Solid-solid phase transitions with space-dependent wells

We investigate a second-order Modica-Mortola functional of the form $$E_{\epsilon }[u]:={\int }_{\mathrm{\Omega }}\frac{1}{\epsilon }W(x,\mathrm{\nabla }u)+\epsilon \mathrm{\mid }{\mathrm{\nabla }}^{2}u{\mathrm{\mid }}^{2}dx,$$ 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 $E_{\epsilon }$ $\mathrm{\Gamma }$-converges to a local interfacial energy defined for suitable laminate-type configurations.