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Level set approach to mean curvature flow in arbitrary codimension. (English) Zbl 0868.35046
The authors develop a level set theory for the mean curvature evolution of surfaces with arbitrary codimension which generalizes the work of L. C. Evans and J. Spruck [J. Differ. Geom. 33, No. 3, 635-681 (1991; Zbl 0726.53029)] and Y.-G. Chen, Y. Giga and S. Goto [J. Differ. Geom. 33, 749-786 (1991; Zbl 0715.35037)] on the codimension one case. The main idea is to surround the evolving surface of codimension \(k\) in \(\mathbb{R}^d\) by a family of hypersurfaces (the level sets of a function) evolving with normal velocity equal to the sum of the \((d-k)\) smallest principal curvatures. This is motivated by the theory of barriers proposed by E. De Giorgi [Barriers, boundaries, motion of manifolds, Lectures held in Pavia, Italy (1994)] to describe mean curvature flow in arbitrary codimensions. The authors show that the level set solutions agree with the classical solutions whenever these exist, and that the varifold solutions of K. A. Brakke [The motion of a surface by its mean curvature, Princeton University Press, Princeton, N.J. (1978; Zbl 0386.53047)] are included in the level set solutions.
Reviewer: J.Urbas (Bonn)

MSC:
35G10 Initial value problems for linear higher-order PDEs
35K25 Higher-order parabolic equations
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
35K55 Nonlinear parabolic equations
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