1. ## Maximum Principle

The maximum principle for harmonic functions asserts:
Let $u \in C^2(U) \cap C(\bar{U})$ be harmonic in $U \subset \mathbb{R}^n$.
Then
1) (Weak maximum principle)

$\max_{\bar{U}}u=\max_{\partial U}u$;

2) (Strong maximum principle)

If $U$ is connected and there exists a point $x_0 \in U$ such that

$u(x_0)=\max_{\bar{U}}u$,

then $u=const$ in $U$.

I wonder if I got this right. The weak principle means that the maximal value on $\bar{U}$ of a harmonic function is necessarily attained in a boundary point, i.e. there are no solely interior points which give maximal values for a harmonic function. Although it may be that there are points $x \in \partial U$ and $y \in U$ such that

$u(y)=\max_{\bar{U}}u=u(x)$,

in which case the shape of $U$ may cause the function to be constant (i.e. in case $U$ is connected and so 2) holds); but if $U$ is any open set nothing specific is known about $u$, i.e. it may rather be non-constant.

I wonder why the conclusion of the strong version holds just for $U$ and not for $\bar{U}$. I mean 1) holds anyway and $u \in C(\bar{U})$. Doesn't this, together with 2), imply that $u$ is constant on the boundary? The proof of the strong version, however, works only with $U$, so that it's no big deal to accept it as it is.