I will discuss periodic homogenization for the Allen-Cahn equation with Neumann boundary conditions. It is well-known that rescaled solutions of the (homogeneous) Allen-Cahn equation converge to generalized solutions of mean-curvature flow. Using a variational approach, I will show that under suitable hypotheses, a similar result holds true for Allen-Cahn equations with periodic reaction terms. An important step of the argument relies on identifying solutions of an associated Eikonal equation. This talk is based on joint work with Rustum Choksi, Irene Fonseca, and Raghavendra Venkatraman.
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