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arXiv · 1806.02408

On the existence of symmetric minimizers

Athanasios Stylianou
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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.

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Athanasios Stylianou. 2018-06-06. On the existence of symmetric minimizers. https://arxiv.org/abs/1806.02408

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