3.
E l l i p t i c O p e r a t o r s and Function Spaces.
I n t h i s l e c t u r e we s h a l l d i s c u s s some of t h e b a s i c spaces
of f u n c t i o n s which a r e used i n a n a l y s i s .
I n a d d i t i o n we s h a l l d i s c u s s
some of t h e fundamental p r o p e r t i e s of e l l i p t i c o p e r a t o r s , f i r s t i n
t h e c a s e of t h e L a p l a c i a n , and then i n g e n e r a l ,
These r e s u l t s ,
e s p e c i a l l y t h e " s p l i t t i n g theorems" a r e of c o n s i d e r a b l e u s e i n proving
c e r t a i n s u b s e t s of t h e f u n c t i o n s p a c e s a r e a c t u a l l y submanifolds.
w i l l f i n d a p p l i c a t i o n i n hydrodynamics and gener a 1 r e l a t i v i t y
This
.
F i n a l l y , we s h a l l c o n s i d e r some elementary p r o p e r t i e s of t h e s p a c e of
maps of one manifold t o a n o t h e r .
We begin then with a d i s c u s s i o n of t h e Sobolev s p a c e s .
Sobolev s p a c e s .
Let
W c Rn
0 be t h e c l o s u r e of
Let
cm
be an open bounded s e t with
-
R
.
Define
c m ( n , Rn)
boundary.
t o be t h e s e t of
-L
f u n c t i o n s from
R
into
on some open s e t i n
(f
of
E
c m ( n , Rm)1
n}
Rn
Rn
t h a t can be extended" t o a
containing
t h e s u p p o r t of
f
R
.
Let
m
C o ( n 2 Rm)
crn
function
=
i s c o n t a i n e d i n a compact s u b s e t
.
To d e s c r i b e t h e Sobolev spaces i n an elementary f a s h i o n ,
we t e m p o r a r i l y i n t r o d u c e some more n o t a t i o n .
*
An
n
multi-index i s
m
This d e f i n i t i o n i s t h e same a s s a y i n g t h a t t h e f u n c t i o n s a r e Con t h e c l o s e d s e t R ( w i t h d i f f e r e n c e q u o t i e n t s taken w i t h i n R)
by v i r t u e of t h e % i t n e y e x t e n s i o n theorem.
See t h e appendix of
Abraham-Robbin [ I ] . The same technique c a n be a p p l i e d t o Sobolev
s p a c e s ; c f . t h e Calderon e x t e n s i o n theorem below and Marsden [ 8 ] .
an ordered s e t of
i s an
n
0
D (u)
=
u
.
For
X
S
(n,
cm(R, Rm)
m
R )
=
k
D u
[
k1
=
If
..., kn)
k = (kl,
+ k2 +
. . . + kn .
~f
by t h e formula
u € Cm(n, Rm)
H:(R,
(or
c I ( ~ Rn))
,
, define
k
2
I ~ u ( x ) ld x .
i --
)
(resp.
(resp.
]kl
then p u t
, define
lulls 2
NOW
non-negative i n t e g e r s .
multi-index,
E cm(R, Rm)
and
n
R ~ ) ) i s d e f i n e d t o be t h e completion of
cO(R, Rm))
under t h e
2
s
norm.
Note t h a t
spaces a r e c a l l e d the Sobolev s p a c e s .
~ , ( n ,R ~ 3
) H'(Q, Rm) ; b u t f o r
1 Is
1
, H;(Q,
These
H'
0
m
0
m
HO(R, R ) = H (R, R )
=
R ~ #) n s ( n , R ~ )a s we
s h a l l s e e below.
There i s another e q u i v a l e n t , and perhaps b e t t e r , d e f i n i t i o n
of t h e SoboPev norm.
so t h a t
maps on
be t h e
-+ Lk(Rn, Rm) where
R~ x Rn
x
k
1 1
k
d u
k
d u :
... x
\
the
Let
and
Rn -+ Rm
kth
t o t a l d e r i v a t i v e of
Lk(Rn, Itm)
u
denotes t h e k - l i n e a r
with t h e s t a n d a r d norm.
Then i f we s e t
times
I ]Is
norms a r e e q u i v a l e n t .
Also n o t e t h a t
HS(n, Rn)
and
This i s a simple e x e r c i s e .
H:(o,
Rn)
a r e H i l b e r t spaces
with the inner product
S o b o l e v Theorem.
(a)
Let
s
>
(n/2)
. Then
+k
k
HS(C2, Rm) c C (0,Rm)
t h e i n c l u s i o n map i s c o n t i n u o u s ( i n f a c t i s compact) when
has the standard
_<k)
Ck
ck(C2, Rm)
t o p o l o g y , ( t h e sup o f t h e d e r i v a t i v e s o f o r d e r
*
(b)
If
s
>
(1-112)
then
m u l t i p l i c a t i o n of components.
