Analysis of the weighted Van der Waals-Cahn-Hilliard model for image segmentation

In the seminal paper The Gradient Theory of Phase Transitions and the Minimal Interface Criterion, 1987, L. Modica ( [13]) proves some conjectures related to the Van der Waals-Cahn-Hilliard theory of phase transitions. This theory intends to overcome the issue of lack of uniqueness of the solution of the initially considered minimization problem - which aims to minimize the total energy of a fluid confined to a bounded container Ω⊂ℝ
ⁿ
and with Gibbs free energy per unit volume, a prescribed function W of the density distribution u - by enforcing that the interface has minimal surface. In that purpose, a family of functionals parameterized by ε > 0 and including a dependency on the density gradient modelling this interfacial energy is introduced, and the asymptotic behavior as ε→0
⁺
of the solutions u
_ε
of the related minimization problem is analyzed through a Γ-convergence result. Motivated by this work, this paper addresses the question of extending this result to the weighted case. It is shown that this new model inherits the fine properties of the original unweighted one with in particular, the validity of the Γ-convergence result, useful for the minimization of weighted perimeter, relevant for image segmentation.