# Symmetry of constrained minimizers of the Cahn-Hilliard energy on the torus

## TL;DR

This paper investigates symmetry properties of minimizers of the Cahn-Hilliard energy on a torus, using Steiner symmetrization and rearrangement techniques, with quantitative estimates in two dimensions.

## Contribution

It provides new sufficient conditions for symmetry of minimizers and demonstrates the use of two-point rearrangements for the Cahn-Hilliard model.

## Key findings

- Established conditions for symmetry via Steiner symmetrization.
- Applied Bonnesen inequality for quantitative sphericity estimates.
- Demonstrated symmetry results for volume-constrained minimizers.

## Abstract

We establish sufficient conditions for a function on the torus to be equal to its Steiner symmetrization and apply the result to volume-constrained minimizers of the Cahn-Hilliard energy. We also show how two-point rearrangements can be used to establish symmetry for the Cahn-Hilliard model. In two dimensions, the Bonnesen inequality can then be applied to quantitatively estimate the sphericity of superlevel sets.