T. Grafke, S. Scholtes, A. Wagner, M. Westdickenberg, Math. Mod. Meth. Appl. Sci. 36 (2026)
Abstract
We explore recent progress and open questions concerning local minima and saddle points of the Cahn-Hilliard energy in $d\ge 2$ 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 $d\ge2$.