## Hopf lemma

**Hopf Lemma**: Suppose , and satisfies that

is uniformly elliptic and , . Suppose has an interior open ball such that and there exists one point satisfies that . satisfies that if . If exists, then

**Proof:** Under some translation, we assume that has radius . Define

Then

Since is uniformly ellptic and , we can choose big enough that

Next consider on .

Firstly for

Secondly since is compact, choose small enough that

So by the maximum principle

Then , hence

**Remark:** This theorem is also true when either of the two cases

(1) with and and

(2) irrespective the sign of .

**Proof:** (1) Since and we can also prove that

So , by the maximum principle

the following step is the same.

(2)Since in , we have

Replacing by , the proof is unchanged.

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