The Gidas--Ni--Nirenberg Theorem

Abstract

We investigate the Gidas--Ni--Nirenberg theorem one of the fundamental foundations in the theory of partial differential equations, it is about the monotonicity and symmetries of positive solutions of nonlinear second order elliptic partial differential equations over some connected, bounded domain and in the entire space, which was established by three mathematicians Gidas, Ni and Nirenberg in 1979. The proof uses the powerful method of moving planes, developed by Alexandrov around 1955. it is applicable in mathematical physics and analysis.