Absolute value of a harmonic function

Question

Let $\Omega \subset R^d$ be open and bounded, let $u\in C^2(\Omega) \cap C(\bar{\Omega})$ be harmonic in $\Omega$. Can we say $$\max_{\bar{\Omega}} |u| \leq \max_{\partial \Omega} |u|$$

Answer 1

Ok, it works: set $M=\max_{\partial \Omega} |u|$ by maximum principle we obtain: $$\max_{\bar{\Omega}} (u) \leq M \quad \text{and} \quad \max_{\bar{\Omega}} (-u) = \max_{\partial \Omega} (-u) \leq M$$ Then the estimate follows.