into fo .
(a .. :; a .. ) with \I:; V and whichta1J J1 0 60
We conclude this section with a brief discussion of a duality theory for positive and negative subspaces. Definition. Given a subset
S C H ,the Q-orthogonal complement S' of
S is
205
- 33 -
R. S. Phillips
I
SI;;.
y; Q(x, y)
=a
for an xES
1.
We denote the closure of S by S .
It is clear that if N(P) it is a negative (positive) subspace, then N(P) is also negative (positive) . Further S I is a closed subspace for any subset S . Likewise
S":l S for any subset S and if So C Sl then
Sio :l SI1 . Lemma 9. If S is a closed linear subspace then S" Proof. From the fact
Q(x, y) = (Wx, y) SI
( 14)
= (WS)l.
and using the added fact that range W SI
W
= S.
we obtain
,
=H
we obtain
= S.l..
However if S is a closed subspace this implies S
= (WSI)J.
• Applying
(11) to SI instead of S gives S" = (W SI) J. ,so that S" = S. Lemma 10. If No and Po are Q-orthogonal complements with No
negative and Po
positive, then
[No, Po
1
is a dual pair of maxi-
mal subspaces. Proof. By Theorem 6 there exist
l N, P 1 is
N:l Nand o
p:J P
0
such that
a dual pair of maximal subspaces. However
N C pi C pi o
=N
0
PeN' C NIo
and
so that the maximality of Nand P requires that N
=P 0
= No and P = Po
Lemma 11. If N is a maximal negative then P = N'
is maximal
positive. Proof. If N is maximal negative then N is closed and by Lemm8
206
- 34 R. S. Phillips
9,
N = Nil
= pi
. In view of Lemma 10, it sufficies to show that P is posi-
tive. If this is not the case then there exists ayE. P such that Q(y, y)
+ Ia
I2
Q(y, y) ~
so that N1 is a negative subspace. Further y ce y € Nt we get Q(y, y)- =
°
f
°
N . For if yEN then sin-
contrary to one choice of y. It follows that
N1 is a proper extension of N which is negative and this is contrary to the fact that N is assumd to be maximal negative. Lemma 12. A dual pair of maximal subspaces
[N, P
J
are necessarily
Q - orthogonal complements. Proof. By Lemma 11, NI is maximal positive and P' is maximal negative. By hypothesis
Pc NI, N C pi , and N, Pare a,lso maximal subspa-
ces. It follows that P = N' and N = pl.
5 - Boundary space theory. One of the principal applications of dissipative operators is to the initialvalue problem for partial differential equations where traditionally the problem of extending an operator has been put in terms of boundary conditions. As we shall see the extension problem takes on a somewhat simpler form in this case and is amenable to a more complete solution. A simple example will serve both as motivation and as a model for the development. Let
H = L2 (0,1)
DL = DM o
= [Y(X); y absolutely continuous; y,y'
E. L2 (0,1) ;
0
y(o) = LoY = yl - Y M z = -z' - z o
207
°= y(l) ]
- 35 -
R. S. Phillips
A simple calculation shows that
(LoY, z) - (y, Moz) = y(l) z(l) - y(O) z(O) = O. Thus Land M are both dissipative with dense domains and contained in o 0 each others' adjoints. The adjoints themselves can be shown to be Du< = DM)( =[ y; y absolutely continuous; y, y' E. L 2(O, 1) o
0
= y' - y.
and ~y
:r7z=-z'-z o
We seek to characterize the maximal dissipative operators L such that Lo C L eLl = M~; in particular DL C DL C DL . The problem can ther~ o
1
fore be simplified by considering the factor space
which is two dimensional space with coordinates [a, b] ,where y G. DL -)[a, bJ means that y(O) = a,y(l) = b. 1
For
yE
It
,L dissipative
(Ly,y)t(y,Ly)= /y(l)\
2
-/y(O)[
2
-2
208
Jo( 1 )yl 2 dx~O.
- 36 -
R. S. Phillips
Actually this requires
Iy(l) /2 - Iy(O) \ 2<.0
; for if
,I y(l) 12 _ / y(O) 12 > 0
then since DLo C DL we can find a sequence {Yn JC DL
y -7[Y(O),Y(I)] and n
as n --j
lim
£I 1
y nOn
I
2
such that
= O. Hence
dx
00 ,
which is impossible, Consequently / y( 1) \2 -/ y(O) / 2 ~ 0 for all y E. DL • Thus relative to the form AA/\
Q(y, z)
A
1\
"
= y(l) z(l) - y(O) z(O) ,II
"
maps into a negative subspace DL . Such subspaces can only be .
A
zero or one dimensional. Since L is maximal dissipative DL must be one A
dimensional, that is of the form y(l)
1\
=0< y(O) when 10< \ ~ 1 . For every
such choice of 0< we obtain a dissipative operator, say L ,with domain of the form , 1\y(l)
A]
= rJ y(O) .
Each operator L of this type is maximal dissipative since by theorem, 5 it would otherwise have a maximal dissipative extension L' such that L C L' C Ll and this as we have just seen is impossible. It is also clear that these operators define all of the maximal dissipative operators between L
o
and Ll . Thus the domain of all of the maximal dissipative operators between L
209
o
- 37 .;
R. S, Phillips
and L1 are determined by and in one-to-one correspondence with the maximal A
A
A
negative subspaces of H relative to the form Q . Now Q is included in A
H by the form Q(x, y)
(15)
on H, which for
2
2
= (x', y ) + (x ,y') + 2(x', y')
x, y E. DL
becomes 1
Q( [x', L 1X']
'~" L 1y'] ) = (x', Ly') + (Llx', y') + 2(x', y')
= x'(I) y'(I) - x'(O) y'(O) . This suggests that we use (15) as a starting point rather then the form Q used in the previous section. For Q(x, y)
=0 .
