We study a non local evolution and define the interface in terms of a local equilibrium condition. We prove that in a diffusive scaling limit the local equilibrium condition propagates in time thus defining an interface evolution which is given by a motion by mean curvature. The analysis extend through all times before the appearance of singularities.
"Fine structure of the interface motion." Differential Integral Equations 12 (2) 207 - 259, 1999. https://doi.org/10.57262/die/1367265630