Sharp-interface limit for weighted Van der Waals-Cahn-Hilliard phase transitions.

This thesis focuses on a weighted formulation of the Van-der-Waals-Cahn-Hilliard model for liquid-liquid phase transitions. We focus on the setting of degenerate weights, namely functions ω that are not bounded by positive constants from below, but can, instead, attain any non-negative value. Our main result is that minimizers of the weighted Van-derWaals-Cahn-Hilliard functionals converge, as the phase-transition parameter vanishes, to minimizers of a sharp-interface perimeter functional.Similar to the classical model, the convergence result is obtained using Γ-convergence. This means that we proceed in three steps. First, we show compactness of the functional domain, second we establish the existence of a lower bound, third, we prove its optimality.The main difference induced by the weaker assumption ω > 0 is that the functionals are finite on a bigger subspace than in other classical and weighted formulations. For this reason, an adaptation of both the Liminf and Limsup proof strategies is required.