Plenary – “Geometric Flows and Phase Transitions in Heterogeneous Media” (I. Fonseca)
We present the first, unconditional convergence results for an Allen-Cahn type bi-stable reaction diffusion equation in a periodic medium. Our limiting dynamics are given by an analog for anisotropic mean curvature flow, of the formulation due to Ken Brakke.
As an essential ingredient in the analysis, we obtain an explicit expression for the effective surface tension, which dictates the limiting anisotropic mean curvature. This allows us to demonstrate the regularity and uniform ellipticity of the limiting surface tension.