Axially symmetric volume constrained anisotropic mean curvature flow
AAxially symmetric volume constrainedanisotropic mean curvature flow
By B
BENNETT P ALMER and W
ENXIANG Z HU Abstract
We study the long time existence theory for a non local flow associatedto a free boundary problem for a trapped non liquid drop. The drop hasfree boundary components on two horizontal plates and its free energy isanisotropic and axially symmetric. For axially symmetric initial surfaceswith sufficiently large volume, we show that the flow exists for all time.Numerical simulations of the curvature flow are presented.
The evolution of interfaces of structured materials is of interest in a widerange of disiplines related to materials science [14]. Structured materialssuch as crystals, polycrystals and liquid crystals have a surface energywhich is anisotropic; their energy density depends on the direction ofthe surface at each point. Over the years, various methods have beendeveloped to track these interfaces, including the phase-field and levelset methods. Here, we will consider a particular free boundary problemutilizing the anisotropic mean curvature flow.The mean (isotropic) curvature flow with constrained volume was con-sidered in [9]. In relation to the free boundary problem considered here,the papers of Athanassenas [3], [4] are particularly relevant . Also, in therecent paper [5], volume preserving mean curvature flow in a Riemanniansetting is studied.Volume constrained anisotropic mean curvature flow for hypersurfaceswas considered in [1], [13]. In these papers, the emphasis is on the evolu-tion of closed convex hypersurfaces.Consider an anisotropic surface energy which assigns to a sufficientlysmooth surface with unit normal ν the value(1) F (Σ) = (cid:90) Σ γ ( ν ) d Σ . The function γ : S → R + is assumed to satisfy a convexity condition :the surface(2) W = ∂ (cid:92) n ∈ S { Y · n ≤ γ ( n ) } , a r X i v : . [ m a t h . DG ] O c t hich is known as the Wulff shape . In this paper, it will be assumed that(W1) W is a smooth,uniformly convex surface of revolution with verticalrotation axis.(W2) W is symmetric with respect to reflection through the horizontalplane z = 0.Although we will not assume it in general, the following condition willalso enter into discussion.(W3) The generating curve of W has non-decreasing curvature (with re-spect to the inward pointing normal) as a function of arc length on { z ≥ } as one moves in an upward direction.The condition (W3) will be referred to as the curvature condition .Because of (W1), the Gauss map of W defines a bijection of W onto S . Therefore quantities defined on W can be expressed unambiguouslyon the sphere. In particular, the principal curvatures of W with respectto the inward pointing normal, µ i , i = 1 , /µ = γ − ν γ (cid:48) , /µ = (1 − ν ) γ (cid:48)(cid:48) + 1 /µ . The uniform convexity means that µ i > ξ on W can be expressed in terms of the position vector ν on S as(3) ξ = 1 µ ν + γ (cid:48) ( ν ) E . Given an oriented embedded surface X : Σ → R with unit normalfield ν , we define the Cahn-Hoffman field as the composition ξ := ξ ◦ ν .The first variation of energy defines the anisotropic mean curvature Λ by(4) δ F = − (cid:90) Σ Λ δX · ν d Σ + (cid:73) ∂ Σ ( ξ × δX · dX ) . Some local expressions for the anisotropic mean curvature are:Λ := − trace Σ Adν = − ( ∇ · Dγ − Hγ ) , A := dξ = ( D γ + γ | S . Here D , D denote the gradient and Hessian operators acting on functionson S . These can always be identified with tensors on Σ by paralleltranslation in R since T p Σ = T ν ( p ) S . Another useful expression isΛ = −∇ · ξ , where ∇· denotes the surface divergence.The problem we consider here is to understand the evolution to equi-librium of a drop of material trapped between two horizontal planes Π i , i = 0 , z = 0 and z = h respectively. The surfaceenergy is assumed to be of the form (1). It will also be assumed that: • the initial surface is axially symmetric, • the generating curve of the initial surface is a graph over the rotationaxis. throughout the evolution, the surfaces meets the planes Π i orthog-onally. • the initial surface extends smoothly to an infinite periodic surfacewith period h .Of course, the third assumption is a consequence of the fourth, but weinclude it for clarity.It is easy to see that the axial symmetry is preserved by the evolutionconsidered below. It is also consistent with the form of the known mini-mizer for sufficiently large volumes and neutral wetting along the interfaceof Σ with the planes, [11]. For small volumes, the drop must disconnectduring the minimization process.It will be shown below that if the initial volume is sufficiently large,then the generating curves of the evolving surfaces are also graphs for alltimes for which the flow is defined.Although we expect that a similar analysis can be carried out in arbi-trary dimensions, we have decided to concentrate on the case of surfacesin R because of its obvious physical significance. We will restrict our attention to axially symmetric surfaces with verticalrotation axis. In addition, the generating curve will always be a graph ofthe form r = r ( z ) and we write the surface as X = ( r ( z ) e iθ , z ) where wehave identified the first two coordinates on R with the complex plane C .The outward pointing normal to the surface is ν = (1 + r z ) − / ( e iθ , − r z ).Consider the flow:(5) ∂X∂t = (Λ − ¯Λ) ν , with the boundary condition(6) ν · E ≡ , on ∂ Σ . From (14), we obtain(7) ∂ t z = (Λ − ¯Λ) ν . It is evident from this formula that, in general, z must depend on t , so,following [7], we write z = z ( ζ, t ) where we choose ζ to lie in the interval[0 , h ]. We then express the immersion as X = X ( ζ, t ) = ( r ( z ( ζ, t ) , t ) , z ( ζ, t )) . We get r z z t + r t = (Λ − ¯Λ) (cid:113) − ν Combining this with (7), we get, using that ν = − r z / √ r z , r t = (Λ − ¯Λ)( (cid:113) − ν − r z ν ) = (Λ − ¯Λ)( 1 √ r z + r z √ r z ) , .e.