Motion of level sets by mean curvature. III
type: Item
instance of: scholarly article
title: Motion of level sets by mean curvature. III
zbMATH Open document ID: 0768.53003
DOI: 10.1007/BF02921385
author: S. Singh
published in: The Journal of Geometric Analysis
publication date: 1992-06-28T00:00:00Z
review text: 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 \textit{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)[.
Mathematics Subject Classification ID: 53A10
Mathematics Subject Classification ID: 35K55
zbMATH DE Number: 33105
zbMATH Keywords: weak solutions of nonlinear PDE
zbMATH Keywords: Hausdorff measure
zbMATH Keywords: mean curvature flow
reviewed by: Ion Mihai
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf02921385
OpenAlex ID: W3023640078
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description: scientific article; zbMATH DE number 33105