The maximum principle for harmonic functions asserts:
Let be harmonic in .
Then
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.