(8) r t = (Λ − ¯Λ) (cid:112) r z . The evolution for r is identical with the evolution for r under the flow( ∂ t X ) ⊥ = (Λ − Λ ) ν , where ⊥ denotes the normal component.Short time existence for (8) is standard, see for example Theorem8.1.1 of [12]. By the equation (12), (13) given below, both Λ and ¯Λ areexpressible in terms of r and its derivatives.The admissible variations for the variational problem are those whichkeep the surface between the planes, δX · E ≡ ∂ Σ , and fix the threedimensional volume enclosed within the surface: (cid:90) Σ δX · ν d Σ = 0 . The infinitesimal generator (Λ − ¯Λ) ν of the evolution (5), (6) clearly sat-isfies these conditions.By (6), along ∂ Σ, the normal ν is contained in the horizontal equatorof S . By (W2), γ (cid:48) ( ν = 0) = 0 holds and so by (3), ξ and X t are parallelalong ∂ Σ. we then get from the first variation formula (4), ∂ t F [Σ t ] = − (cid:90) Σ Λ(Λ − ¯Λ) d Σ = − (cid:90) Σ (Λ − ¯Λ) d Σ , so the flow decreases the anisotropic energy. Remark
When the wetting is not neutral, it appears to be problematicto construct an analogous flow. In this case, the boundary condition forthe minimizer ξ · E ≡ c i (cid:54) = 0 on ∂ Σ ∩ Π i , where c i are non zero constantsrelated to the coupling constants for the wetting energy. This boundarycondition is incompatible with the flow given by (5) maintaining the dropbetween the planes .For the boundary value problem we are considering, the morphology ofthe minimizer depends the initial volume and whether or not the condition(W3) holds. If (W3) holds, it was shown in [11] that for volumes greaterthan or equal to a critical value V , all stable equilibria must be cylinders.Below this value, any minimizer must either disconnect or loose contactwith at least one of the supporting planes.When (W3) does not hold, numerical simulations for a particular classof functionals, [2], show that anisotropic unduloids may occur as stableequilibria for a certain range of volumes. However, for large volumes,only cylinders occur and for sufficiently small volumes, there is no stableconnected surface spanning the two supporting planes.We will next show that of the initial volume is sufficiently large, thesurfaces will not pinch off, i.e. the generating curves are bounded awayfrom the rotation axis.The following lemma uses calibrations to show a minimizing propertyof graphs with Λ = 0. It is well known and its proof is given for complete-ness. emma 2.1. Let Σ be a surface with zero anisotropic mean curvaturewhich can be represented as a graph over a planar domain Ω . Let S be apiecewise smooth oriented surface which is contained in the cylinder Ω × R and which has the same boundary as Σ . Then, F [Σ] ≤ F [ S ] holds. Remark.
In saying that Σ and S share the same boundary, we meanthat Σ − S is the oriented boundary of an oriented 3-chain. Proof.
Let ξ denote the Cahn-Hoffman field of Σ. The condition that Σhas zero anisotropic mean curvature is expressed ∇ · ξ = 0.Because Σ is a graph, we can extend ξ to a field ˜ ξ on Ω × R by making˜ ξ constant on all vertical lines through points in Ω. This field will satisfy˜ ∇ · ˜ ξ , where ˜ ∇ denotes the divergence operator on R . Then, by theStokes’ Theorem, we get F [Σ] = (cid:90) Σ γ ( ν Σ ) d Σ = (cid:90) Σ ξ · ν Σ d Σ= (cid:90) Σ ˜ ξ · ν Σ d Σ = (cid:90) S ˜ ξ · ν S dS ≤ (cid:90) S γ ( ν S ) dS = F [ S ] , where the inequality follows from (2). Corollary 2.1.
Let F be an axially symmetric anisotropic surface energyand let C ⊂ Π i be a circle. Let S be any piecewise smooth compact surfacebounded by C and which is contained in the cylinder over the disc boundedby C . Also, let D be the flat disc bounded by C . Then F [ S ] ≥ F [ D ] = γ ( e ) | D | holds.Proof. This follows immediately from the previous lemma using thefact that the disc has zero anisotropic mean curvature.
Proposition 2.1.
Let Σ be an initial axially symmetric surface enclosinga volume V and intersecting the supporting planes orthogonally. Assumethat (9) F [Σ ] < γ ( e ) V [Σ ] d holds and that the flow (5) exists for all t ∈ [0 , T ) . Then, then no pinchingoccurs. In particular (10) r ≥ c holds for t ∈ [0 , T ) , where γ ( e ) V [Σ ] d − F [Σ ] =: πc . roof. Recall that F [Σ t ] is non increasing. Let C denote the cylinderbetween Π and Π enclosing the same volume V [Σ ] as Σ . Let r C bethe radius of this cylinder. If ρ := r ( ζ, t ) < c for some t < T and some ζ , we can find an annular part of the surface Σ t bounded by two circlesof radii ρ and radius r t > r C . From this piece of Σ t , we form a piecewisesmooth disc type surface by filling the circle of radius ρ with a disc.By the previous corollary, we obtain F [Σ ] + πc > F [Σ t ] + πρ > γ ( e ) πr t ≥ γ ( e ) πr c = γ ( e ) V (Σ ) d , which gives a contradiction. Lemma 2.2.
Assume the conditions (W1), (W2) hold for W and that Σ is an axially symmetric surface intersecting the planes Π i orthogonallyfor which (10) holds. Then there exists c = c ( c ) such that c ≥ | ¯Λ | holds. If, in addition (W3) holds, then we have ≥ ¯Λ ≥ c > −∞ . Proof.
