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Motion of level sets by mean curvature. I. (English) Zbl 0726.53029
Let \(g: \mathbb{R}^n\to\mathbb{R}\) be a continuous function which defines an initial hypersurface \(\Gamma_0\) by \(\Gamma_0=\{x\in\mathbb{R}^n;\ g(x)=0\}\). Consider the parabolic PDE \[ u_t=(\delta_{ij}- u_{x_i}u_{x_j}/| Du|^2)u_{x_ix_j}\text{ in } \mathbb{R}^n\times [0,\infty),\quad u=g\text{ on }\mathbb{R}^n\times \{t=0\}. \] One says that each level set of \(u\) evolves according to its mean curvature. S. Osher and J. A. Sethian [see J. Comput. Phys. 79, No. 1, 12–49 (1988; Zbl 0659.65132)] have introduced various techniques to study the above equation numerically.
In the paper under review, the authors introduce the definition of a weak solution of this equation. Their definition agrees with the classical motion by mean curvature. The existence and uniqueness of a weak solution are proved. Define \(\Gamma_t=\{x\in \mathbb{R}^n;\ u(x,t)=0\}\). \(\{\Gamma_t\}_{t>0}\) is said to be the generalized evolution by mean curvature of the original compact set \(\Gamma_0\). Geometric properties of generalized evolution by mean curvature are obtained.
The last section contains examples of pathologies and conjectures.

MSC:
53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions
58J35 Heat and other parabolic equation methods for PDEs on manifolds
35G10 Initial value problems for linear higher-order PDEs
35K25 Higher-order parabolic equations
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