Scaling of the energy for thin martensitic films

We study the scaling behavior of thin martensitic films. Specifically we consider an elastic energy with two SO(3) invariant wells which are strongly incompatible in the sense of Matos and Šverák, but whose two-dimensional projections may be compatible. We show that in a thin film of thickness h the energy per unit height scales like h. This scaling lies in between the classical membrane theory (where the energy per unit height is of order 1) and the Kirchhoff bending theory, which corresponds to a scaling of h².