A computer assisted proof of the symmetries of least energy nodal solutions on squares

TL;DR

This paper proves symmetry properties of least energy nodal solutions to a nonlinear PDE on squares and examines how these symmetries change when the domain is deformed into rectangles.

Contribution

It provides a rigorous proof of symmetry and symmetry-breaking phenomena for solutions on square and rectangular domains near the linear case.

Findings

Solutions are odd with respect to one diagonal and even with respect to the other near p=2

Symmetry breaks when the domain is a rectangle close to a square

Results connect domain shape to solution symmetry properties

Abstract

In this article, we prove that the least energy nodal solutions to Lane-Emden equation $-{\Delta}u = |u|^{p-2}u$ with zero Dirichlet boundary conditions on a square are odd with respect to one diagonal and even with respect to the other one when $p$ is close to 2. We also show that this symmetry breaks on rectangles close to squares.