(c)
(d)
f
and
-
s
H ~ ( R R, ~ )i s a r i n g u n d e r p o i n t w i s e
(This i s o f t e n c a l l e d t h e Schauder r i n g . )
> h
f E HS(C2, Rm)
( C a l d e r o n E x t e n s i o n Theorem).
f E H ' ( R ~ , Rm)
h a s an extension
fhen f l a n
If
-
f
E
t HS-li
H S ( n , Rm)
.
then
.
R e g a r d i n g ( c ) , s e e P a l a i s [ l ] f o r a d i s c u s s i o n of c o n t i n u o u s
Sobolev c h a i n s ; i . e . ,
t h e d e f i n i t i o n of
HS
for
s
not an integer;
b a s i c a l l y one c a n u s e t h e F o u r i e r t r a n s f o r m o r one c a n i n t e r p o l a t e .
( d ) means t h a t
f
c a n be e x t e n d e d a c r o s s
aR
i n an
H
S
way.
D i f f e r e n t i a b i l i t y p r o p e r t i e s a t t h e boundary p r e s e n t s some
t e c h n i c a l problems b u t a r e v e r y i m p o r t a n t i n hydrodynamics.
i s important t o distinguish
H
:
from
H
S
Thus i t
.
The proof of t h e S o b o l e v Theorem c a n be found i n N i r e n b e r g
[I]; s e e a l s o S o b o l e v [ l ] .
[ l ] and P a l a i s
F o r m o s t o f h y d r o d y n a m i c s we w i l l n e e d
One o f
t h e outstanding problems i n t h e f i e l d i s determining
s.
t o w h a t e x t e n t we c a n r e l a x t h i s c o n d i t i o n o n
many p r o b l e m s ,
tinuities
For
one would l i k e t o a l l o w c o r n e r s and discon-
i n such things
velocity field.
H~
s>(n/2)+1.
P
as t h e density of t h e f l u i d o r t h e
wkjP
=
for t h i s .
spaces a r e often useful
Spaces of S e c t i o n s .
M
Let
Also, l e t
example
E
E
be a f i n i t e dimensional v e c t o r bundle over
.
M
For
may be t h e t a n g e n t b u n d l e , o r a t e n s o r bundle over
n : E +M
Let
be a cornpact m a n i f o l d , p o s s i b l y with boundary.
be t h e c a n o n i c a l p r o j e c t i o n .
M
.
The f o l l o w i n g f a c t i s
u s e f u l and i s obvious from t h e d e f i n i t i o n of a v e c t o r bundle ( s e e
l e c t u r e 1)
.
Proposition.
t h e r e i s a f i n i t e open cover
c h a r t of
M
and
rr
- 1(Ui)
2
, we
x E M
Suppose f o r each
(
U
Ui x R
Of M
m
have
E
i s a map
define, for
s
2
h : M +E
f o r each
0
, H'(E)
d e r i v a t i v e s up t o o r d e r
s
such t h a t
ah
3
i
.
Ui
.
Recall that a section
= id
M .
Informally
E
t o be t h e s e t of s e c t i o n s of
are i n
Rm
such t h a t each
Such a cover i s c a l l e d t r i v i a l i z i n g .
of
ir-l(x)
, we
whose
L2
This makes sense s i n c e i n view of t h e p r o p o s i t i o n , a s e c t i o n
of
E
can l o c a l l y be thought of a s a map from
R~
to
R~
where
n
i s t h e dimension of
on
H'(E)
.
M
S i m i l a r l y , we c a n p u t a H i l b e r t s t r u c t u r e
by u s i n g a t r i v i a l i z i n g c o v e r .
However, s i n c e t h i s
H i l b e r t s p a c e s t r u c t u r e depends on t h e c h o i c e of c h a r t s , t h e norm on
H'(E)
i s n o t c a n o n i c a l , s o we c a l l
S
H (E)
a H i l b e r t i b l e Space ( i e . ,
i t i s a s p a c e on which some c o m p l e t e i n n e r p r o d u c t e x i s t s ) .
one n e e d s some a d d i t i o n a l s t r u c t u r e such a s a
H'(E)
a good norm on
To o b t a i n
connection.
One h a s t o c h e c k t h a t t h e d e f i n i t i o n of
H'(E)
i s independent
of t h e t r i v i a l i z a t i o n and t h i s c a n be done by v i r t u e of compactness
M
of
0
Of c o u r s e t h e Sobolev theorems h a v e a n a l o g u e s f o r
s _> 1
In particular if
h
E
HS(E)
course i f
For
aM
to
s
>
, we
s = 0
.
H'(E)
i t makes s e n s e t o r e s t r i c t a s e c t i o n
T h i s i s by p a r t ( c ) of t h e S o b o l e v Theorem.
,
(1112)
have
.
h
Of
w i l l be c o n t i n u o u s and s o t h i s w i l l be c l e a r .
L2(E)
and r e s t r i c t i o n t o
aM
H:(E)
i n a s i m i l a r way.
For
d o e s n o t make
sense.
