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## Material Information- Title:
- Geometric evolution equations
- Creator:
- Chastain, Stacey E., 1971-
- Publication Date:
- 2000
- Language:
- English
- Physical Description:
- vii, 64 leaves : ; 29 cm.
## Subjects- Subjects / Keywords:
- Boundary conditions ( jstor )
Curvature ( jstor ) Existence ( jstor ) Flow equations ( jstor ) Geometric properties ( jstor ) Image processing ( jstor ) Maps ( jstor ) Mathematical monotonicity ( jstor ) Mathematics ( jstor ) Viscosity ( jstor ) Dissertations, Academic -- Mathematics -- UF ( lcsh ) Mathematics thesis, Ph. D ( lcsh ) - Genre:
- bibliography ( marcgt )
theses ( marcgt ) non-fiction ( marcgt )
## Notes- Thesis:
- Thesis (Ph. D.)--University of Florida, 2000.
- Bibliography:
- Includes bibliographical references (leaves 62-63).
- General Note:
- Printout.
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- Vita.
- Statement of Responsibility:
- by Stacey E. Chastain.
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GEOMETRIC EVOLUTION EQUATIONS By STACEY E. CHASTAIN A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2000 For Mom and Dad ACKNOWLEDGMENTS First and foremost, I would like to thank my advisor, Yunmei Chen, for introducing me to this subject and for her constant guidance and support. Her enthusiasm for teaching and studying mathematics as well as her vast knowledge of PDEs has been an inspiration for me. I would also like to thank my committee members past and present, Bernard Mair, Gerard Emch, Murali Rao, Gang Bao, Peter Hirschfeld, and Jim Dufty for their input and advice. Special thanks to the Department of Mathematics office staff for their assistance throughout my entire graduate career. Finally, my warmest appreciation goes to all my family and friends without whose support and encouragement this work would never have been completed. Missy and Jason, thank you for always being there no matter how far away you were. Scott, thank you for your constant encouragement. TABLE OF CONTENTS ACKNOWLEDGMENTS ............................. iii A BSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi CHAPTERS 1 INTRODUCTION ............................... 1 1.1 Nonlinear Evolution Equations ................... 1 1.2 Problem 1: The Flow of H-Systems ................ 1 1.3 Problem 2: PDE-Based Image Processing ............. 2 2 PRELIMINARIES ............................... 3 2.1 M otivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.2 Sobolov Spaces .......................... .. 3 2.3 H6lder Spaces ... . .... ... .... . ... . ... .. . .. 5 2.4 Global Approximations by Smooth Functions ........... 6 2.5 Embedding Theorems ........................ 7 3 THE FLOW OF AN H-SYSTEM ....................... 9 3.1 Plateau's Problem and H-Systems ..................... 9 3.2 Flow of an H-System ...... ........................ 10 3.3 Notation ............................... 12 3.4 Local Existence ....... ........................... 13 3.5 Some A-priori Estimates ...... ...................... 15 3.5.1 V(MT) Estimates ............................ 15 3.5.2 Regularity ....... .......................... 22 3.5.3 Uniqueness ................................ 29 3.6 Existence up until Time of Energy Concentration ............ 30 3.7 Global Existence .................................. 35 3.8 Behavior of Singularities; ............................ 37 4 MODIFIED MEAN CURVATURE FLOW EQUATION ............. 43 4.1 Evolution of Level Sets by Mean Curvature ................ 43 4.2 Applications to Image Processing ....................... 45 4.3 Anisotropic Diffusion ...... ........................ 46 4.4 Existence and Uniqueness of the Weak Solution ............. 48 4.5 Hausdorff Measure ................................ 49 4.6 Monotonicity .................................... 50 4.7 Consequences of the Monotonicity Formula ................ 54 4.8 Extinction Times ................................. 56 V 5 FURTHER QUESTIONS ........................... 59 5.1 H-systems and Related Topics ................... 59 5.2 PDE-Based Image Processing ......................... 59 REFERENCES ........................................... 62 BIOGRAPHICAL SKETCH ................................... 64 Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy GEOMETRIC EVOLUTION EQUATIONS By Stacey Elizabeth Chastain May 2000 Chairman: Yunmei Chen Major Department: Mathematics This work encompasses two geometric evolution equations with applications to image processing. First we show the existence of a unique solution to the flow of an H-system with Dirichlet boundary condition which is regular up until the first time of energy concentration. If we assume the solution satisfies a certain energy inequality, then the solution exists for all time and is smooth except at finitely many singularities. The behavior of the solutions at these singularities is also discussed. Under certain conformality conditions, the "H" in the steady state version of these systems is in fact the mean curvature of the solution. This equation arose from the study of minimal surfaces with prescribed mean curvature, which in turn came from Plateau's classic "soap bubble" problem. The steady state system was studied in detail by Hildebrandt, Wendt, Brezis and Coron, et al. It was then observed that these H-surfaces (the solutions of the H-systems) could be studied in more generality if they were viewed as asymptotic solutions to the flow of an H-system. Moreover, due to their structure they could be analyzed using similar techniques as those used to study harmonic maps. This is the approach that we take in our analysis. Next we study a modified version of the mean curvature flow equation which can be used as a model for image restoration. This equation is interesting since the level sets of the solution can be used to extract details from an image, while the mean curvature provides for anisotropic diffusion which aids in noise removal. We have introduced a control factor that slows the evolution of the level sets near significant features in the image. We discuss precisely how the model creates this behavior, investigate some of its geometric properties, and compute a bound for the extinction time of the level sets of its solution. CHAPTER 1 INTRODUCTION 1.1 Nonlinear Evolution Equations In this work, we study two different nonlinear evolution equations. The equations I have studied here are not only nonlinear, but also admit additional difficulties such as lack of boundedness, lack of energy inequality, and degeneracy. While these characteristics make them more appropriate to modern applications, they also present added difficulties into the analysis of the equations. The first equation I studied is the flow of an H-system. The main obstacles in studying this system are the unboundedness of it's solutions and lack of an energy inequality. This prevents us from using any standard techniques to analyze the equation. Instead we must rely on very delicate estimates which require extreme care in deriving. For the second part of this exposition I studied a partial differential equation (PDE) based model for image processing, namely the mean curvature flow equation. This model is used to remove noise from an image while preserving its significant features. In addition, the level sets of the solutions can be used to extract features from an image. The model is extremely efficient; however, it is also nonlinear and degenerate, creating further complications in its analysis and implementation. 