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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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[1] Allard, W. On the first variation of a varifold. Ann. Math.95, 417–491 (1972). · Zbl 0252.49028
[2] Brakke, K. A. The Motion of a Surface by its Mean Curvature. Princeton, NJ: Princeton University Press 1978. · Zbl 0386.53047
[3] Burago, Y. D., and Zalgaller, V. A. Geometric Inequalities. New York: Springer-Verlag 1988. · Zbl 0633.53002
[4] Chen, Y.-G., Giga, Y., and Goto, S. Uniqueness and existence of viscosity solutions of generalized mean curvature flow equations. Preprint. · Zbl 0696.35087
[5] Ecker, E., and Huisken, G. Mean curvature evolution of entire graphs.Ann. Math. 130, 453–471 (1989). · Zbl 0696.53036
[6] Evans, L. C., and Spruck, J. Motion of level sets by mean curvature I. J. Diff. Geom.33, 635–681 (1991). · Zbl 0726.53029
[7] Evans, L. C., and Spruck, J. Motion of level sets by mean curvature II. Trans. AMS. To appear. · Zbl 0776.53005
[8] Federer, H. Geometric Measure Theory. New York: Springer-Verlag 1969. · Zbl 0176.00801
[9] Gilbarg, D., and Trudinger, N. S. Elliptic Partial Differential Equations of Second Order. New York: Springer-Verlag 1983. · Zbl 0562.35001
[10] Huisken, G. Flow by mean curvature of convex surfaces into spheres. J. Diff. Geom.20, 237–266 (1984). · Zbl 0556.53001
[11] Korevaar, N. J. An easy proof of the interior gradient bound for solutions to the prescribed mean curvature equation. Proc. Symposia Pure Math.45 (1986). · Zbl 0599.35046
[12] Ladyzhenskaja, O. A., Solonnikov, and Ural’tseva, N. N. Linear and Quasi-Linear Equations of Parabolic Type. Providence, RI: American Mathematical Society 1968.
[13] Lieberman, G. M. The first initial-boundary value problem for quasilinear second order parabolic equations. Ann. Scuola Norm. Sup. Pisa13, 347–387 (1986). · Zbl 0655.35047
[14] Michael, J. H., and Simon, L. M. Sovolev and mean value inequalities on generalized submanifolds ofR n . Comm. Pure Appl. Math.26, 361–379 (1973). · Zbl 0256.53006
[15] Osher, S., and Sethian, J. A. Fronts propagating with curvature dependent speed: Algorithms based on Hamilton-Jacobi formulations. J. Comput. Phys.79, 12–49 (1988). · Zbl 0659.65132
[16] Sethian, J. A. Numerical algorithms for propagating interfaces: Hamilton-Jacobi equations and conservation laws. J. Diff. Geom.31, 131–161 (1990). · Zbl 0691.65082
[17] Mete Soner, H. Motion of a set by the curvature of its boundary. Preprint. · Zbl 0769.35070
[18] Stein, E. M. Singular Intervals and Differentiability Properties of Functions. Princeton, NJ: Princeton University Press 1970. · Zbl 0207.13501
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