Actually
Thus
G(Lo)
G(Ll) = G(Lo)'
x ~ G(Lo)' Y E G(L 1) we note that
is what we shall call a null space and
G(Ll) C G(L o)'.
as can be verified. Finally we note that
This then is the pattern we shall follow in our present development. We begin as before with a regular Hermitian bilinear form Q on H . A closed linear subspace So is called a null space if all x
Q(x, x)
= 0 for
E. So . It follows from the Schwarz inequality that Q(x, y) = 0 for alL x,
y € So . Consequep.tly if we set
S1 = S~
then
So C S1 . The quotient space
will be called a boundary space. In the usual topology for a quotient space, namely,
210
- 38 -
R. S. Phillips
1\
13 denote~ the orthogonal projection =\J3 x l. Moreover flx=foy if and only if
H is again a Hilbert space. In fact if
of S1 on
\~I
S1eso then /\
X - y:' So . Hence H is isomorphic and isometric with S1 also denote the natural map: the context whether
J3 y
81~ 8 1
e
80
by
J3
e8
0
•
We shall
and it will be clearfrom /'
is to be thought of as lying in H or in
81
e8
0 •
Lemma 13 . If x, y E: 8 1 ' then Q(x, y) = Q(fo x,
(i)
fo y)
(ii) Q(~ x, J3Y) = 0
and if
and
for alL y E 8 1
Proof. Now
Q(x, y) = 0 for all
y - j3y E. 80
it follows that
= Q( [~ x + (x - fo x)
Q(x, y)
e8
0 '
then
j3 x = 0 •
x E 80 ,y E 8 1 and since x -
fox
J, [0 y + (y - ~ y)] ) = Q(fo x, fo y) .
This proves (i) and (ii) follows from the continuity of Q. Finally if
Q(fo x, ~ y) = 0 for ali Q(~X, y) = 0
y E 8 1 and by (i) well as
81
e 80
and so
As a consequence
y=
fo y E. 8 1e So
f~r all
then it holds for all
y E. 8 1 . Thus
fi x e 8
0
=81
as
fo x = 0 . Q(x, y)
depends only on the cosets to which x, y be-
"
long. Thus Q induces a bilinear form /Q' on H ,namely 1\
Q( ~x, ~y) = Q(fox, J>Y) •
211
- 39 -
R. S,Phillips A
1\
Lemma 14. Q is a regular hermitian form on H. Proof.
-e
A
is clearly continuous and Hermitian so that Q defines a bounA
ded self-adjoint operator W with the property /\""
1'''11
Q(x, y) = (Wx, y) = Q( fo x, j)y) = (W
fo x, 53 y) .
/\
The last part of Lemma 13 also implies that W is one-to-one. It remains to A
show that W has a bounded inverse. To this end we let jection of H on SI
flfJW -1
W
to SI
e
e
1\
So' Then W can be thought of as a restriction of
So' Let x E. SI
xf SI . Further setting
y 1 = fooW- 1 x E SI
e So
po denote the pro-
e
So . Then W SI -1
W
= S;
implies that
= Yo + Yl when Yo E. So and
x
' we infer from the relation
W So = S/ that fooWY 0 =0
and hence that
n.. W (J. W-1 x = r~ n. Wy 1 =Fo p" (x - Wy ) = (J. x = x. ro)"o 0 j"o Hence
J3oW -1
with the fact that
poW
is a right inverse for
floW
.
on
SI
e So
and this together
A
is one-to-one proves that W is regular.
Corollary. The previous duality theorem holds for the negative and po1\
1\
sitive subspaces of H relative to Q. A
Lemma 15. The mapping: M...o.+ M = J3M defines a one-to-one between 1\
the subspaces of SI which contain So and the subspaces of H. This correspondence preserves (a) negativity, (b) positivity, (c) inclusion, and (d) 1\
Q-orthogonal subspaces correspond to Q-orthogonal subspaces. In particular, subspaces of SI which are maximal negative (maximal positive) relative
tel
subspaces of SI contain So and correspond to maximal negative (maximal
212
- 40 -
R. 8, Phillips A
positive) subspaces of H. Proof. Everything except the last assertion follows from the relation Q(x, y) = Q{J x, j3y)
for
x, y E S1 . If N is maximal negative relative to
subspaces in 81 , then since
Q{8 1, 80 ) = 0 we can adjoin
80
to N without
affecting its negativity, Consequently N must already contain 80 , Lemma 16.If
~ isasubspace
_Proof, Now
80 C Me 8 1 = 80' so that
8o eM'
C
.S'1 and by Lemma 15,
consequence
M' C
nal to M and hence
~
fo -11\M' ~
is maximal positive,
P-1~
81 ' P =
fo-1M,
fi
13M'
and M =
then
C
M'
=fo -l~'CSi'
8 "8 ' :J M' :J S' -= S ,Thus 1 .01 0 ""is orthogonal toj3Nl" M' , As a
fo -1/\M'. Again by Lemma 15, fo -11'M'
Lemma 17 • Suppose Then
of
is Q-orthogo-
f3.-II'M'.