1.2 Problem 1: The Flow of H-Systems One of the classic studies in partial differential equations is that of harmonic maps. In 1985, Struwe proved the existence of harmonic maps into an arbitrary manifold [20]. The techniques used in [20] have since been used to study other general second order evolution equations with variational structure. Two such equations are the Landau-Lifshitz equation and the flow of an H-System. In the following work, we analyzed the flow of the H-System. This flow actually has rich geometric interpretation. If the solution to the steady state H-system satisfies certain conformal conditions, then "H(u)" represents the mean curvature of u. We prove the existence of a unique solution which is regular up until the first time of energy concentration as well as prove global existence and and discuss the behavior of the singularities. 1.3 Problem 2: PDE-Based Image Processing Computer vision is another fascinating subject for which partial differential equations serve as extremely effective models. These models work in a systematic way, performing all necessary tasks simultaneously. They remove noise while retaining and even enhancing significant features. The model we study here arises from evolving level sets of an image by their mean curvature. This model is based on the work of Osher and Sethian in [18]. The level sets of the solution to this equation can be used to identify important features in an image. Embedding the curves as level sets of a surface allows for changes in the topology of the curves without disrupting the evolution. In [9], Evans and Spruck proved various geometric properties as well as the extinction time of level sets to the mean curvature flow equation. We introduce a control factor into this equation to inhibit the evolution of the level sets near the boundaries of significant features in an image. The existence of a viscosity solution to this modified equation has been shown in [4]. We study how this control factor affects the geometric properties as well as the extinction time of the level sets of our solutions. CHAPTER 2 PRELIMINARIES 2.1 Motivation The equations we study here are too complex to arrive at explicit solutions. Instead, we must be satisfied in studying the existence, uniqueness, regularity and various other significant properties of their solutions. The natural spaces to look for and study these solutions in are Sobolov and H6lder spaces. For the ease of the reader, the definitions and properties of these spaces that will be needed in the following chapter are included here. This information can be found in [11] and [17]. 2.2 Sobolov Spaces Throughout this chapter, Q C R'. For convenience, we will use the notation '= x [s,t] and Q As usual, C(Q) will denote the set of infinitely differentiable functions with compact support on Q. Definition 2.2.1 [Weak Derivative] If u, v E L'o(Q) and a = ( a,...,an) where the ai 's, i = 1, ..., n, are non-negative integers, then we say that v is the ath-weak derivative of u, written v = D'u if j uDaodx= (-1)1 1 fn vqdx for all test functions 0 E C'(Q). Fix 1 < p < 0 and let k and 1 be non-negative integers. Define the Sobolov space, Wk (Q) as follows: Wk(Q) - {u: Q -+ R I VIaI < k, D'u exists in the weak sense and D~u E LP(Q)} with corresponding norm EZIQ l p<0o p OC. If p = 2, we write H k(Q) = Wk(Q). We also denote by Hok(Q), the closure of C,(Q) in Hk(Q). In fact, one can show that Hok(Q) {u E Hk(Q)l ID' u = 0 on OaV ja _< k - 1}. On the evolution space Q' define the Sobolov space wit A I hVa + Zr 1 (fo ID-ulPdxdt) , lIUIjW~k(2)- ZIajSk eSS supIDaul + Z a5 Q' l=p<00 p=0o Ilullwk(n) Ejj (fn. o aldx 1)P 5 These spaces have the appropriate structure to invoke some standard results from functional analysis. In particular, Theorem 2.2.2 Wk (Q) and Wk'"(Qt) with the above defined norms are reflexive Banach spaces for any 1 < p < co. For more details, see [11]. 2.3 H6lder Spaces H6lder Spaces are usually the appropriate setting for studying the regularity of the solutions. Although it is not always possible to initially find the existence of a solution in these spaces, one can find a solution in a Sobolov space and use embedding theorems to conclude that the solution is in fact H6lder continuous. To that end, we define the following. Let k denote any positive integer. We will use the standard notation Ck(Q) A {u: Q - R I D'u exists and is continuous V 1yj < k}. Recall that IlJullckp) mnax ID~ul. Iyl For 0 < a < 1, define the space C'(Q) f {u: a- xER I sup lu(x)- u(y)l < } X,YEQ ix - yl with corresponding norm 1u(x) - u(y)l HUico(Q) = sup X'YGQ Ix - y[a We can new define the space Ck+,(Q) {u : Q - R I u c ck(Q) and D'u E C0(Q)V 11- = k} with corresponding norm IlUlCk+o(Q) H ckjIj(n)+ ES IID'1uIjco(). I'yl=k Finally, on the evolution space Q' we can define Ck+, k2 (Q ) =a {u : Qt - R I DtD~u E C(Qt) V2r + s < k and D~u E Ca(Qt), D u e C(Qt)V h1 k} DX S , 8 with corresponding norm I UIIck+a, 4 (n ma I mxD'D'u I+ lD~u lc.(Qt) + J JD 2u u1c, (Qt). 2r+s As with Sobolov spaces, H6lder spaces also afford a nice structure. Theorem 2.3.1 Ck+a(Q) and Ck+a, k+ (Q1) with the above defined norms are Banach spaces. For more details, see [11] and [17]. 2.4 Global Approximations by Smooth Functions If one begins with smooth initial and boundary data, there are sufficient techniques available to determining the existence of a local solution to the nonlinear problems we study here. In order to arrive at our existence results, we approximate our initial and boundary conditions with smooth functions and find solutions to the system using this smooth data. We then use these solutions to approximate a solution to our problem. The real work comes in finding appropriate uniform bounds on our solutions to be able to take the proper limits. Below we provide the approximation theorems that will be needed once our estimates are sufficient. Theorem 2.4.1 Assume Q is bounded and u E Wp(Q) for some 1 < p < o. Then there exists um E CI(Q) n Wp(f ) such that u - u in A. Theorem 2.4.2 Assume Q is bounded, 0Q is C', and u E W k(Q) for some 1 < p < 0o. Then there exists um E Cï¿½c(Q) n Wk(Q) such that um -+u in See [11] for proofs of the above theorems. 2.5 Embedding Theorems The larger the space in which we search for solutions, the easier it will be to find one. Once this course of action is complete, it is then desirable to see how regular the above found solution can be. The standard way to do this is using embedding theorems. In the following, B1, - B2 denotes that B1 is continuously embedded in 132. As in the previous sections, Q C Rn. Theorem 2.5.1 1. Suppose 1 < p < 0c, mp < n, and p < q < 'P . Then n-mp WPI. M (Q) " -*A 2. Suppose 1 < p < oc, mp = n, and p < q < oo. Then wpi+ M(Q) -+ wq(A) Furthermore, if p = 1, then W +M (0) -4 cJ+q(Q). 3. Suppose 1 n, and O < A < m - (<1). Then |

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PAGE 1 GEOMETRIC EVOLUTION EQUATIONS By STAGEY E. GHASTAIN A DISSERTATION PRESENTED TO THE GRADUATE SGHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOGTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2000 PAGE 2 For Mom and Dad PAGE 3 ACKNOWLEDGMENTS First and foremost, I would like to thank my advisor, Yunmei Chen, for introducing me to this subject and for her constant guidance and support. Her enthusiasm for teaching and studying mathematics as well as her vast knowledge of PDEs has been an inspiration for me. I would also like to thank my committee members past and present, Bernard Mair, Gerard Emch, Murali Rao, Gang Bao, Peter Hirschfeld, and Jim Dufty for their input and advice. Special thanks to the Department of Mathematics office staff for their assistance throughout my entire graduate career. Finally, my warmest appreciation goes to all my family and friends without whose support and encouragement this work would never have been completed. Missy and Jason, thank you for always being there no matter how far away you were. Scott, thank you for your constant encouragement. iii PAGE 4 TABLE OF CONTENTS ACKNOWLEDGMENTS iii ABSTRACT vi CHAPTERS 1 INTRODUCTION 1 1.1 Nonlinear Evolution Equations 1 1.2 Problem 1: The Flow of H-Systems 1 1.3 Problem 2: PDE-Based Image Processing 2 2 PRELIMINARIES 3 2.1 