Rigorous Numerics for the Cahn-Hilliard Equation on the Unit Square

Résumé

While the structure of the set of stationary solutions of the Cahn-Hilliard equa tion on one-dimensional domains is completely understood, only partial results are available for two-dimensional base domains. In this pap er, we demonstrate how rigorous computational techniques can b e employed to establish computer assisted existence proofs for equilibria of the Cahn-Hilliard equation on the unit square. Our method is based on results by Mischaikow and Zgliczynski [22], and ´ combines rigorous computations with Conley index techniques. We are able to establish branches of equilibria and, under more restrictive conditions, even the local uniqueness of sp ecific equilibrium solutions. Sample computations for sev eral branches are presented, which illustrate the resulting patterns.