The idea of the proof is essentially taken from [3].We first note that the curvature condition (W3) can be expressed(11) ν ∂ ν µ ≥ , ∀ ν . For an axially symmetric surface whose generating curve is a graph,the anisotropic mean curvature is given by(12) Λ = k µ + k µ = r zz µ (1 + r z ) / − µ r (1 + r z ) / . and its average value is(13) ¯Λ = (cid:82) h Λ r (1 + r z ) / dz (cid:82) h r (1 + r z ) / dz . Recall that µ i are the principal curvatures of the Wulff shape W withrespect to the inward pointing normal so 0 << µ i < ∞ holds. It thenfollows easily that0 ≥ (cid:82) h k µ r (1 + r z ) / dz (cid:82) h r (1 + r z ) / dz = (cid:82) h − µ dz (cid:82) h r (1 + r z ) / dz ≥ c ( c ) . From the boundary condition, we have r z = 0 on ∂ Σ . Note that r zz / (1 + r z ) = (arctan r z ) z . We get (cid:90) Σ k µ d Σ = 2 π (cid:90) h rr zz µ (1 + r z ) dz = 2 π (cid:90) h rµ (arctan r z ) z dz = − π (cid:90) h r z µ (arctan r z ) dz − π (cid:90) h ∂ z ( 1 µ ) r (arctan r z ) dz . ote that, using Proposition (2.1), we obtain0 ≥ − π (cid:82) h r z µ (arctan r z ) dz π (cid:82) h r (1 + r z ) / dz ≥ c ( c ) . First assume that (W3) holds. Note that since ν = − r z / √ r z , we canwrite − π (cid:90) ∂ z ( 1 µ ) r (arctan r z ) dz = − π (cid:90) ∂ ν ( 1 µ ) r (arctan (cid:0) − ν (cid:112) − ν (cid:1) ) dν = − π (cid:90) ( ∂ ν µ µ ) r (arctan (cid:0) ν (cid:112) − ν (cid:1) ) dν ≤ ν and arctan( ν (1 − ν ) − / ) have the same sign. Also,since ∂ ν µ /µ is uniformly bounded, we obtain that0 ≥ (cid:82) Σ k µ d Σ (cid:82) Σ d Σ ≥ c ( c ) , and the result follows by combining this with the previous inequalities.It (W3) is not assumed to hold, then we easily obtain | ( k /µ ) | d Σ ≤ cr dz dθ for a constant c while the integrand in thedenominator is counded below by r . q.e.d Again, the governing evolution equation is(14) ∂ t X = (Λ − ¯Λ) ν . For any smooth variation δX = ψν + T of a surface X , the correspond-ing pointwise variation of the anisotropic mean curvature is(15) δ Λ = J [ ψ ] + ∇ Λ · T .
Here J is the self=adjoint elliptic operator given by(16) J [ u ] = ∇ · A ∇ u + (cid:104) Adν, dν (cid:105) u .
Using that the evolution of X is given by (5), we obtain(17) Λ t − ∇ · A ∇ Λ = (cid:104) dν, Adν (cid:105) (Λ − ¯Λ) . We recall from [11] that the normal ν satisfies the equation(18) ∇ · A ∇ ν j + (cid:104) Adν, dν (cid:105) ν j = −∇ Λ · E j , j = 1 , , . This is a consequence of (15) and the translation invariance of the func-tional. o compute the evolution of the normal, we use that for δX = ψν + T ,one has δν = −∇ ψ + dνT . Since X t = (Λ − ¯Λ) ν , we get(19) ν t = −∇ ΛWe define the parabolic operator P [ f ] = f t − ∇ · A ∇ f . Lemma 3.1.
Define ω := √ r z = 1 / (cid:112) − ν . Then (20) P [ ω ] = − ∇ ω · A ∇ ωω + ωr µ − (cid:104) A dν, dν (cid:105) ω .
Proof.
By combining (18) and (19), we obtain the vector equation P [ ν ] = −∇ Λ . For the surface of revolution, we have ν = ( (cid:113) − ν e iθ , ν ) , where we have identified the space of the first two coordinates with thecomplex plane. It follows from (18) that(21) ∇ · A ∇ ( (cid:113) − ν ) e iθ ) = −(cid:104) dν, Adν (cid:105) ( (cid:113) − ν e iθ − ∇ Λ , where the last term denotes the projection of ∇ Λ onto R .The metric on Σ is dS = (1 + r z ) dz + r dθ := α ⊗ α + α ⊗ α If f ( z ) is a differentiable function, then ∇ f · ∇ θd Σ = df ∧ ∗ dθ = f z dz ∧∗ α /r = f z dz ∧ ( − ω/r ) dz = 0, and therefore ∇ f · ∇ θ = 0. Also ∆ θ dS = d ∗ dθ = d ∗ ( α /r ) = d ( − ω/r ) dz = 0. Finally |∇ θ | dS = dθ ∧ ∗ dθ = dθ ∧ ( − ω/r ) dz = (1 /r ) d Σ so |∇ θ | = 1 /r .Using these formulas to expand out the left hand side of (21) and usingthat ν only depends on z , we have ∇ · A ∇ ( (cid:113) − ν e iθ )= e iθ ∇ · A ∇ ( (cid:113) − ν ) + ie iθ (cid:112) − ν µ ( i |∇ θ | + ∆ θ )= e iθ ∇ · A ∇ ( (cid:113) − ν ) − e iθ (cid:112) − ν r µ . Combining this with (21), we get(22) ∇ · A ∇ ( (cid:113) − ν ) = (cid:112) − ν µ r − (cid:104) Adν, dν (cid:105) (cid:113) − ν − e − iθ ∇ Λ . rom this, it is easy to obtain ∇ · A ∇ ω = ∇ · A ∇ (cid:112) − ν − (1 − ν ) − ∇ · A ∇ ( (cid:113) − ν ) −∇ (1 − ν ) − · A ∇ (cid:113) − ν = − ωµ r + (cid:104) Adν, dν (cid:105) ω + e − iθ ω ∇ Λ − ∇ (1 − ν ) − · A ∇ (cid:113) − ν = − ωµ r + (cid:104) Adν, dν (cid:105) ω + e − iθ ω ∇ Λ + 2 ω ∇ ω · A ∇ ω = ( ∗ ) . From (19), we get ( (cid:112) − ν ) t = − e − iθ ∇ Λ which implies ω t = (1 − ν ) − e − iθ ∇ Λ = ω e − iθ ∇ Λ . Combining this with (*) yields (20)We state the following well known Maximum Principle, [15].
Theorem 3.1.
If the operator u (cid:55)→ ∇ · A ∇ [ u ] is elliptic and h : M × [ t , t ] → R satisfies ( ∂ t − ∇ · A ∇ ) h ≤ , holds, then max M h ( · , t ) ≤ max M h ( · , t ) . In the case we are considering, ellipticity of u (cid:55)→ ∇ · A ∇ [ u ] followsfrom the convexity condition W1. Proposition 3.1.