Motivation 3 2.2 Sobolov Spaces 3 2.3 Holder Spaces 5 2.4 Global Approximations by Smooth Functions 6 2.5 Embedding Theorems 7 3 THE FLOW OF AN H-SYSTEM 9 3.1 Plateau's Problem and H-Systems 9 3.2 Flow of an H-System 10 3.3 Notation 12 3.4 Local Existence 13 3.5 Some A-priori Estimates 15 3.5.1 V{M^) Estimates 15 3.5.2 Regularity 22 3.5.3 Uniqueness 29 3.6 Existence up until Time of Energy Concentration 30 3.7 Global Existence 35 3.8 Behavior of Singularities 37 4 MODIFIED MEAN CURVATURE FLOW EQUATION 43 4.1 Evolution of Level Sets by Mean Curvature 43 4.2 Applications to Image Processing 45 4.3 Anisotropic Diffusion 46 4.4 Existence and Uniqueness of the Weak Solution 48 4.5 Hausdorff Measure 49 4.6 Monotonicity 50 4.7 Consequences of the Monotonicity Formula 54 4.8 Extinction Times 56 iv PAGE 5 V 5 FURTHER QUESTIONS 59 5.1 H-systems and Related Topics 59 5.2 PDE-Based Image Processing 59 REFERENCES 62 BIOGRAPHICAL SKETCH 64 PAGE 6 Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy GEOMETRIC EVOLUTION EQUATIONS By Stacey Elizabeth Chastain May 2000 Chairman: Yunmei Chen Major Department: Mathematics This work encompasses two geometric evolution equations with applications to image processing. First we show the existence of a unique solution to the flow of an H-system with Dirichlet boundary condition which is regular up until the first time of energy concentration. If we assume the solution satisfies a certain energy inequality, then the solution exists for all time and is smooth except at finitely many singularities. The behavior of the solutions at these singularities is also discussed. Under certain conformality conditions, the "H" in the steady state version of these systems is in fact the mean curvature of the solution. This equation arose from the study of minimal surfaces with prescribed mean curvature, which in turn came from Plateau's classic "soap bubble" problem. The steady state system was studied in detail by Hildebrandt, Wendt, Brezis and Coron, et al. It was then observed that these H-surfaces (the solutions of the H-systems) could be studied in more generality if they were viewed as asymptotic solutions to the flow of an H-system. Moreover, vi PAGE 7 Vll due to their structure they could be analyzed using similar techniques as those used to study harmonic maps. This is the approach that we take in our analysis. Next we study a modified version of the mean curvature flow equation which can be used as a model for image restoration. This equation is interesting since the level sets of the solution can be used to extract details from an image, while the mean curvature provides for anisotropic diffusion which aids in noise removal. We have introduced a control factor that slows the evolution of the level sets near significant features in the image. We discuss precisely how the model creates this behavior, investigate some of its geometric properties, and compute a bound for the extinction time of the level sets of its solution. PAGE 8 CHAPTER 1 INTRODUCTION 1.1 Nonlinear Evolution Equations In this work, we study two different nonlinear evolution equations. The equations I have studied here are not only nonlinear, but also admit additional difficulties such as lack of boundedness, lack of energy inequality, and degeneracy. While these characteristics make them more appropriate to modern applications, they also present added difficulties into the analysis of the equations. The first equation I studied is the flow of an H-system. The main obstacles in studying this system are the unboundedness of it's solutions and lack of an energy inequality. This prevents us from using any standard techniques to analyze the equation. Instead we must rely on very delicate estimates which require extreme care in deriving. For the second part of this exposition I studied a partial diflFerential equation (PDE) based model for image processing, namely the mean curvature flow equation. This model is used to remove noise from an image while preserving its significant features. In addition, the level sets of the solutions can be used to extract features from an image. The model is extremely efficient; however, it is also nonlinear and degenerate, creating further complications in its analysis and implementation. 1.2 Problem 1: The Flow of H-Svstems One of the classic studies in partial diff'erential equations is that of harmonic maps. In 1985, Struwe proved the existence of harmonic maps into an arbitrary 1 PAGE 9 2 manifold [20]. The techniques used in [20] have since been used to study other general second order evolution equations with variational structure. Two such equations are the Landau-Lifshitz equation and the flow of an H-System. In the following work, we analyzed the flow of the H-System. This flow actually has rich geometric interpretation. If the solution to the steady state H-system satisfies certain conformal conditions, then " !!(Â«)" represents the mean curvature of u. We prove the existence of a unique solution which is regular up until the first time of energy concentration as well as prove global existence and and discuss the behavior of the singularities. 1.3 Problem 2: PDE-Based Image Processing Computer vision is another fascinating subject for which partial diflferential equations serve as extremely eflFective models. These models work in a systematic way, performing all necessary tasks simultaneously. They remove noise while retaining and even enhancing significant features. The model we study here arises from evolving level sets of an image by their mean curvature. This model is based on the work of Osher and Sethian in [18]. The level sets of the solution to this equation can be used to identify important features in an image. Embedding the curves as level sets of a surface allows for changes in the topology of the curves without disrupting the evolution. In [9], Evans and Spruck proved various geometric properties as well as the extinction time of level sets to the mean curvature flow equation. We introduce a control factor into this equation to inhibit the evolution of the level sets near the boundaries of signiflcant features in an image. The existence of a viscosity solution to this modified equation has been shown in [4]. We study how this control factor affects the geometric properties as well as the extinction time of the level sets of our solutions. PAGE 10 CHAPTER 2 PRELIMINARIES 2.1 Motivation The equations we study here are too complex to arrive at explicit solutions. Instead, we must be satisfied in studying the existence, uniqueness, regularity and various other significant properties of their solutions. The natural spaces to look for and study these solutions in are Sobolov and Holder spaces. For the ease of the reader, the definitions and properties of these spaces that will be needed in the following chapter are included here. This information can be found in [11] and [17]. 2.2 Sobolov Spaces Throughout this chapter, 1^ c M" . For convenience, we will use the notation Jl', = nx[s,t] and = fi*. As usual, C^{^) will denote the set of infinitely differentiable functions with compact support on fl. Definition 2.2.1 [Weak Derivative] Ifu,ve Ll^{Q) and a = (ai,...,aÂ„) where the ai's, i = l,...,n, are non-negative integers, then we say that v is the a^^-weak derivative of u, written v = D'^u if 3 PAGE 11 for all test functions ^ G C^(f)). Fix 1 < p < oo and let k and / be non-negative integers. Define the Sobolov space, W^(n) as follows: W!^{Sl) = {u : R I V|a| < k, D^'u exists in the weak sense and D^u G LP(f))} with corresponding norm E|a| PAGE 12 5 These spaces have the appropriate structure to invoke some standard results from functional analysis. In particular, Theorem 2.2.2 Wp{Q) and Wp'^{Ql) with the above defined norms are reflexive Banach spaces for any 1 < p < oo. For more details, see [11]. 