On the structure of phase transition maps : Density estimates and applications.

Jeudi 11 juin 2015 14:15-15:15 - Nicholas Alikakos - University of Athens

Résumé : The scalar Allen-Cahn (or Ginzburg-Landau) equation is related to Minimal Surfaces and Minimal Graphs via the level sets of its solutions. The 1979 De Giorgi conjecture for the scalar problem has been settled relatively recently in a series of papers by Ghoussoub and Gui (2d), Ambrosio and Cabre (3d), Savin (up to 8th dimension) and Del Pino, Kowlaczyk and Wei (counterexample for 9 dim. and above). The vector Allen-Cahn is related to Plateau Complexes. These are non-orientable minimal objects with a hierarchical structure. After explaining these relationships we focus on vector extensions of the Caffarelli-Cordoba Density Estimates (L. Caffarelli and A. Cordoba, Comm. Pure and Applied Mathematics Volume 48, Issue 1, pages 1–12, January 1995). In particular we establish lower co-dimension density estimates. These are useful for studying the hierarchical structure of certain entire vector solutions. We also give applications to minimal solutions (lower bounds, Liouville theorems)

Lieu : bât. 425 - 113-115

On the structure of phase transition maps : Density estimates and applications.  Version PDF