Singular limits of scalar Ginzburg-Landau equations with multiple-well potentials

Abstract: "We characterize the limiting behavior of scalar phase-field equations with infinitely many potential wells as the density of potential wells tends to infinity. An example of such a family of equations is u [epsilon over t] = [delta]u[superscript epsilon] - 1/[epsilon superscript 1 + α]W(́u[superscript epsilon]/[epsilon superscript 1-α]), where W is a periodic function. We prove that solutions of the above equation converge to solutions of the Mean Curvature PDE for a range of positive values of the parameter α, and we also determine the limiting equation when α = 0. We show that our techniques can be modified to apply to fully nonlinear equations and to other classes of infinite-well equations. We discuss some applications to questions of interaction between wave fronts in dynamic phase transitions."