A minimising movement scheme for the p-elastic energy of curves
aa r X i v : . [ m a t h . A P ] J a n A minimising movement scheme for the p -elasticenergy of curves Simon Blatt, Christopher P. Hopper, and Nicole Vorderobermeier
Abstract.
We prove short-time existence for the negative L -gradientflow of the p -elastic energy of curves via a minimising movementscheme. In order to account for the degeneracy caused by the en-ergy’s invariance under curve reparametrisations, we write the evolvingcurves as approximate normal graphs over a fixed smooth curve. Thisenables us to establish short-time existence and give a lower bound onthe solution’s lifetime that depends only on the W ,p -Sobolev normof the initial data.
1. Introduction
For closed curves γ : R / Z → R n in the W ,p -Sobolev class we shall considerthe energy E ( γ ) = 1 p Z R / Z | κ | p ds + λ Z R / Z ds, (1.1)i.e. the sum of the p -elastic energy E ( p ) ( γ ) = p R R / Z | κ | p ds and a positivemultiple λ > γ = γ ( t, s ) : [0 , T ) × R / Z → R n in the class L ∞ (cid:0) [0 , T ) , W ,p ( R / Z , R n ) (cid:1) ∩ W , (cid:0) [0 , T ) , L ( R / Z , R n ) (cid:1) is said to be a weak solution of the negative L -gradient flow of E if one has Z T Z R / Z h ∂ t γ, ψ i dsdt = Z T δ ψ t E ( γ t ) dt (1.2)for all test functions ψ ∈ C ∞ c ( R / Z × (0 , T ) , R n ), i.e. the curve γ satisfies ∂ t γ = −∇ L E ( γ ) weakly, where δ ψ t E ( γ t ) = ddε E ( γ t + εψ t ) | ε =0 is the firstvariation of the functional E at the curve γ t = γ ( t, · ) in the direction of thetest function ψ t = ψ ( t, · ).While the L -gradient flow of (1.1) has been extensively studied when p = 2, both in the Euclidean (cf. [4,5,9,15]) and manifold constrained (cf. [3, Mathematics Subject Classification.
Key words and phrases. minimising movement, p-elastic energy for curves curves, gra-dient flow, approximate normal graphs.All three authors acknowledge support by the Austrian Science Fund (FWF), GrantP29487. Nicole Vorderobermeier also acknowledges funding by the Austrian MarshallPlan Foundation. p = 2 case. For examplea second order evolution equation has been considered for closed curves andplanar networks (cf. [21,22]) and the asymptotic of the flow has been studiedaway from degenerate point (cf. [24]), however short-time existence for theequation (1.2) has yet to be established when p = 2. The aim of this articleis to address both short and long time existence in the case p > W ,p -Sobolev class. Ourapproach is to rewrite the evolving curves as approximate normal graphs inorder to utilise de Giorgi’s method of minimising movements (cf. [6]).It is well known that the invariance of the energy (1.1) under reparametri-sations of the curve γ leads to an evolution equation (1.2) that fails to bestrongly parabolic (even in the p = 2 case). This characteristic is in com-mon with many other geometric evolution equations. For example the failureof the strong ellipticity of the Ricci tensor is principally due to the secondBianchi identities. For this reason, short-time existence for the Ricci flowwas originally established in [13] by appealing to the Nash-Moser implicit-function theorem (and the earlier exposition in [12]). DeTurck [7] subse-quently showed that the Ricci flow is equivalent to an initial value problemfor a parabolic system modulo the action of the diffeomorphism group of theunderlying manifold. Thus, in a dramatic simplification that bypassed theNash-Moser argument, one can pass from a weakly parabolic to a stronglyparabolic system of equations by an appropriate choice of a 1-parameterfamily of diffeomorphisms. Perelman [23] also exploited the same diffeomor-phism invariance in his gradient flow formalism for the Ricci flow. Versions ofthe DeTurck trick have also been used to obtain short-time existence for themean curvature flow (cf. [2,14]), the Willmore flow (cf. [16]) and the gradientflow of the elastic energy in both the Euclidean and manifold constrainedcases.In seeking to pass from the degenerate flow (1.1) to a strongly parabolicsystem, one can consider a time dependent family of curves γ t = γ ( t, · ) thatare written as a normal graph over a given fixed smooth curve e γ , i.e. a familyof curve of the form γ t = e γ + φ t where φ t = φ ( t, · ) is a perturbation normalto the fixed curve e γ . In this way we obtain an evolution equation of the form Z T Z R / Z h γ, ∂ ⊥ t ψ i dsdt = Z T δ ψ t E ( γ t ) dt (1.3)for all test functions ψ ∈ C ∞ c ((0 , T ) × R / Z , R n ), i.e. the curve γ satisfies ∂ ⊥ t γ = −∇ L E ( γ ) weakly, where the normal velocity ∂ ⊥ t γ is the vectorcomponent of ∂ t γ normal to the fixed curve e γ . Then in order to obtain asolution of (1.2) from a solution of (1.3), one can consider solutions Θ t = In fact the diffeomorphism invariance of the Riemannian curvature tensor naturallyyields the Bianchi identities (cf. [17]). Thus the strongly ellipticity failure of the Riccitensor is due entirely to this geometric invariance. minimising movement scheme for the p -elastic energy of curves 3 Θ( t, · ) of the ordinary differential equation ∂ t Θ( t, x ) = F ( t, Θ( t, x ))Θ(0 , x ) = x, (1.4)where F ( t, y ) = − h ∂ t γ ( t,y ) ,γ ′ ( t,y ) i| γ ′ ( t,y ) | and γ is a solution of (1.3). The existenceof ODE solutions can thus be established on a time interval 0 ≤ t < ε for some ε > x ∈ R / Z . Therefore ifΘ t = Θ( t, · ) is a solution of (1.4) and γ t = γ ( t, · ) is a solution of (1.3),the composition γ t ◦ Θ t is a solution of (1.2). By taking this approach onecan thus establish the existence of solutions for geometric flows with initialdata in the C ,α -H¨older class even though the original equations may beill-defined (see, e.g., [10, 19, 25]). In fact a recent paper by LeCrone, Shaoand Simonett [18] showed how to reduce the regularity of the initial data tothe C ,α -H¨older class.In order to carry out the aforementioned programme, one has to guaranteethat a given initial curve Γ can be written as a normal graph over a fixedsmooth curve e γ . Since it is not possible to write every curve Γ in the W ,p -Sobolev class as a normal graph over a smooth curve, we are spurred on tointroduce the notion of a unit quasi-tangent τ (cf. Definition 2.4) which thendefines an approximate tangential projection P Tτ and an approximate normalprojection P ⊥ τ = I − P Tτ (cf. Definition 2.6). In which case, one can write thecurve Γ as equal to e γ + Φ up to a reparametrisation, i.e. as an approximatenormal graph over a smooth curve e γ with some perturbation Φ orthogonalto τ (cf. Lemma 2.12). Then by applying a minimising movements scheme,it is possible to establish the existence of a family of curves of the form γ t = e γ + φ t , for a suitable perturbation φ t orthogonal to τ , that satisfies ∂ ⊥ t γ = −∇ L E ( γ ) weakly. Indeed, we have: Theorem 1.1 (Existence) . For any given initial curve Γ ∈ W ,p ( R /L Z , R n ) parametrised by arc-length there exists a smooth curve e γ ∈ C ∞ ( R /L Z , R n ) parametrised by arc-length, a quasi-tangent τ to the curve e γ , a finite time T = T ( p, λ, E (Γ)) > and a family of perturbations φ in the class L ∞ (cid:0) [0 , T ) , W ,p ( R /L Z , R n ) (cid:1) ∩ (cid:0) W , ∩ C / (cid:1)(cid:0) [0 , T ) , L ( R /L Z , R n ) (cid:1) which are orthogonal to τ such that the family of curves γ ( t, s ) = e γ ( s ) + φ ( t, s ) , ≤ t < T, satisfies the initial condition γ (0 , · ) = Γ ◦ σ for some reparametrisation σ of R /L Z and Z T Z R /L Z h ∂ ⊥ t γ, ψ i dsdt = − Z T δ ψ t E ( γ t ) dt (1.5) for all test functions ψ ∈ C ∞ c ((0 , T ) × R /L Z , R n ) orthogonal to τ . Note that the time of existence only depends on the energy of the initialcurve. So we are very close to restarting the flow and deduce long time
BLATT, HOPPER, AND VORDEROBERMEIER existence. We discuss in the final section, why this is not as straightforwardas it might seem.By assuming the solution has some additional regularity, one can showthat equation (1.5) holds for all test functions (i.e. our solution solves theoriginal weak form of the desired evolution equation).
