Lee K.K. Lectures on Dynamical Systems, Structural Stability and Their Applications
Подождите немного. Документ загружается.
briefly
introduce
the
notion
of
nonlinear
differential
operators,
the
linearization
and
the
symbol
of
such
operators,
and
the
index
of
a
nonlinear
elliptic
operator.
Recall
that,
let
~=
B
~
M
be
a
em
fiber
bundle
over
M.
Given
b
0
€ B
with
~(b
0
)
= p
0
and
local
sections
s
1
,
s
2
of
B
defined
near
p
0
with
s
1
(p
0
)
= s
2
(p
0
)
= b
0
•
By
choosing
a
chart
at
p
0
€
M,
a
local
trivialization
of
B
near
p
0
,
and
a
chart
near
b
0
in
the
fiber
~-
1
(p
0
),
we
can
define
the
kth
order
Taylor
expansions
of
s
1
and
s
2
at
p
0
•
If
the
Taylor
expansions
are
the
same
for
one
set
of
choices,
they
will
be
the
same
for
any
other
(i.e.,
all
in
the
same
equivalence
class),
then
we
say
that
s
1
and
s
2
have
the
same
k-jet
at
p
0
•
Recall
again,
this
set
of
equivalence
classes
is
denoted
by
Jk(B)b
and
the
equivalence
class
of
s
is
denoted
by
jk(s)P.
Let
J
0
k(B) =
Ub•B
Jk(B)b
and
let
~
0
k:
J
0
k(B)
~
B
maps
J
0
k(B)b
to
b.
It
can
be
shown
that
~ok
has
the
structure
of
a
em
fiber
bundle
over
B
whose
fiber
at
b
0
is,
ek
m=l
L
8
m(MP~'
(Bp.>b1,
all
polynomial
maps
of
degree
less
then
or
equal
to
k
from
MP.
into
(Bp.>b,.
We
define
a~
fiber
bundle
~k:
Jk(B)
~
M
whose
total
space
is
J/(B)
and
the
projection
is
~k
=
~-~
0
k,
i.e.,
~k(jk(s)P
0
)
= p
0
•
If
0
~
n
~
k,
we
can
generalize
the
fiber
bundle
by
defining
~nk
: Jnk(B)
~
J"(B)
and
whose
total
space
is
Jk(B)
and
the
projection
is
given
by
~nk(jk(s)
) = j
(s)
p,
n
p,
.
We
then
have
a
natural
map,
the
k-jet
extension
map,
jk:
Cm(B)
~
~(Jk(B)),
defined
by
jk(s)
(p)
=
jk(s)P.
If
tis
a
vector
bundle
neighborhood
of
s €
Cm(B)
then
Jk(t)
is
a
vector
bundle,
and
in
fact
it
is
a
vector
bundle
neighborhood
of
jk(s)
€
~(Jk(B)).
Furthermore,
jk:
Cm(B)
~
Cm(Jk(B))
restricts
to
a
linear
map
of
Cm(t)
to
~(Jk(t)).
Equipped
with
the
knowledge
of
proper
relationships,
we
recall
that
if
t
and
ry
are
vector
bundles
over
M,
then
a
linear
differential
operator
of
order
k
from
~(t)
to
Cm(ry)
is
a
linear
map
P:
~(t)
~
Cm(ry)
and
it
can
be
factored
as
a
composition
Cm(t)
j.~
Cm(Jk(t))
f*'-+
Cm(ry)
where
f:
Jk(t)
~
ry
is
a
~vector
bundle
morphism
over
M.
By
analogy,
it
is
clear
how
to
define
a
non-linear
differential
operator
of
order
k.
Let
B
1
and
B
2
be
em
fiber
bundles
over
M.
A
mapping
D:
134
cm(s
1
)
~
cm(B
2
)
is
called
a
non-linear
differential
operator
of
order
k
from
cm(B
1
)
to
cm(s
2
if
it
can
be
factored
as
cm(B;)
i~
cm(Jk(B
1
))
f•~
C"'(B
2
)
where
f:
Jk(B
1
)
~
B
2
is
a
C"'
fiber
bundle
morphism
over
M.
Let
us
denote
the
set
of
all
differential
operators
of
order
k
from
c"'(B
1
)
to
C"'(B
2
)
(or
from
B
1
to
B
2
for
simplicity)
by
Dk(B
11
B
2
).
