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 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.
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Notes
If \(k=(2\pi s, 2\pi t)\), then its Euclidian norm is denoted by \(|k|^2=4\pi ^2(s^2+t^2)\).
This computation is carried out for \(\varphi \) smooth in V and the result follows for general \(\varphi \in V\) by a standard density argument.
One inequality comes from (3.5). To prove that 1 is indeed the infimum in (3.6), it is enough to consider the case of dimension \(N=1\): for every \(n\ge 1\), let \(v_n=n\) in \((0,\frac{1}{n})\) and \(v_n=-\frac{n}{n-1}\) in \((\frac{1}{n},1)\). Then the sequence
$$\begin{aligned}u_n=\left( \int _{\mathbb {T}}v_n^3\right) ^{-\frac{1}{3}}v_n\end{aligned}$$is a minimizing sequence in (3.6) yielding the value 1 for the infimum.
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Acknowledgements
The authors thank Xavier Lamy for useful comments. R.I. acknowledges partial support by the ANR Project ANR-14-CE25-0009-01.
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Communicated by Dr. Anthony Bloch.
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Ignat, R., Zorgati, H. Dimension Reduction and Optimality of the Uniform State in a Phase-Field-Crystal Model Involving a Higher-Order Functional. J Nonlinear Sci 30, 261–282 (2020). https://doi.org/10.1007/s00332-019-09573-0
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DOI: https://doi.org/10.1007/s00332-019-09573-0