Suppose \(\Omega\subseteq \mathbb{R}^d\) is a bounded domain and \(u\in C^2(\Omega)\cap C(\bar{\Omega})\) is subharmonic . Then either for every \(x\in \Omega\)
\[ u(x)<\max_{x\in \partial\Omega} u(x) \]or \(u\) is constant.
According to the weak maximum principle we have \(u(x)\le \max_{\partial\Omega}u\). Consider the set \(U=\{x\mid u(x)=M\}\). Since \(u\) is continuous \(U\) is closed.
Assume there is point \(x\) with \(u(x)=\max_{\partial\Omega}u=:M\), i.e. \(U\) is not empty. Using the mean value property there is a suitable ball \(B_r(x)\subseteq \Omega\) with
\[ M=u(x)\le \frac{1}{\lvert B_r(x)\rvert}\int_{B_r(x)} u\le M. \]Therefore, \(U\) is open, since \(B_r(x)\subseteq U\).
Since \(\Omega\) is path connected and therefore connected the only open, closed and nonempty subset of \(\Omega\) is \(\Omega\) itself. In conclusion, \(u\) is constant.
- The strong maximum principle also holds for \(C^0\)-subharmonic functions .