
Maximum Principle
The maximum principle for harmonic functions asserts:
Let $\displaystyle u \in C^2(U) \cap C(\bar{U})$ be harmonic in $\displaystyle U \subset \mathbb{R}^n$.
Then
1) (Weak maximum principle)
$\displaystyle \max_{\bar{U}}u=\max_{\partial U}u$;
2) (Strong maximum principle)
If $\displaystyle U$ is connected and there exists a point $\displaystyle x_0 \in U$ such that
$\displaystyle u(x_0)=\max_{\bar{U}}u$,
then $\displaystyle u=const$ in $\displaystyle U$.
I wonder if I got this right. The weak principle means that the maximal value on $\displaystyle \bar{U}$ 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 $\displaystyle x \in \partial U$ and $\displaystyle y \in U$ such that
$\displaystyle u(y)=\max_{\bar{U}}u=u(x)$,
in which case the shape of $\displaystyle U$ may cause the function to be constant (i.e. in case $\displaystyle U$ is connected and so 2) holds); but if $\displaystyle U$ is any open set nothing specific is known about $\displaystyle u$, i.e. it may rather be nonconstant.
I wonder why the conclusion of the strong version holds just for $\displaystyle U$ and not for $\displaystyle \bar{U}$. I mean 1) holds anyway and $\displaystyle u \in C(\bar{U})$. Doesn't this, together with 2), imply that $\displaystyle u$ is constant on the boundary? The proof of the strong version, however, works only with $\displaystyle U$, so that it's no big deal to accept it as it is.