One d e f i n e s
restrict
h
t
H:(E)
t o order
s
-
+.
to
aM
,h
M
> f , when
we
w i l l vanish, a s w i l l i t s derivatives
Much of t h e t h e o r y goes o v e r f o r
s p e c i f y a m e t r i c on
s
and a c o n n e c t i o n on
M
noncompact, b u t we must
E ; further
M
must be
c o m p l e t e and obey some c u r v a t u r e r e s t r i c t i o n such a s s e c t i o n a l c u r v a t u r e
bounded above; s e e C a n t o r [ 2 ] .
o p e r a t i o n s on D i f f e r e n t i a l Forms.
M
Now, l e t
be a compact o r i e n t e d Riemannian n - m a n i f o l d
w i t h o u t boundary.
As i n L e c t u r e 1, l e t
whose f i b e r a t
x
E
M
M
at
x
E
M
,
to
R
.
For each
forms a g r a d e d a l g e b r a w i t h t h e wedge p r o d u c t .
i s a s p a c e of
H
S
M
be t h e v e c t o r b u n d l e o v e r
c o n s i s t s of k - l i n e a r skew-symmetric maps from
T ~ M, t h e t a n g e n t s p a c e t o
n IIk
@k=0 x
hk
d i f f e r e n t i a l k-forms.
Then
x
,
H'(A k )
The e x t e r i o r d e r i v a t i v e
d
then i s a n o p e r a t o r :
~t d r o p s one d e g r e e of d i f f e r e n t i a b i l i t y b e c a u s e
once; i . e . ,
xEM
differentiates
i s a f i r s t order operator.
*
The s t a r o p e r a t o r
at
d
k
: tIS(* ) + H S ( ~ n - k )
i s g i v e n on
A
k
by
and
where t h e
"+"
o r i e n t e d and
orthogonal a t
i s taken i f the
otherwise,
It-"
x
,
and
7
'
:
dxl A
x1,
... A
..-,x
n
dxn
i s positively
form a c o o r d i n a t e system
i s extended l i n e a r l y a s an o p e r a t o r
h
k
4
hn-k
.
k
a E H'(A )
Now i f
can be taken a s an o p e r a t o r from
The space
product.
hk
$:a
then c l e a r l y
Hs(nk)
to
H ~ ( A ~ ,- so
~ )
Hs(hn-k)
.
x € M
,
c a r r i e s , a t each p o i n t
I t i s the usual business:
E
:
I
an i n n e r
the metric converts covariant
t e n s o r s t o c o n t r a v a r i a n t ones ( i . e . , i t r a i s e s o r lowers i n d i c e s ) and
then one c o n t r a c t s .
el A
... A
If
mk, R1 A
check t h a t i f
oi
... A
,
Rj
P k>
a r e one forms, we have
.
= det[<oli9 P.>]
J
i s t h e volume form on
M
I t i s not hard t o
then
Note t h a t t h e i n n e r product may be d e f i n e d by t h e above
formula.
See F l a n d e r s [ I ] f o r more d e t a i l s on t h e s e m a t t e r s .
the operator
6 : H
s+l
i s an i n n e r product on
k
( A ) ~ H ' ( A ~ - ' ) by
H
O
(A k )
( a , P)
5 = (-1)n(k+l)+l,d7v:
(and hence on
=
H'(A))
+, B>Q .
'M
Proposition.
Proof.
For m E
Note t h a t
k
H'(A )
B E H'(A k+ 1)
k
d ( a A *P) = dQ A *B + ( - 1) a A
= d ~ A >':R
-
A 9:SR
given by
Define
.
There
a M = f~ , by Stokes Theorem, we g e t
Since
=
(day
Rephrasing, one s a y s t h a t
product
d
B) -
.
( a , 6B)
and
6
a r e a d j o i n t s i n t h e ( ,) inner
.
6
The
operator.
o p e r a t o r corresponds t o t h e c l a s s i c a l divergence
T h i s i s e a s i l y seen:
let
X
be a v e c t o r f i e l d on
Then because of t h e Riemannian s t r u c t u r e
-X , where
Q(,v>
proof and D i s c u s s i o n .
respect t o
X
.
corresponds t o a 1-form
.
div(X) = -6(?)
Proposition.
.
.
w
X(v) =
X
M
Let
LXp
be t h e L i e d e r i v a t i v e of
Then by d e f i n i t i o n ,
div(X)p = LXp
p
with
( s e e Abraham [ 2 ] ) .
We have t h e g e n e r a l formula
Now
d(p) = 0
N
since
i s an n-form, so
p
N
e a s i l y checks t h a t
since f o r
k = 1
,
i
(-1)
=
X )
.
n(k+l)+l
Hence
= - I .
a
L p = d ( i p)
X
X
=
d(J;X)
(one
A
The L a p l a c e de Rham o p e r a t o r i s d e f i n e d by
k
A : HS(h ) - t H
Note t h a t
on
,
R~
A(f)
Note
6f = 0
=
2
-e ( f )
k
HS(h )
, then
and
-
= 0
I t i s obvious t h a t i f
dct
To show t h e c o n v e r s e , assume
Sd)cu,
D)
=
+
(6@, 6a)
A form
8 E
0
=
=
Let w E
-
.
d~
.
