On the existence of symmetric minimizers

TL;DR

This paper explores a symmetrization technique called G-averaging for functions under a topological group, enabling the construction of symmetric minimizers for certain functionals, with applications to differential equations.

Contribution

It introduces an abstract G-averaging method that produces G-invariant minimizers under relaxed convexity conditions, with applications to PDEs.

Findings

G-averaging yields symmetric minimizers for specific functionals.

Application to p-Laplace and polyharmonic Poisson problems demonstrates the method.

Open problems suggest further research directions.

Abstract

In this note we revisit a less known symmetrization method for functions with respect to a topological group $G$, which we call $G$-averaging. We note that, although quite non-technical in nature, this method yields $G$-invariant minimizers of functionals satisfying some relaxed convexity properties. We give an abstract theorem and show how it can be applied to the $p$-Laplace and polyharmonic Poisson problem in order to construct symmetric solutions. We also pose some open problems and explore further possibilities where the method of $G$-averaging could be applied to.