Assume that the initial surface satisfies the estimate(9). Then the generating curve is a graph for all time t for which theevolution (5) exists.Proof. As in [3], we use the boundary condition (6) and the assump-tions on the initial curve to extend the surfaces generated by the flow toperiodic surfaces.For any constant c , we obtain from (20)(23)(log( ω ) − ct ) t − ∇ · A ∇ (log( ω ) − ct ) = 1 r µ − c − ∇ ω · A ∇ ωω − (cid:104) A dν, dν (cid:105) . By Proposition (2.1), there is a constant c = c ( t ) such that r ≥ c holds for t ∈ [ t, t ]. Hence, for a suitable constant c , we have (log( ω ) − ct ) t − ∇ · A ∇ (log( ω ) − ct ) ≤
0. It follows from the Maximum Principal,that(24) max ≤ z ≤ d, ≤ t ≤ t (log( ω ) − ct ) ≤ max ≤ z ≤ d,t =0 log ω . From the definition of ω we have from (24) that there exist constants c (Σ ), a (Σ ) such that if the flow exists for t ∈ [0 , T ), then(25) ω ≤ c e a t : or 0 ≤ t < T .Recalling the definition of ω , the result follows. q.e.d.Proposition 3.2. Assume the conditions (W1) and (W2) hold. Assumethat the flow (5) is defined for ≤ t < T . Then the curvatures of thesurfaces X ( t ) remain bounded, i.e. there exists a constant c ( T ) . with (26) k ( t ) + k ( t ) =: || dν || ( t ) ≤ c ( T ) , ∀ t, ≤ t < T where k i ( t ) denote the principal curvatures at time t . First note that k = − ( rω ) − . By Proposition (2.1) we have r ≥ c ( T )and c ( T ) ≥ ω ≥ ω . Itfollows that | k | ≤ c ( T ). Since µ i , i = 1 ,
2, are uniformly bounded belowand above, (0 < a ≤ µ i ≤ /a for some a ∈ R ), it is enough to show theexistence of a bound(27) Λ ≤ c ( T )for 0 ≤ t < T and then (26) will follow.We recall the standard formula:(28) P [ fg ] = fP [ g ] + gP [ f ] − ∇ f · A ∇ g , for sufficiently smooth functions f and g . The endomorphism field A = D γ + γ id is self-adjoint so that the last term is symmetric in f, g .From the last equation (17) and (20) we obtain(29) P [ ω ] = 2 ω µ r − (cid:104) Adν, dν (cid:105) ω − ∇ ωA ∇ ω , (30) P [Λ ] = 2 (cid:104) Adν, dν (cid:105)
Λ(Λ − ¯Λ) − ∇ Λ · A ∇ Λ . For a suitable C function h ( x ) and B ∈ R both of which we determinelater, we get P [ h ( ω )(Λ + B )]= (Λ + B ) h (cid:48) ( ω ) (cid:0) ω µ r − (cid:104) Adν, dν (cid:105) ω − ∇ ωA ∇ ω (cid:1) − (Λ + B ) h (cid:48)(cid:48) ( ω ) ∇ ω · A ∇ ω + h ( ω ) (cid:0) (cid:104) Adν, dν (cid:105)
Λ(Λ − ¯Λ) − ∇ Λ · A ∇ Λ (cid:1) − h (cid:48) ( ω ) ∇ ω · A ∇ Λ . e take h ( x ) := e ax for a constant a to be determined later, and get P [ e aω (Λ + B )]= e aω (cid:18) (Λ + B ) a (cid:0) ω µ r − (cid:104) Adν, dν (cid:105) ω − ∇ ωA ∇ ω (cid:1) − (Λ + B ) a ∇ ω · A ∇ ω + 2 (cid:104) Adν, dν (cid:105)
Λ(Λ − ¯Λ) − ∇ Λ · A ∇ Λ − a ∇ ω · A ∇ Λ (cid:19) = e aω (cid:18) a (Λ + B ) µ r + (cid:2) − a (Λ + B ) + 2Λ − (cid:3) (cid:104) Adν, dν (cid:105) + (cid:2) − a (Λ + B ) ∇ ω · A ∇ ω − ∇ Λ · A ∇ Λ − a ∇ ω · A ∇ Λ (cid:3)(cid:19) = ( ∗∗ ) . We have − a (Λ + B ) + 2Λ − ≤ − a (Λ + B ) + 3Λ + ¯Λ , using 2 xy ≤ x + y . Because of Lemma (13), we have the existence of aconstant c ( T ) with ¯Λ ≤ ( c ( T )) . Hence by choosing a, B >>
0, we getthat the term in (**) which includes the factor (cid:104)
Adν, dν (cid:105) is non positive.Next notice that since the tensor A is positive definite and self adjoint,at each fixed point p ∈ Σ, we have an inner product defined by ( u, v ) = Au · v , u, v ∈ T p Σ. We then have using 2 | ( u, v ) | ≤ ( u, u ) + ( v, v ),2 a |∇ ω · A ∇ Λ | = 4 a | Λ ||∇ ω · A ∇ Λ | = 4 a | Λ || ( ∇ ω , ∇ Λ) ≤ a Λ ( ∇ ω , ∇ ω ) + ( ∇ Λ , ∇ Λ)= 4 a Λ ∇ ω · A ∇ ω + ∇ Λ · A ∇ Λ . This means that both terms in (**) between the square brackets are nonpositive for suitable choices of the constants a and B and so we can con-clude that P [ e aω (Λ + B )] ≤ e aω a (Λ + B ) µ r , holds. Now recall that by Proposition (2.1), we have r ≥ c (Σ ) >
0. Itthen follows that P [ e − Mt e aω (Λ + B )] = e − Mt (cid:18) − Me aω (Λ + B ) + P [ e aω (Λ + B )] (cid:19) ≤ e − Mt (cid:18) − Me aω (Λ + B ) + e aω a (Λ + B ) µ r (cid:19) ≤ , for a sufficiently large constant M . Recalling from Proposition (3.1)that ω ≤ (cid:112) c ( T ), we find that (27) follows from the Maximum Prin-ciple. q.e.d. Higher order regularity
In this section we obtain bounds on higher order derivatives of the surface.In similar problems involving mean curvature flow, these are obtained frombounds on higher order derivatives of the second fundamental form | dν | .In our case, a nice evolution equation for this quantity is unavailable.However, since we are only working with surfaces of revolution, we canuse (30) instead to obtain the desired bounds. Proposition 4.1.
