Abstract
We present an analytical treatment of a three-dimensional variational model of a system that exhibits a second-order phase transition in the presence of dipolar interactions. Within the framework of Ginzburg–Landau theory, we concentrate on the case in which the domain occupied by the sample has the shape of a flat thin film and obtain a reduced two-dimensional, non-local variational model that describes the energetics of the system in terms of the order parameter averages across the film thickness. Namely, we show that the reduced two-dimensional model is in a certain sense asymptotically equivalent to the original three-dimensional model for small film thicknesses. Using this asymptotic equivalence, we analyze two different thin film limits for the full three-dimensional model via the methods of \(\Gamma \)-convergence applied to the reduced two-dimensional model. In the first regime, in which the film thickness vanishes while all other parameters remain fixed, we recover the local two-dimensional Ginzburg–Landau model. On the other hand, when the film thickness vanishes while the sample’s lateral dimensions diverge at the right rate, we show that the system exhibits a transition from homogeneous to spatially modulated global energy minimizers. We identify a sharp threshold for this transition.
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Notes
Recall that \(D + B_\delta = \mathbf \{ {\mathbf {r}} \in {\mathbb {R}}^2 : \text {dist}(\mathbf r, D) < \delta \}\).
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Acknowledgements
This work was supported, in part, by NSF via Grants DMS-1313687 and DMS-1614948. The author would like to thank V. Slastikov for many valuable comments.
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Communicated by A.Malchiodi.
Dedicated to V. V. Osipov on the occasion of his 77th birthday.
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Muratov, C.B. A universal thin film model for Ginzburg–Landau energy with dipolar interaction. Calc. Var. 58, 52 (2019). https://doi.org/10.1007/s00526-019-1493-4
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DOI: https://doi.org/10.1007/s00526-019-1493-4