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

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

Department of Mathematics

University of Maryland, College Park

College Park, MD 20742, USA

rhn@math.umd.edu
Claudio Verdi

Dipartimento di Matematica

Universita di Milano

Milano, 20133, Italy

verdi@paola.mat.unimi.it
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Abstract

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.