  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.
