Résumé
We investigate the singular limit, as $\ep \to 0$, of the Allen-Cahn equation $\ue _t=\Delta \ue+\ep^{-2}f(\ue)$, with $f$ a balanced bistable nonlinearity. We consider rather general initial data $u_0$ that is independent of $\ep$. It is known that this equation converges to the generalized motion by mean curvature --- in the sense of viscosity solutions--- defined by Evans, Spruck and Chen, Giga, Goto. However the convergence rate has not been known. We prove that the transition layers of the solutions $u^\ep$ are sandwiched between two sharp \lq\lq interfaces" moving by mean curvature, provided that these \lq\lq interfaces" sandwich at $t=0$ an $\mathcal O(\ep|\ln\ep|)$ neighborhood of the initial layer. In some special cases, which allow both {\it extinction} and {\it pinches off} phenomenon, this enables to obtain an $\mathcal O(\ep|\ln \ep|)$ estimate of the location and the {\it thickness measured in space-time} of the transition layers. A result on the {\it regularity of the generalized motion by mean curvature} is also provided in the Appendix.