For positive constants σ m = σ m ( T ) , τ m = τ m ( T ) ,there holds (31) P [ |∇ m Λ | ] ≤ − σ m |∇ m +1 Λ | + τ m (1 + |∇ m Λ | ) , τ m = τ m ( |∇ j Λ | ) , j < m for 0 ≤ t ≤ T . Lemma 4.1.
Let f = f ( s ) be a sufficiently smooth function. Then (32) |∇ m f ( s ) ≤ |∇ ms f ( s ) | + E , where E only depends on |∇ j f | , j ≤ m − .Proof. The proof is straightforward and is left to the reader.
Proof of Proposition (4.1)
At a fixed time t , we let ω := ds , ω = rdθ and we write, using the summation convention, d Λ = Λ j ω j , ∇ d Λ = Λ jk ω j ω k , ... Then at any time t , we can write(33) |∇ m Λ | = | Λ j ....j m | (cid:89) k =1 ..m g j k j k , where, although the coframe is fixed, all quantities depend on t . Through-out the evolution, all the surfaces are axially symmetric, so g ii = 1 /g ii .For a normal variation X ( t ) = X + tψν + ... , the first variation of themetric is ∂ t g ij = − ψh ij where h ij are the coefficients of the second fun-damental form. Since all the surfaces are axially symmetric, g ij = δ ij g ii and h ij = δ ij h ii for all t . Using this, we obtain from (33) ∂ t |∇ m Λ | = 4 H (Λ − ¯Λ) |∇ m Λ | + 2Λ j ....j m ( ∂ t Λ j ....j m ) (cid:89) k =1 ..m g j k j k = 4 H (Λ − ¯Λ) |∇ m Λ | + 2 (cid:104)∇ m Λ , ∇ m J [Λ] (cid:105)∼ (cid:104)∇ m Λ , ∇ m ( ∇ · A ∇ Λ) (cid:105) , where ∼ means that the quantities are equal up to terms of orders lessthan or equal to m . In the given frame, for a function f ( s ),(34) ∇ · A ∇ f = 1 rω ∂ s ( rf s µ ) . In particular, since Λ only depends on s , we get(35) ∂ t |∇ m Λ | ∼ ωµ ( ∇ ms Λ)( ∇ m +2 s Λ) + Φ m ( ∇ ms Λ)( ∇ m +1 s Λ) , or suitable functions Φ m which are bounded by lower order derivatives.Since |∇ m Λ | only depends on s , applying (34) gives ∇ · A ∇|∇ m Λ | = 1 rω ∂ s ( r∂ s |∇ m Λ | µ )= 1 rω ∂ s ( 2 rµ (cid:104)∇ m Λ , ∇ s ∇ m Λ (cid:105) )= 2 ωµ (cid:104)∇ m Λ , ∇ s ∇ s ∇ m Λ (cid:105) + 2 ωµ |∇ m +1 s Λ | + 1 rω ( 2 rµ ) s (cid:104)∇ m Λ , ∇ s ∇ m Λ (cid:105) . From the last equation, (35) and Lemma 4.1, we can write P [ |∇ m Λ | ] ∼ − ωµ |∇ m +1 s Λ | + ˜Φ m (cid:104)∇ m Λ , ∇ s ∇ m Λ (cid:105) . for a suitable function ˜Φ m which depends only on lower order derivatives.The second term can be bounded | ˜Φ m (cid:104)∇ m Λ , ∇ s ∇ m Λ (cid:105)| ≤ (cid:15) |∇ m +1 s Λ | + (cid:15) − | ˜Φ m || ∇ m Λ | ∼ (cid:15) |∇ m +1 s Λ | . By applying lemma (4.1), we obtain P [ |∇ m Λ | ] ∼ − σ m |∇ m +1 s Λ | , where for suitable (cid:15) , we can take, using (25),(36) σ m := (cid:0) max( µ ) (cid:113) c ( T ) (cid:1) − . This shows that (31) holds. q.e.d.Theorem 4.1.
There exist constants C m ( T ) such that (37) |∇ m Λ | ≤ C m ( T ) , ∀ t < T . Proof of Th. (4.1).
Denote by S ( m ) the statement (37). The state-ment S (0) is just (27) which was shown above. We assume S ( j ) holds forall j < m .Let Ψ m := |∇ m Λ | . By (28), (29) and (31), we find, for positiveconstants a and BP [ e aω (Ψ m + B )] ≤ e aω (cid:18) − σ m Ψ m +1 + τ m (1 + Ψ m )+(Ψ m + B )( a (cid:2) ω µ r − (cid:104) Adν, dν (cid:105) ω − ∇ ωA ∇ ω (cid:3) − a ∇ ω · A ∇ ω ) − a ∇ Ψ m · A ∇ ω (cid:19) . e can estimate the last term above using2 a |∇ Ψ m · A ∇ ω | ≤ (cid:15) Ψ m +1 + (cid:15) − a ∇ ω · A ∇ ω . We choose (cid:15) < σ m ( T ) and choose B >> − B ∇ ωA ∇ ω + (cid:15) − a ∇ ω · A ∇ ω < ω is bounded by (25).Recalling that ( ω/r ) is also bounded in any finite time interval, so wearrive at P [ e aω (Ψ m + B )] < Ce aω (Ψ m + B ) , for t ≤ T and a suitable constant C = C ( T ). Here we are using theinduction hypothesis since the constant τ m = τ m ( |∇ j Λ | ), j < m , havebeen absorbed into the costant C ( T ). It then follows that P [ e − Ct e aω (Ψ m + B )] < ≤ t ≤ T , so the result follows from the Maximum Principle. q.e.d.Proposition 4.2. If (37) holds then for suitable constants, the principalcurvatures satisfy (38) |∇ m k | ≤ c m , |∇ m k | ≤ c (cid:48) m hold.Proof. For a surface of revolution, the Codazzi equations reduce to( k ) s = ( k − k ) r s /r . It follows by an easy induction argument that an upper bound for |∇ ms k | and hence |∇ m k | , can be obtained from upper bounds on |∇ j k i | , i = 1 , j < m . For this one needs (10) and the fact that derivatives of r of order j have upper bounds which depend on derivatives of k of order ≤ j − r ss = k z s .For a surface of revolution, the anisotropic principal curvatures arethe functions λ j := k j /µ j , j = 1 ,
2. These are the eigenvalues of thedifferential dξ : T Σ → T W . Note that Λ = λ + λ .We have ( k j ) s = ( λ j ) s µ j + λ j µ (cid:48) j ( ν ) ( − k ) z s from which it follows easilythat |∇ ms k j | ≤ (max µ j ) |∇ ms λ j | + E , |∇ ms λ j | ≤ (max(1 /µ j )) |∇ ms k j | + E (cid:48) , where E , E (cid:48) depends on derivatives of the k j ’s of order less than or equalto m − |∇ m λ | ≤ |∇ m Λ | + |∇ m λ |≤ |∇ m Λ | + (max(1 /µ j )) |∇ ms k | + E (cid:48) ≤ |∇ m Λ | + E + E (cid:48) , where E (and E (cid:48) ) only depends on |∇ j k i | , j ≤ m −
1. Using induction,(37) and (26), the result follows. q.e.d. heorem 4.2. Assume that the initial surface satisfies (9). Then theflow exists for all time.Proof.
