Sự tiến hóa của một di chuyển hypersurface theo độ cong trung bình của nó đã được xem xét bởi Brakke [1] theo quan điểm hình học của, và bởi Evans, Spruck [3] theo điểm analysic. Bắt đầu từ một Γ0 bề mặt ban đầu trong R n, các bề mặt | TẠP CHÍ PHÁT TRIỂN KH CN TẬP 11 SÓ 06 - 2008 LEVEL SET EVOLUTION WITH SPEED DEPENDING ON MEAN CURVATURE EXISTENCE OF A WEAK SOLUTION Nguyen Chanh Dinh Danang University of Technology Manuscript Received on February 01st 2007 Manuscript Revised April 28th 2008 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 r0 inRn the surfaces rt evolve in time with normal velocity equals to their mean curvature vector. The surfaces rt 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. Key words Mean curvature flow level set methods evolution equations weak solutions 1. INTRODUCTION Let r0 be a smooth hypersurface which is say the smooth connected boundary of a bounded open subset U of Rn n 2 . As time progresses we allow the surface to evolve by moving each point at a velocity equals to n 1 times the mean curvature vector plus some function F at that point. Assuming this evolution is smooth we define thereby for each t 0 a new hypersurface rt. The primary problem is then to study geometric properties of rt t 0 in terms of r0. We will proceed as follows We select some continuous function u0 Rn R so that its level set is r0 that is r0 x e Rn u0 x 0 . Consider the following problem ut s V uxxl F x Vu in Rn X 0 u u w 7 with initial condition u u0 on Rn X t 0 . Now the PDE says
