Computation of geometric partial differential equations and mean curvature flow


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1 Acta Numerica (2005), pp c Cambridge University Press, 2005 DOI: /S Printed in te United Kingdom Computation of geometric partial differential equations and mean curvature flow Klaus Deckelnick Institut für Analysis und Numerik, OttovonGuerickeUniversität Magdeburg, Universitätsplatz 2, D Magdeburg, Germany Gerard Dziuk Abteilung für Angewandte Matematik, AlbertLudwigsUniversität Freiburg i. Br., HermannHerderStraße 10, D Freiburg i.br., Germany Carles M. Elliott Department of Matematics, University of Sussex, Mantell Building, Falmer, Brigton, BN1 9RF, UK Tis review concerns te computation of curvaturedependent interface motion governed by geometric partial differential equations. Te canonical problem of mean curvature flow is tat of finding a surface wic evolves so tat, at every point on te surface, te normal velocity is given by te mean curvature. In recent years te interest in geometric PDEs involving curvature as burgeoned. Example of applications are, amongst oters, te motion of grain boundaries in alloys, pase transitions and image processing. Te metods of analysis, discretization and numerical analysis depend on ow te surface is represented. Te simplest approac is wen te surface is a grap over a base domain. Tis is an example of a sarp interface approac wic, in te general parametric approac, involves seeking a parametrization of te surface over a base surface, suc as a spere. On te oter and an interface can be represented implicitly as a level surface of a function, and tis idea gives rise to te socalled level set metod. Anoter implicit approac is te pase field metod, wic approximates te interface by a zero level set of a
2 2 K. Deckelnick, G. Dziuk and C. M. Elliott pase field satisfying a PDE depending on a new parameter. Eac approac as its own advantages and disadvantages. In te article we describe te matematical formulations of tese approaces and teir discretizations. Algoritms are set out for eac approac, convergence results are given and are supported by computational results and numerous grapical figures. Besides mean curvature flow, te topics of anisotropy and te iger order geometric PDEs for Willmore flow and surface diffusion are covered. CONTENTS 1 Introduction 2 2 Some geometric analysis 12 3 Definition and elementary properties of mean curvature flow 17 4 Parametric mean curvature flow 19 5 Mean curvature flow of graps 28 6 Mean curvature flow of level sets 36 7 Pase field approac to mean curvature flow 43 8 Anisotropic mean curvature flow 51 9 Fourt order flows 70 Appendix 85 References Introduction A geometric evolution equation defines te motion of a ypersurface by prescribing te normal velocity of te surface in terms of geometric quantities. As well as being of striking matematical interest, geometric evolution problems occur in a wide variety of scientific and tecnological applications. A traditional source of problems is materials science, were te understanding of te strengt and properties of materials requires te matematical modelling of te morpology of microstructure. Evolving surfaces migt be grain boundaries, wic separate differing orientations of te same crystalline pase, or solid liquid interfaces exibiting dendritic structures in undercooled solidification. On te oter and newer applications are associated wit image processing. For example, in order to identify a dark sape in a ligt background in a twodimensional image a socalled snake contour is evolved so tat it wraps around te sape. In tis article we survey numerical metods for te evolution of surfaces wose normal velocity is strongly dependent on te mean curvature of te surface. Te objective is to find a family {Γ(t)} t [0,T ] of closed compact and orientable ypersurfaces in R n+1 wose evolution is defined by specifying te velocity V of in te normal direction ν. An example of a general geometric
3 Computation of geometric PDEs and mean curvature flow 3 evolution equation is V = f(x, ν, H) on Γ(t), (1.1) were f depends on te application and te x dependence migt arise from evaluating on te surface Γ(t) field variables wic satisfy teir own system of nonlinear partial differential equations in R n+1 away from te surface. It is important to note tat, in order to specify te evolution of te surface, it is sufficient to define te normal velocity. Te prototype problem is motion by mean curvature, forwic V = H on Γ(t), (1.2) were H is te sum of te n principal curvatures of Γ(t). We call H te mean curvature rater tan te aritmetic mean of te principal curvatures. Our sign convention is tat H is positive for speres, wit ν being te outward normal. It is well known tat, starting from an initial surface Γ 0, tis equation is a gradient flow for te area functional, E(Γ) = 1dA. (1.3) In applications te area functional is an interfacial energy wit a constant energy density 1. Equation (1.2) may be viewed as an