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Motion of level sets by mean curvature. II. (English) Zbl 0776.53005
In the first paper of this series [J. Differ. Geom. 33, No. 3, 635-681 (1991; Zbl 0726.53029)], the authors defined and studied a generalized notion of evolution via mean curvature. In the paper under review, a fully nonlinear uniformly parabolic equation is studied. Let $$\Gamma_ 0$$ be the smooth connected boundary of a bounded open set $$U\subset\mathbb{R}^ n$$ and $g(x)=\begin{cases} \text{dist}(x,\Gamma_ 0), & x\in\mathbb{R}^ n-\overline U\\ -\text{dist} (x,\Gamma_ 0), & x\in U \end{cases}$ be the signed distance function. Fix $$\delta>0$$ such that $$g$$ is smooth in $V=\{x\in\mathbb{R}^ n;-\delta<g(x)<\delta\}$ and put $$Q=V\times(0,t_ 0)$$, $$\Sigma=\partial V\times[0,t_ 0]$$. A geometric motivation leads to the PDE $\begin{cases} v_ t=F(D^ 2v,v) & \text{ in } Q, \\ | Dv|^ 2=1 & \text{ on } \Sigma, \\ v=g & \text{ on } V\times\{t=0\}.\end{cases}$ The authors solve it and prove that the sets $$\Gamma_ t\{x\in V;\;v(x,t)=0\},\;(0\leq t\leq t_ 0)$$ are smooth hypersurfaces evolving by mean curvature starting from $$\Gamma_ 0$$. As an application, a new and elementary proof of short time existence for the classical motion of a smooth hypersurface evolving according to its mean curvature is given.

##### MSC:
 53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature 35K55 Nonlinear parabolic equations
##### Keywords:
mean curvature flow; uniformly parabolic equation
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##### References:
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