Communications in Mathematical Sciences
Volume 21 (2023)
Gamma convergence for the de Gennes–Cahn–Hilliard energy
Pages: 2131 – 2144
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.
de Gennes–Cahn–Hilliard energy, Gamma convergence, sharp interface limit, surface diffusion
2010 Mathematics Subject Classification
35B40, 35J20, 35J60, 35Q92
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