HS(Ak)
y
k
Here
- cm(A )
Furthermore
summarized by
,
6
,
0
AQ
=
,
0
Am = 0
then
0 = (Acu, cu)
Hk
k
= ( y E C"(A )
=
so the r e s u l t follows.
a M = @).
such t h a t
a F H
s+E
m = do
(A
+
L
2
k-1
be
s e c t i o n s of
C~
a r e mutually
=
.
i s c a l l e d harmonic.
denotes the
Y
6ol = 0
Then
Then t h e r e i s
Cm(hk)
, 66 , and
.
0
.
iff
+
A
,
)
k
y
.
o r t h o g o n a l and s o a r e
uniquely determined.
If
d
i s the usual Laplacian.
0
and
( d a y da)
f o r which
H~+'(A~"+~)
A(y) = 0
.
d6
i s a r e a l valued function
Aol =
&a/ =
The Hodge d e c o m p o s i t i o n theorem ( f o r
heo or em.
f
+
on f u n c t i o n s .
da/
+
If
2
V f = div(grad f )
where
Let cu E
Proposition.
((d6
.
k
(A )
6d
i t i s e a s y t o c h e c k , u s i n g t h e above e x p r e s s i o n s f o r
that
Proof.
s-2
=
0)
,
t h e n t h e above may be
The f a c t t h a t t h e Harmonic forms
from r e g u l a r i t y theorems on t h e L a p l a c i a n .
xk
are a l l
C
m
, follows
This f a c t i s a l s o c a l l e d
w e y l l s lemma o r , i t s g e n e r a l i z a t i o n , F r i e d r i c h ' s theorem.
We s h a l l
d i s c u s s t h i s f u r t h e r below.
The Hodge theorem goes back t o V. W. D , Hodge [ I ] , i n t h e
1930's.
S u b s t a n t i a l c o n t r i b u t i o n s have been made by many a u t h o r s ,
leading up t o t h e p r e s e n t theorem.
See f o r example Weyl [ I ] , and
~ o r r e y - E e l l s[ I ] .
We can e a s i l y check t h a t t h e s p a c e s i n t h e Hodge decomposition
a r e orthogonal.
since
6
For example
i s t h e a d j o i n t of
d
and
d
2
= 0
.
The b a s i c i d e a behind t h e Hodge theorem c a n be a b s t r a c t e d
a s follows.
with
T
2
=
0
We c o n s i d e r a l i n e a r o p e r a t o r
T
.
i s the
I n our c a s e
ignore the f a c t t h a t
t h e a d j o i n t of
T
.
T
Let
T = d
and
E
on a H i l b e r t space
L~
forms.
i s only d e n s e l y d e f i n e d , e t c . )
H = (x
E = Range T
E
E ~ T X= 0 and T*x = 0 )
Range T*
Let
.
E
(We
T*
be
We a s s e r t
@
which, a p a r t from t e c h n i c a l p o i n t s on d i f f e r e n t i a b i l i t y and s o on i s
t h e e s s e n t i a l c o n t e n t of t h e Hodge decomposition.
To s e e t h i s , n o t e , a s b e f o r e t h a t t h e r a n g e s of
T
and
T*
a r e o r t h o g o n a l because
a x , T>'cy> =
Let
@
a 2x ,
y> = 0
be t h e o r t h o g o n a l complement of
kl c C
Certainly
.
a y , x>= 0
Tx = 0
Similarly
x
But i f
,
so
Range T
O
.
Range T;'c
E C ,
for a l l
C c 51
.
y =>T+cx = 0
.
M
.
and hence
C, =
The complete proof of t h e theorem may be found i n Morrey [ I ] .
For more elementary e x p o s i t i o n s , a l s o c o n s u l t F l a n d e r s [ I ] and Warner
[I].
An i n t e r e s t i n g consequence of t h i s theorem i s t h a t
ilk
is
isomorphic t o t h e k t h de Rham cohomology c l a s s ( t h e c l a s e d k-forms mod
the exact ones).
may be w r i t t e n
o u t when
(SP,
6P)
m
=
0
This i s c l e a r s i n c e over
u, = dm
+y .
, each
c l o s e d form
(One can check t h a t t h e
i s c l o s e d ; indeed we g e t
so
M
0 = d6P
so
68
u,
term drops
(dSB, B) = 0
or
6P = 0 .)
The Hodge theorem p l a y s a fundamental r o l e i n incompressible
hydrodynamics, a s we s h a l l see i n l e c t u r e 4 .
I t e n a b l e s one t o i n t r o d u c e
t h e p r e s s u r e f o r a given f l u i d s t a t e .
Below we s h a l l g e n e r a l i z e t h e Hodge theorem t o y i e l d some
decomposition theorems f o r g e n e r a l e l l i p t i c o p e r a t o r s ( r a t h e r than t h e
s p e c i a l c a s e of t h e L a p l a c i a n )
. However,
we f i r s t pause t o d i s c u s s
what h a p p e n s i f a boundary i s p r e s e n t .