Corollary 1.2.
If the solution γ ( t, · ) of Theorem 1.1 belongs to the W ,p -Sobolev class for almost all ≤ t < T , then Z T Z R /L Z h ∂ ⊥ t γ, ψ i dsdt = − Z T δ ψ t E ( γ ) dt for all test functions ψ ∈ C ∞ c ((0 , T ) × R /L Z , R n ) .
2. Minimising movements scheme
It is remarked by De Giorgi [6] that a generalised minimising movementsscheme could provide a formalism for the existence of steepest descent curvesof a functional in a metric space. In order to establish the existence of weaksolutions for (1.2), we need to take care of the twofold degeneracies arisingfrom the invariance of (1.1) under curve reparametrisation and the fact that p >
2. We tackle this issue by writing the evolving curve as an approximatenormal graph over a fixed smooth curve so that we can work with the normalvelocity (rather than the time derivative) of the evolving curve.
For an embedded C k -submanifold M of R n without boundary, the normal bundle ( T M ) ⊥ → M is only of the class C k − . If we define the ‘endpoint’ map E : ( T M ) ⊥ → R n by sending( x, v ) x + v and assume k ≥
2, one can use the inverse function theorem to show thatthere exists a tubular neighbourhood U of M in R n that is the diffeomorphicimage under the C k − -map E of an open neighbourhood of the zero sectionof ( T M ) ⊥ . Moreover, the squared distance function ζ ( x ) = dist( x, M ) isa function in C k ( U ) (cf. [11]) and the Hessian matrix ∇ ζ ( x ) represents theorthogonal projection on the normal space to M at a point x (cf. [1, p. 704]).Of course these results no longer hold in the case k = 1, i.e. when the inversefunction theorem is not applicable. As the normal bundle of an embed-ded W ,p -curve in R n is only of the class W ,p , one cannot directly apply thestandard methods of § quasi-tangent . minimising movement scheme for the p -elastic energy of curves 5 Definition 2.1.
A function η ∈ C ∞ ( R ) is called a mollifier if it satisfies theconditions: (i) η ≥ R , (ii) η ( x ) = 0 for all | x | ≥
1, and (iii) R R η ( x ) dx =1. The associated rescaled mollifier is the function η ε ( x ) = ε η ( xε ) for any ε > γ ∈ W ,p ( R / Z , R n ) parametrised by arc-length.The mollification of γ is defined to be the function γ ε ( x ) = ( γ ∗ η ε )( x ) = Z R γ ( x − y ) η ε ( y ) dy, i.e. the convolution of the given curve γ and the rescaled mollifier η ε .For the mollified curve γ ε we derive the following well-known estimates.Firstly, from the mean value theorem and the Sobolev embeddings, we findthat | γ ε ( x ) − γ ( x ) | = (cid:12)(cid:12)(cid:12)(cid:12)Z R ( γ ( x − y ) − γ ( x )) η ε ( y ) dy (cid:12)(cid:12)(cid:12)(cid:12) ≤ ε k γ ′ k L ∞ ≤ Cε k γ ′ k W ,p (2.1)Likewise, we find that | γ ′ ε ( x ) − γ ′ ( x ) | = (cid:12)(cid:12)(cid:12)(cid:12)Z R η ε ( y )( γ ′ ( x − y ) − γ ′ ( x )) dy (cid:12)(cid:12)(cid:12)(cid:12) ≤ √ ε k γ ′ k C / ≤ C √ ε k γ ′ k W ,p . (2.2)For higher derivatives we can use the Sobolev embeddings, H¨older’s inequal-ity and integration by parts to obtain the L ∞ -bound | γ ( k +2) ε ( x ) | = (cid:12)(cid:12)(cid:12)(cid:12)Z R η ( k ) ε ( y ) γ ′′ ( x − y ) dy (cid:12)(cid:12)(cid:12)(cid:12) ≤ k η ( k ) ε k L q k γ ′′ k L p ≤ Cε − k − q k γ ′′ k L p = Cε − k − p k γ ′′ k L p (2.3)for integers k ≥ p + q = 1.We will use the next lemma to fix the smoothing parameter ε. Lemma 2.2.
If for an
M > we have a curve γ ∈ W ,p ( R / Z , R n ) parametrisedby arc-length which satisfies k γ ′ k W ,p ≤ M , then there exists an ε = ε ( p, M ) > such that the unit tangent τ = γ ′ ε | γ ′ ε | satisfies k τ − γ ′ k L ∞ ≤ . (2.4) BLATT, HOPPER, AND VORDEROBERMEIER
Proof.
Using (2.2) we get k γ ′ ε − γ ′ k L ∞ ≤ C √ ε k γ ′ k W ,p . As the retractionmap Π : x x | x | is locally Lipschitz on R n \{ } , the tangent τ = Π( γ ′ ε ) = γ ′ ε | γ ′ ε | to the mollified curve γ ε satisfies (2.4) for some ε > (cid:3) Corollary 2.3.
If for an
M > we have a curve γ ∈ W ,p ( R / Z , R n ) parametrised by arc-length which satisfies k γ ′ k W ,p ≤ M , then for an ε = ε ( p, M ) > as in Lemma 2.2 the mollified curve γ ε has a unit tangent map τ : R / Z → S n − that is smooth and satisfies k τ ′ k L ∞ , k τ ′′ k L ∞ ≤ C (2.5) for a constant C = C ( p, M ) > . Definition 2.4.
We say τ is unit quasi-tangent to the W ,p -curve γ if it isthe unit tangent to the mollified curve γ ε for some ε = ε ( p, M ) > Definition 2.5.
We denote by P Tv w = h w, v | v | i v | v | the orthogonal projec-tion of w onto the line R v for any vectors v, w ∈ R n . Likewise, we denoteby P ⊥ v w = w − P Tv w the orthogonal projection of w onto the orthogonalcomplement ( R v ) ⊥ of the line R v . Definition 2.6. If τ is unit quasi-tangent to a W ,p -curve γ , we denoteby ( W ,p ) Tτ (resp. ( W ,p ) ⊥ τ ) the set of all w ∈ W ,p ( R / Z , R n ) such that P ⊥ τ w = 0 a.e. (resp. P Tτ w = 0 a.e.).We will now prove the following statement that gives a lower bound onthe thickness of the set of regular curves around γ . Lemma 2.7.
