# Radial and cylindrical symmetry of solutions to the Cahn-Hilliard equation

## TL;DR

This paper classifies entire solutions to the Cahn-Hilliard equation in Euclidean space, establishing conditions under which solutions exhibit radial or cylindrical symmetry, especially when their nodal sets are bounded or cylindrical.

## Contribution

It provides new symmetry classification results for solutions of the Cahn-Hilliard equation with specific nodal set geometries.

## Key findings

- Solutions with bounded or cylindrical nodal sets are radially or cylindrically symmetric under certain conditions.
- The paper extends symmetry results to solutions with particular geometric constraints on their nodal sets.

## Abstract

The paper is devoted to the classification of entire solutions to the Cahn-Hilliard equation $-\Delta u = u-u^3-\delta$ in $\R^N$, with particular interest in those solutions whose nodal set is either bounded or contained in a cylinder. The aim is to prove either radial or cylindrical symmetry, under suitable hypothesis.