Differential and Integral Equations

Quasi-optimal error estimates for the mean curvature flow with a forcing term

G. Bellettini and M. Paolini

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Abstract

We study a singularly perturbed reaction-diffusion equation with a small parameter $\epsilon>0$. This problem is viewed as an approximation of the evolution of an interface by its mean curvature with a forcing term. We derive a quasi-optimal error estimate of order $\mathcal{O}(\epsilon^2| \rm{log}\, \epsilon|^2)$ for the interfaces, which is valid before the onset of singularities, by constructing suitable sub and super solutions. The proof relies on the behavior at infinity of functions appearing in the truncated asymptotic expansion, and by using a modified distance function combined with a vertical shift.

Article information

Source
Differential Integral Equations, Volume 8, Number 4 (1995), 735-752.

Dates
First available in Project Euclid: 20 May 2013

Permanent link to this document
https://projecteuclid.org/euclid.die/1369055609

Mathematical Reviews number (MathSciNet)
MR1306590

Zentralblatt MATH identifier
0820.49019

Subjects
Primary: 35B25: Singular perturbations
Secondary: 35K57: Reaction-diffusion equations 58E12: Applications to minimal surfaces (problems in two independent variables) [See also 49Q05]

Citation

Bellettini, G.; Paolini, M. Quasi-optimal error estimates for the mean curvature flow with a forcing term. Differential Integral Equations 8 (1995), no. 4, 735--752. https://projecteuclid.org/euclid.die/1369055609


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