2.3 Holder Spaces Holder Spaces are usually the appropriate setting for studying the regularity of the solutions. Although it is not always possible to initially find the existence of a solution in these spaces, one can find a solution in a Sobolov space and use embedding theorems to conclude that the solution is in fact Holder continuous. To that end, we define the following. Let k denote any positive integer. We will use the standard notation = {m : 1^ ^ M I D^u exists and is continuous V |7| < k}. Recall that Mc'-in) = 51 Â«ifxiD>|. |7| PAGE 13 6 We can new define the space C*+"(f}) 4 {u : Q ^ K I M G C\n) and L>> G CÂ°(f7)V I7I = k} with corresponding norm lÂ«llc*+Â°(n) ll^llc'cn) + W^^'^Wc-iU)Finally, on the evolution space we can define = {u : Q* ^ R I DlD'^u e C{nl) \/2r + s PAGE 14 7 our initial and boundary conditions with smooth functions and find solutions to the system using this smooth data. We then use these solutions to approximate a solution to our problem. The real work comes in finding appropriate uniform bounds on our solutions to be able to take the proper limits. Below we provide the approximation theorems that will be needed once our estimates are suflftcient. Theorem 2.4.1 Assume Q is bounded and u E Wp{Q) for some 1 < p < oo. Then there exists G CÂ°Â°(Q) n W!^{9) such that u"" in W^{n). Theorem 2.4.2 Assume Q is bounded, dO, is C\ and u G W^{^) for some 1 < p < oo. Then there exists u"" G CÂ°Â°{U) D W^{^1) such that u'^-^u in W^{n). See [11] for proofs of the above theorems. 2.5 Embedding Theorems The larger the space in which we search for solutions, the easier it will be to find one. Once this course of action is complete, it is then desirable to see how regular the above found solution can be. The standard way to do this is using embedding theorems. In the following, Bi ^ B2 denotes that Bi is continuously embedded in B2. As in the previous sections, c R". Theorem 2.5.1 1. Suppose 1 < p < 00, mp < n, and p < q < Then PAGE 15 2. Suppose 1 < p < oo, mp = n, and p < q < oo. Then Furthermore, if p Â— 1, then 3. Suppose \ < p < oo, mp > n, and 0 PAGE 16 CHAPTER 3 THE FLOW OF AN H-SYSTEM 3.1 Plateau's Problem and H-Systems "Given a Jordan curve, F, in and a constant, H, find a surface, u{xi,X2), in of mean curvature H spanning F." This is the statement of Plateau's classic soap bubble problem for surfaces with prescribed mean curvature. One might recall that if = 0, then u represents a minimal surface. Analytically, Plateau's problem can be stated as follows. Let M C be a bounded set with smooth boundary. Define M* = M X [s, t] and M* = A map u e C'^{M,R^) satisfying Au = 2H{u)u:, Auy (3.1) is called an H-surface (supported by M). If u is a conformal representation of a surface 5; i.e., u satisfies the relations then H{u) is the mean curvature of S at u. 9 PAGE 17 10 The existence of /f-surfaces under various boundary conditions have been studied in detail (see [1], [15], [24], [25]). In all of these cases, the mean curvature was assumed to be constant. If one considers the flow of an i/-System, this assumption can be generalized. 3.2 Flow of an H-Svstem In the rest of the chapter, we will study the existence and behavior of the singularities for the heat flow of the /f-system associated with (3.1), dtu = Au Â— 2H{u)ux A Uy, in ' w(0,a;) = uq{x), in M (3-2) u{t,x) = x{x), on{dMf where, uo G H\M), x e Hl{dM), uo{x)\^3m) = x{x) and H G W^{R^). The existence of a global regular solution to (3.2) has been shown by Rey [19] assuming uq G H^{M,R^) n LÂ°Â°(M,M^) and ||-f^||L-(R3)||ito||L-{M) < 1(3.3) The assumption of small initial data is essential for the proof, since this leads to an energy inequality and a boundedness property for the solution of (3.2), which in general does not hold. PAGE 18 11 Moreover, Struwe [21] studied the free boundary problem associated with (3.2): dtu = Au Â— 2H{u)ux A Uy, in u{0,x) = uo{x), in M (3.4) u{t,x) e E, a.e. on (dM)'^ dnu{t,x) _L TÂ„(t,a;)E, a.e. on (dM)'^ where E is a surface diffeomorphic to the sphere, Tu(^t,x)'^ denotes the tangent space to E at u{t,x), and Ug E {u e H^{M) : u{dM) C S}. Assuming H is constant, he showed the existence of a local regular solution to (3.4). The free boundary condition complicates matters; however, since u\dM ^ E the solutions are bounded, a property that in general does not hold. Here we will study the existence and the behavior of the singularities of (3.2). We will permit i/ to be a function of u and will not be restricted to small initial data. We will show the existence of a unique, regular solution to (3.2) up until the first time of an energy concentration. In general, the solution does not satisfy an energy inequality; however, in the event that it does, we can show the existence of a global solution which is smooth everywhere except possibly at finitely many singularities. We will conclude by discussing the behavior of these singularities. The main difficulties we encounter in our discussion is the lack of the energy inequality and the unboundness of the LÂ°Â°-norm of the solution. These issues will be addressed throughout the remainder of the paper. PAGE 19 3.3 Notation Let M and be defined as in section 2.2. Define V{Ml) = e C\[s,tlH\M)) : \V^ul \dM e L\MI)} where the derivatives are taken in the distributional sense. We will denote B^{x) = Br{x) n M for any X e M where Br{x) = {x' G R^llx-x'l < R} If z = {x,t) e M X [0, oo), then for t > i?^ define Qr{z) = Br{x) x{tR\ t) and = x {t R^, t). We will denote the energy of u as D{u;n) = 1 [ \Vu\'^dx and for convenience, D{u; M) = D{u). In addition, define the functional Eh{u) = D{u) + \ I Q{u)ux A Uydx where H{S, U2, U3),J^ H{uu S, U3), J H{uu U2, s)] . PAGE 20 13 Note that the critical points of Eh{u) in H^{M) are weak solutions of the steady state of (3.2). Finally, the quantity e{R)^ sup D{u{t);B^{x)) (3.7) will be instrumental throughout the remainder of the paper. 3.4 Local Existence We will need to employ the following Sobolov type inequality. Lemma 3.4.1 For all u G W^'^{M'^), p > 4, we have that Vm Â€ C"'t(M^) for a = {1 Â— ^) and there exists Cp > 0 such that Theorem 3.4.2 Suppose Uq G C^{M), x e CÂ°Â°(aM) and Uo\dM = XThen there exists an 5 > 0 and a unique solution u G C^+"'^+t(M'^) of (3.2) on M^. Proof: For any p > 4, define an operator K : W^'\M'^) ^ L^iM"^) x Wp'''{M) x Wp~'^{{dMf) by Ku = [dtu Au 2H{u)u^ A Uy, u{x, 0), u{x, t)\^aM)T)- PAGE 21 14 The differential of K at the point v is DK{u)v = {dtVÂ—Av-2H{u)[vxf\Uy+Ux/\Vy]Â—2DH{u)v-UxAUy, v{x, 0), v{x,t)\(^dM)'^)By [17] [IV, Theorem 9.1], there exists a unique solution in Wp'^{M'^) to the corresponding parabolic system dtv Â— Av Â— 2H{u)[vx A Uy + Ux /\ Vy] Â— 2DH{u)v Ux A Uy + g in ^ v{x,0) = h{x) in M (3-8) v{x,t) = k{x), on dM'^ for any {9,h,k) e IJ'{M'^) x Wp~'^{M) x Wp~^{{dMf). Therefore, DK(u) is an isomorphism. Moreover,, evaluating K at uq yields Kuo = {go,uo,x), whereto = -Auo-2H{uo){uo)xA{uo)y G U'{M'^). By the Inverse Function Theorem, there exists a neighborhood Ni about uo in W^'^{M'^) and a neighborhood iV2 about {go,uo,x) in D'{M'^) x Wp~'^M) x iyp~^((aMf ) such that K :Ni^N2 PAGE 22 15 is invertible. Define the function 95= \ 0 in f/o, in Mj. For small enough 5, {gs,UQ,x) ^ -^2, so there exists a unique u e Ni e Wp'^(M^) such that Ku = {gs, uq, x)'i that is, u satisfies dtu Â— Am 2H{u)ux /\Uy + gs, in ' u{0,x) = uo{x), u{t,x) = x{x), in M on {dMf (3.9) Hence, there exists a unique solution, u G Wp'^{M^) to (3.2). By lemma 3.4.1, Vu G CÂ°'t(M''). Finally, applying [17] [IV, Theorem 5.2] to (3.9) gives us that 3.5 Some A-priori Estimates 3.5.1 V(M^) Estimates Lemma 3.5.1 For any smooth bounded domain Q C E^, and function (p e //^(Q) ^ \ PAGE 23 16 Lemma 3.5.1 can be found in [17] [II, Theorem 2.2 and Remark 2.1]. As indicated in [20], using lemma 3.5.1 and a covering argument one can show the following. Refer to (3.5) and (3.6) in section 3.3 for the definitions of D{u) and