the maximum principle for harmonic functions

A harmonic function cannot have a peak or a valley in the interior of its region — its extremes can only occur on the boundary. Think of a soap film on a wire loop: it never bulges up to a high point or dips to a low point in the middle, because any such bump could relax by spreading out. The highest and lowest points of the film are always on the rim. This is the maximum (and minimum) principle, and it is one of the most powerful tools in all of PDE theory.

Precisely, the weak maximum principle says: if u is harmonic on a bounded region and continuous up to the boundary, then the maximum and minimum of u over the closed region are both attained on the boundary. It follows in one line from the mean-value property — if an interior point equalled the maximum, it would equal its sphere average, which is impossible unless every nearby value also equals the maximum, forcing u to be constant. The strong maximum principle sharpens this: if u attains an interior maximum at even one point, then u is constant throughout a connected region. There is no way to touch the ceiling once inside without being flat against it everywhere.

The payoff is uniqueness and stability for the Dirichlet problem, almost for free. If two harmonic functions share the same boundary values, their difference is harmonic with zero boundary values, so its maximum and minimum are both zero — the difference is identically zero and the solution is unique. Stability follows the same way: if two boundary datasets differ by at most epsilon everywhere, the two solutions differ by at most epsilon everywhere inside, so the solution depends continuously on the data. This single principle is the backbone of well-posedness for elliptic equations, and versions of it hold for the heat equation and far beyond.