Maximum principle of harmonic function without mean value formula

Question

Are there any way to prove maximum principle of harmonic functions without the mean value formula? In other words I would like to show $$ \max_{\overline{\Omega}}(f)=\max_{\partial \Omega}(f) $$ for a harmonic function $f$ on a bounded domain $\Omega$ without using the formula $$ f(x)=\frac{1}{V(B(x,r)}\int_{B(x,r)}f(z)dz. $$

Answer 1

Yes, just think of the 1D-case: $f''=0$ on $(a,b)$. The function $f$ attains a maximum in $[a,b]$, and you must exclude that this maximum point lies in $(a,b)$. You first treat the situation where $f''>0$ in $(a,b)$, and then perturb $f$, for example $f(x)+\varepsilon x^2$.

In higher dimension this approach still works, but you must use the concept of positive-definite matrices.

You can read a proof in the book by Gilbarg and Trudinger, Elliptic partial differential equations of second order, Chapter 3.