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Motion of level sets by mean curvature. III. (English) Zbl 0768.53003
The authors initiated the investigation of the generalized motion of sets via mean curvature by the level set method [see J. Differ. Geom. 33, 635- 681 (1991; Zbl 0726.53029)]. Given a smooth hypersurface $$\Gamma_ 0$$, a smooth function $$g: \mathbb{R}^ n \to \mathbb{R}$$, with $$\Gamma_ 0 = \{x \in \mathbb{R}^ n;\;g(x) = 0\}$$ is selected. Next, the weak solution $$u: \mathbb{R}^ n \times [0,\infty) \to \mathbb{R}$$ of the nonlinear PDE $\begin{cases} u_ t = \left(\delta_{ij} - {u_{x_ i}u_{x_ j}\over | Du|^ 2}\right)u_{x_ ix_ j}& \text{in $$\mathbb{R}^ n \times (0,\infty)$$}\\u = g & \text{on $$\mathbb{R}^ n \times \{t = 0\}$$}\end{cases}$ defines the generalized evolution by mean curvature $$\{\Gamma_ t\}_{t \geq 0}$$, by $$\Gamma_ t = \{x \in \mathbb{R}^ n; u(x,t) = 0\},\quad t \geq 0.$$ Let $$H^{n-1}$$ denote the $$(n-1)$$-dimensional Hausdorff measure in $$\mathbb{R}^ n$$. It is shown that for a.e. $$\gamma \in \mathbb{R}$$, the level sets $$\Gamma^ \gamma_ t = \{x \in \mathbb{R}^ n; u(x,t) = \gamma\}$$ are countably $$H^{n-1}$$-rectifiable for a.e. $$t > 0$$. An analog of K. A. Brakke’s “clearing out” lemma [The motion of a surface by its mean curvature (Princeton, NJ, 1978; Zbl 0386.53047)] is proved and applications are given. In the last section, the authors establish the smoothness of the generalized mean curvature flow wherever the motion can be described locally in space and time as the graph of a continuous function.
[For Part II, cf. Trans. Am. Math. Soc. 330, 321-332 (1992)[.

##### MSC:
 53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature 35K55 Nonlinear parabolic equations
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