Gamma Convergence for the de Gennes-Cahn-Hilliard energy

TL;DR

This paper investigates the Gamma convergence of the de Gennes-Cahn-Hilliard energy, revealing that its limit is proportional to the interface area, influenced by the de Gennes coefficient and potential.

Contribution

It provides the first analysis of the Gamma limit for the degenerate de Gennes-Cahn-Hilliard energy, linking it to interface area and transition layer profiles.

Findings

Gamma limit is proportional to interface area

Constant depends on de Gennes coefficient and potential

Transition layer profile determined by double well potential

Abstract

The degenerate de Gennes-Cahn-Hilliard (dGCH) equation is a model for phase separation which may more closely approximate surface diffusion than others in the limit when the thickness of the transition layer approaches zero. As a first step to understand the limiting behavior, in this paper we study the $\Gamma$--limit of the dGCH energy. We find that its $\Gamma$--limit is a constant multiple of the interface area, where the constant is determined by the de Gennes coefficient together with the double well potential. In contrast, the transition layer profile is solely determined by the double well potential.