On the Stationary Cahn--Hilliard Equation: Bubble Solutions

Stationary Solutions for the Cahn-hilliard Equation

We study the Cahn-Hilliard equation in a bounded domain without any symmetry assumptions. We assume that the mean curvature of the boundary has a nongenerate critical point. Then we show that there exists a spike-like stationary solution whose global maximum lies on the boundary. Our method is based on Lyapunov-Schmidt reduction and the Brouwer fixed-point theorem. Résumé. Nous étudions l’équat...

On the Stationary Cahn-hilliard Equation: Interior Spike Solutions

We study solutions of the stationary Cahn-Hilliard equation in a bounded smooth domain which have a spike in the interior. We show that a large class of interior points (the “nondegenerate peak” points) have the following property: there exist such solutions whose spike lies close to a given nondegenerate peak point. Our construction uses among others the methods of viscosity solution, weak con...

Spectral Analysis of Stationary Solutions of the Cahn–Hilliard Equation

For the Cahn–Hilliard equation on R, there are precisely three types of bounded non-constant stationary solutions: periodic solutions, pulse-type reversal solutions, and monotonic transition waves. We study the spectrum of the linear operator obtained upon linearization about each of these waves, establishing linear stability for all transition waves, linear instability for all reversal waves, ...

The Cahn-hilliard Equation

1. Steady states. There are many two component systems in which phase separation can be induced by rapidly cooling the system. Thus, if a two component system, which is spatially uniform at temperature T1, is rapidly cooled to a second sufficiently lower temperature T2, then the cooled system will separate into regions of higher and lower concentration. A phenomenological description of the beh...

The Cahn–Hilliard Equation