The result follows by a standard argument. Assume theflow exists on a finite time interval [0 , T max ). Because of the uniformestimates given by (26) and (38), the flow can also be extended smoothlyto t = T max . Then, by applying the local existence result, the flow can beextended to [0 , T max + (cid:15) ) for some (cid:15) > q.e.d.Remark We show L convergence of the anisotropic mean curvatureto its mean. Note that ∂ t F [Σ t ] = − (cid:90) Σ (Λ − ¯Λ) d Σ , and therefore F [Σ ] ≥ (cid:90) ∞ (cid:90) Σ t (Λ − ¯Λ) d Σ t dt . In particular (cid:90) Σ (Λ − ¯Λ) d Σ t → , as t → ∞ . Based on the different descriptions of the evolving surface, different meth-ods can be used to numerically solve a surface evolution equation. Theseinclude parametric, level set, and phase field methods (see [6]). Eachmethod has its own advantage and disadvantage. We choose the para-metric method, which basically is a front tracking method, namely, thesurface is evolved and tracked according to the surface evolution equation(8).Instead of using fully implicit schemes for the temporal discretizationof equation (8), a semi–implicit backward Euler method is used. Thekey of time forwarding in the semi–implicit scheme is to approximatethe nonlinear terms in the equation by using the previously computedapproximated solution, while the linear terms still need to be solved im-plicitly. Therefore we can avoid solving systems of nonlinear equations(for example, using the Newton’s method) at each time step and thusthe computational cost can be reduced. On the other hand, the implicitfeature will increase the stability of the scheme so the restrictions on timestep sizes can be loosened.As for the spatial discretization, a second–order finite differencemethod can be used. Finite element method using piecewise linear func-tions can also be used, and if the mass is lumped, it will be equivalentto the finite difference method. However, based on our numerical exper-iments with the above two methods, we choose to present the methodof using cubic spline approximations. The spline approximation not onlyprovides a higher order method, it also ensures the continuity of the secondorder derivatives across the spatial nodes, which we think it is important n the approximation of curvature flows. On the other hand, since thecurvature flow we study here is essentially one dimensional, as we will seebelow, when using the spline approximation, the resulting linear systemthat we need to solve is very sparse and the computational cost will be ofthe same order as the cost using finite difference or finite element method. In this section, we discuss numerical methods for simulating the ini-tial boundary value problem (5) and (6). For a representative class ofexamples, we concentrate on the Rapini–Papoular functionals given by γ = 1 + (cid:15)ν , where − ≤ (cid:15) ≤ T >
N, K be positive integers. Let0 = z ≤ · · · ≤ z N ≤ z N +1 = 1 be a partition of the interval [0 ,
1] andlet 0 = t ≤ · · · ≤ t K ≤ = T be an equally spaced partition of the timeinterval [0 , T ].We use the following notation r kn = r ( t k , z n ) , ( r z ) kn = r z ( t k , z n ) , ( r zz ) kn = r zz ( t k , z n ) , Λ kn = Λ( t k , z n ) , where 0 ≤ k ≤ K and 1 ≤ n ≤ N + 1. We also set τ = T /K, Q ( r ( t, z )) = 1 + ( r z ( t, z )) . Using the semi–implicit backward Euler method and the above nota-tions, the discretized evolution equation is(39) r k +1 n − r kn τ = (cid:16) ( r zz ) k +1 n µ [ Q (( r z ) kn )] / − µ [ Q (( r z ) kn )] / − Λ kn (cid:17)(cid:112) Q (( r z ) kn ) , for 0 ≤ k ≤ K and 1 ≤ n ≤ N + 1.For an approximation of the generating curve of the axially surface r ( t, z ), we seek a spline function S ( t, z ) that satisfies equation (39) andthe following properties :(i) S n ( t, z ) = S ( t, z ) (cid:12)(cid:12) [ z n ,z n +1 ] , the restriction of S ( t, z ) on interval[ z n , z n +1 ], 1 ≤ n ≤ N , is a polynomial of degree no more than3.(ii) The first and second order partial derivatives of S with respect to z exist at nodes z , · · · , z N +1 and are continuous at the internal nodes z , · · · , z N .(iii) ∂S∂z ( t, z ) = α = ∂r∂z ( t,
0) and ∂S∂z ( t, z N +1 ) = β = ∂r∂z ( t, α = β = 0 if the contact angles of the surface at the top and bottomplanes are required to be right angles.At time t = t k +1 , 0 ≤ k ≤ K −
1, the unknown for equation (39) isdenoted by a column vector of length 2 N :(40) (cid:126)x k +1 = ( r k +11 , · · · , r k +1 N +1 ; d k +12 , · · · , d k +1 N ) T here r k +1 n = S ( t k +1 , z n ) , d k +1 n = ∂S∂z ( t k +1 , z n ) , ≤ n ≤ N + 1 . To derive a linear system of (cid:126)x k +1 from equation (39), for each 1 ≤ n ≤ N , we adopt the following notations: S kn ( z ) = S n ( t k , z ) = S ( t k , z ) (cid:12)(cid:12)(cid:12) [ z n ,z n +1 ] ( t k , z ) , ≤ k ≤ K,h n = z n +1 − z n ,δ n = r k +1 n − r kn h n ,s = z − z n , z ∈ [ z n +1 , z n ] . The piecewise–defined spline function S ( t k , z ) consists of the functions S kn of the following on the interval [ z n , z n +1 ]:(41) S kn ( s ) = 3 h n s − s h n r kn +1 + h n − h n s + 2 s h n r kn + s ( s − h n ) h n d kn +1 + s ( s − h n ) h n d kn , for 0 ≤ k ≤ K . Following the standard theory about cubic splines, wederive the following linear system(42) 3 h n − r k +1 n − + (cid:16) h n − h n − (cid:17) r k +1 n − h n r k +1 n +1 + h n d k +1 n − + 2( h n − + h n ) d k +1 n + h n − d k +1 n +1 = 0 , ≤ n ≤ N .
