Convergence of Double Obstacle Problems to the Generalized Geometric Motion of Fronts
Ricardo H. Nochetto, C. Verdi · SIAM Journal on Mathematical Analysis · 1995
The connection between the generalized geometric motion of interfaces, interpreted in the viscosity sense, and a singularly perturbed parabolic problem with double obstacle $ \pm 1$ and small parameter $\varepsilon $ 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 $O(\varepsilon )$ 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 $O(\varepsilon )$. The proofs are based on constructing viscosity subsolutions and supersolutions to the double obstacles problem in terms of the signed distance function and approximate traveling waves dictated by formal asymptotics.