Geometry and Dynamics of Phase Transitions in Periodic Media
In this talk we discuss recent progress on homogenization of the Allen-Cahn equation in a periodic medium to Brakke's formulation of anisotropic mean curvature flow. The starting point of our work is the recent Gamma-convergence result of Cristoferi-Fonseca-Hagerty-Popovici, and we combine it with the Sandier-Serfaty scheme for the convergence of gradient flows. Along the way, our analysis also provides resolution to a long-standing open problem in geometry: what do distance functions to planes look like, in periodic Riemannian metrics on $R^n$ that are conformal to the Euclidean one? The corresponding question about distance functions to points, has a long history that goes back to Gromov and Burago. This represents joint work with Irene Fonseca (CMU), Rustum Choksi and Jessica Lin (McGill).
Password: “arizona” (all lower case)