~ o d g et h e o r y f o r m a n i f o l d s w i t h boundary.
T h i s t h e o r y was worked o u t by Icodaira [ I ] , D u f f - S p e n c e r [ I ] ,
( S e e Morrey [2] , C h a p t e r 7 . )
and Morrey [ I ] .
Differentiability across
t h e boundary i s v e r y d e l i c a t e , b u t i m p o r t a n t .
The b e s t p o s s i b l e
r e s u l t s i n t h i s r e g a r d were worked o u t by Morrey.
Also note t h a t
d
6 may n o t be a d j o i n t s i n t h i s c a s e ,
and
because boundary t e r m s a r i s e when we i n t e g r a t e by p a r t s .
Hence we
must impose c e r t a i n boundary c o n d i t i o n s .
i f t h e normal p a r t ,
i n c l u s i o n map.
Let
when
X
one-form
iff
aM
iXp
noc = i*(*a)
Analogously
X
a
Then
a
=
i s p a r a l l e l or tangent t o
X
iff
Then
i p
X
X
n
i s tangent t o
- 1
i s tangent t o
i s normal t o
aM
.
aM 4 M
i s perpendicular t o
b e a v e c t o r f i e l d on
and a l s o t o t h e
i :
0 where
M
aM
.
.
aM
i s tangent or perpendicular t o
volume f o r m ) .
to
.
k
oc E HS(A )
Let
form
aM
i s the
3~ i f
t~ = i"(@) = 0
U s i n g t h e m e t r i c , we know
.
iXp
X
corresponds t o the
(p
i s , a s usual, the
i f and o n l y i f
Similarly
X
X
and
E
i s tangent
i s normal t o
Set
HS(Ak) = ( a E H ' ( A ~ )/ a i s t a n g e n t t o 3 ~ )
k
H ~ ( A
) = {a
aM
k
HS(A ) / a i s p e r p e n d i c u l a r t o
a~)
aM
.
The c o n d i t i o n t h a t
do!
= 0
and
6a/
i s now s t r o n g e r t h a n
= 0
Following Kodaira [ I ] , one c a l l s e l e m e n t s of
xS
, harmonic
La, = 0
fields.
The Hodge Theorem.
k
H'(A )
=
d
s f 1 k- 1
( ~ ~ 1)
cn
a
s+l
S(H,
cn k+l ) ) m
ns(nk)
.
One can e a s i l y check from t h e formula
( d a , l3) = ( a , 6P)
+j
r
cu A *p
aM
t h a t t h e summands i n t h i s decomposition a r e o r t h o g o n a l .
There a r e two o t h e r c l o s e l y r e l a t e d decompositions t h a t a r e
of i n t e r e s t .
Theorem.
where
and d u a l l y
where
S
Cn
a r e t h e c l o s e d forms normal t o
aM
.
D i f f e r e n t i a l O p e r a t o r s and T h e i r Symbols.
Let
E
and
F
be v e c t o r bundles over
M
and l e t
.
c r n ( ~ ), H'(E)
Assume
denote t h e
and
Crn
H~
s e c t i o n s of
i s Riemannian and t h e f i b e r s of
M
E
and
E
F
a s above.
have i n n e r
products .
A k t h o r d e r d i f f e r e n t i a l o p e r a t o r i s a l i n e a r map
D : crn(E) -+ c ~ ( F ) such t h a t i f
x E M
order a t
, then
E
f
D(f)(x) = 0
crn(E) and
.
f
vanishes t o kth
(Vanishing t o k t h o r d e r makes
i n t r i n s i c sense independent of c h a r t s .)
Then i n l o c a l c h a r t s
where
j
mapping
=
( j , .
.
,j
E
to
F
.
Now
D
)
D
h a s t h e form
i s a m u l t i - i n d e x and
h a s an a d j o i n t o p e r a t o r
t h e s t a n d a r d Euclidean i n n e r product
where
of
a
pdx
j s
1A
. .. A
dxn
D*
a
j
D*
cm
function
given i n c h a r t s ( w i t h
on f i b e r s ) by
i s t h e volume element and
The c r u c i a l p r o p e r t y of
is a
a*
j
i s the transpose
is
( g , D;kh) = (Dg, h)
where
(
,)
denotes the
2
inner product,
g E c ~ ( E ), and
h E c;(F)
.
A k t h o r d e r o p e r a t o r i n d u c e s n a t u r a l l y a map
D : H'(E)
+H
S-k
(F)
.
For example we h a v e t h e o p e r a t o r s
k
d : H'(A ) + H ~ - ' ( A ~ + ' )
6 : R'(A~)
.
A : H'(A~)
and
The symbol of
D
5 E
a s s i g n s t o each
TM
;
,a
l i n e a r map
I t i s d e f i n e d by
where
g E c r n ( ~ R)
,
,
dg(x) =
5
i s danger o f c o n f u s i o n we w r i t e
and
m
f E C (E)
nc-(D)
5
of d e g r e e
j:
p l a c e of a
a/ax
k
=
C gij
2
e
.
