  There is a growing interest, inspired by advances in technology, in the low
temperature physics of thin films. These quasi-2D systems show a wide range of
ordering effects including formation of striped states, reorientation
transitions, bubble formation in strong magnetic fields, etc. The origins of
these phenomena are, in many cases, traced to competition between short ranged
exchange ferromagnetic interactions, favoring a homogeneous ordered state, and
the long ranged dipole-dipole interaction, which opposes such ordering on the
scale of the whole sample. The present theoretical understanding of these
phenomena is based on a combination of variational methods and a variety of
approximations, e.g., mean-field and spin-wave theory. The comparison between
the predictions of these approximate methods and the results of MonteCarlo
simulations are often difficult because of the slow relaxation dynamics
associated with the long-range nature of the dipole-dipole interactions. In
this note we will review recent work where we prove existence of periodic
structures in some lattice and continuum model systems with competing
interactions. The continuum models have also been used to describe
micromagnets, diblock polymers, etc.