Eh{u). Lemma 3.5.2 There exists constants c,Rq > 0 such that for any T < oo, any u e V{M'^), any R G (0,it!o] / \Vu\^dxdt < cesssupD{u{t); B^{x))Jmt {x,t)eM'r {[ \V\\^dxdt + R'^ [ \Vu\'^dxdt}. (3.11) Jm'^ Jmt Moreover, for any Xq G M, any R e {0,Ro], any u G V{M'^), and any function rj e CQÂ°{Bji{xo)) depending only on the distance \x-Xo\ and non-increasing as a function of this distance, there holds / \Vu\^r]'^dxdt < cesssupD{u{t); B^{xo))Jm'^ 0 PAGE 24 17 / \dtu\^dxdt + EÂ„{u{T)) = Eh{uo). (3.13) Proof: Taking the derivative of D{u{t)) and using Greene's identity, we obtain di D{u{t))= [ Vu-Vdtudx=-[ Au-^tudxe L^([0,r]). Using this fact and computing d_ di / Q{u)ux A Uydx = 3 / H{u)ux A Uydtudx we find that ^EH{u{t))= [ [-Au + 2H{u)uxAUy]-dtudx = [ \dtu\''dx e L'{[0,T]). Jm Jm Integrating this last equation on [0, T] gives the final result. We will need to employ the following function. Let g G H'^{M) satisfy Ai? = 0, in M (3.14) 5 = X, on dM. By the theory of elliptic equations, there exists a unique solution g e H^{M) to (3.14) such that PAGE 25 18 (3.15) where c > 0 depends only on M. Recall the definition of e(i?) given in (3.7) in section 3.3. Lemma 3.5.5 Suppose Uq Â€ H\M), x e Hl{dM), Uo{x)\(^dM) = x{x) and H G W^{R^). Then there exist constants ci and ci > 0, depending only on H such that for any solution u G V(M^) of (3.2) and any R G (0,i?o] there holds the estimate [ \V'u\'dxdt < c,D{uo) + cir(l + e,R~') sup D{u{t)) + Cir||x|L. JmT Q PAGE 26 19 [ dt{^^^^)dxdt + [ \Au\^dxdt < ||i/||L/ \Vu\^\/\u\dxdt By lemmas 3.5.2 and 3.5.4, the above inequality becomes <-/ |Atxrd2;dt + c / \Vu\^dxdt. < ce D{u{T)) D{uo) + \ f \Au\^dxdt 2 JM-r (R) [ \V\\'^dxdt + ce{R)TR-'^ sup D{u{t)). (3.17) JM^ 0 PAGE 27 20 for all solutions u G V{M'^) of (3.2), for all R G (0,i?o], and for all Xq G M there holds the estimate: D{u{T); B^{xo)) + / \W^u\^dxdt < 2D(u(0); B^j^ixo)) + C2TR-'^c{R), provided e{R) < Â€2Proof: Fix Xq G M. Let 77 G CqÂ°{B2r{xo)) be a non-increasing function of the distance |a; Â— zqI such that r/ = 1 on Bii{xo) and [Vt^I < ;| in B2r{xo). Multiplying (3.2) by -Au rj^ and integrating over we find that / dt{\\yu\^ri^)dxdt+ [ dtu Vu 2r]Vridxdt + f \Au\'^r]'^dxdt < c PAGE 28 21 Since u is a solution to (3.2), \dtu\ < c{\V'^u\ + iVup). Using this fact and lemma 3.5.2, we get that / \dtu\^rfdxdt PAGE 29 22 < / dt{-\Vu\Y)dxdt + I \Au\Wdxdt + cTR-h{R) < ce{R) [ \V\\'^r)'^dxdt + c5 [ \dtu\'^T]'^dxdt + cTR-^e{R) < c{e{R) + e{R)5) [ \V\\Ydxdt + c{5)TR-h{R). Choosing 62 and 5 small enough, if e(i?) < 62 we get f dt{l\Vu\Y)dxdt+ [ \V\\^r]^dxdt < cTR-h{R). Therefore, D{u{T)-B^{xq))+ [ \V\\Wdxdt< [ l\Vu{T)\'^rj^dx < I \\^u{ PAGE 30 23 and for any 0 < a < 1, ll'^^llc'>-t(QM(,o)) < C(l + \\x\\c->{dM)) (3.22) where c = c{R, a). Proof: Let ^ G C, 00/70)2 be a cut-off function such that 0 < ^ < 1, ^ = 1 on Br{xo) X (^0 r^^o + ^^), ^ = 0 outside of Br{xo) x {to R'^,to + R/^), m\ < for |/| < 2 and |6| < Set [/ = <. Then Ut = AUf, ' U{0,x) = 0, in (zo) on B^{xo) x{t = toR^} (3.23) U{t,x) =^{t,x)x{x), ondM X (to-R^to) where / = 2Vm Â• uA{ 2H{u)u^ A u^^^. Applying [17][VII,Theorem 10.4] and using the function g from (3.14) we have that PAGE 31 24 + II|VmP^|Ilp(q^(zo)) + llxllc-(aQ^(xo))) PAGE 32 25 to complete the lemma. To that end, choose 0 < p < f such that pf sup |Vup = max {(^ uf sup \Vu\^} (3.26) 2 Qp{zo) 0<<^ PAGE 33 Then v e C"+2't+i(5i) and it satisfies 26 Vt = Av Â— 2H{v)vx A Vy, on Si v{x, t) ^ + e"^x), if xi + e~23; G dM n (^o)Choosing a= p + e2 we can conclude from (3.28) and (3.30) that sup \Vv\^ < sup |Vup < e"^ sup |Vwp < 4. (3.32) (i,t)e5i (x,t)eQ^-i/2(2i) (x,t)eo^^^_i/2(zo) In addition, (3.27) implies that \Vv\'{0) = e''\Vu\^{zi) = 1. (3.33) In order to arrive at our final contradiction, we only have left to show that there exists a constant C > 0 such that '[ |Vt;| 1 PAGE 34 27 \Vv,f{0) = 1, (3.37) and / \Vvi\'^dxdt 0 as i ^ oo. (3.38) Applying (3.25) to Vi (since Vi satisfies (3.35) and (3.36)) and taking R Â— 1 and r = |, for any 0 < a < 1 we have W^^iWc-iis,) ^ + \\x\\c^{dM))Therefore, there exists a subsequence {fj^} of {vi} and a function v such that V^;i, ^ Vt; inC"''T(5i) (3.39) where 0 < a' < a. Thus, it follows from (3.38) that / \Vv\^dxdt = 0 Jsi 2 which implies that = 0 on ^i. However, (3.37) and (3.39) gives us that |Vu|(0) = 1. Since v 6 ^''^(^i), this is impossible so (3.34) must be true. Therefore, if (3.30) PAGE 35 28 holds, then KC Wvl'^dxdt [ |V^| JSi < e-^C [ \Vufdxdt < e"^C sup / \Vu\'^dxdt < e~^C sup / \Vu\'^dxdt te[to-R'^,to] JB^ixi) Choosing 63 small enough leads to a contradiction, so (3.29) must be true. Therefore, setting (7 = ^ in (3.28) and using (3.29) gives us that f f ) sup |Vup < p") < 4 (3.40) and so l|Vu|LÂ«,(QM(,Â„)) < CR 1 Finally (3.25) yields 4 where C = C{R,a). PAGE 36 29 3.5.3 Uniqueness Lemma 3.5.8 For any u G HliW') it holds that u G L'^{W') and II'"IIZ,4(K2) < 4||u||i2(K2)||VM||^2(K2). The proof of lemma 3.5.8 can be found in [16]. Lemma 3.5.9 Suppose Uq G H^{M), x Â£ H^{dM), uq{x)\i^qm) = x{^) (^nd H G W^{R^). Further, suppose that u,v e V{M'^) are weak solutions to (3.2) with the same initial condition u{0,x) = ^(0, x) and the same boundary condition u{t, x)\dM = v{t,x)\dMThen u = v on M. Proof: Let w = u Â— v. Since u and v are solutions to (3.2), we have \dtw Aw\ < \2H{u)ux Auy 2H{v)vx A Vy\ < 2\H{u) H{v)\\u, A Uy\ + 2\H{v)\\u, A {u v)y\ + 2\H{v)\\{u v)^ A Vy\ < \w\\Vuf + c\Vw\{\Vu\ + \Vv\) Note that since u,v E V{M'^), then for a.e. t G [0,r], u,v Â£ H'^{M) ^ PF^'*(M). If we multiply the above inequality by \w\, integrate over M, and apply lemma 3.5.8 we get < PAGE 37 30 +c{f \Vw\'')'2{[ (|Vu|^ + |Vu|^)^(/ \w\^dxdt)'^ PAGE 38 31 M X (0, T) where T > 0 is characterized by the condition lim sup D{u{t)]B^{x)) > e for all R > 0, with a constant e > 0 depending only on H. Proof: 1. Since Uq G H^{M) and x ^ Hi{dM), there exist sequences v]^ e CÂ°Â°(M) n H\M) and x"^ e CÂ°Â°(aM) n Hl{dM) satisfying u'^{x)\sm = x'^ix) such that wS" ^ uo strongly in F^(M) (3.41) and X"" X strongly in (dM). (3.42) By theorem 3.4.2, there exist 5m > 0 and G F(M*Â™) such that u"" solves (3.2) on M^"" with initial and boundary data Mq* and x"^ respectively. Let e = min{ei,e2,e3} where ci, 62, and 63 are determined from lemmas 3.5.5, 3.5.6 and 3.5.7 respectively. By (3.41), there exists some R> 0 such that for all x e M, D{u^-B^^{x))<\. Then by lemma 3.5.6, if T = 0{R^t) then PAGE 39 32 sup D{u"'{ty,B^{x)) PAGE 40 33 u"" weakly in y(M^). (3.46) In addition, (3.45) implies that Vm"* ^ Vu uniformly on (M'^) (3.47) for any M' C M. 3. We claim that (3.46) implies that u is a weak solution of (3.2) on V{M'^). In fact, by (3.46), as m Â— >^ oo we have that V^"* ^ V\ weakly in ^^(M^), (3.48) dtu"" dtu weakly in ^^(M^), (3.49) and Vu"^ ^ Vu weakly in LÂ°Â°(0, T; L2(M)) (3.50) From (3.48), the Sobolov Embedding Theorem gives us that, VvT^Vu strongly in ^^(0, T; L^(M)) (3.51) PAGE 41 34 and u'^^u strongly in 1^(0, T; LÂ°Â°(M)). (3.52) By (3.48) and (3.49), as m Â— ) oo we have that dtu"" Au"" dtu Au weakly in L'^{M^). So it only remains to show that as m Â— > oo, H{u'^)u'^ Au"^^ H{u)u^ A Uy weakly in L^{M'^). (3.53) By (3.52), as m Â— > oo H{vJ^) H{u) strongly in 1^(0, T; LÂ°Â°(M)). (3.54) Furthermore, (3.51) and (3.50) imply that < ^ strongly in 1^(0, T; L^{M)) and weakly inLÂ°Â°(0,T; L2(M)) (3.55) So (3.53) follows directly from (3.54) and (3.55). Therefore, u e V{M'^) is a solution of (3.2) in the sense of distribution so the equation holds in L^{M'^) and a.e. in fact it is a solution in L'^{M'^) and hence the equation holds a.e. Moreover, from (3.45) and (3.47), Vu G CÂ°'t(M^^). Then by [17][IV,Theorem 5.2] u Â€ C2+"'i+t(Mj) is a classical solution of (3.2). PAGE 42 35 5. This solution can in fact be extended up until the first time of energy concentration. This is due to the fact if (3.6.1) does not hold for any x E M a.t time t = T, then (3.50) holds. Since