Proper ferroelastic phase transitions in thin epitaxial films with symmetry-conserving and symmetry-breaking misfit strains

We study how the ferroelastic domain structure sets in in an epitaxial film of a material with second order proper ferroelastic transition. The domain structures considered are similar to either $a_{1}/a_{2}/a_{1}/a_{2}$ or $c/a/c/a$ structures in perovskite ferroelectrics. If the "extrinsic" misfit strain, not associated with the transition, does not break the symmetry of the high-temperature phase, the phase transition in the film occurs at somewhat lower temperature compared to the bulk. The loss of stability then occurs with respect to a sinusoidal strain wave, which evolves into the domain structure with practically the same geometry and approximately the same period. In the presence of the symmetry-breaking component of the misfit strain ("extrinsic" misfit) the character of the phase transition is qualitatively different. In this case it is a {\em topological} transition between single-domain and multi-domain states, which starts from a low density of the domain walls.