Higher-order phase transitions with line-tension effect

Galvão-Sousa, Bernardo

ESAIM: Control, Optimisation and Calculus of Variations, Tome 17 (2011) no. 3, pp. 603-647.

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 Poincaré, Anal. non linéaire 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

$${ℱ}_{\epsilon }\left(u\right):={\epsilon }^3{\int }_{\Omega }{\left|D^{2}u\right|}^2+\frac{1}{\epsilon }{\int }_{\Omega }W\left(u\right)+{\lambda }_{\epsilon }{\int }_{\partial \Omega }V\left(Tu\right),$$

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 $\epsilon {\lambda }_{\epsilon }^{\frac{2}{3}}\sim 1$.

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