Suppose and are harmonic function in and respectively. For any and with , we have

We give two different proofs. The first is using possion integral formula

Here is the radius of .

WLOG, assume are the origin. And , , . Define

We have to prove . Apply the possion integral formula

where . Noticing that

So

The second proof relies on Green formula. Let , we will prove is independent of . Here we require are still inside the domain of and

Since , , plug in this fact

Define , for any , , then are harmonic in the disc

, on .

By the Green second identity, we know that the end of last equation is 0.

See Gustin’s paper: A bilinear integral identity for harmonic functions. And Gilbarg Trudinger’s book, chapter 2,2.18

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