M' . Thus M' =
is maximal negative in N
~
and let
~ = ~,
•
=.fo -1~ is a negative subspace of
is a positive subspace of 81 ' and both Nand P are maxi-
mal with respect to the subspaces of Hand Q-orthogonal complements.
"
Proof. It follows from Lemma 11 that P is maximal positive. That N is negative and P positive is clear from Lemma 15. By Lemma 16, P =fo
-1.A
N' = (j3
-11\
N)' = N'
and similarly
N = P' . Finally Lemma lO/im-
plies that Nand P form a dual pair of maximal subspaces. Corollary. Any negative (positive) subspace N of 81 which is maximal negative (maximal positive) relative to subspaces of 8 1 is also maximal negative (maximal positive) relative to the subspaces of H. Moreover N' C 8 1 is maximal positive (maximal negative) relative to the subspaces of H. We now have Theorem 7. The mapPing}
-1
defines a one-to-one correspondence
213
- 41 -
R. S. Phillips
between the maximal negative (maximal positive) subspaces of
'H
and the ma-
ximal negative (maximal positive) subspaces of H which lie in Sl • Under A
this correspondence Q-orthogonal complements go into Q-orthogonal complements.
6- Operator theory. We now apply the previous theory to operators in the following way. For H we take H
= Ho x Ho with elements
l
Xl,
x2
J' ....
and inner product
2 2
(x,y) = (x',y')
+ (x ,y )
and we take Q in the form' Q(x,y)
= (x',y2) + (x 2,y') - (Dx',y'):: (Wx,y)
where D is a bounded negative self-adjoint operator. Parenthetically we note that the theory can be extended to the case where D merely negative selfadjoint. In order to show that Q is regular we define the auxiliary operators on H
U
[Y',i]
[ y', _y 2 + Dy'
J
V[Y',iJ It is readily verified that U2:: I, V2 :: -I ,and W::; -V U so that 'vii is re~
gular. The operator U is of interest in its own right. In fact ( 16}
Q(Uy, Uz)
(17)
Q(y, Uz)
= -Q(y, z)
= (y2, z') _ (y', z2),
214
- 42 -
R. S. Phillips
According to (16) the mapping
y~Uy
defines a one-to-one inclusion preser-
ving correspondence between the negative and positive subspaces of H which preserves Q-orthogonality. On the other hand (17) shows that if N = G(L) where
II
is dense, thenU(NI) = G(Lx) .
We shall need the following Lemmas. Lemma 18. If N is a negative subspace with dense domain then N is a graph. Proof. Suppose N has an element of the form andchoose
u=[U1,}JCN
so that
(ul,i)to
y = [ 0, y2] , y2
f0
;
;here we have used the
fact that DN is dense. o ~Q(u+o(y,u+o(y) = Q(u,u)+ 2Reo<.(y2,u l) for all D( • This requires
u
(y2, I.) = 0 which is impossible.
Lemma 19. If S is a subspace with dense domain, then SI is a graph. Proof. If SI were not a graph it would contain an element of the form
making use of the density of DS . Then
u E S , Y €" SI
implies
which is impossible. We come now to the main Theorem: Theorem 8. Let So be a null space with dense domain and set Sl = Sb and'
In this case So and Sl will be graphs of linear operators, say Lo and
215
- 43 -
R. S. Phillips
Ll respectively. There is a one-to-one correspondence between the maxi,. mal negative subspaces
L~ ] of Q (taken relative to Q)and the maximal
dissipative operators [L
J such that
Lo C L eLI'
this correspondence
being defined by (18 ) which is dense in Ho ' The adjoint transformation
M = LX is again maximal dissipative with
dense domain and ca" be described as follows:
D(So)
and
so graphs of operators Mo and Ml respectively. Let M = LX is such that
is maximal positive and
1\
P
D(SI)
are al-
.1'
t:..
= N' . Then P
Mo C Me Ml
and
(19 ) Proof. Since Do;;:
"10
is dense and S
0
is a 'negative' subspace, Lemma
18 shows that So is a graph, say G(Lo)' On the other hand Lemma 19 as,... 1\
serts that Sl is a graph, say G(L 1). The space H is clearly a boundary space so that there exists a one-to-one correspondence between the ma-
.
ximal negative subspaces
fA] LN
1\
of H and the maximal negative subspaces
[N] of H which are contained in Sl ' this correspondence is given by N
=f.; -1(~) . Since N::J So we see that DN is dense and hence that N
is a graph, say G(L) . Clearly
Lo C L C L1
and
DL
is given by (18).
We now show that the operators L defined by the maximal negative subspaces contained in SI constitute all of the maximal dissipative operators between Lo and Ll . In the first place if G(L) is negative then L
216
- 44 -
R. S. Phillins
must be dissipative since (Ly',y')
+ (y', Ly') = Q( fyI, Ly'], [Y', Ly' J )+ (Dy',y').::: 0
Conversely if L is a dissipative extension of L o ' then gative subspace. In fact, given
{y~}
C
DL
with
z'n = y' -
a consequence of DL
o
y~ ~O
G(L)
It
there exists a sequence
in Ho
and
[Y'n~
C
DLo' Thisis
being dense and y' + Dr.. C DL • By Lemma 13 (i) 0
Q( [Y',Ly'], [Y',Ly']) = Q(
so that
y' E.