If for an
M > we have a curve γ ∈ W ,p ( R / Z , R n ) parametrisedby arc-length which satisfies k γ ′ k W ,p ≤ M , then there exists a constant K = K ( p, M ) > and a unit quasi-tangent τ to the curve γ such that thecurve γ + φ satisfies inf x ∈ R / Z h γ ′ + φ ′ , τ i ≥ and hence inf x ∈ R / Z | γ ′ ( x ) + φ ′ ( x ) | ≥ for each φ ∈ ( W ,p ) ⊥ τ with k φ k L ∞ ≤ K . In particular, γ + φ is a regularcurve.Proof. We first note that h γ ′ , τ i = | γ ′ | + h γ ′ , τ − γ ′ i ≥ − | τ − γ ′ | ≥ byLemma 2.2 and the fact that | γ ′ | = 1. Upon differentiating the orthogonalitycondition h φ, τ i = 0, we get h φ ′ , τ i = −h φ, τ ′ i . In which case the estimate(2.5) implies that |h φ ′ , τ i| = |h φ, τ ′ i| ≤ C k φ k L ∞ ≤ whenever k φ k L ∞ ≤ C = K . Thus h γ ′ + φ ′ , τ i ≥ −
14 = 12whenever k φ k L ∞ ≤ K , i.e. γ + φ is a regular curve. As τ is of unit length,we also have | γ ′ + φ ′ | ≥ on R / Z . (cid:3) minimising movement scheme for the p -elastic energy of curves 7 We will now deduce the following lower bound for the L p -norm of thecurvature of a curve e γ + φ in terms of the L p -norm of the second derivativeof φ . This bound extends to our situation the well know analogous resultfor the case of a real normal graph over a smooth curve. Lemma 2.8.
If for an
M > we have a curve γ ∈ W ,p ( R / Z , R n ) parametrisedby arc-length which satisfies k γ ′ k W ,p ≤ M , then for the constant K = K ( p, M ) > from Lemma 2.7 and a unit quasi-tangent τ to the curve γ such that for each φ ∈ ( W ,p ) ⊥ τ with k φ k L ∞ ≤ K | v | ≤ C | P ⊥ γ ′ + φ ′ v | for all v ∈ R n pointing in an approximate normal direction and Z R / Z | φ ′′ | p ds ≤ C (cid:18) Z R / Z | κ γ + φ | p ds (cid:19) for some C = C ( M, p ) .Proof. Since h γ ′ + φ ′ , τ i ≥ from Lemma 2.7 and | γ ′ + φ ′ | ≤ | γ ′ | + | φ ′ | ≤ h γ ′ + φ ′ | γ ′ + φ ′ | , τ i ≥
12 11+Λ . Hence the angle between γ ′ + φ ′ and τ isbounded strictly away from π . In which case we have | v | ≤ C | P ⊥ γ ′ + φ ′ v | for all v ∈ R n pointing in an approximate normal direction.For the second estimate, we recall the curvature formula given by κ γ + φ = P ⊥ γ ′ + φ ′ ( γ ′′ + φ ′′ ) | γ ′ + φ ′ | . Now by the triangle inequality we see that (cid:12)(cid:12) P ⊥ γ ′ + φ ′ ( φ ′′ ) (cid:12)(cid:12) ≤ (cid:12)(cid:12) P ⊥ γ ′ + φ ′ ( γ ′′ + φ ′′ ) (cid:12)(cid:12) + (cid:12)(cid:12) P ⊥ γ ′ + φ ′ ( γ ′′ ) (cid:12)(cid:12) ≤ C ( | κ γ + φ | + | γ ′′ | ) , since | γ ′ + φ ′ | ≤ | γ ′ | + | φ ′ | ≤ P Tτ φ ′′ , wedifferentiate the equation h φ, τ i = 0 twice to get h φ ′′ , τ i = − h φ ′ , τ ′ i−h φ, τ ′′ i . It then follows that | P Tτ φ ′′ | = |h φ ′′ , τ i| ≤ |h φ, τ ′′ i| + 2 |h φ ′ , τ ′ i| ≤ C ( K + Λ) , since both τ ′ and τ ′′ are bounded by Corollary 2.3. In combining both thetangential and normal parts of φ ′′ and using the fact that the angle between γ ′ + φ ′ and τ is bounded strictly away from π , we find that | φ ′′ | ≤ C ( | P ⊥ γ ′ + φ ′ φ ′′ | + | P Tτ φ ′′ | ) ≤ C (1 + | γ ′′ | + | κ γ + φ | )from which the desired integral estimate follows (since k γ ′′ k L p ≤ M byLemma 2.12. (cid:3) Next we show that there exists a good substitute for the nearest neigh-bourhood projection which yields a local tubular neighbourhood. We alsoobtain a lower bound on thickness of the tubular neighbourhood that onlydepends on the W ,p -norm of the curve. BLATT, HOPPER, AND VORDEROBERMEIER
Definition 2.9. If τ is a unit quasi-tangent to a W ,p -curve γ , the ( n − N x = { v ∈ R n : P Tτ ( x ) v = 0 } is called an approximate normal space to γ at a given fixed point x ∈ R / Z .By considering the map H x : B δ ( x ) × N x → R n given by( x, v ) γ ( x ) + P ⊥ τ ( x ) v (2.6)for some 0 < δ <
1, we obtain the following:
Lemma 2.10.
If for an
M > we have a curve γ ∈ W ,p ( R / Z , R n ) parametrised by arc-length which satisfies k γ ′ k W ,p ≤ M , then there existsa sufficiently small constant δ = δ ( p, M ) > and a unit quasi-tangent τ tothe curve γ such that (2.6) maps B δ ( x ) × B δ (0) diffeomorphically onto itsimage and B δ/ (cid:0) γ ( B δ/ ( x )) (cid:1) ⊂ H x (cid:0) B δ ( x ) × B δ (0) (cid:1) . (2.7) Proof.
We first show that H x is a local diffeomorphism by way of the inversefunction theorem. To do so we calculate the partial derivatives ∂H x ∂x = γ ′ ( x ) − h v, τ ′ ( x ) i τ ( x ) − h v, τ ( x ) i τ ′ ( x ) ∂H x ∂v = v + P ⊥ τ ( x ) v − P ⊥ τ ( x ) v. Then from the estimates (2.4) and (2.5) together with the Sobolev embedding W ,p ( B δ ( x ) , R n ) ֒ → C , − p ( B δ ( x ) , R n ) we find that (cid:12)(cid:12)(cid:12) ∂H x ∂x − τ ( x ) (cid:12)(cid:12)(cid:12) ≤ | γ ′ ( x ) − τ ( x ) | + C | v | + | γ ′ ( x ) − γ ′ ( x ) |≤
14 + C | v | + Cδ − p for some constant C = C ( p, M ) >
0. By taking some δ > p and M ), we have (cid:12)(cid:12)(cid:12) ∂H x ∂x − τ ( x ) (cid:12)(cid:12)(cid:12) ≤ x ∈ B δ ( x ) ⊂ R / Z and v ∈ B δ (0) ⊂ N x . Likewise, whenever δ > (cid:12)(cid:12)(cid:12) ∂H x ∂v − v (cid:12)(cid:12)(cid:12) ≤ x, v ) ∈ B δ ( x ) × B δ (0).Let us now assume that τ ( x ) = e without loss of generality. From theabove estimates we see that the Jacobi matrix DH x satisfies k DH x − I k ≤ , (2.8) minimising movement scheme for the p -elastic energy of curves 9 where k · k denotes the operator norm. Therefore DH x is invertible and so H x maps B δ ( x ) × B δ (0) diffeomorphically onto its image by the inversefunction theorem. Moreover, (2.8) implies that | H x ( z ) − H x ( z ) | = (cid:12)(cid:12)(cid:12)(cid:12)Z DH x ( z + θ ( z − z ))( z − z ) dθ (cid:12)(cid:12)(cid:12)(cid:12) ≥ | z − z | − | z − z | = | z − z | . In which case the map H x is bi-Lipschitz and hence injective on B δ ( x ) × B δ (0). From the fact that dist( ∂ ( B δ ( x ) × B δ (0)) , B δ/ ( x ) × { } ) ≥ δ andthe latter bi-Lipschitz estimate we havedist (cid:16) H x (cid:0) ∂ ( B δ ( x ) × B δ (0)) (cid:1) , γ ( B δ/ ( x )) (cid:17) ≥ δ (cid:3) We can now use Lemma 2.10 to show that any W ,p -curve γ can be writtenas an approximate normal graph over a given W ,p -curve e γ whenever thecurves are C -close to each other. Lemma 2.11.
