Chuyển động của tập mức với vận tốc phụ thuộc vào độ cong trung bình: Sự tồn tại nghiệm yếu

Bài viết tiến hành cho phép bề mặt phát triển bằng cách di chuyển từng mặt điểm tại một vận tốc bằng (n -1) lần vectơ độ cong trung bình cộng với một số hàm F tại đó điểm. | TAÏP CHÍ PHAÙT TRIEÅN KH&CN, TAÄP 11, SOÁ 06 - 2008 LEVEL SET EVOLUTION WITH SPEED DEPENDING ON MEAN CURVATURE: EXISTENCE OF A WEAK SOLUTION Nguyen Chanh Dinh Danang University of Technology 1. INTRODUCTION Let Γ0 be a smooth hypersurface which is, say, the smooth connected boundary of a bounded open subset U of R n , 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 Γt . The primary problem is then to study geometric properties of {Γt }t >0 in terms of Γ0 . We will proceed as follows: We select some continuous function u 0 : R n → R so that its level set is Γ0 , that is { } Γ0 = x ∈ R n | u0 ( x) = 0 . Consider the following problem ux ux ut = δ ij − i 2j ∇u u − F ( x) ∇u xi x j in R n × (0, ∞), () with initial condition u = u0 on R n × {t = 0}. () Now the PDE () says that each level set of u evolves according to its mean curvature with forcing term F, at least in regions where u is smooth and its spatial gradient ∇u does not vanish. Similarly, we then define { } Γt := x ∈ R n | u ( x, t ) = 0 () for each time t > 0 . We will show that there is a weak solution of equation () satisfying condition () in the weak sense. AND ELEMENTARY PROPERTIES OF WEAK SOLUTIONS In this section we concern with the definition and some properties of weak solutions of mean curvature evolution PDE (). For this suppose temporarily that u = u ( x, t ) is a smooth function whose spatial gradient ∇u := (u x1 ,., u xn ) does not vanish in some open region Ω of R × (0, ∞) . Assume further that each level set n { } Γt = x ∈ R n | u ( x, t ) = 0 (t ≥ 0) () of u smoothly evolves according to its mean curvature and function F, as described in Section I. {Γ } Let υ = υ ( x, t ) be a smooth unit normal .