G(L) is a ne-
[z'n,Lz~l, [z~, Lz~J ) ~ (Dz~, z~)~O,
is a negative subspace. It follows that every maximal dissipa'
tive operator L extending Lo has a maximal negative subspace containing So for a graph and conversely. In particular the maximal dissipative operators L such that
Lo C L C Ll
have graphs which
const~tute
the maximal
negative subspaces N such that So C N C S1 ' which was to be proved. As to the adjoint operators, we note that
%(So) = DSo
and
U(Sl) = [U(So)
J' . Employing the above arguments we see that both the null
space U(So)
and
U(Sl)
are graphs, say of Mo and Ml respectively.
If G(L) is maximal negative, then G(L)'
ximal positive by Theorem 7 and so G(M) = U [G(L)']
l
=j3-1(p) (hence P= gIl is ma-
U G(L)' ]
is maximal negative. Set
,this will be a graph by Lemma 18. Clearly
l
U(S'l) C U [G(L)'] = U G(M)] C U(S'o) so that
Mo C Me Ml . On the other hand
217
- 45 R. S, Phillips
Q(U [G(M)
J' G(L)) = Q(G(L)'~
from which it follows that dense domain. Finnally
G(L) )
=0
M = LX and hence is maximal dissipative with
P[G(L)'] = ~
is the same as
fo {u [G(M)JJ =~
which proves (19). We now apply this theory to first order symmetric differential systems defined as follows: Let G be a domain in R, y(x) a vector-valued function defined on G toe n
with
inner product
where the
Ai(x)
m
<, >, We consider operators of the form
are continuously differentiable and symmetric matrix-va-
lued functions in G, B(x) is a continuous matrix-valued function in G and
xC:: G ,
(20)
Note that G need not be bounded and that the growth of the coefficients is onfy mildly restricted; in particular the coefficients Ai can go to zero near the boundary
r
of G.
The Hilbert space Ho is defined as the space of vector-valued functions on G with the inner product
(y', z')
=
1
< y'(x), z'(x) > dx •
G
We make one further
assumption besides (20), namely
(21 )
D~-KI
x EG,
,
218
- 46 -
R. S. Phillips
for some constant
K > 0 ; In this case ]):
y~Dy
defines a bounded linear operator on Ho . However this assumption is made only to simplify the exposition, an extended theory is available to handle the case where
]) is not bounded. H = Ho x Ho
We define Q on
by
Q(y, z) = (y', z2) + (y2, z') - (IDy', z') . Set D = [y';y1 continuously differentiable with compact support in Loo
n
L
For
00
y'
.
=L
\
Al ~ y' + By" ()
i:: 1
y E. G(L
G]
00
)
Xl-
we have
:1 I
[Li +] dx - (IDy',y')
G
= here
An =
L Ai ni
< Any',y' >dS
=0
;
11
where n i are the components of the outer normal to
f . The last step is not necessarily meaningful since we have not assumed that f' is sm';ooth enough to have a normal. However since y(x) has
CUffi-
pact support in G, the last equality is readily verified.
219
~
47
~
R. 8. Phillips Let
8
o
denote the closure of G(L
00
). It is clearly a null space with
dense domain and hence the graph (Lemma 18) of some operator, say L . o 8 1 = 8~ is again the graph of an operator, say L 1 . It can be shown that L 1 corresponds to a differential operator of the given type in some lized sense. Defining the boundary space
it = 8 1/8 0
we see that
genera~
~
is a
space of cosets. each coset corresponding to a boundary data. A subspace of H is then a homogeneous boundary condition, a maximal negative ces of ~
subspa~
correspond to the proper boundary condition,for the domainllf
the maximal dissipative
operato~
L· suell that
220
1.0 C L eLI'
- 48 -
References
1.
Dunford and Schwartz, Linear operators, part I, Interscience Publ. (1958).
2.
Friedrichs, Symmetric positive linear differential equations, Comm. Pure Appl: Math., Vol. 11 (1958) 333-418 .
.3.
Hille-and Phillips, Functional analysis-and semi-groups, Amer. Math. Soc. Coll. Publ. (1957).
4.
Lax and Phillips, Local boundary conditions for dissipative symmetric linear differential operators, Comm. Pure Appl. Math., vol. 13 (1960) 427-455.
5.
Lumer and PhilLps, Dissipative operators in a Banach space, Pac. Jr, Math., vol. 11 (1961) 679-698.
6.
Phillips, Dissipative operators and hyperbolic systems of partial differential equations, Trans. Amer. Math. Soc., vol.90 (1959) 193-254.
7.
Phillips, Dissipative operators and parabolic partial differential equations, Comm. Pure Appl. Math., vol. 12 (1959) 249-276,
8.
Phillips, On the integration of the diffusion equation with boundary conditions, Trans. Amer. Math. Soc., vol. 98 (1961) 62-84.
9.
Phillips, The extension of dual subspaces invariant under an algebra, Proc. Int. Symposium on Linear Spaces, Jerusalem (1960) 366-398.
