the comparison principle

Two identical metal plates, one started and edged hotter than the other at every point. Common sense says the hotter setup stays hotter everywhere, for all time — you cannot make a cooler input produce a warmer output through diffusion. The comparison principle turns this common sense into a theorem: bigger data gives a bigger solution, pointwise. It is the maximum principle wearing its most useful hat.

Precisely: suppose u and v both solve the same elliptic or parabolic equation (or u is a subsolution and v a supersolution), and on the boundary (or initial data) you have u <= v. Then u <= v everywhere inside, for all time. The proof is one line of maximum principle: the difference w = u - v solves the same linear equation with w <= 0 on the boundary, so by the maximum principle w <= 0 throughout. A special, enormously useful case is comparing your unknown solution against an explicit barrier you can write down: if a known supersolution v lies above your data, then your solution is trapped below v everywhere, giving an a priori L-infinity bound (an upper bound you have before you have solved anything).

This single idea is the workhorse of qualitative PDE. It gives uniqueness (if u <= v and v <= u then u = v), stability (control the solution's size by the data's size), and a priori bounds that keep nonlinear iterations from blowing up. It underlies the method of sub- and supersolutions (monotone iteration), where you sandwich a solution between a subsolution and a supersolution and squeeze. Honesty note: it is essentially an elliptic/parabolic phenomenon — hyperbolic equations like the wave equation do NOT obey a comparison principle, because their solutions can oscillate freely below and above the data.