Theorem
3.
5.
4
If
1
:5
k,
D
1
(B
1
, B
2
)
!:;;
Dk
(B
11
B
2
) we
can
construct
higher
order
differential
operators
from
the
lower
order
ones
by
repeating
the
jet
extension
map
(or
by
induction
method
through
jet
extension
map)
as
indicated
in
the
following
lemma.
Lemma
3 .
5.
5
Let
j
k:
c"'
(B)
~
c"'
(Jk (B) )
and
j
1
' :
c"'
(Jk (B) )
~
c"'(J
1
(Jk(B)))
be
jet
extension
maps.
Then
j
1
•·jk
is
a
differential
operator
of
order
k+l
from
B
to
J
1
(Jk(B)).
Using
the
definition
of
Dk(B
1
,B
2
)
and
the
above
lemma,
we
can
prove
the
following:
Theorem
3.5.6
If
P
1
€
Dk(B
1
,B
2
),
and
P
2
€ D
1
(B
2
,B
3
),
then
P
2
P
1
€
Dk+t<B
1
,~).
Now
we
shall
see
what
a
nonlinear
differential
operator
really
looks
like
in
local
coordinates.
Let
us
choose
coordinates
x
1
,
•••
,xm
in
a
neighborhood
U
of
p
0
in
M,
coordinates
y
1
,
•••
,yn
in
a
neighborhood
V
of
b
0
in
BP
and
using
a
local
trivialization
to
identify
a
neighborhood
of
b
0
in
B
with
U x V
so
that
~
restricted
to
U x V
is
a
projection
on
the
first
component.
A
section
s
of
B
over
U
with
s(p)
€ V
for
p € U
is
given
by
a map p
~
(p,s(p))
of
U
into
U x V.
Let
s;(P)
=
Y;(s(p)),
then
jk(s)
in
coordinate
form
is
given
by
a map p
~
(p,s(p)
,y;a(jk(s)
(p)))
where
Y;a(jk(s)
(p))
= oas;
(p),
and
as
usual,
Da
= a
lal;ax,
a,
. . .
axma..,,
where
0
:5
I a I
:5
k.
Now
let~·=
n
~
M
be
a
c"'
vector
bundle
over
M
and
let
P
€
Dk(B,"),
say
P = f*jk
where
f:
Jk(B)
~
n
is
a
c"'
fiber
bundle
morphism.
Let
v
1
,
•••
,vr
be
a
basis
of
c"'
sections
of
n
over
u.
If
we
restrict
f
to
the
part
of
J
0
k(B)
over
U x V,
it
is
given
by
fi
(j
=
l,
..
,r)
with
coordinates
(x
1
,
•••
,xm,y
1
••
,yn,Y;a)
by
(x,y,y;a)
~
:r;ri=l
f(x,y,y;a)vi(x).
Then
the
explicit
expression
for
Ps
is
135
Ps(x)
= :t•i=l
fi(x,s
1
(x),
•••
,sn(x),
D
8
11
(x)vi(x).
We
shall
call
the
ordered
r-tuple
of
functions
fi(x,y,y;)
the
parametric
expressions
for
the
operator
P
near
b
0
relative
to
the
choices
made.
By
utilizing
the
linear
structure
in
~.
it
is
possible
to
single
out
certain
vector
subspaces
of
the
vector
space
Dk(B,~)
of
kth
order
differential
operators
from
B
to
~
which
are
polynomial
functions
of
certain
derivatives.
These
classes
play
an
important
role
in
nonlinear
analysis,
particularly
in
calculus
of
variations.
Let
B
be
a
c-
fiber
bundle
over
M,
t a
c-
vector
bundle
over
M
and
P E
Dk(B,t).
Let
wand
1
are
integers
with
w >
o,
and
0 S u <
k.
We
say
that
P
is
a
polynomial
of
weight
< w
with
respect
to
derivatives
of
order
> u
(denoted
by
P E
Dkw;u(B,t))
if
for
each
parametric
representation
of
P
as
above,
and
each
of
the
functions
fi(x,y,y;
11
)
can
be
written
as
a sum
of
functions
of
the
form
G(x,y,y/)Y
1
ll•
•••
y
1
/f
where
all
lrl
S
u,
all
IP;I
>
u,
and
IP
1
1 +
•••
+
IPq~
S w.
