## Properties of Green function

As we all know, Green’s function , where is the fundamental solution of Laplacian operator

and is solved for each by

Then for any , satisfies

is bounded. is the first kind of Green function of .

- for all , .
- Let be a bounded and locally integrable function in . Then

as .

Let be a non-negative harmonic function in , such that , then

*We omit the proof of this lemma. The importance of this lemma is sup is bounded by inf with a constant independent of radius.*

Fix . Denote . Recall that is a harmonic function in , and as , there exists with such that and .

we can cover by finitely many balls centered at with radius . By using the lemma iteratively, we know that on the .

Since is a harmonic function in and on , by the maximum principle

in .

This implies

When , we must have , then , because .

So . Let . We have

as .

This implies when .

GT’s chapter 2.3.

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