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

SIAM J. Math. Anal., 26 (1995), 1514-1526.

Ricardo H. Nochetto
Department of Mathematics
University of Maryland, College Park
College Park, MD 20742, USA

Claudio Verdi
Dipartimento di Matematica
Universita di Milano
Milano, 20133, Italy


The connection between generalized geometric motion of interfaces, interpreted in the viscosity sense, and a singularly perturbed parabolic problem with double obstacle $\pm 1$ and small parameter $\eps$ 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 $\O(\eps)$ 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. The proof is based on constructing sub and supersolutions to the double obstacle problem in terms of the signed distance function and approximate traveling waves dictated by formal asymptotics. If the underlying viscosity solution satisfies a nondegeneracy property, namely its gradient does not vanish, then our results yield a linear rate of convergence $\O(\eps)$ for interfaces.