We
abbreviate
Dkw;o(B,t)
to
Dkw(B,t)
and
elements
of
this
space
are
referred
as
polynomial
differential
operators
of
order
k
and
weight
s w.
As a
special
case,
elements
of
Dk
k;k·l
( B,
t)
,
i.e.
,
the
f i
of
a
parametric
representation
are
linear
in
derivatives
of
order
k,
are
called
quasi-linear
differential
operators
of
order
k
from
B
to
t.
It
is
evident
from
the
definition
that
D
0
(B,t)
~
Dko;u(B,t)
for
all
k
and
u <
k.
Nonetheless,
it
is
not
immediately
clear
that
Dkw;u(B,t)
form
a
subspace
of
Dk(B,t)
with
positive
dimension
if
w > o.
The
next
theorem
shows
how
to
construct
lots
of
operators
in
Dk
w;u
( B,
t)
.
Theorem
3.5.7
Let
B
be
a
c-
fiber
bundle
over
M,
t a
c~
vector
bundle
over
M
and
P E
Dk(B,t).
In
order
that
P E
Dkw;u(B,t),
it
is
sufficient
that
for
each
b
0
E B
there
exist
at
least
one
parametric
expression
for
P
near
b
0
satisfying
the
conditions
in
the
above
definition
of
polynomial
differential
operators
of
weight
S w.
It
should
be
noted
that
in
order
to
prove
the
theorem,
136
it
suffices
to
assume
that
B
is
a
vector
bundle.
This
is
not
surprising.
After
all,
Theorem
3.5.7
is
a
local
statement.
With
a
choice
of
coordinates,
there
are
many
formulas
and
results
of
polynomial
differential
operators.
Since
some
of
them
will
involve
some
results
in
functional
analysis,
which
we
have
neither
the
time
nor
the
space
to
discuss,
we
shall
only
recommend
a
few
references
for
further
reading
[Palais
1965,
Ch.4;
Palais
1968;
Berger
1977].
Let
V
be
a
function
which
associates
to
each
~
vector
bundle
t
over
a
compact
n-dim
~
manifold
a
complete,
normable,
topological
vector
space
V(t)
which
includes
~(t).
Axiom A
(for
V)
Let
M
and
N
be
compact
n-dim
~
manifolds
and
let
~:
M
~
N
be
a
diffeomorphism
of
M
into
N.
If
tis
a
vector
bundle
over
N,
then
s
~
s·t
is
a
continuous
linear
map
of
V(t)
into
V(~*t).
In
usual
examples,
this
map
is
onto.
If
M
~
N
and
~
is
inclusion,
then
the
Axiom
says
that
restriction
is
continuous
from
V(t)
to
V(t/M)
and
the
onto-ness
expresses
the
possibility
of
extending
s e
V(t/M)
to
an
element
of
V(t).
Axiom B
If
t
is
a
vector
bundle
over
a
compact
c~
m-dim
manifold
M
then
V(t)
~
C
0
(t)
and
the
inclusion
map
is
continuous.
Furthermore,
if
n
is
another
vector
bundle
over
M
and
f:
t
~
~
is
a
~
fiber
preserving
map,
then
f.:
C
0
(t)
~
C
0
(~)
restricts
to
a
continuous
map
V(f):
V(t)
~ V(~).
Let
t
be
a
~
vector
bundle
over
a
compact
~
m-manifold
M,
and
let
v(k)(t)
=
{s
E
ck(t)
I
jk(s)
E
V(Jk(t))}.
Thus
jk
is
a
continuous
linear
injection
of
VCk>
(t)
into
the
Banach
space
V(Jk(t))
and
VCk>(t)
become
a
normable
topological
vector
space
if
we
topologize
it
by
requiring
that
jk
be
a
homeomorphism
into.
Let
Vk
(
t)
to
be
the
completion
of
Vck>
(
t)
so
that
jk
extends
to
a
continuous
linear
isomorphism
of
Vk(t)
onto
a
closed
linear
subspace
of
V(Jk(t)).
With
these
axioms
stated
and
notations
defined,
we
are
ready
to
discuss
briefly
the
linearization
and
the
symbol
of
a
differential
operator.