analogue for surfaces of te parabolic eat equation u t u =0. On te oter and, anoter geometric equation is V = Γ(t) H on Γ(t), (1.4) were Γ(t) is te Laplace Beltrami or surface Laplacian operator on Γ(t). Tis can be viewed as an analogue of te spatially fourt order parabolic equation u t + 2 u = Approaces In order to solve a surface evolution equation analytically or numerically, we need a description of Γ(t). Eac coice of description leads to a particular nonlinear partial differential equation defining te evolution. Tus te computational metod depends strongly on te way we coose to describe te surface. For tis article we sall focus on four possible approaces. Parametric approac. Te ypersurfaces Γ(t) aregivenas Γ(t) =X(,t)(M), were M is a suitable reference manifold (fixing te topological type of Γ(t)) and X : M [0,T) R n+1 as to be determined. Here X(p, t), for p M, is Γ
4 4 K. Deckelnick, G. Dziuk and C. M. Elliott Figure 1.1. A dumbbellsaped twodimensional surface parametrized over te unit spere. Figure 1.2. A lemniscate, parametrized over te unit circle. te position vector at time t of a point on Γ(t). If we are interested in closed curves in te plane ten M can be te unit circle S 1, wereas if Γ(t) isa twodimensional surface ten M could be te unit spere S 2 (see Figures 1.1 and 1.2). Geometrical quantities are easily expressed as derivatives of te parametrization so tat evolution laws suc as (1.2) may be translated into nonlinear parabolic systems of PDEs for te vector X. Wit tis approac tere is no notion of te surface being te boundary of an open set and aving an inside and outside, so selfintersection is perfectly natural for smoot parametrizations and is not necessarily associated wit singularities. For example in te plane a figure of eigt curve can be smootly mapped onto te unit circle one to one (Figure 1.2). At te crossing point te curve as two smootly evaluated normals and curvatures wic depend on te parametrization. A parametrized curve evolving by mean curvature can evolve smootly from tis configuration. Graps. We assume tat Γ(t) can be written in te form Γ(t) ={(x, u(x, t)) x Ω}, were Ω R n and te eigt function u : Ω [0,T) R astobe found. We sall see tat te law (1.2) leads to a nonlinear parabolic PDE for u. Clearly, te assumption tat Γ(t) is a grap is rater restrictive; owever, tecniques developed for tis case ave turned out to be very useful in understanding more general situations. Since te eigt is a smoot function we can view Γ(t) as dividing Ω R into two sets, namely te regions above and below te grap.
5 Computation of geometric PDEs and mean curvature flow 5 Figure 1.3. Level lines of a level set function (rigt) for te figure of eigt curve (left). Figure 1.4. Grap of a level set function for te figure of eigt curve, cut at te zero level. Negative part left and positive part (grapically enlarged) rigt. Level set metod. We look for Γ(t) as te zero level set of an auxiliary function u : R n+1 [0, ) R, tat is, Γ(t) ={x R n+1 u(x, t) =0}. Te law (1.2) now translates into a nonlinear, degenerate and singular PDE for u. Clearly intrinsic to tis approac is te notion of Γ(t) being a dividing surface between te two regions were te level set function is positive and negative. Tus we ave te notion of inside and outside. In order to describe a figure of eigt by a level set function it is necessary to ave te level set function positive and negative, as sown in Figures 1.3 and 1.4. Pase field approac. Te pase field approac is based on an approximation of te sarp interface by a diffuse interface Γ ɛ (t) ={x R n+1 1+Cɛ u ɛ (x, t) 1 Cɛ} of widt O(ɛ), across wic te pase field function u ɛ as a transition from approximately one bulk negative value 1 to approximately a second positive bulk value +1. Te zero level set of te pase field function approximates te surface. Just as in te level set metod tere is te notion
6 6 K. Deckelnick, G. Dziuk and C. M. Elliott of a material interface separating an inside and outside and in te basic implementation interface selfintersection and topological cange are andled automatically. Te bulk values of te pase field function correspond to te minima of a omogeneous energy function wit two equal double wells. Interfacial energy is assigned to te diffuse interface via te gradient of te pase field function. For motion by mean curvature te evolution is defined as a semilinear parabolic equation of reaction diffusion or Ginzburg Landau type. Frequently in applications matematical models are derived wic, from te beginning, involve diffuse interfaces and pase field functions. Comments Conceptually te grap formulation is te simplest and most efficient. It involves solving a scalar nonlinear parabolic equation in n space dimensions and directly computes te surface. However, tere are many circumstances were te surface is not a grap. Furtermore, even if te initial surface is a grap it is possible tat over te course of te evolution tat property migt be lost, despite te surface evolving smootly. Tis