Optimal rate of convergence to the motion by mean curvature with a driving force

Abstract

We consider a singularly perturbed parabolic problem with a small parameter $ \varepsilon>0 $. This problem can be regarded as an approximation of the motion of a hypersurface by its mean curvature with a driving force. In this paper we derive a rate of convergence of an order $ \varepsilon^2 $ for the motion of a smooth and compact hypersurface by its mean curvature with a driving force. We also consider the special case of a circle evolving by its curvature and show that our rate is optimal.