Invoking the boundary conditions, the above system can be written as(43) M (cid:126)x = (cid:126)b , where (cid:126)x is given as in equation (40), (cid:126)b is a column vector of length N − (cid:126)b = (cid:2) − h α, , · · · , , − h N β (cid:3) T , and M = [ M , M ] is an ( N − × (2 N ) matrix with M = h (cid:16) h − h (cid:17) − h h (cid:16) h − h (cid:17) − h . . . . . . . . .3 h N − (cid:16) h N − h N − (cid:17) − h N ( N − × ( N +1) , and M = h + h ) h h h + h ) h . . . . . . . . . h N − h N − + h N ) h N − h N h N − + h N ) ( N − × ( N − . n the other hand, from the equation (39), we can derive the rest ofthe equations needed for solving (cid:126)x . For convenience, we use( µ ) kn = µ ( r ( t k , z n )) , ( µ ) kn = µ ( r ( t k , z n )) , Q kn = 1 + ( r z ( t k , z n )) , and ξ kn = 1( µ ) kn Q kn τ, η kn = − (cid:16) µ ) kn r kn + Λ kn (cid:112) Q kn (cid:17) τ for 0 ≤ k ≤ K and 1 ≤ n ≤ N + 1. Then, the system (39) can be writtenas(44) r k +1 n = ( r zz ) k +1 n ξ kn + η kn + r kn , ≤ n ≤ N + 1 . Since from equation (41), we have r k +1 n = 6 δ k +1 n − d k +1 n +1 − d k +1 n h n , ≤ n ≤ N ; r k +1 n = − δ k +1 N + 4 d k +1 N +1 + 2 d k +1 N h N , n = N + 1 , using the boundary conditions, we can write (44) as(45) M (cid:126)x = (cid:126)b , where (cid:126)b is a column vector of length N + 1 given by (cid:126)b = h ( η k + r k ) − αξ k h ( η k + r k )... h N − ( η kN − + r kN − ) h N ( η kN + r kN ) − βξ kN h N ( η kN +1 + r kN +1 ) + 4 βξ kN +1 , and M = [ M , M ] is an ( N − × (2 N ) matrix with M = h + 6 ξ k h − ξ k h . . . . . . h N + 6 ξ kN h N − ξ kN h N − ξ kN +1 h N h N + 6 ξ kN +1 h N ( N +1) × ( N +1) , and M = ξ k ξ k ξ k . . . . . .4 ξ kN − ξ kN − ξ kN − ξ kN +1 ( N +1) × ( N − . Combining systems (43) and (45), we finally obtain the linear system M(cid:126)x = (cid:126)b, where M = (cid:20) M M M M (cid:21) , (cid:126)b = (cid:34) (cid:126)b (cid:126)b (cid:35) . As we can see, the system (46) is very sparse and can be solved by stan-dard sparse solvers. We also would like to mention that the computationof Λ kn is done by using Gaussian quadratures. Some of the results from our numerical experiments will be presentedbelow to illustrate the evolution the curvature flow that we have studied.In all of our simulations, we let z ∈ [0 , Example 5.1.
In this experiment, (cid:15) = 0 . , T = 3 , N = 500 , τ = 10 − .The initial profile of r is chosen to be the cubic Hermite interpolant thatsatisfies r (0 ,
0) = 0 . , r (0 ,
1) = 0 . , r z (0 ,
0) = r z (0 ,
1) = 0 . The results are shown in Figure 1:(a) r ( t, z ), the profiles of the generating curve of the surface at differenttimes are shown in plot (a). The initial profile is eventually evolvesto a cylinder and remains thereafter.(b) The snapshots of the surfaces in three dimensional space at differenttimes are also shown in plot (b).(c) The history of the values of the energy functional F (in equation(1)) is shown in plot (c), and it can be seen that the energy isdecreasing as the surface evolves under equation (8) until it remainsnearly unchanged, which numerically implies that a minimum of theenergy has been reached and the minimizer is corresponding to thethe surface of a cylinder.(d) The history of the values of the volume enclosed by the surface isshown in plot (d). Although it is seen that the volume is not pre-served at the beginning of the evolution process, the error (relativeto the initial volume) is within 0 . kn as in our semi–implicit scheme,which numerically violates the law of volume preserving curvatureflow (equation (8)), unless in the later stage of the evolution, thesurface nearly has constant anisotropic mean curvature.According to Theorem 5.1 of [11], the threshold of stability for cylin-ders is(47) µ (0) µ (0) 1 r ≤ π h . or γ = 1 + (cid:15)ν and (cid:15) = 0 .
2, the cylinder is stable provided r ≥ √ (cid:15)π ≈ . . This experiment numerically verifies the above stability analysis. In othersimilar experiments with larger volume fractions, we have also observedthat no pinching has occurred. (a) (b)(c) (d)Figure 1: Example 5.1. Snapshots of surface profiles, values of energy functional,and volumes. (cid:15) = 0 . r (0 ,
0) = 0 . r (0 ,
1) = 0 . Example 5.2.
