A note on the recovery sequence in the double gradient model for phase transitions

We investigate the $\mathrm{lim\hspace{0.17em}sup}$ inequality in the double gradient model for phase transitions governed by a Modica–Mortola functional with a double-well potential in two dimensions. Specifically, we consider energy functionals of the form $$E_{\epsilon }(u,\mathrm{\Omega })={\int }_{\mathrm{\Omega }}(\frac{1}{\epsilon }W(\mathrm{\nabla }u)+\epsilon \mathrm{\mid }{\mathrm{\nabla }}^{2}u{\mathrm{\mid }}^2)dx$$ for maps $u\in H^2(\mathrm{\Omega };{\mathbb{R}}^2)$, where $W$ vanishes only at two wells. Assuming a bound on the optimal profile constant – namely the cell problem on the unit cube – in terms of the geodesic distance between the two wells, we characterise the limiting interfacial energy via periodic recovery sequences as $\epsilon \to 0^{+}$.