u e V{M^), we also have that dtu G L'^{0, T; L'^{M)). Therefore, u Â€ C{[0,T]; H\M)). In particular, u{T) e H\M). Therefore, the above argument guarantees the existence of a solution to (3.2) using u{T) and the new initial data and we can continue this solution to a larger time interval. We can repeat this argument and continue the solution up until the first time of energy concentration, that is, when t = Ti. Moreover, the solution is regular on M x (0,Ti]). Remark 3.6.2 If x ^ C^-^''{dM), we can choose Uq* G CÂ°Â°(A/) so that itQ^laM Â— X and (3.41) holds. Then by lemma 3.5.7, ll^^llcÂ°-t(Q|(zo)) ^ + \\x\y+->{aM)) 4 where C = C{R,a). In this case, we can conclude that u e C^'*""'^"'"f (M^) for any 0 PAGE 43 36 and 0 < k < K , characterized by the condition, limsupZ)(u(0;5fi (4)) > ^ for all R> 0, with e depending only on H. Proof: Theorem 3.6.1 guarantees the existence of a unique, regular solution to (3.2) on M X (0, Ti) where Ti > 0 is the time of the first singularity. By the additivity of the energy and (3.56), there can only be finitely many singularities at that time, {{x\,Ti)}\l^. Therefore, for any R 6 (0,i2o] and any M' C M -iJ^^^B^{x\) we have that D(u(Ti),M') < limmfD(M(T),M') < lmiinfD(u(T)) T,-^^,D{u{nB^{x\)) < L>(u(0)) Lie. Letting R^Q, M' ^ M, one can conclude that D{u{T{)) < D{u{Q)) Lie. (3.57) Since D{u{T{)) < D(u(0)), we have that u{T{) e H\M). Therefore, w(Ti) can be used as new initial data for (3.2) and another application of theorem 3.6.1 implies that the solution, u, can be continued to a larger time interval, [0,72]. This process can be repeated at each time u has a singularity, thus obtaining a solution on [0,oo). Furthermore, (3.56), (3.57), and the fact that D(u(0)) < oo, imply that PAGE 44 37 there are only finitely many times {Tfcjj^i at which u can attain singularities. Hence, u can only have finitely many singularities on [0, oo). 3.8 Behavior of Singularities Theorem 3.8.1 Let u : M x (0,Ti] be a solution to (3.2) obtained in theorem 3.6.1 with smooth boundary data, satisfying (3.56). Suppose {xo,Ti), xq G M, is a point such that for all R> 0 where e is determined in theorem 3.4-2. Then the following holds. (i) If xq G M, then there exist sequences tm Ti, x^ ^ Xq, and Rm \ 0 and a non-constant map u Â€ iZ/^g fi Cjg^iW') such that as m ^ oo, the resettling sequence \imsnpD{u{t)-B^{xo)) > I (3.58) '^m(^) Â— ui^RjjiX + Xjji, tjji^ converges strongly to u{x) in Hj^^ n C/^ 7oc(^^) (i'l^du satisfies An = 2H{u)ux A Uy in R^. Au = 2H{u)ux A Uy in < u{x) = x{xo) on dRl PAGE 45 38 where 1^ = {{x,x)\y > Â—a}. Proof: First we will prove (ii). Let {xq, Ti) be a point satisfying (3.58) where xq e dM and Ti > 0. By the finiteness of the singular set, there exists a 5 > 0 such that u e CÂ°Â°((Bf (:ro) x [T, 5\T,])\{x,,T{)). Let {Rm} be a sequence of real numbers such that Rm \ 0. By lemma 3.5.7, there exists a sequences {xm} and {t^} such that Xm xq, Tm Ti and sup sup / \Du\'^{y,t)dy = e^ / \Du\'^{x,Tra)dx. (3.59) te[Ti-S^,Ti] lesf (xo) JB^^ix) JB^J^^m) Fix 0 < < 2c2l{R) where C2 is from lemma 3.5.6. Define Bm = {xe W\Rn,X + Xme (xq)} and Then Wm{x, t) = u{RmX + X^, Rl,t + T^). : jB^ X [-C^ 0] ^ satisfies the equation dtWm = 2H{Wm){Wm)x A {Wm)y OU Bm X [-C^, 0] (3.60) Wm{x,t)=x{R^X + Xm) ioTRmX + XmEdM PAGE 46 39 dxdt By lemma 3.5.4, we have that [ [ < / \dtu\'^dxdt Â— > 0 as m ^ oo. So, for all t Â€ [-C2,0], / \dtWm\^{x,t)dx Â— > 0 as m ^ oo. (3.61) JBm By lemma3.5.6 and (3.59), for any f 6 [r^ C^R^, nn], e = / \Du\^{x,Tm)dx < 2 / \Du\\x, t)dx + C2{Tm r)R-hiR) < 2 / \Du\^{x,t)dx + -. Therefore, / \Du\'^{x,t)dx>-. So we can conclude that, / \DwJ'{x,t)dx= [ \DuWx,Rl^t + Tm)dx JBm JBm PAGE 47 40 Therefore, for all t e [-C^,0], >[ \Du\'^{x,Rl,t + Tm)dx>^. / \Dwm\'^{x,t)dx>^. (3.62) Finally, by (3.59) we can see that sup sup / \Dwm\'^{y,t)dy = (3.63) te[-c^,o] xeBm JBi{x)nBm sup sup / \Du\'^{y,t)dy ^ -C^Rl ,Tm] xeB^ (xÂ„) Jb^ (x) te[TÂ„-c^RlÂ„Tm] xeB^ixo) Jb^^{x If '^'^^(g^'^^) ^ oo then fi^ ^ as m ^ oo. Let x G and y G B^{x). For m is sufficiently large, we have that Rmy + G M so by lemma 3.5.7 and (3.63) we get that sup 1 1 L>U;,ji 1 1 (Bi {x)r>Bm ) < c i6[-C2,0] which gives us sup \\DwJ^\(.2+c,^2^ PAGE 48 41 / \Dwm\'^{x,r]m)dx>^ (3.66) and I \DWm{-, Vm) I lcf+"(K2) < ^ (3.67) Therefore, there exists a function u : Â— > and a subsequence of Wjn{-, ijm) such that Wm{; Vm) u strongly in Hl^ n Cl^{^ , R^). (3.68) Set ^ = r;^ on both sides of (3.60). Letting m ^ oo and using (3.65) and (3.68) we can conclude that u satisfies the equation Au = 2H{u)ux A Uy in R^. (3.69) Moreover, u is non-constant by (3.66) and (3.68).. Therefore, taking tm = T^Vm + Tm we have that Vm{x) = Wm{x, T]m) = u{RmX + Xm, tm) is the sequence desired in the theorem. The proof of xq G M follows an analogous argument to that we have just presented, so it will be omitted for brevity. It only remains to show (m) case 2. Suppose ^'^^(^--^^^ a as m ^ oo. Then B^^Rl = {{x^'),x^^))\x^')>-a}. PAGE 49 42 Note that on the set {x^'^^ = -a}, we have that RmX + Xm^ Xq. Also, for all x such that RmX + Xm e dM, Wm{x,t) = x{RmX + x^). So by lemma 3.5.7 and (3.63) we can conclude that Therefore, sup ||lOm||c2+Â°(Bi(x)nSÂ„) < C. ie[-c2,o] sup WWrnWcf+^iRl) < Cholds. Employing a similar argument as used in the first case, we can obtain a smooth, non-constant map u : -^R^ and a subsequence of Wm{x,rim) such that Wm{-,Vm)^u strongly in H^^^ n C/o,(E^, R^) and u satisfies the equation Am = 2H{u)ux A Uy in u{x)=x{xo) on^R^.n PAGE 50 CHAPTER 4 MODIFIED MEAN CURVATURE FLOW EQUATION 4.1 Evolution of Level Sets by Mean Curvature Consider a hypersurface, c{t) = {xi{t), ...,Xnit)), evolving in the direction opposite its unit normal, N, with speed equal to its mean curvature, k, that is, c'{t) = -kN. This evolution may not be smooth as the hypersurface may "pinch off' and change topology as i -)Â• oo. To compensate for this, we can embed c{t) into a surface, u, so that c{t) is the zero level set of u at time t. Let w : R" -> [0, oo) be defined such that c{t) = {xeW I u{x,t) = 0}. Notice that on this level set of u, the chain rule gives us SILiWxi(a:i)f + Mt = 0. 43 PAGE 51 44 In fact, Ut = -Vu Â• c'{t) --Vu Â• -kN = Vu Â• div Wu\div Vu Vu Therefore, the evolution of the level sets can be studied by considering the mean curvature flow where I{x) is a smooth surface such that c(0) = {x G E" | I{x) = 0}. This equation has been studied in detail by Evans and Spruck in [7]-[10]. They proved the existence of a unique, smooth solution as well as various geometric properties and the finite extinction time of the level sets of the solution. In the following, we introduce a control factor into the right hand side of (4.1). The existence of a unique viscosity solution to our modified equation was proved in [4]. We study some geometric properties of the level sets of this solution as well as prove that although the control factor may slow the evolution of these level sets, they will still eventually shrink to a point. < in W X [0, oo) (4.1) u{x,0) = I{x) in R' PAGE 52 45 4.2 Applications to Image Processing In order to understand the applications of the mean curvature flow equation to image processing, we need some basic definitions. Definition 4.2.1 Let C M" and TZ C R. An intensity map is a map u : ^ ^ TZ where u{x) represents the intensity of the image at the point x ^ Q. For 2 or 3 dimensional images, we usually take the image domain to be f2 = [0, 1]^ or [0, 1]^ respectively, and the possible range of intensity values to be 7?. = [0, 1]. Most PDE-based image processing models employ Neumann boundary conditions. This enables us to consider the image to be defined on all of R" by reflection. The evolution of a curve as described in section 4.1 can be used to identify or extract objects from an image. In order to do this, the curve must be slowed down or halted at the recognition of an edge (or object boundary) in the