Proof that Laplace equation permits no local maxima

Question

I glanced at a proof of the uniqueness theorem for Laplace's equation which implicitly relied on the non-existence of local maxima in the solution, and realized I didn't know how to prove that very basic result. It's something I've always taken for granted and never bothered to investigate. I don't have extensive formal education in PDEs, but ought to be able to prove at least that much. Any hints are appreciated.

I did a quick search on Google and the StackExchange, and didn't find anything, but I may have missed something.

Thanks

Answer 1

It is a consequence of the mean value property of harmonic functions: if $\Delta f=0$, $$ f(x,y) = \frac{\int_{\partial B} f(u,v)\,d\mu}{\int_{\partial B} 1\,d\mu}$$ where $B$ is any ball centered at $(x,y)$ inside the domain of $f$. It follows that the extremal values of a non-constant harmonic function are always attained on the boundary of the domain.