If for an
M > we have a curve γ ∈ W ,p ( R / Z , R n ) parametrised by arc-length that satisfies k γ ′ k W ,p ≤ M , then there existsa sufficiently small constant ρ = ρ ( p, M ) > and a unit quasi-tangent τ to the curve γ such that for each curve e γ ∈ W ,p ( R / Z , R n ) satisfying k γ − e γ k C ≤ ρ we have some φ ∈ ( W ,p ) ⊥ τ and a reparametrisation σ of R / Z for which e γ ◦ σ = γ + φ. Proof.
Firstly, choose δ > R / Z is compact, thereexists points x , . . . , x ℓ in R / Z such that the balls B δ/ ( x ), . . . , B δ/ ( x ℓ )cover R / Z . Let the mappings H x j for j = 1 , . . . , ℓ be defined by (2.6) and letΠ x j : B δ/ (cid:0) γ ( B δ/ ( x j )) (cid:1) → R / Z be the corresponding retraction maps givenby Π x j = π ◦ H − x j where π : B δ ( x j ) × N x j → R / Z sends ( x, v ) x , i.e. the projection onto thefirst coordinate. We can then set σ ( x ) = Π x j ( e γ ( x )) (2.9)for any x ∈ B δ ( x j ) in order to get a well-defined C -mapping. Furthermore,from the inverse function theorem applied to Π x j and the estimate (2.8) wesee that σ ′ ( x ) > ρ > σ is bi-Lipschitz).In addition, by setting e φ = e γ − γ ◦ σ we see from (2.6) that e φ belongs to( W ,p ) ⊥ τ . Therefore γ ◦ σ is a regular curve equal to e γ − e φ . In order to changethe roles of γ and e γ , we apply the inverse function theorem to σ to justifythe reparametrisation e γ ◦ σ − = γ ◦ σ ◦ σ − + e φ ◦ σ − = γ + φ , where we set φ = e φ ◦ σ − ∈ ( W ,p ) ⊥ τ . (cid:3) Using the above lemmata we can write every W ,p -curve γ as an approx-imate normal graph over a smooth curve e γ . Be aware that from now on tillthe end of this article we consider normal graphs over the curve ˜ γ instead of γ. Lemma 2.12.
Let γ ∈ W ,p ( R / Z , R n ) be a curve parametrised by arc-length.For every ε > there exists a smooth curve e γ ∈ C ∞ ( R / Z , R n ) parametrisedby arc-length with k γ − e γ k W ,p ≤ ε , a unit quasi-tangent τ to the curve ˜ γ and some φ ∈ ( W ,p ) ⊥ τ such that γ ◦ σ = e γ + φ (2.10) for a reparametrisation σ of R / Z .Proof. Firstly, there exists a smooth curve e γ ∈ C ∞ ( R / Z , R n ) parametrisedby arc-length such that k ˜ γ − γ k W ,p ≤ ε by the density of C ∞ ( R / Z , R n ) in W ,p ( R / Z , R n ). Moreover, we have k γ − e γ k C ≤ Cε = ρ by the Sobolev embeddings. Thus by taking some ε > φ ∈ ( W ,p ) ⊥ τ and a reparametri-sation σ of R / Z such that γ ◦ σ = e γ + φ . (cid:3) The representation of γ by a normal graph φ over e γ we obtain fromLemma 2.12 satisfies the following C estimates. These enable us to controlthe second derivative of φ by the curvature of γ using Lemma 2.8. Corollary 2.13.
For the decomposition (2.10) there exists a constant
C > depending on an upper bound M on k γ ′ k W ,p and p such that k φ k L ∞ ≤ C k γ − e γ k L ∞ and k φ ′ k L ∞ ≤ C (1 + k γ ′ − e γ ′ k L ∞ ) . Proof.
From the construction of σ given by (2.9) we see that | σ ( x ) − x | = | σ ( x ) − σ ◦ σ − ( x ) | ≤ k σ ′ k L ∞ | x − σ − ( x ) | and | x − σ − ( x ) | = | Π x j ( e γ ( x )) − Π x j ( γ ( x )) | ≤ (cid:0) max j k D Π x j k L ∞ (cid:1) | e γ ( x ) − γ ( x ) | , since there exists some ball B δ ( x j ) such that x = Π x j ( e γ ( x )). As we have | φ ( x ) | = | γ ( σ ( x )) − e γ ( x ) | ≤ | γ ( σ ( x )) − γ ( x ) | + | γ ( x ) − e γ ( x ) | and | γ ( σ ( x )) − γ ( x ) | ≤ k γ ′ k L ∞ | σ ( x ) − x | , it follows that k φ k L ∞ ≤ (1 + C k γ ′ k W ,p ) k γ − e γ k L ∞ by the Sobolev embeddings. In addition, we have k φ ′ k L ∞ = k ( γ ◦ σ ) ′ − e γ ′ k L ∞ ≤ k γ ′ − e γ ′ k L ∞ + k γ ′ k L ∞ + k ( γ ◦ σ ) ′ k L ∞ ≤ k γ ′ − e γ ′ k L ∞ + C k γ ′ k W ,p from the uniform bi-Lipschitz property of σ and the Sobolev embeddings. (cid:3) minimising movement scheme for the p -elastic energy of curves 11 After breaking thereparametrisation invariance of (1.1) by way of the approximate normalgraphs, it is now a straight forward matter to prove the short-time existenceof solutions for the minimising movement scheme.Let us first consider an initial curve Γ ∈ W ,p ( R /L Z , R n ) of length L parametrised by arc-length. In the following it will be essential that allestimates only depend on an upper bound on the energy of this curve.We first note that an upper bound on the energy also implies a lowerbound on the length, since by Fenchel’s theorem together with H¨older’sinequality we have2 π ≤ Z R /L Z | κ | ds ≤ L − p (cid:18) Z R /L Z | κ | p ds (cid:19) p so that L p − ≥ (2 π ) p p E ( p ) (Γ) . By scaling the results of Section 2.2, we can drop the assumption that thecurve is of unit length and recover all the previous estimates concerningapproximate normal graphs (with proviso that the relevant constants nowdepend on λ and the energy bound). In particular, we say that the unitvector field τ is quasi-tangent to a W ,p -curve γ of length L whenever τ ( · L )is quasi-tangent to the curve γ ( · L ).Now for the initial curve, the result of Lemma 2.12 implies that thereexists a smooth curve e γ parametrised by arc-length, a unit quasi-tangent τ to the curve e γ and a perturbation Φ ∈ ( W ,p ) ⊥ τ such that Γ ◦ σ = e γ + Φ.Moreover, by combining the norm bounds of Lemma 2.12 with Corollary 2.13and the Sobolev embeddings, we see that k Φ k L ∞ ≤ µ and k Φ ′ k L ∞ ≤ W for some sufficiently small constant µ = µ ( p, λ, E (Γ)) > W = W ( p, λ, E (Γ)) > t < t < t < · · · we seek to definethe curves γ t j = e γ + φ t j (2.11)with the initial case γ t = e γ + Φ. The time differences t j +1 − t j = h areset to be equal to a fixed parameter h > φ t j +1 for the next time step as theminimiser φ t j +1 = argmin φ ∈ V (cid:26) E ( e γ + φ ) + 12 h Z R /L Z | P ⊥ γ ′ tj ( e γ + φ − γ t j ) | | γ ′ t j | dx (cid:27) , where the class of admissible perturbations is given by V = V ( µ, W ) = { φ ∈ ( W ,p ) ⊥ τ : k φ k L ∞ < µ, k φ ′ k L ∞ < W } . The following lemma states that these discrete-time solutions can be con-structed for at least a short time.
