Lecture 18: Regularity of L harmonic functions Part III
1 Finishing the proof
In order to ﬁnish the proof from last time we need a lemma.
Lemma 1.1
Let φ be a positive and increasing function on the positive reals, and let α, c be positive constants.
For all 0 < γ < α there is � > 0 such that
φ ( r ) ≤ c
�� r
�
α s
�
+ � φ ( s ) (1) for 0 < r < s implies
φ ( r )
≤ c
�
� r
�
γ s
φ ( s ) (2) for some constant c
�
.
Proof Choose 0 < τ < 1 such that �
≤
τ
α
.
Then
φ ( τ s )
≤ c ( τ
α
+ � ) φ ( s )
≤
2 cτ
α
φ ( s ) .
Therefore
φ ( τ k s )
≤
(2 cτ
α
) k
φ ( s ) .
Pick γ so that 2 cτ
α
−
γ
≤
1 and we have
(3)
(4)
φ ( τ k s )
≤
τ kγ
φ ( s ) .
(5)
When r = τ k s this is precisely what we wanted with c
�
= 1.
If instead τ k +1 s then
≤ r
≤
τ k s
φ ( r )
≤
φ ( τ k s )
≤
τ kγ
φ ( s )
≤
1
� r
�
γ
φ ( s )
τ s
(6) which is what we needed.
Note that by taking τ very small we can get γ as close to α we like.
as
1
Now we apply this to what we were doing last time.
Let φ ( r ) =
�
B r
( x
0 that
)
� u

2
.
We showed
φ ( r )
≤
�� r
� n s
+ k

A ij
−
A ij
( x
0
)

�
φ ( s ) (7) our
By picking lemma.
s
Pick we
0 can get
< β < 1

A ij and
−
A ij set γ
( x
=
0
)
 as small as we like so we will be able to apply n
−
2 + 2 β .
By our lemma there is a constant k
� with
φ ( r )
≤ k
�
� r
� n
−
2+2 β
φ ( s ) .
s
(8)
In our old notation this is
�
B r
( x
0
)
� u
 ≤ k
�
� r
� n
−
2+2 β
� s
B s
( x
0
)
� u

2
, (9) and Holder continuity follows by Morrey’s lemma.
2