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 [Ann. Inst. Henri Poincare, Anal. non lineaire 4 (1987) 487–512], and in a different form by Alberti et al. in [Arch. Rational Mech. Anal. 144 (1998) 1–46] 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 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 .