Maximum principles need domain and boundary hypotheses

The weak principle requires boundedness and continuity on the full closure; it allows disconnected open sets. Its conclusion is an attained boundary maximum for subharmonic functions. The strong harmonic principle instead uses connectedness and an attained global interior extremum; neither boundedness nor a boundary trace is needed. These are the distinct scopes of Weak maximum principle for the laplacian and Strong maximum principle for harmonic functions.

At a boundary point, Hopf boundary point lemma for the laplacian needs an interior tangent ball, a strict interior inequality, continuity on its closure, and existence of the supplied outward directional derivative. It gives a positive outward derivative at a maximum; negating the function reverses the sign at a minimum.

Unbounded sets can be treated by Maximum principle with limsup control at infinity if the same finite upper bound controls the boundary and the limsup at infinity. No open-mapping or general boundary-normal theorem is asserted here.