Optimal interface error estimates for the mean curvature flow

Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 21 (1994), 193-212.

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

rhn@math.umd.edu

Maurizio Paolini
Dipartimento di Matematica
Universita di Milano
Milano, 20133, Italy

paolini@galileo.dmf.bs.unicatt.it

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

verdi@paola.mat.unimi.it

Abstract

We study a double obstacle problem for a singularly perturbed reaction diffusion equation with a small parameter $\eps>0$. This problem is viewed as an approximation for the classical mean curvature flow. We derive an optimal error estimate of order $\O(\eps^2)$ for the interfaces, which is valid before the onset of singularities. The proof is based on constructing suitable sub and supersolutions, and using a modified distance function that incorporates a shape correction.