Interactions between Homogenization and Phase-Transition Processes

We study the behavior of nonconvex functionals singularly perturbed by a possibly oscillating inhomogeneous gradient term, in the spirit of the gradient theory of phase transitions. We show that a limit problem giving a sharp interface, as the perturbation vanishes, always exists, but may be inhomogeneous or anisotropic. We specialize this study when the perturbation oscillates periodically, highlighting three types of regimes depending on the speed of oscillations. In the two extreme cases, a separation of scale effect is described.