Phase Transition with the Line‐Tension Effect

Abstract.

We make the connection between the geometric model for capillarity with line tension and the Cahn‐Hilliard model of two‐phase fluids. To this aim we consider the energies$F_\eps(u):=\eps\int_\O |Du|^2 + {1\over\eps} \int_\O W(u) + \lambda\int_{\bd\O} V(u)$ where u is a scalar density function and W and V are double‐well potentials. We show that the behaviour of F _ε in the limit ε→0 and λ→∞ depends on the limit of ε log λ. If this limit is finite and strictly positive, then the singular limit of the energies F _ε leads to a coupled problem of bulk and surface phase transitions, and under certain assumptions agrees with the relaxation of the capillary energy with line tension. These results were announced in [ABS1] and [ABS2].