  Motivated by solid-solid phase transitions in elastic thin films, we perform
a Gamma-convergence analysis for a singularly perturbed energy describing
second order phase transitions in a domain of vanishing thickness. Under a
two-wells assumption, we derive a sharp interface model with an interfacial
energy depending on the asymptotic ratio between the characteristic length
scale of the phase transition and the thickness of the film. In each case, the
interfacial energy is determined by an explicit optimal profile problem. This
asymptotic problem entails a nontrivial dependance on the thickness direction
when the phase transition is created at the same rate as the thin film, while
it shows a separation of scales if the thin film is created at a faster rate
than the phase transition. The last regime, when the phase transition is
created at a faster rate than the thin film, is more involved. Depending on
growth conditions of the potential and the compatibility of the two phases, we
either obtain a sharp interface model with scale separation, or a trivial
situation driven by rigidity effects.
