Phase transition with the line-tension effect

We make the connection between the geometric model for capillarity with lin e tension and the Cahn-Hilliard model of two-phase fluids. To this aim we c onsider the energies F-epsilon(u) := epsilon integral(Omega) \Du\(2) + 1/epsilon integral(Omega) W(u) + lambda integral(partial derivative Omega) V(u) where u is a scalar density function and W and V are double-well potentials . We show that the behaviour of F-epsilon in the limit epsilon --> 0 and la mbda --> infinity depends on the limit of epsilon log lambda. If this limit is finite and strictly positive, then the singular limit of the energies F -epsilon leads to a coupled problem of bulk and surface phase transitions, and under certain assumptions agrees with the relaxation of the capillary e nergy with line tension. These results were announced in [ABS1] and [ABS2].