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.


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