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Optimal interface error estimates for the mean curvature flow. (English) Zbl 0886.35079
The singularly perturbed reaction-diffusion equation with a smooth double equal well potential provides an approximation for an interface evolving by mean curvature. Such an equation was introduced by S. M. Allen and J. W. Cahn, showing its relevance in phase transitions. That equation can be interpreted as a variational approach to the mean curvature flow. It is important for computational purposes, and of theoretical interest as well, to examine the effect of substituting the smooth potential by \[ \Psi(s)= \begin{cases} 1-s^2, &\quad\text{if }s\in [-1,1],\\ +\infty, &\quad\text{if }s\not\in [- 1,1]. \end{cases} \] The goal of this paper is to prove an optimal error estimate, valid before the onset of singularities, for the distance between the mean curvature flow and the approximate interface. This estimate is optimal and improves the results obtained by X. Chen [J. Differ. Equations 96, No. 1, 116-141 (1992; Zbl 0765.35024)] for the regular potential and by X. Chen and C. M. Elliott [Proc. R. Soc. Lond., Ser. A 444, No. 1922, 429-445 (1994; Zbl 0814.35044)] for the double obstacle potential.
Reviewer: V.Arnăutu (Iaşi)

MSC:
35K57 Reaction-diffusion equations
31A35 Connections of harmonic functions with differential equations in two dimensions
35B25 Singular perturbations in context of PDEs
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