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Motion of hypersurfaces and geometric equations. (English) Zbl 0729.53010
J. Math. Soc. Japan (to appear).
This paper introduces a simple algebraic condition which we call strongly geometric so that it characterizes an evolution equation with the property that each level surface of a solution moves under the same law with the speed depending on the normal and curvature tensors of the surface. If an evolution equation is geometric and degenerate parabolic, it is shown that the equation is strongly geometric. It turns out that a level surface approach by Y.-G. Chen and the authors applies to a wide class of evolution equations of hypersurfaces. For example the approach yields a unique global weak solution for an arbitrary initial surface provided that the equation is degenerate parabolic and the speed of the motion grows linearly in curvature tensors.
Reviewer: Y.Giga (Sapporo)

MSC:
53A07 Higher-dimensional and -codimensional surfaces in Euclidean and related \(n\)-spaces
35G25 Initial value problems for nonlinear higher-order PDEs