Uniqueness and existence of viscosity solutions of generalized mean curvature flow equations.

*(English)*Zbl 0696.35087A 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 |