On the existence of symmetric minimizers

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.