On the Structure of Phase Transition Maps
Speaker: Nicholas Alikakos, University of Athens
Location: Warren Weaver Hall 1302
Date: Thursday, February 11, 2016, 11 a.m.
The scalar Allen-Cahn equation models coexistence of two phases, and is related to Minimal Surfaces. The 1979 De Giorgi conjecture for the scalar problem has been settled, not so recently, in a series of papers (Ghoussoub and Gui (2d), Ambrosio and Cabre (3d), Savin (up to 8d) and Del Pino, Kowalczyk and Wei (counterexample for 9d and above)).
The vector Allen-Cahn equation models coexistence of three or more phases and is related to Plateau Complexes. These are non-orientable minimal objects with a hierarchical structure. The analog of the De Giorgi question in the vector case is open.
After stating an existence theorem for equivariant solutions under a reflection group, we focus on vector extensions of the Caffarelli-Cordoba Density Estimates. In particular, we establish lower codimension density estimates. These are useful for studying the hierarchical structure of vector solutions.