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 two-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.

中文翻译：

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涉及高阶泛函的相场-晶体模型中均匀态的降维和最优性

我们研究一种由自由能函数描述的相场晶体模型，该函数涉及周期性参数和固定质量约束下的阶跃参数的二阶导数。我们证明了渐近薄膜形式的\（\ Gamma \）收敛结果，导致了二维模型的简化。对于简化模型，我们证明了均匀状态的全局极小值的充要条件。我们还证明了Ohta-Kawasaki模型的相似结果。