Lecture Seven: Consequences of Cacciopolli
1
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 B2r be a ball in Rn , and let L be a uniformly elliptic second order
operator with
Lf = � · A�f,
and λ|v|2 ≤ v · Av ≤ Λ|v|2 . There are positive constants c, and d depending only on the
dimension and the ratio Λλ such that
�
�
u2 ≥ (1 + c)
u2
(1)
Br
B2r
and
�
�
2
|�u|2
|�u| ≥ (1 + d)
(2)
Br
B2r
for all u with Lu = 0 on B2r .
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 B2m s then
�
�
2
m
u ≥ (1 + c)
u2 .
(3)
B 2m s
Bs
m
log(1+c)
We can rewrite (1 + c)m = em log(1+c) = 2 log 2 . Define α =
then
� �α �
�
t
2
u2 .
u ≥
s
Bs
Bt
log(1+c)
log 2
> 0, and let t = 2m s.
(4)
It runs out that a slightly weaker version of (4) holds for all t, not just when t/m is a power
of 2.
1
Proposition 1.2 Let t > s > 0 be real numbers. If u is L harmonic on Bt then
� �α �
�
t
2
u2 .
u ≥
2s
Bs
Bt
Proof Take 2m s ≤ t ≤ 2m+1 s. Since u2 is positive we have
�
�
u.
u≤
B 2m s
(5)
(6)
Bt
We can estimate this bound by (4), so
2mα
�
�
u≤
Bs
u.
(7)
Bt
Also
mα
2
(m+1)α −α
=2
s
≥2
−α
� �α
t
.
s
Plugging this into (7) gives
�
�
2
u ≥
Bt
t
2s
�α �
u2
(8)
Bs
as required.
We can do exactly the same calculation for the Dirichlet energy to give
Proposition 1.3 Let t > s > 0 be real numbers and let β =
on Bt then
�
2
�
|�u | ≥
Bt
t
2s
�β �
|�u|2 .
Bs
There is a nice corollary to this
Corollary 1.4 Take u : Rn → R an L harmonic function. If
�
u2 < ∞
Rn
then u = 0 on Rn .
2
log(1+d)
log 2
. If u is L harmonic
(9)
Proof Suppose u is not identically zero, say u(x0 ) �= 0. Then
�
u2 = � > 0,
B1 (x0 )
so
�
�
2
u =
Bt (x0 )
t
2s
�α
�,
which goes to infinity as t gets large.
Now we will prove one of the two inequalities at the start of this lecture. Recall from
the proof of the Cacciopolli inequality last time that if u is L harmonic on B2r then
� �2 �
�
Λ
2
2
φ2 |�u|2
u |�φ| ≥
4
λ
B2r
B2r
(10)
for all φ ≥ 0 with φ = 0 on the boundary. Also note that
|�(φu)|2 = |u�φ + φ�u|2 ≤ 2|u�φ2 | + 2|φ�u|2
by Cauchy­Schwarz. Putting these together gives
�
2
|�(φu)|
B2r
�
2|u�φ2 | + 2|φ�u|2
B2r
� � �
��
Λ 2
≤ 2 4
+1
u2 |�φ|2 .
λ
B2r
≤
(11)
(12)
Notice that φu is zero on the boundary of B2r , so the Dirichlet­Poincare applies, and we
have
�
�
c(n)
2
|�(φu)|2 ,
(13)
(φu) ≤
r2 B2r
B2r
so
��
� � �
�
c(n)
Λ 2
2
(φu) ≤ 2 4
+1
u2 |�φ|2 .
(14)
r2 B2r
λ
B2r
Pick
�
φ(x) =
1
2r−|x|
r
if |x| ≤ r;
if r < x ≤ 2r
as usual to give
c(n)
r2
�
2
u2 ≤ 2
r
Br
��
� � �
Λ 2
4
+1
u2 .
λ
B2r \Br
3
(15)
Therefore
⎛
⎞
⎝1 + �� c(�n)
�⎠
Λ 2
2 2λ + 1
as expected.
4
�
2
�
u ≤
Br
B2r
u2
(16)