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

We investigate the $\limsup$ 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 $$E_\varepsilon(u, \Omega) = \int_\Omega \left( \frac{1}{\varepsilon} W(\nabla u) + \varepsilon |\nabla^2 u|^2 \right) dx$$ for maps $u \in H^2(\Omega; \mathbb{R}^2)$, 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 $\varepsilon \to 0^+$.