would lead to gradient blowup of te solution of te grap equation. Tere is te possibility tat te solution of a numerical discretization exists globally and appears to be stable even toug tere is no solution to te continuous equation. Te parametric approac is also direct. It is conceptually more advanced tan te grap approac and one as to solve in n space dimensions a system of n + 1 parabolic equations. If te surface is a grap ten te parametric approac is less efficient tan solving for te eigt of te surface. On te oter and it is more widely applicable. In te case of a closed curve one can use periodic boundary conditions on te unit interval in order to solve over te circle. A closed twodimensional surface can be approximated by a polyedral surface. A parametrized surface does not see an inside or outside. From te point of view of differential geometry tis may not be an issue. However, wen te surface separates two pases, or two materials, or two colours, tere are significant issues. For example, consider using two colours in Figure 1.2 in order to define te curve as te interface between te coloured regions. Black may be used to colour te inside of bot loops and wite to colour te te rest of te plane, but if black is used inside just one loop ten te oter loop is lost. Tus, in order to use te parametric approac wit tis initial condition, one eiter tinks of a parametrization wic traverses te curve witout a crossing, but wit a single selfintersection, or regards tem as being two separate closed curves wic touc at one point. Tese coices lead to differing evolutions for mean curvature flow. Contrary to te parametric approac, te level set metod as te capability of tracking topological canges (like pincingoff or merging) of Γ(t) in an automatic way. In te basic implementation of te metod topological
7 Computation of geometric PDEs and mean curvature flow 7 cange is noting special and is observed in postprocessing te computational output. Tis is because, in principle, zero level sets of continuous functions can exibit tese features. However, from te matematical point of view tere are issues of existence of solutions of te degenerate partial differential equations tat te level set approac generates. In te case of motion by mean curvature tere is te notion of a viscosity solution wic yields a unique evolution from any continuous function. Te example of te lemniscate discussed in te context of te parametric approac introduces a new idea in te level set approac of fattening of te interface. Te level set for tis example develops an interior wose boundary yields bot of te described solutions. Selfintersection, merger and pincoff can all be simulated by tis approac. Tis advantage, owever, needs to be offset against te fact tat te problem now becomes (n + 1)dimensional in space. Te pase field approac can also andle topological cange, selfintersection, merger and pincoff witout doing anyting special. It is te one approac wic in its conception involves an approximation. Te fact tat it involves a new parameter ɛ is bot an advantage and a disadvantage. Te parabolic equations are in principle easy to solve but possess a certain computational stiffness due to te tickness of te diffuse interface. However, in many applications pase field models arise naturally and te ɛ parameter allows us to resolve singularities in a way wic may be viewed as being pysically motivated. From bot te matematical and pysical points of view it is widely applicable in a rational way, wereas te use of te level set metod is frequently ad oc. In general, te coice of one or te oter approac will depend on weter one expects topological canges in te flow Applications In wat follows we list some problems in wic a law of te form (1.1) or generalizations of it arise. Grain boundary motion Grain boundaries in alloys are interfaces wic separate bulk crystalline regions of te same pase but wit differing orientations. Associated wit te grain boundary is a surface energy wic gives rise to a termodynamic restoring force. For a constant surface energy density tis is simply te surface tension force proportional to te mean curvature and te resulting evolution law is just (1.2). Frequently tere is also a driving force causing motion of te grain boundary. Surface growt Te growt of tin films on substrates is tecnologically important. For example, epitaxy is a metod for growing single crystals by te deposition
