Existence and properties of certain critical points of the Cahn-Hilliard energy

TL;DR

This paper investigates the energy landscape of the Cahn-Hilliard model on a torus, establishing the existence and properties of critical points such as local minimizers and saddle points in the large system size regime.

Contribution

It proves the existence of a droplet-shaped local energy minimizer and a saddle point with a quantifiable energy barrier, using advanced variational and geometric methods.

Findings

Existence of a droplet-shaped local minimizer.

Existence of a saddle point with a specific energy level.

Quantitative bounds on the energy barrier.

Abstract

The Cahn-Hilliard energy landscape on the torus is explored in the critical regime of large system size and mean value close to $-1$. Existence and properties of a "droplet-shaped" local energy minimizer are established. A standard mountain pass argument leads to the existence of a saddle point whose energy is equal to the energy barrier, for which a quantitative bound is deduced. In addition, finer properties of the local minimizer and appropriately defined constrained minimizers are deduced. The proofs employ the $\Gamma$-limit (identified in a previous work), quantitative isoperimetric inequalities, variational arguments, and Steiner symmetrization.