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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.

53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature
35K55 Nonlinear parabolic equations
Full Text: DOI
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