Solid-solid phase transitions with space-dependent wells

We investigate a second-order Modica-Mortola functional of the form $$E_\varepsilon[u] := \int_\Omega \frac{1}{\varepsilon} W(x, \nabla u) + \varepsilon|\nabla^2 u|^2 \, dx,$$ which models solid--solid phase transitions in heterogeneous media. We neglect the assumption of frame indifference in the elastic energy density $W$ while allowing for space-dependent, pointwise-compatible wells. Under suitable assumptions on $W$ and regularity conditions on the associated rank-one connection, we prove that $E_\varepsilon$ $\Gamma$-converges to a local interfacial energy defined for suitable laminate-type configurations.