Harmonic function and maximum modulus principle

Question

Does Maximum modulus principle hold true for harmonic functions? I think we need open mapping theorem for this condition to hold true?

Answer 1

Yes it does. Once you have the mean value theorem, it is straightforward to prove the (strong) maximum principle.

By mean value theorem i mean

$u(x_0)=\int_{\{|x-x_0|=r\}} u(z)dSz$

for every $r$ such that the ball is contained in your domain.

Answer 2

Yes it holds for harmonic functions, and indeed in $n$ dimensions.