Symmetry of Solutions to Variational Problems for Nonlinear Elliptic Equations via Reflection Methods

Abstract

We discuss some recent results on symmetry of solutions of nonlinear partial differential equations. We focus on elliptic and degenerate elliptic boundary value problems of second order with variational structure and the simple looking case where the underlying domain is radially symmetric. In this setting, we study solutions which are given as minimizers of constrained minimization problems or have low Morse index, and we examine which amount of symmetry of the data is inherited by these solutions. We highlight how the answer to this general question depends on specific assumptions on the data. The underlying techniques collected in this survey are elementary as they solely rely on hyperplane reflections and well known analytical and topological tools, but they yield surprisingly general results in situations where classical methods do not apply.