8 8 K. Deckelnick, G. Dziuk and C. M. Elliott of atoms and molecules on to a growing film surface. Tere are numerous pysical mecanisms operating at differing time and lengt scales wic affect te growt process. A simple model would ave a driving force representing te deposition flux of atoms onto te surface wic migt be in te normal direction or in a fixed vertical direction parallel to a beam of arriving atoms. Image processing One of te most important problems in image processing is to automatically detect contours of objects. We essentially follow te exposition of Aubert and Kornprobst (2002). Suppose tat M R n+1 (n = 1 or 2) is a given object and let I(x) =χ Ω\M (x) be te caracteristic function of Ω \ M. Te function 1 g(x) = 1+ I σ (x) 2, were I σ is a mollification of I, will be small near te contour of M. It is terefore natural to look for minimizers of te functional J(Γ) = g(x)da Γ were Γ is a curve in R 2 or a surface in R 3. Te corresponding L 2 gradient flow leads to te following evolution law: find curves/surfaces (moving snakes ) Γ(t) suc tat V = (gν)= gh g ν on Γ(t). Here, t plays te role of an artificial time; clearly tis law fits into te framework (1.1). Stefan problem for undercooled solidification Consider a container Ω R n+1 (n = 1 or 2) filled wit an undercooled liquid. Solidification of te liquid follows te nucleation of initial solid seed wit caracteristic diameter larger tan te critical radius. Te seed will ten grow into te liquid. A matematical model for tis situation is te Stefan problem wit kinetic undercooling, in wic te solid liquid interface is described by a curve/surface Γ(t) and as to be determined togeter wit te temperature distribution. Here te interior of Γ(t) is te solid region Ω S (t) and te exterior is te liquid region Ω L (t). Using a suitable nondimensionalization te problem ten reads: for a given initial pase boundary Γ 0 and initial temperature distribution Θ 0 =Θ 0 (x) (x Ω), find te nondimensional temperature Θ = Θ(x, t) and te pase boundary Γ(t) (t >0), suc tat te eat equation is satisfied in te bulk, tat is, Θ t Θ = 0 in Ω \ Γ(t),
9 Computation of geometric PDEs and mean curvature flow 9 togeter wit te initial value Θ(, 0) = Θ 0 in Ω. On te moving boundary te following two conditions are satisfied: V = 1 [ ] Θ on Γ(t), (1.5) ε l ν Θ+ε V β(ν)v + σh γ =0 onγ(t). (1.6) Here, [ Θ/ ν] denotes te jump in te normal derivative of te temperature field across te interface and ε l is te constant measuring te latent eat of solidification. Equation (1.6) is te Gibbs Tomson law; ε V,σ are nondimensional positive constants measuring te strengt of te kinetic undercooling and surface tension wic depress te temperature on te solid liquid interface from te scaled equilibrium zero melting temperature. Furtermore, H γ is an anisotropic mean curvature associated wit a surface energy density, γ(ν), depending on te orientation of te normal. Tere may also be anisotropy, β(ν), in te kinetic undercooling. Note tat (1.6) can be rewritten as ε V σ β(ν)v = H γ 1 σ Θ on Γ(t). If we consider Θ as being given, tis equation again fits into our general framework (1.1) provided we allow for a coefficient in front of V and a generalized notion of mean curvature. Figure 1.5 from Scmidt (1996) sows a simulation in wic te free boundary was described by te parametric approac resulting in a sarp interface model. One can see te free boundary forming a dendrite. For Figure 1.5. Evolution of a dendrite wit sixfold anisotropy. Timesteps of te free boundary (left) and adapted grid for te temperature at one timestep (rigt).
10 10 K. Deckelnick, G. Dziuk and C. M. Elliott results concerning treedimensional dendrites and more information about te algoritm we refer to Scmidt (1996). Figure 1.6 from Fried (1999) illustrates a possible effect of using a level set metod for te free boundary in tis problem. Dendrites may seem to merge. But if a smaller timestep is used te dendrites stay apart. For more information about a level set algoritm for dendritic growt we refer to Fried (1999, 2004). Surface diffusion and Willmore flow Te following laws do not fit into (1.1), but we list tem as examples of important geometric evolution equations in wic te normal velocity depends on iger derivatives of mean curvature. Te surface diffusion equation V = Γ H (1.7) models te diffusion of mass witin te bounding surface of a solid body. At te atomistic level atoms on te surface move along te surface owing to a driving force consisting of a cemical potential difference. For a surface wit constant surface energy density te appropriate cemical potential in tis setting is te mean curvature H. Tis leads to te flux law ρv = div Γ j, were ρ is te mass density and j is te mass flux in te surface, wit te constitutive flux law (Herring 1951, Mullins 1957) j = D Γ H. Here, D is te diffusion constant. From tese equations we obtain te law (1.7) after an appropriate nondimensionalization. In order to model te Figure 1.6. A possible effect of te use of a level set metod. Growing dendrites: merging (left) for large timestep size and staying apart (rigt) for smaller timestep size.
