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Optimal rate of convergence to the motion by mean curvature with a driving force. (English) Zbl 1171.35009

The author studies the rate of convergence of hypersurfaces associated with solutions of a singularly perturbed parabolic equation with double well potential \(\Psi(r)=(1-r^2)^2/4\) and forcing term \(g\), which approximates the mean curvature flow with driving force: \[ \begin{cases} u_t^\epsilon -\Delta u^\epsilon + \frac{1}{\epsilon^2}\psi(u^\epsilon)=\frac{c_0}{\epsilon}g, \;\;\mathrm{in} \;(0,T)\times \mathbb{R^N}, \cr u^\epsilon (0,x)=u_0^\epsilon(x), \;\;x \in \mathbb{R^N}.\end{cases} \] Here \(\epsilon\) is a parameter, \(\psi(r)=\Psi'(r)\), \(c_0\) = constant, and \(u_0^\epsilon\) is a bounded, continuous function.
It has been shown that the interface \(\Gamma^{\epsilon}(t)=\{ x\in {\mathbb R}^n: u^{\epsilon}(t,x)=0 \}, 0\leq t \leq T,\) converges to a hypersurface \(\Gamma(t)_{0\leq t \leq T} \) moving by the mean curvature flow as \(\epsilon \downarrow 0\). The author investigates the optimal rate of convergence before singularities form. He shows that the Hausdorff distance satisfies the estimate
\[ sup_{t\in[0,T]} d_H (\Gamma(t),\Gamma^\epsilon(t))\leq L(T) \epsilon^2, \]
where \(L\) is a constant depending on \(T\). This improves current estimates and is optimal in the case of radial symmetry. Techniques in the proof include the construction of sub and super-solutions of the equation, suggested by formal asymptotics.

MSC:

35B25 Singular perturbations in context of PDEs
35K55 Nonlinear parabolic equations
35B40 Asymptotic behavior of solutions to PDEs
35C20 Asymptotic expansions of solutions to PDEs
35K20 Initial-boundary value problems for second-order parabolic equations
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