137
Let
B
1
and
B
2
be
C"
fiber
bundles
over
a
compact
n-dim
C"
manifold
M
and
let
P:
C"(B
1
)
-+
C"(B
2
)
be
an
element
of
Dlc(B
1
,B
2
),
let
us
say
P =
f.jlc
where
f:
Jlc(B
1
)
-+
B
2
is
a
fiber
bundle
morphism
over
M.
Let
V
satisfy
Axiom
A
and
B
so
that
P
extends
to
a
C"
map
of
Vlc(B
1
)
into
V(B
2
).
If
s E C"(B
1
),
then
d.f.
is
a
linear
map
of
(Vk(B
1
) )
8
into
(V(B
2
)
)ps•
It
can
be
shown
that
there
exists
A(P)
8
E
Diffk((B
1
)
8
,
(B
2
)p
8
),
the
"linearization
of
Pats",
such
that
A(P)
8
:
C"((B
1
)
8
)
-+
C"( (B
2
)p
8
)
extends
to
dP
8
•
Furthermore,
A(P)
s
depends
only
on
P
but
not
on
V.
The
linear
differential
operator
A(P)
8
has
a
symbol
ak(A(P)
8
).
This
symbol
is
a
function
on
the
cotangent
bundle
of
M,
T*(M),
and
for
(v,x)
E
T*(M),
a(A(P)
5
)
(v,x)
is
a
linear
map
of
the
fiber
of
the
tangent
space
of
(B
1
)x
at
s
(X),.
( ( B
1
)
x>
sex>,
into
the the
fiber
of
( (B
2
)
x>
PsCx>.
Furthermore,
ak(A(P)
8
)
(v,x)
depends
only
on
jk(s)
(x)
and
v,
thus
we
can
denote
it
by
alc(P)
(jk(s)
(x),(v,x)).
In
other
words,
ak(P)
is
a
function
defined
on
the
total
space
of
the
fiber
product
Jk(B
1
)x,.T*(M),
which
we
denote
by
T\(B
1
).
There
are
also
two
natural
maps
,.
:
T*
k ( B
1
)
-+
B
1
and
f
1
:
T*
k ( B
1
)
-+
B
2
defined
by
7r(jk(s)(x),(v,x))
=
s(x)
and
f
1
(jk(s)(x),(v,x))
fjlc(s)
(x)
=
Ps(x).
But
these
give
rise
to
two
C"
vector
bundles
over
T\(B
1
),
i.e.,
,.*(Tf(B
1
))
and
f
1
*(Tf(B
2
))
and
their
fibers
at
(j~(s)(X),(v,x))
are
((B
1
)x>scx>
and
((Bz>x>Pscx>
respectively.
Thus
a(P)
is
an
element
of
Hom(7r*(Tf(B
1
)),
f
1
*
(Tf
( B
2
) ) ) ,
and
this
vector
bundle
homomorphism
over
T\
( B
1
)
is
called
the
Ck~th
order)
symbol
of
the
differential
operator
P.
Theorem
3.5.8
Given
s E C"(B
1
),
oi
cs>f
is
a
C"
vector
bundle
homomorphism
of
Jk(B
1
)
8
)
into
(B:)Ps
and
hence
defines
an
element
A(P)
8
E
Diffk((B
1
)
8
,
(B
2
)p
8
)
called
the
linearization
of
P
at
s.
If
t
is
a
vector
bundle
neighborhood
of
s
in
B
1
(so
we may
identify
(B
1
)
5
with
t),
then
for
a E
C"(t),
A(P)
8
a(x)
depends
only
on
jk(s)
(x)
and
jk(a)(x).
Moreover,
A(P)
5
(a)(x)
=
d/dtlt=o(P(s+ta)(x)).
Corollary
3.5.9
Lets
E C"(B
1
),
t
1
a
vector
bundle
neighborhood
of
s
in
B
1
and
t
2
a
vector
bundle
neighborhood
of
Ps
in
B
2
•
Then
for
a E
C"(t
1
),
(P(s+ta)
-
Ps)/t
converges
138
in
the
c-
topology
to
A(P)
5
(a)
as
t
~
o.
We
shall
now
derive
an
expression
for
A(P)
5
in
local
coordinates.