I f there
D
i s a polynomial
For example, i f
aLf- -t ( l o w e r o r d e r t e r m s )
axi&
then
=
o b t a i n e d by s u b s t i t u t i n g each
' i n t h e h i g h e s t o r d e r term.
D(f)
f(x)
t o d e n o t e t h e dependence on
By w r i t i n g t h i s o u t i n c o o r d i n a t e s one s e e s t h a t
expression i n
,
F
j
in
.
(
j
i s f o r each
ij
a map o f
Ex
Fx) . F o r r e a l v a l u e d
to
f u n c t i o n s , t h e c l a s s i c a l d e f i n i t i o n of a n e l l i p t i c o p e r a t o r i s t h a t
the above q u a d r a t i c form be d e f i n i t e .
T h i s c a n be g e n e r a l i z e d a s
follows:
is called elliptic i f
D
i s a n isomorphism f o r e a c h
5
5 B O
We have now s e e n a l l t h r e e c l a s s i c a l t y p e s of p a r t i a l
d i f f e r e n t i a l equations:
elliptic:
parabolic:
TO s e e t h a t
AQ
t y p i f i e d by
a~
at
- = AU
t y p i f i e d by
A : J3"(hk)
4
= @
llse2(nk)
i s e l l i p t i c one u s e s t h e
facts that
and
(1)
the symbolof
d
is
o5
=
(2)
t h e symbol of
6
is
5
= i
(3)
t h e symbol i s m u l t i p l i c a t i v e :
E
517
c
CT
5
(D O D ) = o ( D ) o g (D )
1 2
5
2
.
The R e g u l a r i t y Theorem and S p l i t t i n g Theorems.
Theorem.
order
f
E
k
Let
.
M
Let
be compact w i t h o u t boundary.
f
E
L2(E)
and suppose
Let
D ( f ) E H'(F)
D
be e l l i p t i c of
.
Then
.
I-I~+~(E)
One c a n a l l o w b o u n d a r i e s i f t h e a p p r o p r i a t e boundary c o n d i t i o n s
a r e used.
See Nirenberg [ I ] .
g e t Weyl's lemma:
Af
0
=
As a s p e c i a l c a s e of t h i s theorem we
=> f
cm .
is
The proof of t h e theorem i s t o o i n t r i c a t e t o go i n t o h e r e ;
s e e P a l a i s [ I ] o r Yosida [ I ] .
I t i s important t o note t h a t t h i s s o r t
ck
of r e s u l t i s c e r t a i n l y f a l s e i f we u s e
spaces
, 0 < cy <
ck*
1 would be s u i t a b l e .
Let
Theorem. (Fredholm A l t e r n a t i v e )
H'(F)
(D* : H'(F)
-t
that either
H
S-k
D
=
.
(E))
s p a c e s , a l t h o u g h Holder
D(H
s+k
D
be a s above.
Then
( E ) ) O k e r D*
Indeed t h i s h o l d s t r u e i f we merely assume
h a s i n j e c t i v e symbol.
D*
The proof of t h i s l e a n s h e a v i l y on t h e r e g u l a r i t y theorem.
The main t e c h n i c a l p o i n t i s t o show t h a t
the f a c t that
lfls+k
one shows t h a t t h e
k e r D*
, just
L2
+
S
H
s+k
o r t h o g o n a l complement of
splitting via regularity.
One c o u l d u s e , e . g . :
i s closed.
D
D(H
D
=
(One u s e s
elliptic.)
s+k
T h i s y i e l d s an
)
Then
i s in
L2
splitting
The s p l i t t i n g i n c a s e
h a s i n j e c t i v e symbol r e l i e s on t h e f a c t t h a t
case, e l l i p t i c .
)
[ l ~ f l l ~, )f o r
a s i n t h e Hodge argument.
and we g e t an
D
_< const(llflls
D(H
d
D;:'D
is, in this
t o g e t t h e Hodge theorem.
For d e t a i l s on t h i s , s e e Berger-Ebin [ I ] .
I n l a t e r a p p l i c a t i o n s ( s e e l e c t u r e s 4 and 10) we w i l l u s e t h i s
r e s u l t i n t h e f o l l o w i n g way.
w i l l be d e f i n e d by c o n s t r a i n t s
C e r t a i n s e t s i n which we a r e i n t e r e s t e d
f(x) = 0
.
The r e l a t i o n
v g, T X f * v
w i l l be a d i f f e r e n t i a l o p e r a t o r .
hence
To show i t i s s u r j e c t i v e (and
f - l ( ~ ) i s a submanifold) we c a n show
with i n j e c t i v e symbol.
For then
(
T
)
ker(Txf)* = 0
,
so
i s injective
T f
X
itself w i l l
be o n t o .
Manifolds of Maps.
History.
The b a s i c i d e a was f i r s t l a i d down by E e l l s [ I ] i n 1958.
He c o n s t r u c t e d a smooth manifold o u t of t h e c o n t i n u o u s maps between
two m a n i f o l d s .
c a s e of
C
k
I n 1961, Smale and Abraham worked o u t t h e more g e n e r a l
mappings.
