We consider a standard functional in the mesoscopic theory of phase transitions, consisting of a gradient term with a double-well potential, and we add to it a bulk term modeling the interaction with a periodic mean zero external field. This field is amplified and dilated with a power of the transition layer thickness epsilon leading to a nontrivial interaction of forcing and concentration when epsilon --> 0. We show that the functionals Gamma-converge after additive renormalization to an anisotropic surface energy, if the period of the oscillation is larger than the interface thickness. Difficulties arise from the fact that the functionals have nonconstant absolute minimizers and are not uniformly bounded from below.
|Number of pages
|Interfaces and Free Boundaries
|Published - 2006