11 Computation of geometric PDEs and mean curvature flow 11 underlying structure of te solid body bounded by Γ, anisotropic surface diffusion is important, tat is, V = Γ H γ, (1.8) wit H γ denoting te anisotropic mean curvature of te surface Γ as it is introduced in (8.15). A similar evolution law is Willmore flow, V = Γ H + H Γ ν H3 on Γ(t), (1.9) wic arises as te L 2 gradient flow for te classical bending energy E(Γ) = 1 2 Γ H2 da. Apart from applications in mecanics and membrane pysics tis flow as recently been used for surface restoration and inpainting Outline of article Tis article is organized as follows. In Section 2 we present some useful geometric analysis, in particular te notion of mean curvature. Te basic mean curvature flow is defined in Section 3 and some elementary properties are described. Te next four sections consider in turn basic approaces for numerical approximation. In Section 4 we consider te parametric approac. We start wit te classical curve sortening flow and present a semidiscrete numerical sceme as well as error estimates. Next, we sow ow to apply te above ideas to te approximation of igerdimensional surfaces. A crucial point is to construct numerical scemes wic reflect te intrinsic nature of te flow. Section 5 is concerned wit graps. We prove an error bound for a semidiscrete finite element sceme tereby sowing te virtue of working wit geometric quantities. A fully discrete sceme along wit stability issues is discussed afterwards. In Section 6 we introduce te level set equation as a way of andling topological canges. We briefly discuss te framework of viscosity solutions wic allows a satisfactory existence and uniqueness teory. For numerical purposes it is convenient to regularize te level set equation. We collect some properties of te regularized problem and clarify its formal similarity to te grap setting. Te approximation of mean curvature flow by pase field metods is considered in Section 7. Even before numerical discretization tere is te notion of approximation of a sarp interface by a diffuse interface of widt O(ɛ). Te pase field approac depends on te notion of a diffuse interfacial energy composed of quadratic gradient and omogeneous free energy terms involving a pase field function. Te coice of double well energy potential is discussed. We recall some analytical results as well as a convergence analysis for a discretization in space by linear finite elements. We finis tis section by discussing te discretization in time togeter wit te question of stability. In Section 8 we introduce te concept of te anisotropy γ togeter wit its relevant
12 12 K. Deckelnick, G. Dziuk and C. M. Elliott properties and subsequently generalize te ideas of te previous sections to tis setting. Finally, Section 9 is concerned wit fourt order flows: we present discretization tecniques for bot surface diffusion and Willmore flow. For te convenience of te reader we ave included a long list of references, wic are related to te subject of tese notes, but not all of wic are cited in te text. 2. Some geometric analysis Te aim of tis section is to collect some useful definitions and results from differential geometry. We refer to Gilbarg and Trudinger (1998) and Giga (2002) for a more detailed exposition of tis material Hypersurfaces A subset Γ R n+1 is called a C 2 ypersurface if for eac point x 0 Γ tere exists an open set U R n+1 containing x 0 and a function u C 2 (U) suc tat U Γ={x U u(x) =0}, and u(x) 0 forallx U Γ. (2.1) Te tangent space T x Γ is ten te ndimensional linear subspace of R n+1 tat is ortogonal to u(x). It is independent of te particular coice of function u wic is used to describe Γ. A C 2 ypersurface Γ R n+1 is called orientable if tere exists a vectorfield ν C 1 (Γ, R n+1 )(i.e., ν C 1 in an open neigbourood of Γ) suc tat ν(x) T x Γand ν(x) =1for all x Γ. In wat follows, we sall assume tat Γ R n+1 is an oriented C 2 ypersurface. We define te tangential gradient of a function f, wic is differentiable in an open neigbourood of Γ by Γ f(x) = f(x) f(x) ν(x) ν(x), x Γ. Here denotes te usual gradient in R n+1. Note also tat Γ f(x) is te ortogonal projection of f(x) ontot x Γ. It is straigtforward to sow tat Γ f only depends on te values of f on Γ. We use te notation Γ f(x) =(D 1 f(x),...,d n+1 f(x)) (2.2) for te n + 1 components of te tangential gradient. Obviously Γ f(x) ν(x) =0, x Γ. If f is twice differentiable in an open neigbourood of Γ, ten we define
13 Computation of geometric PDEs and mean curvature flow 13 te Laplace Beltrami operator of f as n+1 Γ f(x) = Γ Γ f(x) = D i D i f(x), x Γ. (2.3) 2.2. Oriented distance function A useful level set representation of a ypersurface can be obtained wit te elp of te distance function. Let Γ be as above and assume in addition tat Γ is compact. Te Jordan Brouwer decomposition teorem ten implies tat tere exists an open bounded set Ω R n+1 suc tat Γ = Ω. We assume tat te unit normal field to Γ points away from Ω and define te oriented (signed) distance function d by dist(x, Γ), x R n+1 \ Ω d(x) = 0, x Γ dist(x, Γ), x Ω. It is well known tat d is globally Lipscitzcontinuous and tat tere exists δ>0 suc tat i=1 d C 2 (Γ δ ), were Γ δ = {x R n+1 d(x) <δ}. (2.4) Every point x Γ δ can be uniquely written as x = a(x)+d(x)ν(a(x)), x Γ δ, (2.5) were a(x) Γ. Furtermore, d(x) =ν(a(x)),x Γ δ, wic implies in particular tat d(x) 1 in Γ δ. (2.6) Figure 2.1. Grap (rigt) of te oriented distance function for te curve (left).
