Gamma Convergence and Applications to Phase Transitions

1. Liquid–Liquid Phase Transitions Consider a fluid confined into a container Ω ⊂ R . Assume that the total mass of the fluid is m, so that admissible density distributions u : Ω→ R satisfy the constraint ∫ Ω u(x) dx = m. The total energy is given by the functional u 7→ ∫ Ω W (u(x)) dx, where W : R→ [0,∞) is the energy per unit volume. Assume that W supports two phases a < b, that is, W is a double-well potential, with {t ∈ R : W (t) = 0} = {a, b}. Then any density distribution u that renders the body stable in the sense of Gibbs is a minimizer of the following problem