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We study front propagation problems for forced mean curvature flows and theirphase field variants that take place in stratified media, i.e., heterogeneousmedia whose characteristics do not vary in one direction. We consider phasechange fronts in infinite cylinders whose axis coincides with the symmetry axisof the medium. Using the recently developed variational approaches, we providea convergence result relating asymptotic in time front propagation in thediffuse interface case to that in the sharp interface case, for suitablybalanced nonlinearities of Allen-Cahn type. The result is established by usingarguments in the spirit of $\Gamma$-convergence, to obtain a correspondencebetween the minimizers of an exponentially weighted Ginzburg-Landau typefunctional and the minimizers of an exponentially weighted area typefunctional. These minimizers yield the fastest traveling waves invading a givenstable equilibrium in the respective models and determine the asymptoticpropagation speeds for front-like initial data. We further show thatgenerically these fronts are the exponentially stable global attractors forthis kind of initial data and give sufficient conditions under which completephase change occurs via the formation of the considered fronts.

Front propagation in geometric and phase field models of stratified media