Numerics and analysis of Cahn--Hilliard critical points

TL;DR

This paper investigates the properties of local minima and saddle points in the Cahn--Hilliard energy landscape for dimensions two and higher, using numerical and analytical methods to understand critical points in large systems.

Contribution

It combines the String Method with convexity analysis to study critical points of the Cahn--Hilliard energy, providing new insights into their structure and behavior.

Findings

Numerical insights into minima and saddle points in 2D.

Adaptation of convexity analysis for level sets in higher dimensions.

Enhanced understanding of critical points in large systems.

Abstract

We explore recent progress and open questions concerning local minima and saddle points of the Cahn--Hilliard energy in $d\geq 2$ and the critical parameter regime of large system size and mean value close to $-1$. We employ the String Method of E, Ren, and Vanden-Eijnden -- a numerical algorithm for computing transition pathways in complex systems -- in $d=2$ to gain additional insight into the properties of the minima and saddle point. Motivated by the numerical observations, we adapt a method of Caffarelli and Spruck to study convexity of level sets in $d\geq 2$.