# Symmetry-breaking in a generalized Wirtinger inequality

## TL;DR

This paper provides elementary proofs for symmetry and asymmetry properties of minimizers in a generalized Wirtinger inequality, clarifying the conditions under which solutions are symmetric or asymmetric.

## Contribution

It offers a new, computer-free proof of symmetry and asymmetry results, applicable to both global and local minimizers in the inequality.

## Key findings

- Complete characterization of symmetry and asymmetry regions
- Elementary proofs avoiding computer assistance
- Applicability to both global and local minima

## Abstract

The search of the optimal constant for a generalized Wirtinger inequality in an interval consists in minimizing the $p$-norm of the derivative among all functions whose $q$-norm is equal to~1 and whose $(r-1)$-power has zero average. Symmetry properties of minimizers have attracted great attention in mathematical literature in the last decades, leading to a precise characterization of symmetry and asymmetry regions. In this paper we provide a proof of the symmetry result without computer assisted steps, and a proof of the asymmetry result which works as well for local minimizers. As a consequence, we have now a full elementary description of symmetry and asymmetry cases, both for global and for local minima. Proofs rely on appropriate nonlinear variable changes.