Numerics and analysis of Cahn–Hilliard critical points
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We explore recent progress and open questions concerning local minima and saddle points of the Cahn–Hilliard energy in d≥ 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≥ 2.
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