Gamma convergence for the de Gennes–Cahn–Hilliard energy

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.

Keywords

de Gennes–Cahn–Hilliard energy, Gamma convergence, sharp interface limit, surface diffusion

2010 Mathematics Subject Classification

35B40, 35J20, 35J60, 35Q92

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The work of the first author was partially supported by the U.S. National Science Foundation through grant DMS-1815746. The work of the third author was partially supported by the U.S. National Science Foundation through grant DMS-2012634.

Received 28 October 2022

Accepted 20 February 2023

Published 15 November 2023