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 s
�
α
�
+ � 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 ≤ r ≤ τ k s then
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 as we like.
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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)
By picking s we can get || A ij
− A ij
( x
0
) || as small as we like so we will be able to apply our lemma.
Pick 0 < β < 1 and set γ = 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.
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