Báo cáo nghiên cứu khoa học: " LEVEL SET EVOLUTION WITH SPEED DEPENDING ON MEAN CURVATURE: EXISTENCE OF A WEAK SOLUTION" pps

TẠP CHÍ PHÁT TRIỂN KH&CN, TẬP 11, SỐ 06 - 2008 Trang 13 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 01 st , 2007, Manuscript Revised April 28 th , 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 0 Γ in n R , the surfaces t Γ evolve in time with normal velocity equals to their mean curvature vector. The surfaces t Γ 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 0 Γ be a smooth hypersurface which is, say, the smooth connected boundary of a bounded open subset U of n R , 2≥n . As time progresses we allow the surface to evolve by moving each point at a velocity equals to )1( − n times the mean curvature vector plus some function F at that point. Assuming this evolution is smooth, we define thereby for each 0>t a new hypersurface t Γ . The primary problem is then to study geometric properties of {} 0> Γ t t in terms of 0 Γ . We will proceed as follows: We select some continuous function RRu n →: 0 so that its level set is 0 Γ , that is { } 0)(| 00 =∈=Γ xuRx n . Consider the following problem uxFu u uu u ji ji xx xx ijt ∇− ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∇ −= )( 2 δ in ),,0( ∞× n R (1.1) with initial condition 0 uu = on {} .0=× tR n (1.2) Now the PDE (1.1) 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 { } 0),(|: =∈=Γ txuRx n t (1.3) Science & Technology Development, Vol 11, No.06 - 2008 Trang 14 for each time 0>t . We will show that there is a weak solution of equation (1.1) satisfying condition (1.2) in the weak sense. 2.DEFINITION 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 (1.1). For this suppose temporarily that ),( txuu = is a smooth function whose spatial gradient ), ,(: 1 n xx uuu = ∇ does not vanish in some open region Ω of ),0( ∞× n R . Assume further that each level set { } )0(0),(| ≥=∈=Γ ttxuRx n t (2.1) of u smoothly evolves according to its mean curvature and function F, as described in Section I. Let ),( tx υ υ = be a smooth unit normal vector field to { } 0≥ Γ t t in Ω , and F=F(x) be a continuously differentiable function on n R . Then υυ )( 1 1 div n − − is the mean curvature vector field. Thus, if we fix ,,0 Ω ∩ Γ ∈ ≥ t xt the point x evolves according to the differential equation [] ⎩ ⎨ ⎧ = +−= .)( )),(())(()),(()( xtx ssxsxFssxdivx υυυ & (2.2) These equations say that each level set t Γ of u evolves along normal vector direction with velocity equal to its mean curvature plus function F. As )()( tssx s ≥ Γ ∈ , we have 0)),(( =ssxu , and so [] ).),(()),(()),(())(()),(()()()),((0 ssxussxssxusxFssxdivussxu ds d t +⋅∇+⋅∇−== υυυ Setting t s = , we discover )).,(),()((),)(()),(),((),( txtxuxFtxdivtxtxutxu t υ υ υ ⋅ ∇ − ⋅∇= Choosing u u ∇ ∇ =: υ it follows that uFu u uu uF u u divuu ji ji xx xx ijt ∇− ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∇ −=∇− ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∇ ∇ ∇= 2 δ at ),( tx . (2.3) 2.1. Weak solutions We consider now the level set evolution equation TẠP CHÍ PHÁT TRIỂN KH&CN, TẬP 11, SỐ 06 - 2008 Trang 15 uFu u uu u ji ji xx xx ijt ∇− ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∇ −= 2 δ in ).,0( ∞× n R (2.4) with initial condition 0 uu = on {} .0=× tR n (2.5) Definition 2.1. A function )),0(( ∞×∈ n RCu