the weak maximum principle

Stretch a soap film across a wire loop. The film never bulges up above the highest point of the wire or sags below the lowest point — its peaks and valleys all sit on the rim, not in the middle. The weak maximum principle is the same fact for the solution of an elliptic equation: it cannot reach a value bigger (or smaller) than what it already takes somewhere on the boundary. The largest and smallest values live on the edge.

Precisely: if u solves Laplace's equation Laplacian u = 0 (or satisfies the subharmonic inequality Laplacian u >= 0) on a bounded region with boundary, then the maximum of u over the closed region equals the maximum of u over just the boundary. The word weak means it only promises the max is attained somewhere on the boundary — it does not forbid the same maximal value from also appearing inside. The proof idea is short: add a tiny upward-curving bump like epsilon times |x|^2, whose Laplacian is strictly positive, so the bumped function can have no interior maximum (an interior max forces Laplacian <= 0); push that interior max to the boundary, then let epsilon go to zero. A subharmonic function (Laplacian u >= 0) obeys only the maximum half; for full harmonic u you get the minimum half too by applying the same argument to -u.

This is your first and cheapest piece of qualitative information: it bounds the solution everywhere using only the data on the boundary, without solving anything. From it flow uniqueness (two solutions with the same boundary data must coincide, since their difference is harmonic with zero boundary values, hence zero) and stability (a small change in boundary data changes the solution by at most the same small amount in the interior). The strong maximum principle sharpens it further by ruling out an interior maximum unless the solution is constant.