Let \(\Omega\subseteq \mathbb{R}^d\) be open and \(V\subseteq \Omega\) a bounded domain with \(\bar{V}\subseteq \Omega\). Then there is a constant \(C=C(\Omega,V)>0\) such that every non-negative harmonic function \(u\) satisfies
\[ \sup_V u\le C\inf_V u. \]
Proof
This is an implication of the mean value property
.
We find for every point \(x\in V\) an estimate for an arbitrary point \(y\in B_r(x)\subseteq \Omega\) by covering \(B_r(x)\) with \(B_{2r}(y)\subseteq \Omega\) and applying the mean value property.
We cover \(V\) with finitely many balls with fixed radius and we find for every \(x\) and \(y\) a path which lies in a finite sequence of balls. Applying the above estimate on every ball and using a sup-inf argument implies the Harnack inequality.