is a weak subsolution of (2.4) provided that if ϕ −u has local maximum at point ),0(),( 00 ∞×∈ n Rtx for each )( 1+∞ ∈ n RC ϕ , then 00 2 00 at ( , ) if (x , ) 0, ij ij xx tij xx Fxt t ϕϕ ϕδ ϕ ϕ ϕ ϕ ⎧ ⎛⎞ ⎪ ≤− −∇ ⎜⎟ ⎜⎟ ∇ ⎨ ⎝⎠ ⎪ ∇≠ ⎩ and () 00 n 00 at ( , ) for some R with 1, if (x , ) 0. ij tijijxx xt t ϕδηηϕ ηηϕ ⎧ ≤− ⎪ ⎨ ∈≤∇= ⎪ ⎩ Definition 2.2: A function )),0(( ∞×∈ n RCu is a weak supersolution of (2.4) provided that if ϕ −u has local minimum at point ),0(),( 00 ∞×∈ n Rtx for each )( 1+∞ ∈ n RC ϕ , then 00 2 00 at ( , ) if (x , ) 0, ij ij xx tij xx Fxt t ϕϕ ϕδ ϕ ϕ ϕ ϕ ⎧ ⎛⎞ ⎪ ≥− −∇ ⎜⎟ ⎜⎟ ∇ ⎨ ⎝⎠ ⎪ ∇≠ ⎩ and () 00 n 00 at ( , ) for some R with 1, if (x , ) 0. ij tijijxx xt t ϕδηηϕ ηηϕ ⎧ ≥− ⎪ ⎨ ∈≤∇= ⎪ ⎩ Definition 2.3: A function )),0(( ∞×∈ n RCu is a weak solution of (2.4) provided u is both a weak subsolution and a supersolution of (2.4). For more details of this kind of solutions, we refer to [3,4,5]. As preliminary motivation for these definitions, suppose u is a smooth function on ),0( ∞× n R satisfying Science & Technology Development, Vol 11, No.06 - 2008 Trang 16 uFu u uu u ji ji xx xx ijt ∇− ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∇ −≤ 2 δ wherever 0≠∇u . Our function u is thus a classical subsolution of (2.4) on {} 0≠∇u . Suppose now .0),( 00 = ∇ txu Assume additionally that there are points ),(),( 00 txtx kk → for which , ).2,1(,0),( = ≠ ∇ ktxu kk Then ,),()(),()(),( kkkkkxx k j k iijkkt txuxFtxutxu ji ∇−−≤ ηηδ for ),( ),( : kk kk k txu txu ∇ ∇ = η . Since , ),2,1(1 =≤ k k η we may if necessary pass to a subsequence so that ηη → k in n R with .1= η Passing to the limits above, we have ),()(),( 0000 txutxu ji xxjiijt ηηδ −≤ . If, on the other hand, there do not exist such points { } ∞ =1 ),( k kk tx , then 0=∇u near ),( 00 tx , and so 0 2 =∇ u and u is a function of t only, near ).,( 00 tx Moving to the edge of the set {} 0=∇u , we see that u is a nonincreasing function of t. Thus ),()(),( 0000 txutxu ji xxjiijt ηηδ −≤ for any n R∈ η . Further motivation for our definition of weak solution, and, particular, an explanation as to why we assume 1≤ η in the definition will be found in Section III. 2.2. Properties of weak solutions Theorem 2.1. (i) Assume k u is a weak subsolution of (2.4) for k=1,2, and uu k → locally uniformly on ),0( ∞× n R . Then u is a weak subsolution of (2.4). (ii) An analogous assertion holds for weak supersolutions and solutions. Theorem 2.2. Assume u is a weak solution of (2.4) and RR → Ψ : is continuous. Then )(: uv Ψ= is also a weak solution of (2.4). The proofs of these theorems can be done similarly in [3,4]. TẠP CHÍ PHÁT TRIỂN KH&CN, TẬP 11, SỐ 06 - 2008 Trang 17 3.EXISTENCE OF WEAK SOLUTIONS 3.1. Preliminaries In this section we consider the existence of weak solution of the mean curvature flow equation (2.4) with initial condition (2.5). A weak solution will be obtained by passing to limits of classical solutions of an approximate problem. We will assume that for the moment at least, 0 u is smooth. Our intention is to approximate (2.4), (2.5) by the partial differential equation 2/1 2 2 2 2 )( ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ +∇− ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎝ ⎛ +∇ −= ε ε δ εε ε εε ε uxFu u uu u ji ji xx xx ijt in ),,0( ∞× n R (3.1) with initial condition 0 uu = ε on { } .0=× tR n (3.2) for .10 << ε 3.2. Solution of the approximate equations We now investigate the approximations (3.1), (3.2) analytically. To do so, let first 2/10 << σ , consider the PDE () 2/1 2 2 ,,,,, ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ +∇−∇= ε σεσεσεσεσε uFuuau ji xxijt in ),,0( ∞× n R (3.3) with initial condition 0 , uu = σε on {} .0=× tR n (3.4) where .,1,,)1(:)( 2 2 , njiRp p pp pa n ji ijij ≤≤∈ + −+= ε δσ δε The smooth bounded coefficients { } σε , ij a satisfy also the uniformly parabolicity condition, namely, we have , , 2 jiij a ξξξσ σε ≤ for all , n R∈ ξ for each n Rp ∈ , therefore, by the classical PDE theory, there exists unique smooth solution σε , u in ),0( ∞× n R satisfying 0 , uu = σε on { } 0=× tR n . We now consider the approximate equation in the bounded sub-domain of ),0( ∞× n R , i.e., we consider the problem Science & Technology Development, Vol 11, No.06 - 2008 Trang 18 {} ⎪ ⎪ ⎩ ⎪ ⎪ ⎨ ⎧ =×= × ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ +∇− ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎝ ⎛ +∇ −+= ,0 ],,0()()1( 0 , 2/1 2 2 ,, 2 2 , ,, , tBonuu TBinuxFu u uu u ji ji xx xx ijt σε σεσε σε σεσε σε ε ε δσ (3.5) where B is a closed ball of radius 0>r centered at original, and 0>T . Now we want to prove estimates for σεσεσε ,,, ,, uuu t ∇ in the domain ],0[ TB × . Lemma 3.1. Let σε , u be a solution of (3.5). Then we have the estimate ,),( , MtCetxu t +≤ λσε for all ],,0[),( TBtx × ∈ (3.6) where .||sup2:, 3 2 :|,|sup2: 0 FM r M uC BB === λ Proof. Let RR n →: ϕ be a function defined by ,|| 2 1 2:)( 22 ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ −= xrmx ϕ where 0 2 sup 1 : u r m B = , we see .||,, 23 xmnmmx jijii xxxxix −=−=Δ−= ϕϕϕϕϕ We define by ,)(:),( Mtextxv t += λ ϕ we have ,,,)( ttt xx t t mneevevMexv ii λλλλ ϕϕλϕ −=Δ=Δ=+= .|| 3233 t xxxx t xxxx exmevvv jijijiji λλ ϕϕϕ −== Therefore, () () () .01 2 3 ||1 2 3 || )1(|| 2 1 2 )1( )1(:)( 22 2/1 2 2 2 2 323 22 2/1 2 2 2 2 2/1 2 2 2 2 , > ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ −−+≥ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ −−+≥ +∇+ +∇ −++ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ −= +∇+ +∇ +Δ+−= +∇+ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ +∇ −+−= tt t tt xxxx t xx xx ijt meMrnrmexMnr vF v exm mneexrm vF v vvv vv vFv v vv vvL jiji ji ji λλ λ λλ σε λλ ε ε σλ ε ε σ ε ε δσ TẠP CHÍ PHÁT TRIỂN KH&CN, TẬP 11, SỐ 06 - 2008 Trang 19 On the other hand, ||sup||sup 2 3 2 1 2||sup 1 || 2 1 2||sup 1 )()0,( 00 22 0 2 22 0 2 uurru r xru r xxv BBBB ≥= ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ −≥ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ −== ϕ Therefore, ),(0)( ,,, σεσεσε uLvL => and )0,()()0,( 0 , xvxuxu ≤= σε in B. By the classical maximum principle for the parabolic equation, we discover .),(),( , MtCetxvtxu t +≤≤ λσε The proof of the estimate for ),( , txu σε − is similar as above, therefore, we get .),(|),(| , MtCetxvtxu t +≤≤ λσε Lemma 3.2. Let σε , u be a solution of (3.5). Then we have the estimate Ctxu t TB ≤ × |),(|max , ],0[ σε where C is a constant depending only on .||sup|,|sup|,|sup|,|sup 0 2 00 Fuuu BBBB ∇∇ Proof. Differentiate the equation in (3.5) with respect to t, we have () . 