10. F.Riesz and Sz.Nagy, Functional analysis, Ungar Pub!., New York (1955).
221
CENTRO INTERNAZIONALE MATEMATICO ESTIVO
(C. I. M. E. )
L UIG! AMERIO
ALMOST-PERIODIC EQUATIONS IN HILBERT SPACES
ROMA - Istituto Matematico dell 'Universita
223
ALMOST-PERIODIC EQUATIONS IN HILBERT SPACES by Luigi Amerio
1. - The present lecture concerns the existence of almost-periodic (a. p. ) so-
lutions of a. p. linear functional equations in Hilbert spaces. We shall consider vibrating membrane variational equation and,afterwards, the general case of abstract almost-periodic equations (including partial differential equations whose coefficients depend on time t, besides space coordinates). The main purpose is the extension to such equations of Bohr-Neugebaueds and Favard's theorems(l). As it is well known, Bohr-Neugebauer's theorem, for ordinary differential equations with constant coefficients, asserts that if the known term is a. p., the I). every bounded solution is a. p. Favard's theorems concern the general case, of ordinary differential systems whose coefficients and known terms depend almost-periodically on t . For these systems the boundedness of a solution does not imply, in general, its almost-periodicity: but it is possible to prove the existence of one a. p. solution, precisely
;t (t) , by assuming the existence of a bounded solution x (t) : o
x(t) is characterized by the minimax property that the functional ,JA(x) = Sup t
€
J
II x(t)"
J=(-oo<,t ~+oo},
defined in the set of all bounded solutions, takes in ~(t) its minimum value. It is essential, in the proof, to assume a suitable asymptotic behaviour
for the bounded eigensolutions of homogeneous translated systems and their limits : the norms of such eigensolutions must have strictly positive infimuIDs. The circumstance that, for these problems, boundedness is related to almost-periodicity can be extended to some classical partial differential equations. In this case the notion of almost-periodicity must be taken in the sen225
- 2L. Amerio se of the theory developed by Bochner (2) for vector valued functions, with values in a Banach space Y. The original theory, created by Bohr, was extended by Bochner to such functions and it is important, for what follows, to point a notable difference, concerning integration of a. p. functions, between numerical and abstract case. Let f(t) be an a. p. function, from J to Y , and put
J t
F(t) =
f(t) dt
o
If Y is an
euc~~dean
space, then F(t) bounded
=* F(t)
a. p. (theo-
rem of Bohr). For the general case (Y arbitrary Banach space) the same thesis was proved by Bochner under the more severe hypothesis that F(t) has a relatively compact range RF . It was recently proved by Amerio (3) that, although Bochner's hypothesis cannot be substituted in the general case by Bohr's hypothesis,nevertheless this possibility exists if Y is a uniformly convex space: hence Bohr's theorem can be extended, in particular;; to Hilbert spaces: Y Hilbert space, f(t)
~,
F(t) bounded
=} F(t) ~
This property, which belongs to the solutions of the very simple abstract differential equation
y' = f(t), gives rise, in a natural way, to a clas-
sification of Banach spaces, with respect to the connection between boundedness and almost-periodicity of solutions of differential equtions: this connection exists, as we shall see, for some problems interesting continuum mechanics.
2. - a) Let us eonsider the motion of a vibrating membrane
n
boundary, under an external force f, which depends on t, t
e
with fixed J, accor-
ding an arbitrary law. To describe the motion of such a membrane we have to solve the varia-
226
- 3 -
L. Amerio
tional vibrating membrane equation (deduced from Hamilton's principle) :
S{(x'(t)' h'(t))
(2, 1)
J
L
2 - (x(t), h(tU 1 + (f(t), h(t~ 2} dt H
=0 .
L
o
Equation (2. 1) is the weak form of the wave differential equ'\tfon. The notations are well known. Let us assume that (l is a bounded, open and connceted set of the m-dimensional euclidean space Rm; then (y l' y2) 2' (xl' x2)HI mean scalar products in L 2 and
H~ (Hilbert
spaces);
,,l: 2 2 0 precisely: L = L (11) is the space of all complex functions, square integrable in (1 , y ={ ~ (
IIY~
2
r= ( f 1' ... ,
S) ;
{~IY (r )/2 dfl}
=
~m)
eel}
1/2;
L
(n )(the set of all functions
HI = HI (f1) C L 2 is the closure of C 1
o
0
x = [x(
t ) ; J~n }
0
,continuous with their first derivatives and with com-
pact support) i~ the n~rm
II xiII ={j (l:.
l"'pm
a'k(1)
'0)( (S) '?JX (I) ~~+
a(Slx(r)x(n)
H :n j, k J ..r~ o Here C( means the conjugate of the complex number
the coefficients a 'k( }) = ak ,( ~), a( J J,
f)
0(
~ 1/2
dn. J ;
.
moreover,
~ 0 are bounded and measurable in
.Q , and it is 1,;.,m
E
j,k
J€Q} = l h(t, ! ); J€o
In equation (2. 1), x(t) = {x(t,! ); cement .of the membrane, h(t)
= I f(t,
f );
tEO} is the known external force.
means the unknown displa-
1 is a test function,
f(t) =
It is assumed that x(t) and 1 h(t) are (strongly) continuous from J to H ,that their (strong) derivatives o
227
- 4L. Amerio x'\t) and h'(t) are continuous from J to L 2 and that f(t) is locally Boch-
ner-integra5~e
(
111 f(t)1 't
2 dt ( + 00 , L
V compact
A ).
Moreover h(t) has a compact support and equation (2. 1) must be satisfied for every test function h(t). One ca.n prove that there exists, in J , one and only one solution x
=
= x(t) of (2. 1), satisfying the arbitrary initial conditions x(O)
=x
o
E HI 0
x'(O)
'
= xl
E
L
2
.
Let us consider now the cartesian product E = HI X L 2, whose ele1 0 2 ments are the pairs X = {xo' with xo 6 Ho ' xl I: L ,and norms
xd'
Ilx~E = Hxo~ ~:1 + IIx1~~2}
1/2
o E is a Hilbert space: the energy space . .I
Let x
= x(t) be a solution of (2. 1) : then X(t) = {x(t), x'(t)} is a
function from J to E (we shall still call X(t) a solution) and it results IIX(t)R E
.