Lemma 2.14.
There exists a finite time
T > depending only on p , λ and E (Γ) such that the solutions γ t j = e γ + φ t j exist for a series of discrete times t < t < t < · · · < t N < T .Proof. We seek to establish the existence of the perturbations φ t j +1 that areminimisers of the functionals F j ( φ ) = E ( e γ + φ ) + 12 h Z R /L Z | P ⊥ γ ′ tj ( e γ + φ − γ t j ) | | γ ′ t j | dx over the admissible class V . To do so we proceed by an induction argumentwith an initial base case φ t = Φ given by the decomposition of the initialcurve Γ. Indeed, let us assume there exist minimisers φ t i +1 of F i over theclass V for i = 0 , , . . . , j − F i ( φ t i +1 ) ≤ F i ( φ t i ) for i = 0 , , . . . , j − φ t i is a competitor),we note that E ( γ t j ) ≤ E ( γ t ) = E (Γ)and 12 h Z R /L Z | P ⊥ γ ′ ti ( γ t i +1 − γ t i ) | | γ ′ t i | dx ≤ E ( γ t i ) − E ( γ t i +1 ) . In which case Lemma 2.8 implies that the L p -norm of γ ′′ t j is uniformlybounded by a constant which depends only on p and E (Γ). In addition,we have 1 h Z R /L Z | γ t i +1 − γ t i | dx ≤ C (cid:0) E ( γ t i ) − E ( γ t i +1 ) (cid:1) . Then by summing up the latter inequalities, we get the a priori estimate j − X i =0 h Z R /L Z | γ t i +1 − γ t i | dx ≤ C (cid:0) E ( γ t ) − E ( γ t j ) (cid:1) . (2.12)We also recall from H¨older’s inequality that k γ t − γ t j k L ≤ j − X i =0 k γ t i +1 − γ t i k L √ h √ h ≤ j − X i =0 h Z R /L Z | γ t i +1 − γ t i | dx ! j − X i =0 h ! ≤ C p E ( γ t ) p t j (2.13)and from the Gagliardo-Nirenberg interpolation inequality we get k γ ′ t − γ ′ t j k L ∞ ≤ C k γ ′′ t − γ ′′ t j k αL p k γ t − γ t j k − αL with α = p p − . Since Lemma 2.8 implies that the L p -norm of the secondderivatives of γ and γ t j are uniformly bounded, we conclude that k γ ′ t − γ ′ t j k L ∞ ≤ C ( p t j ) − α minimising movement scheme for the p -elastic energy of curves 13 for a constant C > p , λ and E (Γ). Furthermore, there existsa sufficiently small T > p , λ and E (Γ) such that k γ ′ t j k L ∞ ≤ k γ ′ t k L ∞ + k γ ′ t j − γ ′ t k L ∞ ≤ W + C ( p t j ) − α < W (2.14)whenever 0 < t j < T . Since k γ t − γ t j k L ∞ ≤ C k γ ′′ t − γ ′′ t j k βL p k γ t − γ t j k − βL with β = p p − by the Gagliardo-Nirenberg interpolation inequality, we alsohave k γ t j k L ∞ < µ (2.15)whenever 0 < t j < T .In fact we can show that the same estimates hold for a suitably chosenminimising sequence. Let us assume that ( φ n ) is a minimising sequence forthe functional F j in the class V , i.e. F j ( φ n ) → inf φ ∈ V F j ( φ ) and note that F j is bounded from below by construction. As φ t j is still a competitor, wecan assume without loss of generality that F j ( φ n ) ≤ F j ( φ t j ) = E ( γ t j ) ≤ E ( γ t )for all n ∈ N . In which case we can repeat the argument from the above toobtain the bound k γ ′ t − γ ′ n k L ∞ ≤ C ( p t j +1 ) − α with γ n = e γ + φ n . It then follows that k γ ′ n k L ∞ < W (2.16)for all 0 < t j +1 < T . Compactness . As a consequence of Lemma 2.8, the minimising sequence( φ n ) is uniformly bounded in W ,p ( R /L Z , R n ). It then follows that thereexists a weakly converging subsequence in W ,p ( R /L Z , R n ) which we alsodenoted by ( φ n ). In addition, the Rellich-Kondraˇsov compactness theoremimplies that the subsequence ( φ n ) is strongly convergent in C ( R /L Z , R n ).Let us denote the limit of this sequence by φ . Since we have already establishthat k φ n k L ∞ < µ and k φ ′ n k L ∞ < W , it follows that k φ k L ∞ < µ and k φ ′ k L ∞ < W . Therefore the limit φ also belongs to V . Lower semi-continuity . Let us finally prove that F j ( φ ) ≤ lim inf n →∞ F j ( φ n ) . As the L -term in the functional F j converges by the theorem of Rellich-Kondraˇsov and the angle between τ and γ ′ t j is uniformly bounded strictlyaway from π , it suffices to show that E ( p ) ( e γ + φ ) ≤ lim inf n →∞ E ( p ) ( e γ + φ n ) . (2.17) Note that the length term λ R R /L Z ds appearing in the considered energy E ,cf. (1.1), can be dropped as well due to the convergence of the sequence ( φ n )in C ( R /L Z , R n ). In order to prove (2.17) we use the curvature formula for κ e γ + φ n to rewrite E ( p ) ( e γ + φ n ) = Z R /L Z (cid:12)(cid:12)(cid:12) P ⊥ e γ ′ + φ ′ n ( e γ ′′ + φ ′′ n ) (cid:12)(cid:12)(cid:12) p | e γ ′ + φ ′ n | p | e γ ′ + φ ′ n | ds as the expression E ( p ) ( e γ n ) = Z R /L Z (cid:12)(cid:12)(cid:12) P ⊥ e γ ′ + φ ′ ( e γ ′′ + φ ′′ n ) (cid:12)(cid:12)(cid:12) p | e γ ′ + φ ′ | p | e γ ′ + φ ′ | ds + I + I + I , where I = Z R /L Z (cid:18)(cid:12)(cid:12) P ⊥ e γ ′ + φ ′ n ( e γ ′′ + φ ′′ n ) (cid:12)(cid:12) p − (cid:12)(cid:12) P ⊥ e γ ′ + φ ′ ( e γ ′′ + φ ′′ n ) (cid:12)(cid:12) p (cid:19) | e γ ′ + φ ′ n || e γ ′ + φ ′ n | p ds I = Z R /L Z (cid:18) | e γ ′ + φ ′ n | p − | e γ ′ + φ ′ | p (cid:19) (cid:12)(cid:12) P ⊥ e γ ′ + φ ′ ( e γ ′′ + φ ′′ n ) (cid:12)(cid:12) p | e γ ′ + φ ′ n | ds I = Z R /L Z (cid:12)(cid:12)(cid:12) P ⊥ e γ ′ + φ ′ ( e γ ′′ + φ ′′ n ) (cid:12)(cid:12)(cid:12) p | e γ ′ + φ ′ n | p (cid:18) | e γ ′ + φ ′ n | − | e γ ′ + φ ′ | (cid:19) ds. The terms I , I and I vanish in the limit due to the convergence of thesequence ( φ n ) in C ( R /L Z , R n ) and the uniform bound on the W ,p -normof φ n . Moreover, the expression I ( e γ + φ n ) = Z R /L Z (cid:12)(cid:12)(cid:12) P ⊥ e γ ′ + φ ′ ( e γ ′′ + φ ′′ n ) (cid:12)(cid:12)(cid:12) p | e γ ′ + φ ′ | p | e γ ′ + φ ′ | ds p defines a norm equivalent to that of the W ,p -norm. In which case (2.17)follows from the lower semicontinuity of norms under weak convergence. (cid:3) For later reference, let us also state the following a priori estimate for thepiecewise linear interpolations that results from (2.12) and (2.13).
