Higher-order phase transitions with line-tension effect

The behavior of energy minimizers at the boundary of the domain is of great importance in the Van de Waals-Cahn-Hilliard theory for fluid-fluid phase transitions, since it describes the effect of the container walls on the configuration of the liquid. This problem, also known as the liquid-drop problem, was studied by Modica in [21], and in a different form by Alberti, Bouchitte, and Seppecher in [2] for a first-order perturbation model. This work shows that using a second-order perturbation Cahn-Hilliard-type model, the boundary layer is intrinsically connected with the transition layer in the interior of the domain. Precisely, considering the energies $$ \mathcal{F}_{\varepsilon}(u) := \varepsilon^{3} \int_{\Omega} |D^{2}u|^{2} + \frac{1}{\varepsilon} \int_{\Omega} W (u) + \lambda_{\varepsilon} \int_{\partial \Omega} V(Tu), $$ where $u$ is a scalar density function and $W$ and $V$ are double-well potentials, the exact scaling law is identified in the critical regime, when $\varepsilon \lambda_{\varepsilon}^{{2/3}} \sim 1$.