Hence
11 X(t)U
= {lx(t)1 2 1 + Ilx'(t)11 2 2} 1/2 . H L o
~
brating membrane!1
represents the total energy, at the time t , of the vi.
If f(t) E 0 (that is for homogeneous wave equation) the principle of
conservation of energy holds:
II X(t)11 E = const. For the inhomogeneous equation, we can prove the following minimax theorem (4) (whose generalisation will be considered at
§
Assume that (2. 1) admits a bounded solution and let 228
4).
rf
be the set of
- 5-
L. Amerio all (necessarily bounded) solutions. Put )!(x)
II X(t)11 E' p.,
Sup
=
t
eJ
=
Inf f(x)
X E. Pf
there exists one and only one solution, ~(t), such that '"
)J. (x)
tV
=f
.
We shall call 'Jt(t) the minimal solution of (2. 1) : hence ~(t) is the solution which minimizes the supremum of the energy, in J . b) The E-almost-periodicity of the solutions of the homogeneous wave equation (f(t) I' 0) has been proved, under more and more general hypotheses, .
by d1fferent authors: Muckenhaupt von Neumann
(7)
,Sobolev
(8)
(5)
(for m = 1), Bochner
,La:dyzenskaya
(9)
(6)
,Bochner and .
. Let us recall, what is mte-
resting from physico-mathematical point of view, that Bochner has deduced from the principle of conservation of energy the almost-periodicity of a solution X(t), if its range RX is relatively compact. In other words, the solution X(t) has the following behaviour: taken an arbitrary €
>
0 , there exists a relatively dense set
II X(t+'L
Sup t
e
~ E. ,
) - X(t)/I
J
{T J£ such that
E
Afterwards, Sobolev succeeded in eliminating compactness hypothesis: he proved in fact (under suitable smoothness conditions for the boundary of
n. ) that every solution of the
homogeneous equations is a. p. At last, the
same thesis has been proved by Ladyzenskaya, without any smoothness condition. c) Let us consider now the hon-homogeneous wave equation, and assume that f(t) is L 2 - a. p. In this hypothesis, Zaidman(10) has extended Bochner's result, by proving that every solution X(t), whic.h has a relatively .
. (11)
compact range, 1S a. p. Afterwards, AmerlO tion with bounded energy is a. p. : hence
229
has proved that every solu-
- 6L. Amerio
f(t)
~
,
=> X(t)
RX bounded
~.
This statement, which constitutes an extension to the wave equation of Bohr-Neugebauer 1s theorem, can be proved by two different methods: the main tool, in the first method, is a suitable extension (12) to monotone sequences of real a. p. functions of Dini's theorem on monotone sequences of continuous functions, in a compact set. The second method (13) can be applied to much more general cases: one proves, first, that every bounded solution X(t) is weakly a. p.(w. a. p. Hthat is the scalar product (X(tL G)Eis. a. p. ,
V GeE). AfterwarL.s, one proves that the range RX is relatively compact. It follows that X(t) is (strongly) a. p., because of a general theorem .
on a. p. funchons
(14)
Contributions and generalisations, concerning the problem treated (12) (13) . (15). (16) in . ' ,have been subsequently gIven by Bochner ,ZaIdman , Prouse (17). For C -almost-periodicity of solutions (in connection with recent '1 (18) on regulansa ' t '" . ) see Vagh·(19) f 11n resu It s o lOn or wave equatlOn I .
3. - We expose now some recent result s of Amedo (20), concerning the extension of Favard's theorems to linear second order a. p. functional equations in Hilbert spaces : in particular, to the weak form of hyperbolic equations, with coefficients depending on t (besides space coordinates). In agreement with ge· on t hese equatlOns . (21) , we renounce to't·eh · . 0 f t he.sonera l t heorles contmUlty lutions x(t), and we assume only that they are locally square-integrable. Parallely, we are led to consider vector valued a. p. functions in the sense of Stepanov. In this connection, it is worth recalling Bochner's interpretation of Stepanov almost-periodicity (loc. cit. at (2)). a) Let Y be a Hilbert space, y = f(t)Y -a. p. _S2, from J to Y: hence f(t) €
L 2(!:::. , Y), '¢ compact
there exists a relatively dense set
11
,and, taken an arbitrary
{"}E ' such that 230
E>
0,
- 7L. Amerio .,
UA. Ilf(t+'rt1 )-f(t+-rt )11Y2d,}t,
Sup tEJ
Eo
J
A.= [- ~~ ~, ~] .
Let us consider now the Hilbert space L 2(y) = L 2(
fl
,Y); g
~ L 2(y)
0 0 0
means that
g = {g( '1
) ; 'Vf~ A.}
2 E L (
hence
I gA Put
2 Lo(Y)
f (t)
U
=
II g(, lIIy2
~o
= ff(tt.,);
IJ.
0' Y),
d~}i
'1 € 110 }
f (t) is a function from J to L 2(y) and it results o "f(t+T ) - f(t)R
2 =
L (Y)
{J ,
·0
II f(t+ 't'
t'l ) - f(t+ 'I
)/1 ~
d~}\ .
Hence: f(t)
e
L 2(
6 , Y) Vl:J., -=>
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
fdx),
0(
I) holds, it is
X I (tl,
; 1:'
" ) ~
If> (x,
t' ), and
X.\
(t) sa-
tisfies the equation Q>,. (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
is the adjoint of the operator A(r). Luckily, here we can make
the change of variable t-r = s, thereby getting dv YL = _A1f< (s)v ds
(12)
-
s
'
~
O.
