Lecture 18: Regularity of L harmonic functions Part III 1

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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

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