Evans stated the strong maximum principle as follows: $U\subset\mathbb{R}^n$ a bounded and open set. If $u\in C^2(U)\cap C(\overline{U})$ is harmonic within $U$.
Then,
- $\max_{\overline{U}}u=\max_{\partial U}u$
- if $U$ is in addition connected and there exists a point $x_0\in U$ such that $u(x_0)=\max_{\overline{U}}u$ then $u$ is constant within $U$.
I understand the proof of $2$. But why does this already imply 1?
Best Answer
The maximum is attained, because $\overline U$ is compact. And there are only two possibiliies: Either the maximum is attained at some interior point or it is not attained for any interior point (in which case it has to be on the boundary).
Now 2. says that $u = \mathrm{const}$ in the first case. So in particular, we must have $\max_{\overline U} u = \max_{\partial U} u$.
In the second case this equality is also true, trivially.