We
can
suppose
M =
D",
and
we
can
replace
B
1
and
B
2
with
vector
bundle
neighborhoods
of
s
and
Ps,
which
we
can
identify
with
MxRm
and
MxRq
respectively.
Then
a
section
s
of
B
1
is
given
by
n
real
functions
of
x =
(X
1
1
•••
,
Xn)
E
D"
1
S
(X)
= ( s
1
(X)
,
•••
,
Sm
(X)
)
and
similarly
Ps
is
given
by
q-real
functions
of
x,
Ps(x)
=
((Ps)
1
(x),
•••
,(Ps)q(x))
and
when
we
say
P E
Dk(B
1
,B
2
)
we
mean
that
there
exist
q
of
c-
functions
fi
(X,Y;rY/
1
)
(i
=
1,
••.
,m,
and
a
range
over
n-multi-indices
with
lal S
k)
such
that
(Ps)
i
(x)
=
fi
(x,
s;
(x)
,
pCis;
(x))
•
Recall
that
the
f;s
defined
earlier
are
called
the
parametric
representation
of
P.
Now
A(P)
8
E
Diffk(B
1
,B
2
).
Similarly,
there
are
q
c--functions
gj(X,Y;rYt>
such
that
if
a =
(a
1
,
••
,am)
is
a
c--section
of
B,
then
(P)
8
(a)(x)
=
(g
1
(x,a;(x),pCia;(x)),
•.•
,
gq(x,a;(x),pCia;(x))).
Since
A(P)
5
is
a
linear
differential
operator,
gi
(x,
Y;,
Y;
01
)
are
functions
linear
in
(Y;,
Y;
01
) •
That
is,
there
exists
c--functions
of
x,
A;i
and
Ai
01
;
such
that
gj(X,Y;rY;cz)
=I:;
A;j(X)Y; +
I:cz,i
Ajcz,;(X)Y;cz~
•
The
problem
is
to
express
these
A;J
and
A
1
01
;
in
terms
of
the
fi
and
s.
As_
we
shall
see
in
the
following
theorem,
the
answer
isA;l(x)
=
(af/aY;)(x,s;(X),pCis;(X))
and
AjCI,i
=
caf/aY;
01
)(X,S;(X),pCis;(X)).
More
precisely,
we
have
the
following
theorem.
Theorem
3.5.10
Let
P E
Diffk(B
1
,B
2
)
be
given
parametrically
by
P(s)
=
(f
1
(x,s;(x),pCis;(x)),
•••
,
fq(x,s;(x),pCis;(x))),
then
A(P)
5
is
given
parametrically
by
A ( P)
8
(a)
(X)
= ( g
1
(X
1
a;
(X)
1
pCia;
(X)
)
1
••
1
gq
(X
1
a;
(X)
1
pCia;
(X)
) )
1
where
gq(X,Y;rY;
01
) =
I:;(af/aY;)(x,s;(x),pCis;(x))Y;
+
I:
01
,;(af/aY;
01
) (X,S;(X) ,pCis;(X)
)Y;CI•
Let
us
recall
the
definition
of
the
symbol
of
a
linear
differential
operator.
Let
t
1
and
t
2
be
c-
vector
bundles
over
M
and
let
P E
Diffk(t
1
,t
2
).
If
vis
a
cotangent
vector
of
M
at
x,
the
symbol
of
P
at
(v,x)
is
a
linear
map
ak(P)
(v,x)
of
(t
1
)x
into
(t
2
)x
defined
as:
choosing
any
c"'
function
g
on
M
such
that
g(x)
=
O,
and
dgx = v
and
given
e
139
e
(t
1
)x
choose
any
c!'
section
f
of
t
1
such
that
f(x)
=
e.
Then
ak
(P)
(v,
x) e =
(k!)
·lp
(gkf)
(x)
•
It
is
immediate
from
the
definition
of
the
symbol
that
if
P =
F.jk
where
F
is
a
vector
bundle
homomorphism
of
Jk(t
1
)
into
t
2
,
then
ak(P)
(v,x)
depends
on
F
only
through
its
value
at
x.
Indeed,
ak(P)
(v,x)e
=
Fx(jk(gkf)
(x)jk!)
where
g
and
f
as
before.
Let
us
now
restate
the
definition
of
the
symbol
of
P e Dk(a
1
,a
2
)
in
a
more
compact
form.
