Convergence of double obstacle problems to the generalized geometric motion of fronts

The connection between the generalized geometric motion of interfaces, interpreted in the viscosity sense, and a singularly perturbed parabolic problem with double obstacle +/- 1 and small parameter epsilon is examined. This approach retains the local character of the limit problem, because the noncoincidence set, where all the action takes place, is a thin transition layer of thickness O(epsilon) irrespective of the forcing term. Zero-level sets are shown to converge past singularities to the generalized motion by mean curvature with forcing, provided no fattening occurs. If the underlying viscosity solution satisfies a nondegeneracy property, namely, its gradient does not vanish, then our results yield interface error estimates and layer width estimates of order O(epsilon). The proofs are based on constructing viscosity subsolutions and supersolutions to the double obstacle problem in terms of the signed distance function and approximate traveling waves dictated by formal asymptotics.