In this experiment, (cid:15) = 0 . , T = 3 , N = 500 , τ = 10 − .The initial profile of r is chosen to be the cubic Hermite interpolant thatsatisfies r (0 ,
0) = 0 . , r (0 ,
1) = 0 . , r z (0 ,
0) = r z (0 ,
1) = 0 . he results are shown in Figure 2. The initial volume is about 0 . . × − %. (a) (b)(c) (d)Figure 2: Example 5.2. Snapshots of surface profiles, values of energy functional,and volumes. (cid:15) = 0 . r (0 ,
0) = 0 . r (0 ,
1) = 0 . Example 5.3.
In this experiment, (cid:15) = 0 . , T = 3 , N = 500 , τ = 10 − .The initial profile of r is chosen to be the cubic Hermite interpolant thatsatisfies r (0 ,
0) = 0 . , r (0 ,
1) = 0 . , r z (0 ,
0) = r z (0 ,
1) = 0 . The only difference between this example and Example 5.2 is the valuesof r (0 , xample 5.4. In this experiment, (cid:15) = 0 . , T = 4 , N = 1000 , τ = 10 − .The initial profile of r is chosen to be the cubic Hermite interpolant thatsatisfies r (0 ,
0) = 0 . , r (0 ,
1) = 0 . , r z (0 ,
0) = r z (0 ,
1) = 0 . The result is shown in Figure 4.
Example 5.5.
In this experiment, (cid:15) = 0 . , T = 4 , N = 500 , τ = 10 − .The initial profile of r is chosen to be the cubic Hermite interpolant thatsatisfies r (0 ,
0) = 0 . , r (0 ,
1) = 0 . , r z (0 ,
0) = r z (0 ,
1) = 0 . The results are shown in Figure 5.
Through these examples 5.2 to 5.5, we see that for small volume frac-tions, namely, when inequality (9) is not satisfied, the stability of cylindersis lost and singularity may develop, and therefore the flow may not exitfor all time.We present two more examples: In Example 5.6, (cid:15) <
0, in Example5.7, a different initial profile is used. The results are shown in Figure 6and Figure 7, respectively.
Example 5.6.
In this experiment, (cid:15) = − . , T = 2 , N = 500 , τ = 10 − .The initial profile of r is chosen to be the cubic Hermite interpolant thatsatisfies r (0 ,
0) = 0 . , r (0 ,
1) = 0 . , r z (0 ,
0) = r z (0 ,
1) = 0 . Example 5.7.
In this experiment, (cid:15) = 0 . , the initial profile is given by r (0 , z ) = 1 + 14 cos(8 πz ) . References [1] Andrews, Ben Volume-preserving anisotropic mean curvature flow.Indiana Univ. Math. J. 50 (2001), no. 2, 783827.[2] Arroyo, Josu, Koiso, Miyuki and Palmer, Bennett; Stability of nonliquid bridges, Preprint 2009.[3] Athanassenas, Maria Volume-preserving mean curvature flow of ro-tationally symmetric surfaces. Comment. Math. Helv. 72 (1997), no.1, 5266.[4] Athanassenas, Maria Behaviour of singularities of the rotationallysymmetric, volume-preserving mean curvature flow. Calc. Var. Par-tial Differential Equations 17 (2003), no. 1, 116.[5] Cabezas-Rivas, E. and Miquel, V; Volume preserving mean curvatureflow of revolution hypersurfaces between two equidistants.
6] K. Deckelnick, G. Dziuk, and C. M. Elliott,
Computation of geo-metric partial differential equations and mean curvature flow , Acta.Numer. (2005), pp. 1–94[7] Ecker, Klaus Regularity theory for mean curvature flow. Progressin Nonlinear Differential Equations and their Applications, 57.Birkh¨auser Boston, Inc., Boston, MA, 2004.[8] Giga, Yoshikazu Surface evolution equations. A level set approach.Monographs in Mathematics, 99. Birkhuser Verlag, Basel, 2006.[9] Huisken, Gerhard The volume preserving mean curvature flow. J.Reine Angew. Math. 382 (1987).[10] Koiso, Miyuki; Palmer, Bennett Geometry and stability of surfaceswith constant anisotropic mean curvature. Indiana Univ. Math. J. 54(2005), no. 6, 1817–1852.[11] Koiso, Miyuki; Palmer, Bennett Stability of anisotropic capillarysurfaces between two parallel planes. Calc. Var. Partial DifferentialEquations 25 (2006), no. 3, 275–298.[12] Lunardi, Allesandra, Analytic Semigroups and Optimal Regularityin Parabolic Problems , Birkh¨auser Basel; 1 edition (May 12, 2003)[13] McCoy, James A. Mixed volume preserving curvature flows. Calc.Var. Partial Differential Equations 24 (2005), no. 2, 131154.[14] Taylor, Jean, Some mathematical challenges in materials science,Bulletin of The American Mathematical Society , vol. 40, no. 01,pp. 69-88, 2002[15] Protter, M. H., Weinberger, H. F., Maximum Principles in Differen-tial Equations. New York-Berlin-Heidelberg-Tokyo, Springer-Verlag1984.Bennett P
ALMER
Department of MathematicsIdaho State UniversityPocatello, ID 83209U.S.A.E-mail: [email protected] Z HU Department of MathematicsIdaho State UniversityPocatello, ID 83209U.S.A.E-mail: [email protected] (cid:15) = 0 . r (0 ,
0) = 0 . r (0 ,
1) = 0 . (cid:15) = 0 . r (0 ,
0) = 0 . r (0 ,
1) = 0 .
1. 25igure 5: Example 5.5. Snapshots of the 3D surfaces. (cid:15) = 0 . r (0 ,
0) = 0 . r (0 ,
1) = 0 .
2. 26a) (b)(c) (d)Figure 6: Example 5.6. Snapshots of surface profiles, values of energy functional,and volumes. (cid:15) = − . r (0 ,
0) = 0 . r (0 ,
1) = 0 . (cid:15) = 0 . r (0 , z ) = 1 + cos(8 πz ) //