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

We investigate the 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 \[ E_ε(u, Ω) = ∫_Ω( 1/ε W(∇ u) + ε |∇^2 u|^2 ) dx \] for maps u ∈ H²(Ω; R²), 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 ε → 0⁺.