�
�
�
�
1
Lecture
Seven:
Consequences
of
Cacciopolli
Consequences
of
Cacciopolli
In
this
lecture
we
continue
to
generalise
some
of
our
results
about
the
Laplacian
to
uni
formly
elliptic
second
order
operators.
We
start
by
stating
two
results
without
proof.
Proposition
1.1
Let
B
2
r
be
a
ball
in
R
n
,
and
let
L
be
a
uniformly
elliptic
second
order
operator
with
Lf
=
�
·
A
�
f,
and
λ

v

2
A
v
≤
Λ
v
2
.
There
are
positive
constants
c
,
and
d
depending
only
on
the
≤
v
·
dimension
and
the
ratio

λ

such
that
Λ
2
2
u
≥
(1
+
c
)
u
(1)
B
2
r
B
r
and
�
�
2
2
≥
(1
+
d
)
�
u

�
u

(2)
B
2
r
B
r
for
all
u
with
Lu
=
0
on
B
2
r
.
This
is
very
similar
to
a
result
for
harmonic
functions
from
Lecture
5,
but
we
will
not
give
the
proof
here.
Instead
we
will
work
on
some
of
the
consequences.
it
is
clear
from
(1)
that
if
u
is
L
harmonic
(ie
Lu
=
0)
on
B
2
m
s
then
2
2
u
≥
(1
+
c
)
m
u .
(3)
B
2
m
s
B
s
log(1+
c
)
m
log(1+
c
)
=
2
m
log
2
We
can
rewrite
(1
+
c
)
m
=
e
.
Define
α
=
log(1+
c
)
>
0,
and
let
t
=
2
m
s
.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview.
Sign up
to
access the rest of the document.
 Spring '08
 Miracle
 Laplace operator, Laplace's equation, Harmonic functions, harmonic function

Click to edit the document details