Symmetry Properties for Solutions of Higher-Order Elliptic Boundary Value Problems

Abstract

Goals of this lecture-series: The lecture series is intended to give a glimpse into different methods that allow to conclude that solutions of higher-order nonlinear boundary value problems have the same symmetries as the underlying domain. In the second-order case this has been a very prominent area of research for the last half-century. For higher-order cases the theory has not yet been developed up to a comparable level. We will show in an exemplary way how different techniques from nonlinear analysis and geometry can be used to answer symmetry questions. In particular, we will discuss symmetry properties of solutions to (−Δ)mu = f(x, u) either on \(\mathbb {R}^n\) or on balls with additional Dirichlet boundary conditions. We will give examples for three different methods: methods based on contraction mapping methods based on the moving plane method an example of an overdetermined 4th order problem using Newton’s inequalities