image. Furthermore, the model we are studying here can also be used to retain and enhance significant features in an image while removing noise and obstructions. Therefore, we need to formally define an "object boundary" or "edge". Intuitively, we think of "edges" as the boundaries of the significant objects in an image. At these locations, the intensity of the image should change significantly. Definition 4.2.2 If u is the associated intensity map of an image, then an edge is any place where |Vw| achieves a local maximum. Evans and Spruck proved in [9] that the level sets of (4.1) will eventually shrink to a point. In order to use these curves to extract significant features we need to introduce a control factor that inhibits the evolution near the edges of the image. To that end, if I{x) is an image, one can identify its edges by looking for large changes in intensity; that is, local maxima of |V/|. We do not necessarily want to assume that / is diff"erentiable; in fact, one would hope that / is not even continuous PAGE 53 46 at an edge. Therefore, we will use the local maxima of \VGc * /| to identify edges, where GÂ„{x,t) = ;^^=fr exp(^^) is the Gaussian filter. This enables us to compute the gradient without any prior knowledge about the derivatives of /. Now that we have a mechanism for locating the edges of an image, we can construct a function that will automatically detect these edges. Any function, g, such that 1, away from an edge 0, close to an edge will do just that. Our choice is Introducing g{x) into the right hand side of the mean curvature flow equation (4.1) provides a mechanism for slowing or stopping the evolution of the level sets as they approach object boundaries. In the remainder of the chapter, we study the effect of the control factor, g, on this evolution. 4.3 Anisotropic Diffusion In addition to the evolution of the level sets of its solutions, the mean curvature flow equation, (4.1), also yields a special type of difl"usion which is very useful in image restoration. Let ^ = = ^^^{-Uy,u^). Notice that the right hand side of (4.1) can be simplified to PAGE 54 47 |vÂ«MM^) = |vÂ«MÂ»(p^,^) "^y^xx '^^x^y^xy ~^ ^x^yy " |Vu|2 Notice that the second order term is simply u^^, the second directional derivative of u in the direction of ^ || Vu^. Therefore, (4.1) performs a diffusion only in the direction perpendicular to Vu. This means that diffusion will only be performed in the direction tangential to the edges. This should theoretically prevent blurring across edges. Moreover, the control factor g helps regulate the amount and location of this diffusion. Since \^u\div {g{x)^^ = g{x)\Vu\div (j^) + ^9 Vm, the diffusion term is inhibited near the edges, which gives us another mechanism for preserving significant features. In addition, Vg Â• Vu is a hyperbolic term which allows for shocks. These shocks encourage discontinuities at the edges, which is of course what we would hope for. I PAGE 55 48 4.4 Existence and Uniqueness of the Weak Solution Consider the problem ut = \Vu\div [gix)^^ in K" x [0,oo) u{x, 0) = I{x) on R" (4.3) where 9{x) = l + k\VG^*I\ In [4], Chen, Vemuri and Wang showed that there exists a unique bounded viscosity solution to the approximate problem ul = {\Vu'\'^ + e^)-2div(g(x) ^^^^] in R"xfO,oo) u'{x,0)=I{x) on R". (4.4) provided / is Lipschitz continuous on R" . Furthermore, they proved that as e -> 0, ^ u locally uniformly on R" x [0, oo) where u denotes the weak solution to (4.3) in the sense of viscosity [5]. In this setting, we can study the extinction time of the level sets of u as well as some of their geometric properties. The extinction time problem can be reformulated as finding the time at which the measure of the level sets is equal to zero. Since this is always the case if we use Lebesgue measure to determine the size of an n 1dimensional object in R", we need to use use a different measure for defining the extinction of the level sets. PAGE 56 49 4.5 Hausdorff Measure We are accustomed to assigning measures to n-dimensional subsets of K" , the most common of these being Lebesgue measure. However, it is sometimes convenient to be able to consider measures on m-dimensional subsets of E" (m < n). The following information can be found in Federer [12]. In 1918, F. Hausdorff introduced an m-dimensional measure on K" which can be defined on all subsets of R" . Moreover, his definition is consistent with the usual notion of area for m-dimensional submanifolds of R" . The definition is as follows. For any subset 5 C R" , define the diameter of S to be diam{S) = sup \x Â— y\. x,yeS Let am be the Lebesgue measure of the closed unit ball B(0, 1) C R"*. For small S > 0 define {Sj}j%^ to be a countable covering of A such that Sj C R" and diam{Sj) < 5 for j = 1,2,3.... Then for any A C , m < n, define its m-dimensional Hausdorff measure to be n''{A)^nm inf f a^f^J^II^Y. For any a e ^ we define the density of ^ at a to be the quantity We then say that a is a point of density for the m-dimensional Hausdorff measure if e"'(R'",a) 7^0. It is natural to restate the level set extinction problem as finding the time at which there are no points of density on the zero level set of the solution to (4.3). PAGE 57 50 4.6 Monotonicity In order to further discuss the geometric properties of the level sets of the solution to (4.3) we first need the following monotonicity estimate. Let be a solution of (4.4). Theorem 4.6.1 (Version 1) For all t G [0,T], Proof : Fix S > 0 and define / g{x)\Vu\dx < / g{x)\VI\dx. 0(x) ^ e-''(i+l-l')^ for all X eW. Define Note that \x\ ^^'\t) ^ [ g{x)ip^\Vuf + e^)Ux. (1 + 1X1^)2 Using the notation i PAGE 58 we can estimate 51 da; = / (f)'^div g{x) -/ = -/ -/ ^ -/ VR"x / ^R"x / / JR + + R"x{f {t {t {t M"x{( (|Vn-|2 + e2)i uldx 2q{x)(b r-u)dx 2g{x)(j)V(f>-Vu'H'dx (P^H'YilVuf + e^y^dx g{x)(f)\H'f{\Vu'\^ + e'')Ux g{x)\V(l)\''{\Vuf + e''f2dx g{x)cl)^{\Vuf + e^)'^dx Therefore, By Gronwall's inequality, [ g{x)(l>'{\W\' + Â£'y^dx PAGE 59 52 Letting e, (5 Â— > 0, we can conclude that / g{x)\Vu\dx< / g{x)\VI\dx for any t > 0. D To prove the that the level sets will vanish as well as derive a bound on their extinction time, we need to use the following alternate monotonicity estimate. For any point Zq = {xq, to) G M" x [0, oo), define G,,{x,t) ^ _i -exp(-Â— -) V^47r(fo t) 4(to t) if t < to and (x, t) = 0 otherwise. Theorem 4.6.2 (Version 2) For all t G [0,T], / g{x)\Vu\G,,dx PAGE 60 Then m'it) = / g{x) I G.,dx + / g{x){\Vuf + e^)HG^o)tdx Â— Â— [ div i q(x) r I ^^^G^n + / ^|io Vnf + e^f^G.^dx / six) Jj" u\VG,,dx Therefore, / ({t} 2(^0 t)' JK"x{i} + / '^\h-t\g{x){\Vu^f + e^Y^G,,dx + / ^^7^^MVu'WG,,dx jR-^x{t} JK"x{f} ^ c{t} |2 PAGE 61 By Gronwall's inequality, 54 I Letting e Â— > 0, we can conclude that / 9{x)\Vu\G,,dx PAGE 62 55 By lower semi-continuity and the Lebesgue Dominated Convergence Theorem, / g{x)\Vv\dx < lim inf / g{x)\Vv''\dx < lim / g{x)\VvQ\dx lim g{x)\VvQ\dx / g{x)\Vvo\dx So V also satisfies a monotonicity property: / g{x)\\/v\dx< g(x)\\/vo\dx One of the consequences of the monotonicity formula is a certain " doubling up" {l,s > 0 0,s > 0 and ^-(s) = < . Set Vq = ^+(Â«o) 0,s<0 I l,s<0 and Vq = ^_(uo). Then / \Vv\dx < [ g{x)\Vv\dx < / g{x)\Vvo\dx -/ g{x)\Vv+\dx+ j g{x)\Vv^\d: < -?^"-'(ronR") + -?^"-i(ronE") C C PAGE 63 56 where c = 1 > 0. (4.6) 1 + ||-^||lÂ°Â°(R") This implies that Ft has finite perimeter for any t > 0. In addition, if Ft is the closure of an open set (this may happen in the case that Ff develops an interior), then we can derive a bound for the size of dTt, namely Again, consider the level set Fj and the sequence of smooth functions {"^k} defined in the previous section. As in the last section, since u is a weak solution of (4.3) and is continuous, v'^ = ^fc('u) is a weak solution with initial condition Vq = ^fc(/). Therefore, monotonicity {Version 2) holds for u*; that is, We will now show that even with the control factor g{x), although the motion of the level sets is inhibited, the level sets will still vanish in a finite time interval. This is primarily due to the fact that 0 < c < g{x) < 1 where c is defined in (4.6). First we show that if only a small portion of the initial level set lies within a ball, then there exists a finite time at which the evolved level set no longer intersects a smaller concentric ball. This result is analogous to Brakke's "clearing out" lemma. 