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

where Vu is the (spatial) gradiant of u. Here VM/|VW| is a unit normal to a level surface of u, so div(Vw/|Vw|) is its mean curvature unless Vu vanishes on the surface. Since ut/\Vu\ is a normal velocity of the level surface, (1.3) implies that a level surface of solution u of (1.3) moves by its mean curvature unless Vu vanishes on the surface. We thus call (1.3) the mean curvature flow equation in this paper. The motion of a closed (hyper)surface in R by its mean curvature has been studied by many authors [1], [3], [4], [8], [10], [12], [14], [15]. Such a motion is also important in the singular perturbation theory related to

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