We shall call (12) the adjoint equation to (1). The family of evolution
opera'~
tors which give the solution of (12),' call them P ad(s, q), are easily obtained in terms of (3), as follows. Consider the family of operators p1! (t, s), t (AB)-t = B~ A~ , the operators
P~ (t, s)
(3), but
~
> s.
Because
do not satisfy the evolution equation
*
t
P (s, r) P (t, s) = P (t, r), (13) P:jt (s, s) -- I .
We now set P ad(s, q)
~
= P(q, s).
It is then easy to verify that Pad' thus de-
fined] is the family of evolution operators which
solv~s
(12).
We can now state the "natural" solutions of (12). The expression to be considered is the following : (14)
lim t7iXJ
P d(t, O)P(t, O)f
a
We shall see that, under suitable general conditions, this limit will exist. We shall derive the existence of the limit in the sense of dominated convergence,
247
- 6-
but other similar theorems for other types of convergence are easily stated and proved; in a sense, the theorem we prove below is best possible (cf. Burkholder, Annals of Math. Stat., 1962).
2. ANALYTIC PRELIMINARIES. Before we state our main theorem concerning the convergence of (14) and that of its discrete analog, we must review some of the analytic tools which will be essential steps in the proof. These are the following:
( rJ...) Dourly stochastic operators. Let (S,
L: 'f ) be a probabi-
lity space, namely, a measure space of measure one. A doubly stochastic operator P is one that has the following properties:
(1) P1 = 1 , where 1
is the function identically equal to one; (2) ~ 1 = 1 ., where p. is the adjoint of P; (3) if f
2!
° in Lp(S, £ 'f
},
then Pf ~ 0, that is, P is
a positive operator. Doubly stochastic operators are bounded operators of
r. ,
norm one in all L (S, J-l ), 1 ~ P ~ C'
z.. ,(' )is an atomic measure space with a finite number of
atoms having equal measure. In this case we obtain the classical case of doubly stochastic matrices which are being thoroughly studied. More generally, if the atoms do not necessarily have equal measure, the notion of doubly sto-
chastic operators and their study is related to the extensive theory of matrices with the unimodular property. It would be interesting to extend this theory to the case of a non-atomic measure space, in view of the difficulty of extending the Birkhoff-von Neumann theorem to this case. Probabilistically, doubly stochastic operators are simply Markov processes with discrete parameters and with an invariant measure, for whic'l the adjoint process also has an invariant measure. Such processes occur in the
248
- 7-
G. B. Rota
most disparate connexions. ( ~) Conditional expectation. A projection E in L 2(S,
Z
,r
which is doubly stochastic is called a conditional expectation. There is an extensive theory and representation of such operators (cf. the texts of Doob and Loeve, or the author's paper in Padova Rendiconti, 1959); however, we shall only need the following property: conditional expectation preserves dominated convergence. In other words if f
n
- d~ f , then Ef -d!> 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 t
'
2:. f>'< , f
let (a) Y be a closed subspace of L1 (S
f<,)
pQ ) ,
~
be a measure space, and "'" '
I'DQ ) which is
also
closed under the lattice operations (that is a closed Banach sublattice of L 1(S
()lJ ,
LlI-' fltQ ) (b) Z be a subspace of L,>(J (S ~ ,2:,.,.. '/~
Then (a) There exists a Y
(b) There exists a
(J
-subfield
= L 1(S (J
.0,
Z
of the
(l-field
r~
jIoQ
such that
Lpo, flO )
-subfield
l: ' of i 254
()
such that the closure of Z
).
- 13 G. B. Rota
in L1 (S
Z
pa
f>o ,
/ ' ""
Lemma 2. to L1 (S
II
and
2... = /1 f II
p<j ,
Tf)l
is L1 (S
)
/Xl,
'£. I
;-r JKJ
).
P ) in-
Let T be a linear operator from L1 (S,2:
'>a , / ' 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!">
,
1 f p !: KJ •
We shall make use of these lemmas presently. First, notice that there is a natural mapping of L 1(S, given by f(s)
z: 'fl
) into L 1(S ~ ,~(Xj ,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\ ,~I
),
n
f)t:J )
is mapped into a subspace of the form
= 0,1,2, ... ,
the orthogonal projection of L 2(S"..
where
,6~ 'f
QQ
Z] )
0
=6
into L 2(S
. Let <Xl
En be
,2:./VI 'f~).
The crucial part of the proof consists in verifying that for f in L (S .'0 , !. , II /)oIJ ), E E f = PIP2 ... P P*' P 11 1 ... P.f<1 f. Once this is p 0 n n n nverified, the proof is completed by (a) remarking that E is a martingale,
n
(b) applying the martingale convergence theorem; (c) remarking that the conditional expectation operator E
o
preserves dominated convergence in L . P
255
- 14 G. B. Rota
Let us first compute E. f , remembering that f is in n
Lp (8 t><J ,~
'f r>cI
).
Actually, it can be assumed that f E Lp (l L 2, sin-
ce this subspace is dense. Let
*
j( P P l ' " ...AI P'l: f(s ) = g(s ) n n n n-
(21)
we shall prove that E f(s ,sl"") = g(s ). We have to show that g is the non unique function such that
f
h f dr'1fI 8~
(22)
for all h ~ L
Kl
(8
ItI.
