Maximum Principles in Differential Equations

Abstract

The aim of this thesis is to investigate the maximum principle, which is one of the most important tools employed in differential equations. Specifically, we explore the maximum principle for linear second-order elliptic and parabolic partial differential equations and its applications. The maximum principle for linear elliptic equations shows that a solution of a function attains its maximum or minimum on the boundary of the appropriate region. However, in the case of linear parabolic equations, this principle establishes that a solutions of a function attains its maximum or minimum on a certain part of the boundary, the so-called the parabolic boundary. We establish the comparison principle for elliptic and parabolic equations and for the corresponding weak solutions of Laplace and heat equations. The maximum principle is used to show the comparison principle.