4.8 Extinction Times PAGE 64 57 Theorem 4,8.1 Suppose Co = W'^iVo) < oo. Given p> 0, there exists constants a,r] > 0, possibly depending on Co, such that if 7{"-i(ron52p(0) PAGE 65 58 Choose OL = a{Co,n) > 0 and rj = 77(0;) to be small enough so that the right hand side is less than C2. Thus the result holds. Finally we can construct a bound on the extinction time of {rt}t>oTheorem 4.8.2 Let T = sup{t > 0 | 7^ 0}. Then there exists a constant C3 such thatO< T < Ci?{"-nro)^. Proof : IfW-^iTo) = 00 then we are done. Assume that 0 < n^'-^iVo) < 00. Let p > 0 satisfy ?^"~^(ro) = r]p^~^ where r] is determined in theorem 4.8.1. Then for all xo G M", n''-\TonB2,{xo)) PAGE 66 CHAPTER 5 FURTHER QUESTIONS 5.1 H-systems and Related Topics Many interesting problems arise from the analysis of the flow of H-systems. Here we proved the existence of a solution which is regular up until the first time of energy concentration. Global existence was shown assuming an energy inequality holds. It would be interesting to determine if there are weaker sufficient conditions that provide for global existence. Another issue to consider is the nature of the singularities. For harmonic maps it has been shown that the singularities can only occur on the interior of the region. However, it is still not known whether the singularities of the Landau-Lifshitz equation or those of the flow of H-systems occur in the interior or on the boundary. Finally, we can also investigate this problem in higher dimensions. There has been extensive study of solutions of harmonic maps from E" -> ; however, little is known of solutions to the latter equations in this setting. 5.2 PDE-Based Image Processing It is curious that using (4.2) as the control factor in (4.3) slows the evolution of its solutions, yet the level sets still shrink to a point. This is due to the fact that 1 1 ~ 1 + A;|VG,*/| l + ||/||L<Â»(Rn) ^ 59 PAGE 67 60 If we update g at each time step; that is define = l + k\VG..u\ ^ Â°' then (4.7) may fail in the proof of theorem 4.8.1 and it is not clear that the level sets will eventually vanish. It would be interesting to see if in fact the level sets do not shrink to a point and if so, determining the location at which they stabilize. There is also much analysis to be done on other existing models. Consider the flow Ut = div{g{x)^~^^) X{u I) where = i + ivk*/r Since the equation is in divergence form, one can look for solutions in the space of functions of bounded variation. Furthermore, one can also see that this equation is equivalent to the gradient flow min / g{x)\Vu\dx. Jn In this latter setting, if existence of a solution is known one can ask about the long term behavior of this solution. Building on this model, consider the equation Ut = \^u\div{g{x)^^) X\Vu\{u I). This equation is no longer in divergence form. Due to this fact, only the existence of a viscosity solution has been shown and a better solution is not possible. Nonetheless, PAGE 68 61 we can still investigate geometric properties of the solution as we have done in this work. PAGE 69 REFERENCES Brezis, H., and Coron, J., "Multiple Solutions of //-Systems and Rellich's Conjecture" Comm. Pure Appl. Math. no. 2, 149-187 (1984). Chang, K.C., "Heat flow and boundary value problems for harmonic maps" Ann. Inst. Henri Poincare Anal. 6, 363-395 (1989). Chen, Yunmei, "Existence and Singularities for the Dirichlet Boundary Value Problem of Landau-Lifshitz Equations" preprint Chen, Vemuri, and Wang, "Image Denoising and Segmentation via Nonlinear Diffusion" Computers and Mathematics with Applications no. 39 131-149 (2000) Crandall, M.G., Ishii, H., Lions, RL., "User's Guide to Viscosity Solutions of Second Order Partial Differential Equations" Bulletin of the American Mathematical Society 27 no. 1, 1-67 (1992) Eells, J., and Sampson, J.H. "Harmonic mappings of Riemannian manifolds" Am. J. Math. 86, 109-160 (1964). Evans, L.C., and Spruck, J. "Motion of level sets by mean curvature I" J. Diff. Geom. 33 635-681 (1991). Evans, L.C., and Spruck, J. "Motion of level sets by mean curvature H" Transactions of the AMS no. 1, 321-332 (1992). Evans, L.C., and Spruck, J. "Motion of level sets by mean curvature HI" The Journal of Geometric Analysis 2, no. 2, 121-150 (1992). Evans, L.C., and Spruck, J. "Motion of level sets by mean curvature IV" The Journal of Geometric Analysis 5, no. 1 77-114 (1995). Evans, L.C., Partial Differential Equations. American Mathematical Society (1998) Federer, H., Geometric Measure Theory. New York, SpringerVerlag (1969) Guisti, E., Minimal surfaces and functions of bounded variation. Monogr. Math. 80, Birkhauser, Basel-Boston-Stuttgart, (1984) Hamilton, R., Harmonic maps of manifolds with boundary. (Lect. Notes Math., vol. 471) Berlin, Heidelberg, New York, Springer (1975) Hildebrandt, S., "On the Plateau Problem for Surfaces with Prescribed Mean Curvature" Comm. Pure Appl. Math. 23 97-114 (1970). 62 PAGE 70 63 [16] Ladyzenskaja, O.A., The mathematical theory of viscous incompressible flow. (2nd edition) Math, and Appl. 2, Gordon and Breach, New York, etc. (1969). [17] Ladyzenskaja, O.A., Solonnikov, V.A. &: Ural'ceva, N.N., Linear and Quasilinear Equations of Parabolic Type. Trans. Math. Monographs 23, AMS (1968). [18] Osher, S., and Sethian, J. A. "Fronts propagating with curvature dependent speed: Algorithms based on Hamilton-Jacobi formulations", J. Comput. Phys. 79, 12-49 (1988) [19] Rey, Olivier, "Heat flow for the equation of surfaces with prescribed mean curvature" Math. Ann. 291, 123-146 (1991). [20] Struwe, M., "On the evolution of harmonic mappings of Riemannian surfaces" Commun. Math. Helv. 60, 558-581 (1985). [21] Struwe, M, "The existence of surfaces of constant mean curvature with free boundaries" Math. Helv. no. 1-2, 19-64 (1988). [22] Struwe, M, Geometric Evolution Problems 1992.?? [23] Struwe, M., Variational Methods . Springer Verlag 1990. [24] Struwe, M., "Large i/-Surfaces Via the MountainPass Lemma" Math. Ann. 270, 441-459 (1985). [25] Wente, H. C, "An Existence Theorem for Surfaces of Constant Mean Curvature" J. Math. Ann. Appl. 26, 318-344 (1969). PAGE 71 BIOGRAPHICAL SKETCH Stacey Chastain was born in Fort Riley, Kansas, on October 8, 1971. She graduated from Livingston High School in Livingston, New Jersey, in 1989. She received a Bachelor of Science degree from the University of Florida in 1993. She then completed a Master of Science degree from the University of Florida in 1995 after which she began further studies under the direction of Yunmei Chen. 64 PAGE 72 I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Yunmei Chen, Chairman Professor of Mathematics I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Gerard Emch Professor of Mathematics I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully ^adequate, in scope and quality, as a dissertation for the degree of Doctor oÂ£ Phil Bernard Mair Professor of Mathematics I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Murali Rao Professor of Mathematics I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doct^^r of^hilosophy. Peter Hirschreld Professor of Physics This dissertation was submitted to the Graduate Faculty of the Department of Mathematics in the College of Liberal Arts and Sciences and to the Graduate School and was accepted as partial fulfillment of the requirements for the degree of Doctor of Philosophy. May 2000 Dean, Graduate School |