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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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

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

First available in Project Euclid: 20 May 2013

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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


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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