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

Abstract

We investigate the $\mathrm{limsup}$ 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 \[ E_\varepsilon(u, Ω) = \int_Ω\left( \frac{1}{\varepsilon} W(\nabla u) + \varepsilon |\nabla^2 u|^2 \right) dx \] for maps $u\in H^2(\hat{I}\text{c◯};{\mathbb{R}}^2)$, 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 $\epsilon \to 0^{+}$.

28 pages, 2 figures