Minimizers for the Cahn-Hilliard energy functional with the Flory-Huggins potential under strong anchoring conditions

TL;DR

This paper investigates the minimizers of the Cahn-Hilliard energy with Flory-Huggins potential under strong anchoring, revealing bifurcation phenomena and supported by numerical simulations.

Contribution

It provides a theoretical and numerical analysis of boundary-mediated bifurcations in the energy minimizers under strong anchoring conditions.

Findings

Bifurcation phenomena depend on boundary conditions, transition layer thickness, and temperature.

Numerical simulations confirm theoretical predictions about minimizer behavior.

Transition layer thickness and temperature significantly influence energy minimizers.

Abstract

In this paper, we theoretically and numerically study the minimizers for the Cahn-Hilliard energy with the Flory-Huggins potential under the strong anchoring condition, i.e., the Dirichlet boundary condition. We reveal bifurcation phenomena mediated by the boundary condition, the transition layer thickness, and the temperature of the system. Numerical simulations are also presented to approximate the minimizers of this energy by solving a gradient-flow equation, namely the Allen-Cahn equation constrained with strong anchoring conditions and random initial data. The effects of varying the transition layer thickness and temperature are presented to confirm the theoretical analysis.