Uniqueness and existence of viscosity solutions of generalized mean curvature flow equations. (English) Zbl 0696.35087

Chen, Yun-Gang; Giga, Yoshikazu; Goto, Shun’ichi

J. Differ. Geom. 33, No. 3, 749-786 (1991).

A weak formulation is introduced for motions of a hypersurface including the motion by mean curvature so that one tracks its evolution even after the time when there appear singularities. A unique global weak solution is constructed for an arbitrary intial hypersurface to various motions including motion by mean curvature and its anisotropic version. The hypersurface is regarded as a level set of a function solving a singular degenerate parabolic equation with special invariance called geometric. The theory of viscosity solutions is extended to our singular equations.

Reviewer: Y.Giga

MSC:

35K60	Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations
35D05	Existence of generalized solutions of PDE (MSC2000)
53A10	Minimal surfaces in differential geometry, surfaces with prescribed mean curvature

Keywords:

viscosity solutions