the strong maximum principle

The weak principle says the highest value of a harmonic function sits somewhere on the boundary. The strong principle goes much further and says: it sits ONLY on the boundary. If a harmonic temperature ever touched its maximum at a single interior point, the only way that could happen is if the temperature were the same flat value everywhere. There is no such thing as a lone hot spot in the middle of a harmonic function — a peak inside forces the whole thing to be constant.

Precisely: let u satisfy Laplacian u >= 0 (subharmonic) on a connected open region. If u attains its maximum at some interior point, then u is constant throughout the region. The intuition comes from the mean-value property: a harmonic function at any point equals the average of its values over a small surrounding sphere. If the centre were a strict maximum, the average over the sphere would have to be strictly less than the centre value — contradicting equality. So a strict interior maximum is impossible; the only escape is that the surrounding values are all equal to the peak, and connectedness then spreads that equality across the whole region. The same statement holds for solutions of general uniformly elliptic equations, not just the Laplacian.

This rigidity is the backbone of elliptic uniqueness and of comparison arguments: it says solutions cannot have hidden interior extrema, so their entire behaviour is controlled from the boundary. Combined with the Hopf boundary-point lemma — which controls the derivative where the maximum is attained on the boundary — it gives the standard route to uniqueness for Neumann and Robin problems and to symmetry results via the moving-plane method. Honesty note: it needs connectedness and an elliptic (or parabolic) operator; for a general PDE, or on a disconnected domain, an interior maximum can perfectly well occur without forcing a constant.