Maximum Principle and Its Application to Elliptic Partial Differential Equations

The most important tool that investigates most properties of solutions and equations of second order elliptic and parabolic type is the maximum principle. This has types, and all these types enable us to obtain valuable information about the properties of solutions and perhaps the equations themselves. Originating from the classical theory of harmonic functions, it has been generalized and extended to a wide range of elliptic partial differential equations and plays a critical role in proving uniqueness, comparism theorems, a priori estimates, and regularity results.