T h e i r n o t e s a r e p r e t t y much u n a v a i l a b l e , b u t
t h e 1966 survey a r t i c l e by E e l l s [ 2 ] i s a good r e f e r e n c e .
c a s e i s found i n a 1967 a r t i c l e by E l l i a s s o n [ I ] .
The
HS
T h i s i s a l s o found
i n P a l a i s [ 4 ] where i t i s done i n t h e more g e n e r a l c o n t e x t of f i b e r
bundles
.
Making t h e manifold o u t of t h e
Ck
diffeomorphism group on
a compact manifold w i t h o u t boundary was done i n d e p e n d e n t l y by Abraham
( s e e E e l l s [ Z ] ) and L e s l i e [ I ] around 1966.
The
H'
a paper by Ebin [ I ] and one by Omori [ I ] around 1968.
c a s e i s found i n
Ebin a l s o
showed t h a t t h e volume p r e s e r v i n g diffeomorphisms form a m a n i f o l d .
F i n a l l y Ebin-Marsden [ I ] worked o u t t h e manifold s t r u c t u r e f o r t h e
HS
diffeomorphisms, t h e s y m p l e c t i c and volume p r e s e r v i n g d i f f eomorphisms
f o r a compact manifold w i t h smooth boundary.
Other p a p e r s on m a n i f o l d s of maps i n c l u d e t h o s e of Saber [ I ] ,
111,
L e s l i e [ 2 , 31, Omori [ 2 ] , Gordon
Penot [ 2 , 3 1 , and Graff [ I ] .
Some f u r t h e r r e f e r e n c e s a r e given below.
Local S t r u c t u r e .
Let
and
M
w i t h o u t boundary.
dimension of
(U,
map
v)
N
.
containing
41 o f ocp - 1
Let
n
Say
f
4
be t h e dimension of
E
H'(M, N)
and any c h a r t
m
: cp(U)
be compact m a n i f o l d s and assume
N
i s in
R'
M
, and
E
i f f o r any
m
( V , I))
f(m)
at
R
HS(cp(U), R )
.
is
N
R
the
and any c h a r t
M
in
,
N
the
T h i s can be shown t o
be a w e l l d e f i n e d n o t i o n , independent of c h a r t s f o r
>
s
(n/2)
.
The
<
(n/2)
b a s i c f a c t one needs i s t h a t by t h e Sobolev Theorem we have
0
H'(M, N) c C (M, N)
.
Things a r e n o t a s n i c e , however, f o r
s
.
I t i s p o s s i b l e f o r a map t o have a ( d e r i v a t i v e ) s i n g u l a r i t y which i s
L2
i n t e g r a b l e i n one c o o r d i n a t e system on
another.
So f o r
s
<
(n/2)
Hence, from now on we assume
, H'(M,
s
>
N)
and n o t be i n t e g r a b l e i n
c a n n o t be d e f i n e d i n v a r i a n t l y .
(n/2)
I n order t o f i n d charts i n
N
.
H'(M, N)
determine t h e a p p r o p r i a t e modeling space.
Let
we f i r s t need t o
f
E
H'(M, N)
modeling s p a c e , should i t e x i s t , must be isomorphic t o
whatever t h a t i s .
for
T ~ H ' ( M ,N)
= p ; then
The
T ~ H ' ( M ,N )
,
So a way t o begin i s t o f i n d a p l a u s i b l e c a n d i d a t e
.
If
P
i s any manifold and
be c o n s t r u c t e d by c o n s i d e r i n g any smooth curve
c(0)
.
ct(0)
E
T P
P
p E P
c
then
P
in
T P
P
can
such t h a t
(see lecture 1 ) .
With t h i s i n mind, l e t us c o n s i d e r a c u r v e
c
f
.-
1-1, 1[ + H ' ( M , N
such t h a t
i s a curve i n
N
of t h i s curve a t
so the map
if
Now i f
0
4
,
N .)
,
m E M
( i . e . , f o r each
cf(t) : M
therefore
i.e.,
.
c (0) = f
f
Now
t
then t h e f u n c t i o n
t 1-1, 1[ , c f ( t ) E H'(M, N) and
N
.
'
,
c f ( 0 ) ( m ) = f(m)
so t h e d e r i v a t i v e
an element of
( d / d t ) ~ ~ ( t ) ( m ) l ~i s= ~
m b ( d / d t ) ~ ~ ( t ) ( m ) l maps
~=~ M
n
TN -+ N
c f ( t ) (m)
t
to
TN
.
