Dimension reduction and optimality of the uniform state in a Phase-Field-Crystal model involving a higher order functional

Abstract

We study a Phase-Field-Crystal model described by a free energy functional involving second order derivatives of the order parameter in a periodic setting and under a fixed mass constraint. We prove a $\Gamma$-convergence result in an asymptotic thin-film regime leading to a reduced 2-dimensional model. For the reduced model, we prove necessary and sufficient conditions for the global minimality of the uniform state. We also prove similar results for the Ohta-Kawasaki model.