Maximum Principles for Differential Equations

Abstract

maximum principle is one among the most useful and best known tools used in the study of partial differential equations. This principle is a generalization of the well known fact of calculus that any function u(x) which satisfies the inequality u'' > 0 on an interval $[c,d]$ achieves its maximum value at one of the endpoints of the interval. We express this by saying that solutions of the inequality u'' > 0 satisfy a maximum principle.The study of partial differential equations usually begins with a classification of equations into different types. The equations frequently studied are those of the type elliptic, parabolic and hyperbolic. Solutions of these partial differential equations are also classified as classical solution, weak or generalized solution and viscosity solution. This work is developed in three chapters based on this classifications.The first chapter treats the maximum principles of classical solutions of elliptic and parabolic type. In Chapter 2 we establish the maximum principle of the generalized solution for second-order elliptic operators and the third and last chapter treats the so called viscosity solutions.