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 Γ-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.

Asymptotic analysis for a second-order curved thin film

“…Adding such a higher-order term was also necessary in order to induce the existence of some curve energy terms (see [10]). Second-order terms appear also in some Phase Field Crystal models where they control the variations represented by the first-order term (see [11]). We also cite some works involving higher-order terms related to phase transitions [12][13][14][15][16][17].…”

“…Also, the limit obtained being lower semi-continuous with respect to the considered topology, the existence of minimizers is thus guaranteed (see [42]). This method was used in many works in the context of dimension reduction such as Ignat and Zorgati [11], Acerbi et al [43], Babadjian et al [44], Braides et al [45], Fonseca and Francfort [46], Le Dret and Raoult [47,48], and Zorgati [49][50][51].…”

Asymptotic analysis for a second order curved thin film

“…Also, the limit obtained being lower semi-continuous with respect to the considered topology, the existence of minimizers is thus guaranteed (see [18] ). This method was used in many works in the context of dimension reduction such as [1,5,21,25,28,29,41,42,43,2].…”