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

## TL;DR

This paper investigates a Phase-Field-Crystal model with higher order derivatives, establishing a dimension reduction via $ ext{Gamma}$-convergence and characterizing conditions for the uniform state to be globally minimal.

## Contribution

It provides a $ ext{Gamma}$-convergence analysis for a 3D model to a 2D reduced model and determines the optimality conditions for the uniform state, including for the Ohta-Kawasaki model.

## Key findings

- Dimension reduction from 3D to 2D via $ ext{Gamma}$-convergence.
- Necessary and sufficient conditions for the uniform state's global minimality.
- Extension of results to the Ohta-Kawasaki model.

## 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.