Corollary 2.15.
The piecewise linear interpolations φ ( h ) ( t, · ) = φ t j + t − t j h (cid:0) φ t j +1 − φ t j (cid:1) , t j ≤ t ≤ t j +1 , satisfies the estimates k φ ( h ) t ′′ − φ ( h ) t ′ k L ≤ C √ t ′′ − t ′ and Z t ′′ t ′ Z R /L Z | ∂ t φ ( h ) ( t, s ) | dsdt ≤ C (cid:0) E ( γ t ′ ) − E ( γ t ′′ ) (cid:1) for any ≤ t ′ < t ′′ < T < ∞ . minimising movement scheme for the p -elastic energy of curves 15 Remark . We thus obtain a piecewise linearly interpolated solution γ ( h ) t = e γ + φ ( h ) t , ≤ t < T, (2.18)for the minimising movements scheme.
3. Weak solutions
In order toimprove the regularity of the approximations, we derive the Euler-Lagrangeequations related to the minimising movement scheme.We recall the following expression (cf. [9, Lemma 2.1]) for the first varia-tion of the p -elastic energy, namely δ ψ E ( p ) ( γ ) = Z R /L Z | κ | p − h κ, δ ψ κ i ds + 1 p Z R /L Z | κ | p h ∂ s γ, ∂ s ψ i ds (3.1)where δ ψ κ = (cid:0) ∂ s ψ (cid:1) ⊥ − h κ, ∂ s ψ i ∂ s γ − h ∂ s γ, ∂ s ψ i κ , cf. Proposition A.1. Thefirst variation of the length term appearing in the definition of the energy E , cf. (1.1), is given by δ ψ (cid:16) λ Z R /L Z ds (cid:17) = λ Z R /L Z h ∂ s γ, ∂ s ψ i ds. (3.2)Combining (3.1) and (3.2) with the fact that ∂ s ψ = 1 | γ ′ | ∂ x ψ, where | γ ′ | = | ∂ x γ | , we get ∂ s ψ = 1 | γ ′ | ∂ x (cid:16) | γ ′ | ∂ x ψ (cid:17) = 1 | γ ′ | ∂ x ψ − | γ ′ | D γ ′ | γ ′ | , γ ′′ E ∂ x ψ so that δ ψ E ( γ ) = Z R /L Z | κ | p − | γ ′ | h κ, ∂ x ψ i dx + R ( ψ ) , where R ( ψ ) has the form R ( ψ ) = Z R /L Z h b, ∂ x ψ i dx for some b ∈ L ∞ L , where as a notational shorthand L ∞ L stands for L ∞ (cid:0) [0 , T ) , L ( R / Z , R n ) (cid:1) .On the other hand, solutions of the minimising movement scheme solve h ∂ t γ, P ⊥ τ ψ i = − δ ψ E ( γ ) (3.3)for all ψ ∈ (cid:0) W ,p (cid:1) ⊥ τ . Therefore we conclude that Z R /L Z | κ | p − | γ ′ | h κ, ∂ x ψ i dx + e R ( ψ ) = 0 , (3.4) where e R ( ψ ) = Z R /L Z h b, ∂ x ψ i dx + Z R /L Z h P ⊥ τ ( ∂ t γ ) , ψ i dx. To deduce regular-ity from the equation above, we consider a smooth local orthonormal basis ν , . . . , ν n − for our approximate normal spaces. If ψ is a test function thatis decomposed into the form ψ = n − X i =1 ψ i ν i such that the scalar functions ψ i vanish away from the neighbourhood, wefind that ∂ x ψ = n − X i =1 (cid:16) ∂ x ψ i ν i + 2 ∂ x ψ i ∂ x ν i + ψ i ∂ x ν i (cid:17) . Therefore the evolution equation for the approximation yields n − X i =1 Z R /L Z | κ | p − | γ ′ | ∂ x ψ i h κ, P ⊥ τ ν i i dx = Q ( h ) , (3.5)where Q ( h ) = Z R /L Z h b t , ∂ x ψ i + h c t , ψ i + h P ⊥ τ ( ∂ t γ ( t, · )) , ψ i dx. The following lemma helps us to deduce regularity from this form of theequation.
Lemma 3.1 ( L -estimates) . Let I = ( a, b ) be an open subset of R . If thereexist functions u , f and F in L loc ( I ) such that Z I (cid:0) u∂ x ϕ + F ∂ x ϕ (cid:1) dx = Z I f ϕdx for all ϕ ∈ C ∞ c ( I ) , then u ( x ) = Z xa (cid:18) F ( y ) + Z ya f ( z ) dz (cid:19) dy + m ( x − a ) + d with d = lim x ց a u ( x ) and m ( b − a ) = lim x ր b u ( x ) − (cid:18)Z I (cid:18) F ( y ) + Z ya f ( z ) dz (cid:19) dy + d (cid:19) . Moreover, the function u ∈ W , ( I ) with k u k W , ≤ C ( k f k L + k F k L ) . minimising movement scheme for the p -elastic energy of curves 17 Proof.
