Advances in Differential Equations

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

Katsuyuki Ishii

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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.

Article information

Source
Adv. Differential Equations, Volume 12, Number 5 (2007), 481-514.

Dates
First available in Project Euclid: 29 April 2013

Permanent link to this document
https://projecteuclid.org/euclid.ade/1367241434

Mathematical Reviews number (MathSciNet)
MR2321563

Zentralblatt MATH identifier
1171.35009

Subjects
Primary: 35K55: Nonlinear parabolic equations
Secondary: 35B25: Singular perturbations 53C44: Geometric evolution equations (mean curvature flow, Ricci flow, etc.)

Citation

Ishii, Katsuyuki. Optimal rate of convergence to the motion by mean curvature with a driving force. Adv. Differential Equations 12 (2007), no. 5, 481--514. https://projecteuclid.org/euclid.ade/1367241434


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