14 14 K. Deckelnick, G. Dziuk and C. M. Elliott 2.3. Mean curvature Let us next turn to te notion of mean curvature. By assumption, ν is C 1 in a neigbourood of Γ so tat we may introduce te matrix H jk (x) =D j ν k (x), j,k =1,...,n+1,x Γ. (2.7) It is not difficult to sow tat (H jk (x)) is symmetric. Furtermore, n+1 n+1 H jk (x)ν k (x) = D j ν k (x)ν k (x) = 1 2 D j ν 2 (x) =0, k=1 k=1 since ν =1onΓ. Tus,(H jk (x)) as one eigenvalue wic is equal to zero wit corresponding eigenvector ν(x). Te remaining n eigenvalues κ 1 (x),...,κ n (x) are called te principal curvatures of Γ at te point x. We now define te mean curvature of Γ at x as te trace of te matrix (H jk (x)), tat is, n+1 n H(x) = H jj (x) = κ j (x). (2.8) j=1 Note tat (2.8) differs from te more common definition H = 1 n+1 n j=1 H jj. From (2.7) we derive te following expression for mean curvature, H(x) = Γ ν(x), x Γ, (2.9) were Γ f = n+1 j=1 D jf j denotes te tangential divergence of a vectorfield f. In particular we see tat H>0ifΓ=S n and te unit normal field is cosen to point away from S n, i.e., ν(x) =x. Wile te sign of H depends on te coice of te normal ν, te mean curvature vector Hν is an invariant. A useful formula for tis quantity can be obtained by coosing f(x) =x j,j {1,...,n+1} in (2.3) and observing tat D i x j = δ ij ν j ν i. We ten deduce wit te elp of (2.9) tat n+1 Γ x j = D i (ν j ν i )= ( Γ ν)ν j Γ ν j ν = Hν j, so tat i=1 j=1 Γ x = Hν on Γ. (2.10) Letusnextfixapoint x Γ and calculate H( x) for various representations of te surface Γ near x. Level set representation. Suppose tat Γ is given as in (2.1) near x. Clearly, we ten ave ν(x) =± u(x) u(x)
15 Computation of geometric PDEs and mean curvature flow 15 for x U Γ. If te plus sign applies we obtain H = Γ u u = u u = 1 u n+1 i,j=1 ( δ ij u x i u xj u 2 ) u xi x j. (2.11) In te special case tat u(x) =d(x), were d is te oriented distance function to Γ, we obtain in view of (2.6) Grap representation. Suppose tat H(x) = d(x), x Γ. (2.12) U Γ={(x, v(x)) x Ω}, were Ω R n is open, x =(x 1,...,x n )andv C 2 (Ω). Defining u(x, x n+1 ) = v(x) x n+1 we see tat U Γ is te zero level set of u and te above considerations imply tat ( ) v(x) H(x, v(x)) =, (x, v(x)) U Γ, (2.13) 1+ v(x) 2 were is te gradient in R n and te unit normal is cosen as ν = ( v, 1) 1+ v 2. Parametric representation Suppose tat tere exists an open set V R n and a mapping X C 2 (V,R n+1 ) suc tat U Γ=X(V ), rank DX(θ) =n for all θ V. Te vectors X θ 1 (θ),..., X θ n (θ) ten form a basis of T x Γatx = X(θ). We define te metric on Γ by g ij (θ) = X (θ) X (θ), i,j =1,...,n θ i θ j and let g ij be te components of te inverse matrix of (g ij ). We ten ave te following formulae for te tangential gradient of a function f (defined in a neigbourood of Γ) and te mean curvature vector Hν: were g =det(g ij ). n ij (f X) X Γ f = g, (2.14) θ i,j=1 j θ i Hν = 1 n ( g ij g X ) (2.15) g θ i θ j i,j=1
16 16 K. Deckelnick, G. Dziuk and C. M. Elliott 2.4. Integration by parts Let us assume in tis section tat Γ is in addition compact. Te formula for integration by parts on Γ is (cf. Gilbarg and Trudinger (1998)) D i f da = fhν i da i =1,...,n+1, (2.16) Γ Γ were da denotes te area element on Γ and f is continuously differentiable in a neigbourood of Γ. Applying (2.16) wit = fd i g, summing from i =1,...,n+1 and taking into account tat Γ ν i ν = 0, we obtain Green s formula, Γ f Γ g da = f Γ g da. (2.17) Γ Γ In particular, we deduce from (2.10) Hν φ da = Γ x Γ φ da, (2.18) Γ were φ is continuously differentiable in a neigbourood of Γ wit values in R n+1 and Γ x Γ φ = n+1 i=1 Γx i Γ φ i. Tis relation will be very important for te numerical treatment of mean curvature flow. Te above formulae can be generalized to surfaces wit boundaries by including an appropriate integral over Γ Moving surfaces In tis section we sall be concerned wit surfaces tat evolve in time. A family (Γ(t)) t (0,T ) is called a C 2,1 family of ypersurfaces if, for eac point (x 0,t 0 ) R n+1 (0,T)witx 0 Γ(t 0 ), tere exists an open set U R n+1, δ>0 and a function u C 2,1 (U (t 0 δ, t 0 + δ)) suc tat U Γ(t) ={x U u(x, t) =0} and u(x, t) 0,x U Γ(t). (2.19) Suppose in addition tat eac Γ(t) is oriented by a