Tf (m)
and c o v e r s
f
,
i s t h e c a n o n i c a l p r o j e c t i o n , t h i s diagram
commhtes:
d
c l ( 0 ) = --c ( t )
f
d t f
t=O
I
where
c;(O) (m)
=
d
z c f ( t ) (m) t=O
I
Making t h e i d e n t i f i c a t i o n
d
d
( z c f ( t ) t=O) (m) = ;j;cf( t ) (m) t=O
l
c;(O)
I
i s a good c a n d i d a t e f o r t h e t a n g e n t t o
c
at
f
9
.
f
With t h e above m o t i v a t i o n , l e t us d e f i n e
S
TfH (M, N)
=
s
(X E H (My TN)InNox = f )
Note t h i s i s a l i n e a r s p a c e , f o r i f
we c a n d e f i n e
aVf
+ Xf
( a E R)
V
f
and
a s t h e map
Xi
.
are in
mbaVf(m)
T£H'(M, N)
+ Xf(m)
,
where
Vf(m)
and
Xf(m)
are i n
T
f
a s a model f o r
H ~ ( MN)
,
.
.
N
(m>
near
f
I t i s t h i s s p a c e which we u s e
expN "
.T
P
To show t h i s we need t h e map
Recall that i f
v
P
E T N
general
T N
P
expv
t h e r e i s a unique geodesic
P
whose t a n g e n t v e c t o r a t
N
p
is
v
Then
P
exp ( v )
P P
o
v
through
P
o
=
v
(1)
p
in
N
.
However, s i n c e
N
such t h a t i f
P
d e f i n e t h e map
.
TN
then
P
E
v
T N
expf :
T h i s map c a n be e x t e n d e d t o a map
e x p ( v ) = exp ( v )
TP (M, N)
S
P
P
I H'(M,
N)
X b exp
We a s s e r t t h a t
a netghborhood of
f
in
candidate for a chart i n
-
expf
s p i t e of t h e u s e of t h e map
m e t r i c on
or
N
cS , by
.
exp
.
N)
,
0
in
expv
is
exp : TN + N
.
taking
.
In
With t h i s map we
maps t h e l i n e a r s p a c e
H'(M, N)
H'(M,
X
0
P
.
i s compact
and w i t h o u t boundary, i t i s g e o d e s i c l y c o m p l e t e and hence
d e f i n e d on a l l of
0
to
f
T ~ H ' ( M ,N)
I t s h o u l d be remarked t h a t i n
t h e s t r u c t u r e i s i n d e p e n d e n t of t h e
u s i n g s t a n d a r d p r o p e r t i e s of
HS
c a s e and t o show t h a t t h e change of c h a r t s i s
n e e d s t h e f o l l o w i n g lemma.
o-Lemma.
cm
exp ; M i l n o r [ I ] .
w e l l d e f i n e d ( i . e . , maps i n t o t h e r i g h t s p a c e s ) and i s smooth, one
--L o c a l
onto
and h e n c e i s a
The a s s e r t i o n i s e a s y t o c h e c k i n c a s e t h i n g s a r e
For t h e
p
P
i s a diffeomorphism from some neighborhood of
o n t o a neighborhood
.
p E N
for
-t
( L e f t Composition of M w ,
Let U
-
be a bounded
H S ( ~Rrn)
,
h : Rn + R m
be cm .
-
d e f i n e d by
m ( f ) = hof
is cm
R'
h
This conclusion i s not true i f
map.
and g : ?I + N
4
is
N
T
g(p)
c : 1-1, 1 [ E M
Then
T g(v )
P
P
=
, which
wh
HS-
1
, ThoX
and
a r e manifolds
N
let
1
( d / d t ) g ( c ( t ) ) t=O
.
Tfwh : X
i s t h e map
i s , a t best, in
H'(u,
M
H
and
c(0)
v
=
p
E T M
P
and
cl(0)
=
v
P
Applying t h i s procedure t o
Wh
X E T ~ H ' ( u , R ~ )t h a t t h e
.
H ThoX
s- 1
( U , R ~ )and
R ~ )a t
P
cS
or
i s determined i n t h e u s u a l way:
be a c u r v e such t h a t
t h e t a n g e n t space of
If
HS
have
and u s i n g t h e c h a i n r u l e , we f i n d f o r
t a n g e n t of
i s merely an
, we
p E M
then f o r
C1
4
map.
h
The problem can be seen i n t h i s way.
Tpg : T M
P
s
n
wh : H ( U , R )
, and
p e n set in
wh(f)
T h i s n e c e s s i t y of d i f f e r e n t i a t i n g
But s i n c e
TW
Th
i s only
d o e s e m a p into
.
h
i s a crucial difference
between composition on t h e l e f t and composition on t h e r i g h t .
The e x a c t proof of t h e w-lemma may be found i n Ebin [ I ] and
t h e o t h e r r e f e r e n c e s above.
t o Sobolev [ I ] p. 2 2 3 .
I n f a c t , t h e r e s u l t e s s e n t i a l l y goes back
See a l s o Marcus-Mizel [ I ] , and Bourguinon
-
Brezis [ I ] .
Using t h e w-lemma, i t i s now r o u t i n e t o check t h a t
y i e l d s smooth c h a r t s on
S
H ( M , N)
.
expf
For o t h e r methods of o b t a i n i n g
c h a r t s , s e e P a l a i s [ 4 ] , Penot [ 3 ] and K r i k o r i a n [ I ] .