Let
a
1
and
a
2
be
c!'
fiber
bundle
over
M
and
let
P e
Dk(a,a
2
),
say
P =
f.·
jk
where
f:
Jk(a
1
)
-+ a
2
is
a
c"'
fiber
bundle
morphism.
Let
T;(a
1
)
be
the
fiber
bundle
over
M
which
is
the
fiber
product
of
Jk(a
1
)
and
T*(M)
and
one
defines
the
maps
1f
and
f'
of
T k * ( a
1
)
into
a,
and
a
2
respectively
by
1f(jk(s)x,(v,x))
=
s(x)
and
f'(jk(s),(v,x)
=
f(jk(s)x)
=
Ps(x).
The
symbol
of
P,
ak(P),
is
then
defined
to
be
the
element
of
Hom(1f*Tf(a
1
)
,f'*Tf(a
2
))
given
by
ak(P)
(jk(s)x,
(v,x))
=
ak(Ak(P)
5
)
(v,x)
where
Ak(P)
8
is
the
linearization
of
P
at
s.
(See
Theorem
3.5.8
for
the
definition
of
A(P)
8
).
The
next
theorem
describes
how
to
compute
ak(P)
in
local
coordinates.
Theorem
3.5.11
Let
M = o", a
1
=
MxRm,
a
2
=
MxRq
and
P e
Dk
(a
1
,
a
2
)
be
given
by
Ps(x)
=
(f
1
(x,s
1
(x)
,pas;
(x)),
...
,
fq(x,s;
(x)
,Pas;
(x)))
where
fi(X,Y;rY;a)
are
r!'-function
of
x =
(x
1
,
•••
,xn),
Y;
(y
1
,
•••
,ym)
and
Y;a
(i
=
1,
••
,m
and
a
ranges
over
all
n-multi-
indices
with
1 S lal S
k).
Since
each
fiber
of
Tf(a
1
)
and
Tf(a
2
)
is
canonically
isomorphic
to
Rm
and
Rq
respectively,
ak(P)
(jk(s)x,(v,x))
is
given
by
a
(q
x
m)-
matrix
a;j(x,jk(s)x,
(v,x)).
If
v =
:E
V;dX;
then
this
matrix
is
given
explicitly
by:
a;j(x,jk(s)x,(v,x))
=
:Eial=k
va(df/dY;a)(x,s;(X),p<Zs;(X))
where
ya
= v
1
a
.•.
vna
•
Let
t
1
and
t
2
be
c"'
vector
bundles
over
M
and
let
P =
f.-
jk e
Diffk(t
1
,t
2
)
where
f:
Jk(t
1
)
-+ t
2
is
a c"'
vector
bundle
morphism.
It
is
natural
to
ask
what
is
the
connection
between
ak(P) E Hom(p*t
1
,p*t
2
),
where
p:
T*
(M)
....
M
is
the
natural
projection,
and
ak(P) e Hom(1f*Tf(t
1
)
,1f*Tf(t
2
)),
where
140
1r:
Tk*(t
1
)
-+
t
1
is
the
natural
map.
Now
if
g:
T/(t
1
)
=
Jk(t
1
)~T*(M)
-+
T*(M)
is
the
natural
projection,
then
g*p*(t
1
)
= 7r*Tf(t
1
)
and
g*p*(t
2
)
=
f*Tf(t
2
),
so
we
can
regard
ak(P)
as
an
element
of
Hom(g*p*(t
1
)
,g*p*(t
2
)),
when P
is
considered
as
a
nonlinear
operator.
The
composition
with
g maps Hom(p*t
1
,p*t
2
)
into
Hom(g*p*(t
1
)
,g*p*(t
2
))
where
"ak(P)
=
ak(P)·g"
expresses
the
relation
between
the
nonlinear
ak(P),
the
left
hand
side,
and
the
linear
ak(P),
the
right
hand
side,
through
the
composition
of
g.
The
whole
point
is
that
if
P
is
linear,
then
A(P)
8
= P
for
all
s E CU(t
1
),
so
A(P)
8
is
independent
of
s,
and
ak(P)
(jr(s),
(v,x))
= ak(A(P)
8
)
(v,x)
=
ak(P)
(v,x)
does
not
depend
on
jk(s),
i.e.,
ak(P)
is
lifted
from
T*(M).
