$\Gamma$--convergence of the Allen-Cahn energy with an oscillating forcing term

Abstract

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 modelling the interaction with a periodic mean zero external field. This field is amplified and dilated with a power of the transition layer thickness $\eps$ leading to a nontrivial interaction of forcing and concentration when $\eps \to 0$. 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 non constant absolute minimizers and are not uniformly bounded from below.