Pattern formation of the stationary Cahn-Hilliard model

Abstract

We investigate critical points of the free energy Eε(u) of the Cahn–Hilliard model over the unit square under the constraint of a mean value ü. We show that for any fixed value ü in the so-called spinodal region and to any mode of an infinite class, there are critical points of Eε(u) having the characteristic symmetries of that mode provided ε > 0 is small enough. As ε tends to zero, these critical points have singular limits forming characteristic patterns for each mode. Furthermore, any singular limit is a stable critical point of E0(u)). Our method consists of a global bifurcation analysis of critical points of the energy Eε(u) where the bifurcation parameter is the mean value ü.