Let
us
now
introduce
the
notion
of
an
elliptic
differential
operator.
An
element
P f Dk(8
1
,8
2
)
is
called
an
elliptic
differential
operator
of
order
k
from
8
1
to
8
2
if
for
all
(jk(s),
(v
,x))
E
Tk*
(8
1
)
with
v +
o,
ak(P)
(jk
(s),
(v
,x)):
T((8
1
)x>s<x>-+
T((8
2
)x>Ps<x>
is
a
linear
isomorphism.
We
will
denote
the
set
of
all
such
elliptic
differential
operators
by
Elpk(8
1
,8
2
).
Note
that
Elpk(8
1
,8
2
)
is
nonempty
only
if
8
1
and
8
2
have
the
same
fiber
dimension.
Recall
that
if
t
1
and
t
2
are
CU
vector
bundles
over
M
then
the
set
Ellk(t
1
,t
2
)
of
k-th
order
linear
elliptic
differential
operators
from
t
1
to
t
2
is
the
subset
of
P f
Diffk(t
1
,t
2
)
such
that
ak(P)
(v,x)
is
a
linear
isomorphism
of
(t
1
)x
onto
(t
2
)x
for
all
(v,x)
E
Mx*
with
v +
0.
Obviously,
Ellk
( t
1
,
t
2
)
s;
Elpk ( +
1
,
+
2
) •
Theorem
3.5.12
If
P E
Elpk(8,8
2
)
then
A(P)
8
E
Ellk(
(8
1
)
8
,
(8
2
)p
8
)
for
all
s E
CU(8
1
).
Conversely,
if
each
element
of
Jk(8
1
)
is
the
k-jet
of
a
global
section
of
8
1
,
then
P E
Elpk(8
1
,8
2
)
provided
each
linearization
of
P
is
elliptic.
This
theorem
gives
some
conditions
for
the
linearization
of
nonlinear
elliptic
differential
operators.
In
the
remainder
of
this
section,
we
shall
give
a
definition
and
some
properties
of
the
analytic
index
of
a
nonlinear
141
elliptic
operator.
For
a
given
~
fiber
bundles
B
1
and
B
2
over
a
compact
em
manifold
M
without
boundary,
we
will
associate
to
each
P E
Elpr(B
1
,B
2
)
an
element
i
8
(P)
€
K(~(B
1
))
called
the
analvtic
index
of
P.
If
we
choose
a
"base
point"
s
0
for
~(B
1
)
then
we
have
a
canonical
augmentation
map
dim:
K(~(B
1
))
~
Z
and
dim(i
8
(P))
turns
out
to
be
the
usual
analytic
index
of
the
linearized
elliptic
operator
A(P)
8
:
~(
(B
1
)
8
)
~
~(
(B
2
)p
8
),
i.e.,
dim(Ker(A(P)
8
))
-
dim(coKer(A(P)
8
)).
Note
that,
in
general,
when
~(B
1
)
is
not
homotopically
trivial,
i
8
(P)
can
carry
more
information
about
P
than
does
its
"dimension".
Let
X
and
Y
be
c
1
Hilbert
manifolds
and
let
f:
X
~
Y
be
a c
1
map.
we
say
f
is
a
Fredholm
map
if
df:
TX
~
f*TY
is
a
Fredholm
bundle
morphism
over
X
and
in
this
case
we
define
ind(f)
€ K(X)
by
ind(f)
=
ind(df).
More
generally
if
j:
n
~
X
is
a
continuous
map
we
say
f
is
a
Fredholm
map
relative
to
i
if
df·j:
j*Tx
~
j*f*TY
is
a
Fredholm
bundle
morphism
(i.e.,
if
dfj(wl:
TXi<w>
~
TYfi<w>
is
a
Fredholm
operator
for
each
w €
n),
and
we
define
ind(f,j)
E
K(n)
by
ind(f,j)
~
ind(df·j).
Note
that
if
f:
X
~
Y
is
a
Fredholm
map
then
clearly
f
is
also
a
Fredholm
map
relative
to
any
j:
n
~X
and
then
ind(f,j)
=
j*ind(f)
where
j*:
K(X)
~
K(n)
is
the
functorial
K(j)
•
In
order
to
define
K
for
an
arbitrary
space
X,
the
basic
requirement
is
that
K
should
be
a
functor
from
spaces
and
homotopic
classes
of
maps
to
abelian
groups.
