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We present an analytical treatment of a three-dimensional variational modelof a system that exhibits a second-order phase transition in the presence ofdipolar interactions. Within the framework of Ginzburg-Landau theory, weconcentrate on the case in which the domain occupied by the sample has theshape of a flat thin film and obtain a reduced two-dimensional, non-localvariational model that describes the energetics of the system in terms of theorder parameter averages across the film thickness. Namely, we show that thereduced two-dimensional model is in a certain sense asymptotically equivalentto the original three-dimensional model for small film thicknesses. Using thisasymptotic equivalence, we analyze two different thin film limits for the fullthree-dimensional model via the methods of $\Gamma$-convergence applied to thereduced two-dimensional model. In the first regime, in which the film thicknessvanishes while all other parameters remain fixed, we recover the localtwo-dimensional Ginzburg-Landau model. On the other hand, when the filmthickness vanishes while the sample's lateral dimensions diverge at the rightrate, we show that the system exhibits a transition from homogeneous tospatially modulated global energy minimizers. We identify a sharp threshold forthis transition.

A universal thin film model for Ginzburg-Landau energy with dipolar interaction