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 isanyopen 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.