Energy barrier and $\Gamma$-convergence in the $d$-dimensional Cahn-Hilliard equation

TL;DR

This paper analyzes the energy landscape of the d-dimensional Cahn-Hilliard equation on a flat torus, deriving sharp estimates of energy barriers and a $ extGamma$-limit for the rescaled energy gap, especially near local minimizers.

Contribution

It provides a quantitative description of the energy barrier and establishes a $ extGamma$-convergence result for the energy gap in a specific parameter regime.

Findings

Sharp estimate of the energy barrier size.

$ extGamma$-limit of the rescaled energy gap.

Analysis near local but not global minimizers.

Abstract

We study the d-dimensional Cahn-Hilliard equation on the flat torus in a parameter regime in which the system size is large and the mean value is close---but not too close---to -1. We are particularly interested in a quantitative description of the energy landscape in the case in which the uniform state is a local but not global energy minimizer. In this setting, we derive a sharp leading order estimate of the size of the energy barrier surrounding the uniform state. A sharp interface version of the proof leads to a $\Gamma$-limit of the rescaled energy gap between a given function and the uniform state.