The maximum principle for harmonic functions asserts:
Let be harmonic in .
1) (Weak maximum principle)
2) (Strong maximum principle)
If is connected and there exists a point such that
then in .
I wonder if I got this right. The weak principle means that the maximal value on 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 and such that
in which case the shape of may cause the function to be constant (i.e. in case is connected and so 2) holds); but if is any open set nothing specific is known about , i.e. it may rather be non-constant.
I wonder why the conclusion of the strong version holds just for and not for . I mean 1) holds anyway and . Doesn't this, together with 2), imply that is constant on the boundary? The proof of the strong version, however, works only with , so that it's no big deal to accept it as it is.