Let us first set w ( x ) = F ( x ) + Z xa f ( y ) dyv ( x ) = Z xa w ( y ) dy and note that v ∈ W , ( I ) with v ′ = w . Then integration by parts impliesthat Z I v ( x ) ∂ x ϕ ( x ) dx = − Z I v ′ ( x ) ∂ x ϕ ( x ) dx = − Z I (cid:18) F ( x ) ∂ x ϕ ( x ) + (cid:16) Z xa f ( y ) dy (cid:17) ∂ x ϕ ( x ) (cid:19) dx = − Z I ( F ( x ) ∂ x ϕ ( x ) − f ( x ) ϕ ( x )) dx. Therefore Z I ( u − v ) ∂ x ϕ dx = 0for all ϕ ∈ C ∞ c ( I ). In which case u − v is an affine function from which theconclusion easily follows. (cid:3) We can now use the latter lemma to establish:
Theorem 3.2 (Higher regularity) . If γ ( h ) t is a solution to the minimisingmovements scheme given by (2.18) , there exists a constant C > indepen-dent of h such that (cid:13)(cid:13) | κ | p − P ⊥ τ κ (cid:13)(cid:13) L ([0 ,T ) ,W , ) ≤ C. (3.6) In particular, we have κ uniformly bounded in L L q and γ ′ uniformly boundedin W ,q for all ≤ q < ∞ .Proof. This higher regularity result directly follows from the application ofLemma 3.1 to our evolution equation for the minimising movement schemeapproximations. In particular, from Corollary 2.15 we see that γ ( h ) t satisfies Z T Z R /L Z | ∂ t γ ( h ) ( t, s ) | dsdt ≤ CE ( γ ) . Applying Lemma 2.8 to (3.5) together with a covering argument hence yields (cid:13)(cid:13)(cid:13) | κ | p − | γ ′ | P ⊥ τ κ (cid:13)(cid:13)(cid:13) L ([0 ,T ) ,W , ) < C. Since ( γ ( h ) ) ′ is uniformly bounded in W , and W , is a Banach algebra,this implies k| κ | p − P ⊥ τ κ k L ([0 ,T ) ,W , ) < C. (cid:3) We will use the following resultin order to obtain the convergence of solutions. This result is crucial for thecontrol of the terms involving the energy.
Theorem 3.3.
Let γ n = γ + φ n be a sequence bounded in L ∞ W ,p ∩ C L such that | κ n | p − κ n is uniformly bounded in L W , . Then there exists asubsequence γ n j such that the curvatures κ n j converge in L W ,p . The proof of this theorem relies on the following interpolation estimate.
Lemma 3.4.
There exists a constant C > depending on p such that forany W ,p -curves γ and γ with curvatures κ and κ we have k κ − κ k L p ≤ C ( k| κ | p − κ k L W , + k| κ | p − κ k L W , ) k γ ′ − γ ′ k L L ∞ . If these curves are furthermore approximate normal graphs over e γ as for thesolutions to the minimising movement scheme, we get k κ − κ k L p ≤ C ( k| κ | p − P ⊥ τ κ k L W , + k| κ | p − P ⊥ τ κ k L W , ) k γ ′ − γ ′ k L L ∞ . where now C = C ( λ, p, E (Γ)) . Proof.
First note that Z | κ − κ | p ds ≤ C Z (cid:0) | κ | p − κ − | κ | p − κ (cid:1) ( κ − κ ) ds (cf. [8, §
1, Lemma 4.4]). Then integration by parts and H¨older’s inequalityimply that Z | κ − κ | p ds ≤ − C Z ∂ s (cid:0) | κ | p − κ − | κ | p − κ (cid:1) ( ∂ s γ − ∂ s γ ) ds ≤ C (cid:0)(cid:13)(cid:13) | κ | p − κ (cid:13)(cid:13) W , + (cid:13)(cid:13) | κ | p − κ (cid:13)(cid:13) W , (cid:1) k γ ′ − γ ′ k L ∞ . So by integrating over time and using H¨older’s inequality again we get
Z Z | κ − κ | p dsdt ≤ C (cid:0)(cid:13)(cid:13) | κ | p − κ (cid:13)(cid:13) L W , + (cid:13)(cid:13) | κ | p − κ (cid:13)(cid:13) L W , (cid:1) k γ ′ − γ ′ k L L ∞ . For the second estimate we proceed in a similar way. We apply Lemma 2.8to improve the first inequality to Z | κ − κ | p ds ≤ C Z (cid:0) | κ | p − P ⊥ τ κ − | κ | p − P ⊥ τ κ (cid:1) ( κ − κ ) ds. Integrating by parts then yields Z | κ − κ | p ds ≤ − C Z ∂ s (cid:0) | κ | p − P ⊥ τ κ − | κ | p − P ⊥ τ κ (cid:1) ( ∂ s γ − ∂ s γ ) ds ≤ C (cid:0)(cid:13)(cid:13) | κ | p − P ⊥ κ (cid:13)(cid:13) W , + (cid:13)(cid:13) | κ | p − P ⊥ τ κ (cid:13)(cid:13) W , (cid:1) k γ ′ − γ ′ k L ∞ . (cid:3) minimising movement scheme for the p -elastic energy of curves 19 Proof of Theorem 3.3.
Using a diagonal argument and the compact embed-ding W ,p ֒ → L , we get a subsequence γ n j converging in L for all times t ∈ Q ∩ [0 , T ) (and hence for all 0 ≤ t < T due to the uniform bound in C L ). This result, together with the uniform bound on the W ,p -Sobolevnorm and interpolation estimates, implies that γ n j → γ ∈ C α ([0 , T ) , W , ∞ )with α = p − p − . Thus γ n j converge to γ in L W ,p by Lemma 3.4. (cid:3) Proof of Theorem 1.1.
From the construction in Section 2.3 there exists asolution γ ( h ) t to the minimising movement scheme given by (2.18) for all0 ≤ t < T up to some positive final time T that depends only on p , λ andthe energy E (Γ) of the initial data. We think of this solution as solvinga discrete version of the negative L -gradient flow of E . Theorem 3.3 andCorollary 2.15 can then be applied to get a subsequence that converges in L W ,p such that ∂ t γ ( h ) weakly converges in L . Now in order to show thatthe limit satisfies the desired evolution equations, we use the fact that thesolutions of the minimising movement scheme satisfy Z T Z R /L Z h ∂ ⊥ t γ ( h ) t , ψ i dsdt = Z R /L Z δ ψ t E ( γ ( h ) t ) dt (3.7)for all test functions ψ ∈ C ∞ c ((0 , T ) × R /L Z , R n ).Let us now take a sequence h n → γ ( h n ) converge to a family of curves γ in L W ,p such that ∂ t γ ( h n ) converges to ∂ t γ weakly in L ([0 , T ) , R /L Z ). As γ ′ ( h n ) converges strongly to γ ′ in L , we see that the weak convergence of ∂ t γ ( h n ) to ∂ t γ in L implies Z T Z R /L Z h ∂ ⊥ t γ ( h n ) t , ψ i dsdt → Z T Z R /L Z h ∂ ⊥ t γ t , ψ i dsdt. (3.8)Convergence for the right-hand side of (3.7) is also straight forward. If wedenote by κ n the curvature of γ ( h n ) t and integrate (3.1), we find that Z T δ ψ t E ( γ ( h n ) t ) dt = Z T Z R /L Z | κ n | p − h κ n , δ ψ κ n i dsdt + 1 p Z T Z R /L Z | κ n | p h ∂ s γ ( h n ) , ∂ s ψ i dsdt + λ Z T Z R /L Z h ∂ s γ ( h n ) , ∂ s ψ i dsdt. Since κ n converges to κ in L L p ([0 , T ) × R /L Z ) and ∂ s γ ( h n ) converges to ∂ s γ uniformly, the second term on the right-hand side of the latter equationconverges to the corresponding term for γ in lieu of γ ( h n ) . One can deducethe same fact for the first term via the formula δ ψ κ n = (cid:0) ∂ s ψ (cid:1) ⊥ − h κ n , ∂ s ψ i τ − h ∂ s ψ, τ i κ n , since it implies that δ ψ κ n converges to δ ψ κ in L L p ([0 , T ) × R /L Z , R n ).Therefore we get Z T δ ψ t E ( γ ( h n ) ) dt → Z T δ ψ t E ( γ ) dt. (3.9)In which case equations (3.7), (3.8) and (3.9) imply that Z T Z R /L Z h γ t , ∂ ⊥ t ψ i dsdt = − Z T δ ψ t E ( γ t ) dt. (cid:3) Using the fact thatthe unit tangent belongs to W ,p we can finally prove Corollary 1.2 underthe conditions of Theorem 1.1. Proof of Corollary 1.2.