For
compact
spaces,
such
a
functor
is
naturally
isomorphic
to
the
Grothendieck
group
of
vector
bundles
and
is
representable,
i.e.,
K(X)
should
be
naturally
equivalent
to
[X,C],
the
homotopy
classes
of
maps
of
X
into
a
homotopy
abelian
H-space
c.
This
uniquely
determines
C
up
to
homotopy
type,
i.e.,
C = Z x
BG
where
BG
means
the
classifying
space
of
G,
and
G
is
either
~
O(n)
or
li~
U(n)
depending
on
whether
we
mean
K
0
or~·
By a
theorem
proved
by
Janich
[1965],
and
Atiyah
independently,
there
is
a
nice
choice
of
c
for
our
purpose.
Let
H
be
a
separable,
infinite
dimensional
Hilbert
space
and
let
Fred(H)
be
the
space
of
Fredholm
operators
on
142
H,
i.e.,
bounded
linear
maps.
LetT:
H
~
H
be
such
a map
such
that
Ker
T
and
coker
T
are
finite
dimensional.
T(M)
is
then
automatically
closed
in
H
so
we
can
take
eckerT
=
T(H)~
=
Ker(T*).
Fred(H)
is
topologized
as
a
subspace
o~
the
space
of
all
bounded
operators
on
H
with
the
usual
norm
IITII
=
sup{
IITxll
I
llxll
=
1}.
If
X
is
a
topological
space,
then
a
continuous
map
f:
X
~
Fred(H)
is
called
admissible
if
Ker(f)
=
{(v,x)
E H x
xl
v E
Ker
f(x)}
and
coker(f)
=
{(v,x)
E H x
xl
v E Im
f(x)~}
are
vector
bundles
over
X
under
the
obvious
projection.
Then
it
is
known
that
if
X
is
a
compact
space
andrE
[X,Fred(H)]
then
r
has
an
admissible
representative
g
and
the
element
ind(r)
=
[Ker(g)]
-
[Coker(g)]
of
the
Grothendieck
group
K(X)
of
vector
bundles
over
X
is
well
defined
and
independent
of
the
choice
of
admissible
g,
and
ind:
[X,Fred(H)]
~
K(X)
is
a
bijection.
Furthermore,
Fred(H)
is
homotopy
abelian
H-space
under
usual
operator
composition,
making
[X,Fred(H)]
an
abelian
group,
and
ind
is
a
group
isomorphism.
Now
let
X
be
a
paracompact
space
and
let
B
1
and
B
2
be
Hilbert
space
bundles
over
X
with
GL(H),
the
general
linear
group
of
Hilbert
space
with
the
norm
topology
as
structural
group.
A
Hilbert
bundle
morphism
f:
B
1
~
B
2
is
a
Fredholm
bundle
morphism
if
fx:
(B
1
)x
~
(B
2
)x
is
a
Fredholm
map
for
each
x E
X.
In
this
case,
by
a
theorem
of
Kuiper
which
states
that
GL(H)
is
constructable,
thus
there
exist
bundle
isomorphisms
g:
X X H
~
B
1
and
h:
B
2
~
X X H.
Then
X
~
hxfxgx
is
a map
hfg:
X~
Fred(H).
By
the
contractability
of
GL(H),
it
follows
that
g
and
h
are
well
determined
up
to
homotopy
and
hence
the
homotopy
class
of
hfg
is
a
well
determined
element
of
[X,Fred(H)]
= K(X)
and
we
denote
by
ind(f).
Now
we
are
ready
to
define
the
index
of
a
nonlinear
elliptic
differential
operator.
Definition
and
Theorem
3.5.13
Let
M
be
a
compact
n-dim
~
manifold
without
boundary,
let
B
1
and
B
2
are
~
fiber
bundles
over
M,
and
let
P E
Elpr(B
1
,B
2
).
Let
k >
n/2+r,
then
P:
~(B
1
)
~
~(B
2
)
extends
to
a
~
map
of
Hilbert
manifolds
PCk>:
L
2
k(B
1
)
~
L
2
k-r<B
2
).
Then
PCk>
is
a
Fredholm
map
relative
143