A sharp estimate of the extinction time for the mean curvature flow (Q2455436)

Let \((\Omega_t)_{t\geq 0}\) be a family of bounded open sets in \({\mathbb R}^n\) \((n\geq 2)\). If the boundary \(\Gamma_t=\partial\Omega_t\) is always a smooth \((n-1)\)-dimensional hypersurface, it is said to be moving by mean curvature, if \(V=H\) on \(\Gamma _t\), where \(V(x,t)\) and \(H(x,t)\) denote, respectively, the inward normal velocity and \((n-1)\) times the mean curvature of \(\Gamma_t\) at a point \(x\in\Gamma_t\). It is well known that \(\Gamma_t\) shrinks to a point in a finite time \(t^*\) defined as \[ t^*=t^*(\Gamma_0)=\inf\{t\geq 0:\Omega_t= \emptyset\} \] and called the extinction time. The main theorem of the article is as follows: Let \(\Omega\) be a smooth mean convex (i.e., of nonnegative mean curvature) bounded open set in \({\mathbb R}^n\) and let \(\Gamma_0=\partial\Omega\). If \(\Gamma_t\) denotes the evolution of \(\Gamma_0\) by mean curvature and \(\Gamma_t=\partial\Omega_t\) where \(\Omega_t\) is a smooth mean convex bounded open set in \({\mathbb R}^n\), then: \[ 0\leq t^*\leq\frac{1}{2(n-1)}\left[\frac{ \lambda^n (\Omega)}{\omega_n}\right]^\frac{2}{n}, \] where \(\lambda^n(\Omega)\) stands for the Lebesgue measure of \(\Omega\) and \(\omega_n\) for that of the unit ball in \({\mathbb R}^n\). The same result holds if the open set \(\Omega \subseteq {\mathbb R}^n\) is convex and bounded, or if it is bounded and \(\partial\Omega\) is the hypersurface obtained by rotating a function \(f:[a,b]\rightarrow {\mathbb R}\) with \(f(a)=f(b)=0\) having only one stationary point in \([a,b]\), as it is proven later on. In connection with the main theorem the authors also provide a pointwise comparison result between the solution of a nonlinear degenerate elliptic Dirichlet problem and the solution of a suitable symmetrized problem by making use of the so-called symmetrization techniques developed by \textit{G. Talenti} [Ann. Sc. Norm. Super. Pisa, Cl. Sci. (4) 8, 183--210 (1981; Zbl 0467.35044); Boll. Unione. Mat. Ital. (6) B 4, 917--949 (1985; Zbl 0602.35025)], and \textit{N. S. Trudinger} [Ann. Inst. Henri Poincaré, Anal. Non Linéaire 11, No. 4, 411--425 (1994; Zbl 0859.52001); J. Reine Angew. Math. 488, 203--220 (1997; Zbl 0883.52006)]. Finally, a numerical example is presented to compare the new estimate for \(t^*\) with the previous ones in the literature.