An energy barrier and gamma limit in the Cahn Hilliard energy landscape

We study the d-dimensional Cahn-Hilliard equation on the flat torus in the parameter regime in which the system size is large and the mean value is close---but not too close---to -1. In this setting, the uniform state is a local but not global energy minimizer, and we are interested in a quantitative description of the surrounding energy landscape. 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 that connects the Cahn Hilliard energy with a heuristic description proposed by Biskup, Chayes, and Kotecky. This work is joint with Michael Gelantalis (RWTH).