In abuse of notation, let τ = γ ′ | γ ′ | ∈ W ,p ( R /L Z , R n )be the unit tangent and the vectors ν , . . . , ν n − be a smooth local or-thonormal basis of our approximate normal space. Due to the fact thatany ψ ∈ C ∞ c ( R /L Z , R n ) can be written as ψ = ψ τ + n − X i =1 ψ i ν i with functions ψ i ∈ W ,p ( R /L Z , R n ), we find that Z T Z R /L Z h ∂ ⊥ t γ, ψ i dsdt = n − X i =1 Z T Z R /L Z h ∂ ⊥ t γ, ψ i ν i i dsdt = − Z T δ ψ t E ( γ t ) dt, since both δ ψ τ E ( γ ) = 0 and h ∂ ⊥ t ψ , τ i = 0. (cid:3)
4. Epilogue
Although the minimising movement scheme leads in a rather straight forwardway to the short-time existence of weak solution for our gradient flow, thereare three key questions one would like to resolve, namely:(1) Are weak solutions unique and do they have long-time existence for0 ≤ t < ∞ ?(2) Can one use test functions for the gradient flow that are not orthog-onal to a quasi-tangent?(3) Does our notion of solution depend on the choice of the referencecurve and the approximate normal directions?For long-time existence it looks as if one could, in principle, restart the flowand the above short-time existence result to get an eternal solution. Howeverone should be aware that this solution might have kinks which our methodscannot rule out. If one has uniqueness and some way of modifying the ap-proximate normal, long-time existence would be possible. Our Corollary 1.2 minimising movement scheme for the p -elastic energy of curves 21 is a first indication that a more fastidious regularity theory is needed in orderto resolve the above issues.The question of uniqueness seems to be completely open. For the morestandard non-homogeneous evolution equations involving the p -Laplace op-erator, papers discussing uniqueness have only appeared rather recently. Inparticularly, the method used to prove uniqueness in [3] breaks down for ourcurvature equations. Appendix A. First variation for the p -elastic energy Recall that for closed curves γ : R / Z → R n in the W ,p -Sobolev class the p -elastic energy is given by E ( p ) ( γ ) = p Z R / Z | κ | p ds. For the convenience of the reader, we give further details on the derivationof its first variation. The upcoming statement is proven along the line of[9, Lemma 2.1], for which we identify the arclength element by ds = | ∂ x γ | dx and the arclength derivative by ∂ s = | ∂ x γ | − ∂ x . Proposition A.1.
The first variation of the p -elastic energy E ( p ) for γ ∈ W ,p ( R / Z , R n ) in direction of ψ ∈ W ,p ( R / Z , R n ) is given by δ ψ E ( p ) ( γ ) = Z R / Z | κ | p − h κ, δ ψ κ i ds + p Z R / Z | κ | p h ∂ s γ, ∂ s ψ i ds. where δ ψ κ = (cid:0) ∂ s ψ (cid:1) ⊥ − h κ, ∂ s ψ i ∂ s γ − h ∂ s γ, ∂ s ψ i κ .Proof. We first observe δ ψ E ( p ) ( γ ) = p Z R / Z δ ψ ( | κ | p ) ds + p Z R / Z | κ | p δ ψ ( ds ) . By applying the notation from above and the chain rule, we get δ ψ ( | κ | p ) = ddε (cid:2) | ∂ s ( γ + εψ ) | p (cid:3) ε =0 = h p |h ∂ s ( γ + εψ ) , ∂ s ( γ + εψ ) i| p − h ddε ∂ s ( γ + εψ ) , ∂ s ( γ + εψ ) i i ε =0 = p | κ | p − h δ ψ ( κ ) , κ i and δ ψ ( ds ) = ddε [ | ∂ x ( γ + εψ ) | dx ] ε =0 = h | ∂ x ( γ + εψ ) | h ∂ x ( γ + εψ ) , ∂ x ψ i dx i ε =0 = h ∂ x γ | ∂ x γ | , ∂ x ψ | ∂ x γ | i| ∂ x γ | dx = h ∂ s γ, ∂ s ψ i ds. Similarly, we achieve δ ψ ( κ ) = ddε h | ∂ x ( γ + εψ ) | ∂ x (cid:16) | ∂ x ( γ + εψ ) | ∂ x ( γ + εψ ) (cid:17)i ε =0 = h − | ∂ x ( γ + εψ ) | h ∂ x ( γ + εψ ) , ∂ x ψ i ∂ x (cid:16) | ∂ x ( γ + εψ ) | ∂ x ( γ + εψ ) (cid:17)i ε =0 + h | ∂ x ( γ + εψ ) | ∂ x ( − | ∂ x ( γ + εψ ) | h ∂ x ( γ + εψ ) , ∂ x h i ∂ x ( γ + εψ ) + | ∂ x ( γ + εψ ) | ∂ x h ) i ε =0 = −h ∂ x γ | ∂ x γ | , ∂ x ψ | ∂ x γ | i | ∂ x γ | ∂ x ( ∂ x γ | ∂ x γ | ) − | ∂ x γ | ∂ x (cid:16) h ∂ x γ | ∂ x γ | , ∂ x ψ | ∂ x γ | i ∂ x γ | ∂ x γ | (cid:17) + | ∂ x γ | ∂ x ( ∂ x ψ | ∂ x γ | )= −h ∂ s γ, ∂ s ψ i κ − ∂ s ( h ∂ s γ, ∂ s ψ i ∂ s γ ) + ∂ s ψ and hence by rearranging and the Leibniz rule δ ψ ( κ ) = ∂ s ψ − h ∂ s γ, ∂ s ψ i ∂ s γ − h ∂ s γ, ∂ s ψ i ∂ s γ − h ∂ s γ, ∂ s ψ i ∂ s γ = P ⊥ ∂ s γ ( ∂ s ψ ) − h ∂ s γ, ∂ s ψ i κ − h κ, ∂ s ψ i ∂ s γ. (cid:3) References [1] L. Ambrosio and H. M. Soner,
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A varifold perspective on the p -elastic energy of planar sets (Febru-ary 27, 2019), available at .[25] G. Simonett, The Willmore flow near spheres , Differential Integral Equations (2001), no. 8, 1005–1014.(Simon Blatt) Departement of Mathematics, Paris Lodron Universit¨at Salzburg,Hellbrunner Strasse 34, 5020 Salzburg, Austria
Email address , Simon Blatt: [email protected] (Christopher P. Hopper)
Departement of Mathematics, Paris Lodron Univer-sit¨at Salzburg, Hellbrunner Strasse 34, 5020 Salzburg, Austria
Email address , Christopher P. Hopper: [email protected] (Nicole Vorderobermeier)
Departement of Mathematics, Paris Lodron Univer-sit¨at Salzburg, Hellbrunner Strasse 34, 5020 Salzburg, Austria
Email address , Nicole Vorderobermeier:, Nicole Vorderobermeier: