Hopf maximum principle

The Hopf maximum principle is a maximum principle in the theory of second order elliptic partial differential equations and has been described as the "classic and bedrock result" of that theory. Generalizing the maximum principle for harmonic functions which was already known to Gauss in 1839, Eberhard Hopf proved in 1927 that if a function satisfies a second order partial differential inequality of a certain kind in a domain of Rⁿ and attains a maximum in the domain then the function is constant. The simple idea behind Hopf's proof, the comparison technique he introduced for this purpose, has led to an enormous range of important applications and generalizations.