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.
"Optimal rate of convergence to the motion by mean curvature with a driving force." Adv. Differential Equations 12 (5) 481 - 514, 2007. https://doi.org/10.57262/ade/1367241434