'
f.
=
hg
df~
8"0
~ n' fi oc ). But once (22) has been verified for
h ranging over a dense subset of L
&>IJ
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 ),
the class of functions
twice continuously differentiable in H , with A
(1)
2
where f(t) is given in LI
oc
= f(t)
(J; H)
if the relatiop (2)
J (u(t),
J
cp "(t) + A* Cf (t) )dt =
J
holds, for every
(f,
'f
)dt
J
Cf E:
K A•.
We give a sufficient condition on A (precisely on Art ), which ensures . the existence of (at least) one solution of (2), when f(t) is given in 2
LI
oc
(J; H) without any other assumption at the infinity.
This is not, naturally, a well-posed problem.
261
- 2
~
S. Zaidman
Definition.
We say that A satisfies condition S if.the family of operators
(A* + A 2) -1 depending of the complex parameter M for all
A
A • is bounded in norm by
outside j intervals of length s, on a .double sequence of verti-
cal lines in the complex plane,
( Re).. = 0' n ~ + oo}
()" n ~ - oo} Here j is an integer
0, s a real
~
*
and
l Re ).
=
0, and the location of
the j -intervals may vary with the line. (Comp. Definition, pag. 131, in Agmort-Nirenberg
[1J ).
We have t1:e following Theorem 1.
!i
A is a linear rclosed operator in H with DA dense in H,
which satisfies condition S, th.en, for every f(t) E
L~
(J; H), there is at oc least a function U(t) E Lloc(J; H), solution of (2) for every cp e KJ! . 2'
We prove this result in our paper
[7J ' by using an adaptation
of a method given firstly by Malgrange in his paper on existence for general.
" partial differential equations [6] . (See also the Ch. III in the book of Hormander [5J ). In the proof of various lemmas we need also some tecniques given in Agmon-Nirenberg
(1] .
One sees readily that every self -adjoint operator satisfies the condition S.
§ 2.
After these general considerations, we restrict ourselves to so-
me more special cases. Firstly, when A is self-adjoint and regularize the weak solution
giv~n
~ ~
I,
P real,
we can
in the Theorem 1, and obtain, in fact,
the following
262
- 3-
S. Zaidman
Theorem 2. If u(t) E
Let A be self-adjoint and
L~oc (J; u'(t),
~~I,
and f(t) given
H) satisfies (2) for every u"(t) E
L~oc (J; H)
,
Cf €
e L~oc(J;
H).
KA' then one has
u(t) E DA 2
almost every where in J , and A u(t) E Ll
(J; H). oc Moreover, the relation (1) holds, almost-every where on
t E. J.
This is proved (and also in a somewhat more precise, quantitative, version), in our paper [8] , by using the tecniques of Agmon-Nirenberg . ( [1] - Ch. IV). An immediate corollary of the Theorem 2 , is that u(t) and u'(t) are continuous in H. This permits us to consider the problem of the bounded or almost-periodic solution!) of (1), when f(t) is bounded or almostperiodic from t E J to H. One proves easily (as indicated to us by Agmon), the Theorem 3. In the conditions of the Theorem 2, if f(t) is continuous and bounded from t E J to H, any bounded (from t E J to H) solution u(t) of (1), has also a bounded derivative. Now, consider f(t) almost-periodic in Bochner's sense [4] from t E J to H. We have the following Theorem 4.
~
A is self-adjoint and
function from t €
~
0, and f(t) is an almost-periodic
J to H, then any bounded solution (on J) u(t) of the e-
quation (1), is almost-periodic with its derivative, This theorem is proved in [8] by using a method of orthogonal projections which reduces (essentially) our problem to the problem of integration of Hilbert space valued almost-periodic functions. Here there is an important theorem of Amerio
[2,
3] , which says that bounded integrals of
almost-periodic functions in uniformly convex spaces are also almost-perio-
263
- 4S. Zaidman
die. The results expressed in the preeeeding theorems generalize an early paper of the author on almost-periodicity for the Poisson equation
[<9] .
264
- 5-
S. Zaidman
BIBLIOGRAPHY
[1] S. Agmon- L. Nirenberg. Properties of solutions of ordinary differential equations in Banach spaces, Comm. Pure Appl. Math., May 1963. [2) L. Amerio,
Sull'integrazione delle funzioni quasi-periodiche a valori in uno spazio hilbertiano, Rend. Acc. Naz. Lincei, 28, fasc. I, (1960).
[3) L. Amerio,
Sull'integrazione delle funzioni quasi-periodiche astratte, Ann. di Matelll. vol. LIll, 371-382 (1961).
[4] S. Bochner,
Abstrakte Fast-periodische Funktionen, Acta Math., vol. 61, 149-184, 1933.
" (5J L. Hormander,
Linear Partial Differential Operators, SpringerVerlag, 19fp.
(6) B. Malgrange,
Existence et approximatiqn des solutions des equations aux cterivees partielles et des equations de convolution, Ann. Inst. Fourier, Grenoble, 271-355 (1955-56).
[7J S. Zaidman,
Un teorema di esistenza globale per alcune equazioni differenziali astratte, Ricerche di Matem. (1964).
[ 8] S. Zaidman,
Soluzioni quasi-periodiche per alcune equazioni differenziali in spazi hilbertiani, Ricerche di Matem. (1964).
[ .9 J S. Zaidman,
Quasi-periodicita per l'equazione di Poisson, Rend. Acc. Naz. Lincei, voL. 34, Marzo 1963.
265