unit normal field ν(,t) C 1 (Γ(t), R n+1 ) and tat ν C 0 ( 0<t<T Γ(t) {t}, Rn+1 ). Te normal velocity at a point (x 0,t 0 )(x 0 Γ(t 0 )) is ten defined as V (x 0,t 0 )=φ (t 0 ) ν(x 0,t 0 ), were φ C 1 ((t 0 ɛ, t 0 + ɛ), R n+1 ) satisfies φ(t 0 )=x 0 and φ(t) Γ(t) for t t 0 <ɛ. It can be sown tat V (x 0,t 0 ) is independent of te particular coice of φ. Let us calculate V (x 0,t 0 ) for various representations of Γ(t). Level set representation. Let u be as in (2.19); as above we ten ave ν = ± u u. If te plus sign applies and φ C1 ((t 0 ɛ, t 0 + ɛ), R n+1 ) satisfies φ(t 0 )=x 0 as well as φ(t) Γ(t) for t t 0 <ɛ,weave 0= d dt u(φ(t),t)= u(φ(t),t) φ (t)+u t (φ(t),t), Γ
17 Computation of geometric PDEs and mean curvature flow 17 and ence V (x 0,t 0 )= u t(x 0,t 0 ) u(x 0,t 0 ). (2.20) Grap representation. Suppose tat U Γ(t) ={(x, v(x, t)) x Ω}, were Ω R n is open and v C 2,1 (Ω (t 0 δ, t 0 + δ)). Applying te formula for te level set case to u(x, x n+1,t)=v(x, t) x n+1,weobtain for te unit normal field ν = v t V = 1+ v 2 ( v, 1) 1+ v 2. (2.21) 2.6. Transport teorem for integrals Consider a family (Γ(t)) t (0,T ) of evolving ypersurfaces wic satisfies te assumptions made above and suppose in addition tat eac surface Γ(t) is compact. We are interested in te time derivative of certain volume and area integrals. Lemma 2.1. Let g C 1 (Q), were Q is an open set containing Γ(t) {t}. 0<t<T Suppose in addition tat eac surface Γ(t) is te boundary of an open bounded subset Ω(t) R n+1.ten d g g dx = dt Ω(t) Ω(t) t dx + gv da, (2.22) Γ(t) d g g da = dt Γ(t) Γ(t) t da + g gv H da + V da. (2.23) Γ(t) Γ(t) ν Proof. See te Appendix. 3. Definition and elementary properties of mean curvature flow Te purpose of tis section is to introduce motion by mean curvature and to describe some basic features of tis flow. Consider a C 2,1 family of ypersurfaces (Γ(t)) t [0,T ] R n+1 togeter wit a coice ν of a unit normal. Definition 1. We say tat (Γ(t)) t [0,T ] moves by mean curvature if V = H on Γ(t). (3.1)
18 18 K. Deckelnick, G. Dziuk and C. M. Elliott Here, V denotes te velocity of Γ(t) in te direction of ν and H is mean curvature. As we sall see later, te above equation gives rise to a parabolic equation, or a parabolic system, for te function(s) describing te surfaces Γ(t), to wic an initial condition Γ(0) = Γ 0 (3.2) as to be added. If Γ(t) as a boundary, ten also suitable boundary conditions need to be specified. In order to give a first idea of tis flow we look at te wellknown example of te srinking spere. Let Γ(t) = B r(t) (x 0 ) R n+1, oriented by te unit n r(t) outer normal ν(x) = x x 0 r(t). Ten, V = r (t), H = moves by mean curvature provided tat r (t) = n r(t) on Γ(t), so tat Γ(t). Te solution of tis ODE is given by r(t) = r0 2 2nt, 0 t< r2 0 2n, were Γ 0 = B r0 (x 0 ). Note tat Γ(t) srinks to a point as t r2 0 2n. Te main feature of mean curvature flow is its areadecreasing property, wic is a consequence of te following result. Lemma 3.1. Let Γ(t) be a family of evolving ypersurfaces satisfying V = H on Γ(t) and assume tat eac Γ(t) is compact. Ten V 2 da + d Γ(t) =0, dt were Γ is te area of Γ. Γ(t) Proof. Tis follows immediately from coosing g 1 in (2.23) and te evolution law (3.1). Since te law (3.1) gives rise to a second order parabolic problem we expect existence of a smoot solution locally in time for a smoot initial ypersurface Γ 0. Furtermore, maximum and comparison principles are available wic can be used to sow tat two smoot compact solutions wic are initially disjoint will stay disjoint (see, e.g., Ecker (2002)). Using te srinking spere as a comparison solution, it follows in particular tat if Γ(t), 0 t<t is a smoot solution wit Γ 0 B r0 (x 0 ), ten Γ(t) B r0 2 2nt(x 0)for0 t<min(t, r2 0 2n ). In general, solutions will develop singularities in finite time before tey disappear, but tere are certain initial configurations for wic tey stay smoot until tey srink to a point. Teorem 3.2. Let n 2 and assume tat Γ 0 R n+1 is a smoot, compact and uniformly convex ypersurface. Ten (3.1) and (3.2) ave a smoot solution on a finite time interval [0,T) and te Γ(t) converge to a
