LEVEL SET EVOLUTION WITH SPEED DEPENDING ON MEAN CURVATURE: EXISTENCE OF A WEAK SOLUTION

Abstract

Evolution of a hypersurface moving according to its mean curvature has been considered by Brakke [1] under the geometric point of view, and by Evans, Spruck [3] under the analysic point of view. Starting from an initial surface Γ0 in R n , the surfaces Γt evolve in time with normal velocity equals to their mean curvature vector. The surfaces Γt are then determined by finding the zero level sets of a Lipchitz continuous function which is a weak solution of an evolution equation. The evolution of hypersurface by a deposition process via a level set approach has also been concerned by Dinh, Hoppe [4]. In this paper, we deal with the level set surface evolution with speed depending on mean curvature. The velocity of the motion is composed by mean curvature and a forcing term. We will derive an equation for the evolution containing the surfaces as the zero level sets of its solution. An existence result will be given.