The Maximum Principle and the Moving Plane Method

Abstract

In this article we discuss the maximum principle and it’s application to the study of symmetry of solutions of nonlinear partial differential equations, which was one of the main research topics of Louis Nirenberg. The central question in symmetry that we discuss is the following if a domain Ω ⊂ ℝN and the given boundary data on ∂Ω have some symmetry, for example radial symmetry, axial symmetry or symmetry with respect to some hyperplane, then when we can say that positive solution of a given nonlinear partial differential equation on Ω inherit these symmetries.