19 Computation of geometric PDEs and mean curvature flow 19 point as t T. If one rescales te surfaces in suc a way tat te enclosed volume remains fixed, one as convergence against a spere as t T. Proof. See Huisken (1984). Te case n = 1 is usually referred to as curve sortening flow. Teorem 3.3. Assume tat Γ 0 R 2 is a smoot embedded closed curve. Ten (3.1) and (3.2) ave a smoot embedded solution on a finite time interval [0,T), wic srinks to a round point as t T. Proof. Gage and Hamilton (1986) proved tis result for convex Γ 0 ; subsequently Grayson (1987) sowed tat a smoot embedded closed curve remains smoot and embedded and becomes convex in finite time. If te initial curve is not embedded, cusplike singularities may develop (see Figures 4.2 and 4.3). Te papers of Angenent (1991), Altsculer and Grayson (1992) and Deckelnick (1997) propose various metods of ow to continue te solution past suc a singularity. Te analogue of Teorem 3.3 for surfaces does not old, as can be seen by coosing a suitable dumbbellsaped initial surface wic develops a pincoff singularity before it srinks to a point (see Figure 4.5 and Grayson (1989)). Tis pincoff leads to a cange of te topological type of Γ(t), so tat te parametric approac in wic te topological type is fixed will develop a singularity tat is difficult to andle. Tus te question arises weter it is possible to introduce a notion of solution tat is capable of following te flow troug a singularity. Several suc notions ave been proposed and analysed starting wit te pioneering work of Brakke (1978) on varifold solutions, wic uses tools from geometric measure teory. In tis context we also mention te surface evolver program of Brakke (1992). Level set and pase field metods constitute two completely different approaces wic take an Eulerian point of view. We sall discuss tese in more detail in Sections 6 and Parametric mean curvature flow As is mentioned above, in te parametric approac one cooses a suitable reference manifold M R n+1 (of te topological type of te evolving ypersurfaces Γ(t)) and ten looks for maps X(,t):M R n+1 (0 t<t)suc tat Γ(t) =X(,t)(M). To fix ideas, let us assume tat M is a compact ypersurface witout boundary. If we can find X in suc a way tat X (p, t) = H(X(p, t))ν(x(p, t)) (p, t) M (0,T), (4.1) t ten V = H on Γ(t) follows by taking te dot product wit te normal ν. In order to understand (4.1) let F :Ω R n+1 be a local parametrization
20 20 K. Deckelnick, G. Dziuk and C. M. Elliott of M definedonanopensetω R n and set ˆX(θ, t) =X(F (θ),t), (θ, t) Ω (0,T). Recalling (2.15), te equation (4.1) ten turns into ˆX 1 n ( (θ, t) ĝ ĝ ij ĝ ˆX ) =0 (4.2) t θ i θ j i,j=1 were ĝ ij (θ, t) = ˆX θ i ˆX θ j and ĝ ij, ĝ are as above. Tus (4.2), and ence (4.1), is a nonlinear parabolic system, wic is not defined at points (θ, t) were ĝ(θ, t) = 0. In order to close tis system, an initial condition X(p, 0) = X 0 (p),p M needs to be prescribed, were X 0 : M R n+1 is a parametrization of te initial surface Γ Curve sortening flow Mean curvature evolution in te onedimensional case is usually referred to as curve sortening flow. In te case of closed curves, a convenient coice of a reference manifold is M = S 1, wic can be parametrized globally by F (θ) =(cosθ, sin θ),θ [0, 2π]. In te following, for simplicity, let us identify ˆX(θ, t) andx((cos θ, sin θ),t). Tus, (4.2) becomes X t 1 ( ) Xθ =0 X θ X θ θ ini (0,T), (4.3) X(, 0) = X 0 in I, (4.4) were I =[0, 2π]. In addition, X as to satisfy te periodicity condition X(θ, t) =X(θ +2π, t) 0 t<t, θ R. (4.5) Suppose tat X : R [0,T] R 2 is a smoot solution of (4.3) (4.5), in particular X θ > 0inI [0,T]. If we multiply (4.3) by X θ, take te dot product wit a test function ϕ Hper(I; 1 R 2 )={ϕ H 1 (I; R 2 ) ϕ(0) = ϕ(2π)} and integrate over I, we obtain X θ ϕ θ X t ϕ X θ + =0 forallϕ H 1 I I X θ per(i; R 2 ). (4.6) We use (4.6) in order to discretize in space. For simplicity let θ j = j (j =0,...,N) be a uniform grid wit grid size =2π/N and let { } S = ϕ C 0 (I; R 2 ) ϕ [θj 1,θ j ] P1 2,j =1,...,N; ϕ (0) = ϕ (2π) be te space of piecewise linear continuous functions wit values in R 2.Te spatial discretization of (4.3) is ten given by X θ ϕ θ X t ϕ X θ + =0 forallϕ S. (4.7) X θ I I
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