2 )1( 2/1 2 2 , ,, , 2 2 2 , ,,,,2 2 ,,,,, , 2 2 , ,, , ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ +∇ − ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ +∇ − ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ +∇+ − ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎝ ⎛ +∇ −+= εε ε ε δσ σε σεσε σε σε σεσεσεσεσεσεσεσεσε σε σε σεσε σε u uuF u u uuuuuuuuu u u uu u ii jj jijijiji ji ji txx xx txtxxxtxxxtx xtx xx ijtt This equation is linear with respect to t u , then we may apply the classical maximum principle, we have |,)0,(|sup|),(|sup ,, ],0[ ⋅≤ × σεσε t B t TB utxu and () .)()1()0,( 2/1 2 2 0 0 2 2 0 00 , ε ε δσ σε +∇− ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎝ ⎛ +∇ −+= uxFu u uu xu ji ji xx xx ijt Since 10 << ε and 2/10 << σ , .|),(|sup , ],0[ Ctxu t TB ≤ × σε By the transformation , 1 , uu ε σε a we see Science & Technology Development, Vol 11, No.06 - 2008 Trang 20 2/1 2 2 1 1 )1( ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ +∇− ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ +∇ −+= uFu u uu u ji ji xx xx ijt δσ . (3.7) Lemma 3.3. Let u be a solution of (3.7). Then we have the estimate MCtxuC eCtxue 22 1 ),( |),(| ≤∇ , for all ],,0[),( TBtx × ∈ where 21 ],0[ ,|;),(|sup: CCtxuM TB× = are constants dependent only on |)(|sup xF B and |)(|sup xF B ∇ . We derived estimates for σεσεσε ,,, ,, uuu t ∇ in the bounded domain ],0[ TB × . We note that n jiij Rpa L L ∈≤ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + − ξξξξ ε σε ,)(||1 ,2 22 2 provided Lp ≤|| . The estimates for σεσεσε ,,, ,, uuu t ∇ are uniform in 2/10 << σ . Consequently, uniqueness of the limit implies for each multi-index α : εασεα uDuD → , locally uniformly as 0→ σ , for a smooth function ε u solving approximate equation. 3.3. Passage to limits Theorem 3.4. Assume RRu n →: 0 is a continuous function. Then there exists a weak solution u of (2.4),(2.5). Proof. Suppose first 0 u is smooth. Employing estimates in Lemmas 3.1, 3.2, 3.3, we can extract a subsequence 101 }{}{ << ∞ = ⊂ ε ε ε uu k k so that, as uu k k →→ ε ε ,0 uniformly in ],0[ TB × for some Lipschitz function u in ],0[ TB × . Since r and T are arbitrary, we can extend r and T to infinity so that uu → ε locally uniformly in ),0[ ∞× n R for a locally Lipschitz continuous function u in ),0[ ∞× n R . We assert now that u is a weak solution of (2.4), (2.5). For this, let )( 1+∞ ∈ n RC ϕ and suppose ϕ −u has a strict local maximum at a point ),0[),( 00 ∞×∈ n Rtx . As uu k → ε uniformly near ϕ ε − k utx ),,( 00 has a local maximum at a point ),( kk tx , with ),(),( 00 txtx kk → as ∞→k . (3.8) Since k u ε and ϕ are smooth, we have ϕϕϕ εεε 22 ,, DuDuu kkk tt ≤=∇=∇ at ).,( kk tx (3.9) TẠP CHÍ PHÁT TRIỂN KH&CN, TẬP 11, SỐ 06 - 2008 Trang 21 Since k u ε is a solution of , 2/1 2 2 2 2 ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ +∇− ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎝ ⎛ +∇ −= kxx k xx ijt kk ji k k j k i k uFu u uu u ε ε δ εε ε εε ε we have () 2/1 2 2 2 2 kxx k xx ijt F ji ji εϕϕ εϕ ϕϕ δϕ +∇− ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ +∇ −≤ at ).,( kk tx (3.10) Suppose first 0),( 00 ≠∇ tx ϕ . Then 0),( ≠ ∇ kk tx ϕ for k large enough. We consequently may pass to limits in (3.10), recalling (3.9) to deduce ϕϕ ϕ ϕϕ δϕ ∇− ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∇ −≤ F ji ji xx xx ijt 2 at ).,( 00 tx (3.11) Next, assume instead 0),( 00 = ∇ tx ϕ . Set () , |),(| ),( : 2/1 22 kkk kk k tx tx εϕ ϕ η +∇ ∇ = so that (3.10) becomes ( ) ji xx k j k iijt ϕηηδϕ −≤ at ).,( kk tx (3.12) Since 1|| ≤ k η , we may assume, upon passing to a subsequence and re-indexing if necessary, that ηη → k in n R for some 1|| ≤ η . Sending k to infinity in (3.10), we discover ( ) ji xxjiijt ϕηηδϕ −≤ at ).,( 00 tx (3.13) If ϕ −u has a local maximum, but not necessary a strict local maximum at ),,( 00 tx we repeat the argument above with ),( tx ϕ replaced by ,)(||),(:),( ˆ 4 0 4 0 ttxxtxtx −+−+= ϕϕ again to obtain (3.11) and (3.13). Consequently, u is a weak subsolution of (2.4),(2.5). That u is a weak suppersolution follows analogously. Suppose at last 0 u is only continuous. We select smooth functions ∞ = 10 }{ k k u so that 00 uu k → locally uniformly on n R . Denote by k u the solution of (2.4),(2.5) constructed above with initial function k u 0 . According to the stability of the weak solutions[3,4] the limit ∞→ = k k uulim exists locally uniformly in ),0[ ∞× n R , according to Theorem 2.1 u is a weak solution of (2.4), (2.5). Science & Technology Development, Vol 11, No.06 - 2008 Trang 22 CHUYỂN ĐỘNG CỦA TẬP MỨC VỚI VẬN TỐC PHỤ THUỘC VÀO ĐỘ CONG TRUNG BÌNH: SỰ TỒN TẠI NGHIỆM YẾU Nguyễn Chánh Định Đại học Đà Nẵng TÓM TẮT: Chuyển động của siêu mặt theo độ cong trung bình đã được xem xét bởi Brakke[1] theo quan điểm hình học, và bởi Evans, Spruck[3] theo quan điểm giải tích. Bắt đầu từ mặt 0 Γ trong n R , các mặt t Γ chuyển động theo thời gian với vận tốc bằng độ cong trung bình của chúng theo hướng pháp tuyến ngoài. Các mặt t Γ sau đó được xác định bằng cách tìm các tập mức không của một hàm liên tục Lipschitz, là một nghiệm yếu của phương trình chuyển động. Chuyển động của siêu mặt bởi một quá trình tụ hạt qua cách tiếp cận tập mức cũng đã được nghiên cứu bởi Dinh, Hoppe[4]. Trong bài báo này, chúng tôi xem xét phương trình chuyển động mặt với vận tốc phụ thuộc vào độ cong trung bình.Vận tốc của quá trình chuyển động được kết hợp bởi độ cong trung bình và một ngoại lực. Chúng tôi đưa ra một phương trình chuyển động mà nghiệm của nó chứa mặt chuyển động dưới dạng tập mức không. Một kết quả tồn tại sẽ được đưa ra. REFERENCES [1]. Brakke A., The motion of a surface by its mean curvature, Princeton Univ. Press, Princeton, NJ, (1978). [2]. Gilbarg D. and Trudinger N. S., Elliptic partial differential equations of second order, 2 nd ed., Springer-Verlag, Berlin, (1983). [3]. Evans L. C. and Spruck J., Motion of level set by mean curvature I, J. Diff. Geom., 33, 635-681, (1991). [4]. Nguyen Ch. D. and Hoppe R. H. W, Amorphous surface growth via a level set approach, J. Nonlinear Anal. Theor. Meth. Appl., 66, 704-722, (2007). [5]. Nguyen Chanh Dinh, On the uniqueness of viscosity solutions to second order parabolic partial differential equations, J. Science and Technology, University of Danang, 2(14), 53-57, (2006). . consider the existence of weak solution of the mean curvature flow equation (2.4) with initial condition (2.5). A weak solution will be obtained by passing to limits of classical solutions 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. KH&CN, TẬP 11, SỐ 06 - 2008 Trang 13 LEVEL SET EVOLUTION WITH SPEED DEPENDING ON MEAN CURVATURE: EXISTENCE OF A WEAK SOLUTION Nguyen Chanh Dinh Danang University of Technology (Manuscript