Anisotropic curvature flow of immersed networks

We consider motion by anisotropic curvature of a network of three curves immersed in the plane meeting at a triple junction and with the other ends fixed. We show existence, uniqueness and regularity of a maximal geometric solution and we prove that, if the maximal time is finite, then either the length of one of the curves goes to zero or the $L^2$ norm of the anisotropic curvature blows up.


Introduction
The aim of this work is to study motion by anisotropic curvature of a network of three curves in the plane. This evolution corresponds to a gradient flow of the anisotropic length of the network, which is the sum of the anisotropic lengths of the three curves. Since multiple points of order greater than three are always energetically unstable (see [17,10]), it is natural to consider networks with only triple junctions, the simplest of which is a network with three curves meeting at a common point.
The isotropic version of this problem attracted a considerable attention in recent years (see for instance the extended survey [13] and references therein). In particular, the short time existence for the evolution has been first proved by L. Bronsard and F. Reitich in [5], and later extended in [14,11] where it is shown that, at the maximal existence time, either one curve disappears or the curvature blows up.
The main result of this paper, contained in Theorem 5.1, is the extension of the result in [14] to the smooth anisotropic setting. More precisely, we show that at the maximal existence time of the geometric solution (see Definition 2.6), either the length of one curve goes to zero or the L 2 norm of the anisotropic curvature blows up. In the latter case, we also provide a lower bound on the blow up rate of the curvature (see Lemma 5.2).
A relevant technical issue in this paper is due to the fact that, in the case of networks, the evolution is governed by a system of PDE's rather than by a single equation, hence it is difficult to use the maximum principle, which is usually the main tool to get estimates on the geometric quantities for curvature flows. As a consequence, following [14] in order to control these quantities we rely on delicate integral estimates and interpolation inequalities.
A challenging open problem is the extension of such result to the nonsmooth (including crystalline) anisotropic setting, as it was done in [6,15] for the case of closed planar curves. In the case of networks, the dependence of the integral estimates on the anisotropy, makes such extension problematic.
Let us point out that, in the paper [3], the authors proved a short time existence result for the crystalline evolution of embedded networks, under a suitable assumption on the initial data which allows to reduce the evolution equation to a system of ODE's. We also recall that in the papers [9,2] the authors discuss existence of global weak solutions for the evolutions of embedded networks by anisotropic curvature flow.
The paper is organized as follows: In Section 2 we introduce the notation and define the relevant geometric object that we shall use throughout the paper. In Section 3 we prove a short time existence result for the evolution following the approach in [5,14]. In Section 4 we show the existence and uniqueness of a maximal geometric solution and we prove that, at the maximal time, either the length of one curve tends to zero or the H 1 norm of the anisotropic curvature blows up. Finally, in Section 5 we refine this conclusion by showing that, if the H 1 norm blows up, then also the L 2 norm of the anisotropic curvature blows up. We conclude the paper with an Appendix containing some technical result which are used in the paper.

Anisotropies
Let us recall some definitions and properties of anisotropy maps (see for instance [4]).
Definition 2.2. The set W ϕ := {ϕ 1} is called Wulff shape. We say that ϕ is crystalline if W ϕ is a polygon.
In the following, we shall restrict ourselves to the case of smooth and elliptic anisotropies.
Observe that the homogeneity property of a norm ϕ yields Dϕ(p)·p = ϕ(p) and D 2 ϕ(p)p = 0 for any p = 0, two facts that we will use repeatedly in our computations.

The Geometric Problem
For basic definitions of networks see for instance [13, § 2]. We consider networks S of curves parametrized by regular maps u i : [0, 1] → R 2 , i = 1, 2, 3, such that u i (1) = P i (with P i ∈ R 2 given) and u i (0) = u j (0), for i, j ∈ {1, 2, 3}, that is the curves are parametrized in such a way that the origin is mapped to the triple junction.
ϕ denotes the anisotropic curvature of the curve σ i ) and Here λ i 0 denotes a further geometric quantity, whose expression is formulated in (2.26) below. In particular we see that λ i 0 is given as a linear combination of ψ(θ i )κ i and ψ(θ i±1 )κ i±1 .

(2.19)
A solution to such problem is called geometric solution.
Remark 2.2 (Anisotropic angle condition). The boundary condition at the triple junction is the anisotropic version of the Herring condition (cf. [13,Def. 2.5]) and is derived by considering the first variation of E(S) : (where here and in the following we write ds instead of ds i , the meaning being clear from the context) and (2.20) is immediately deduced. Note that the vectors ξ i := Dϕ • (ν i ) appearing in (2.20) belong to the boundary of the Wulff shape, i.e., ξ i ∈ ∂W ϕ , i = 1, 2, 3. We can state that the angles at which the tangent planes to ∂W ϕ at ξ i can meet are bounded away from zero and π: indeed in one of these two limit cases, the three vectors must be in shape of a Y (possibly with two vectors coinciding), but we get a contradiction using the symmetry and convexity of the Wulff shape.
Since ν i is normal to the tangent plane at ξ i = Dϕ • (ν i ) ∈ ∂W ϕ , this means that there exists a positive constant C depending on ϕ • such that In turns this implies the existence of a postive constant a 0 depending on ϕ • such that For the notion of geometric solution it is enough to specify the normal velocity. To attack the problem analytically, we actually consider the system . Note that the presence of tangential components λ i is necessary to allow for movements of the triple junction. In principle there is some freedom in the choice of these maps, but the freedom is restricted only to the points in the interior of the interval of definition. Indeed we show below in Section 2.0.4 that λ i , i = 1, 2, 3 are fixed by the problem at the boundary. More precisely we show that at the boundary we can express λ i as a linear combination of the geometric quantities ψ(θ i )κ i and ψ(θ i±1 )κ i±1 .
Among all possible choices of tangential components λ i , we highlight one specific flow that will play an important role in our discussion: Definition 2.7. A solution as in Definition 2.6 such that u i t , i = 1, 2, 3, evolves according to (2.13) is called Special Flow.
The Special Flow provides a well posed problem that we can attack analytically. We shall use the Special Flow to derive short time-existence of a geometric solution, and to show its uniqueness and smoothness.

Behavior of a generic tangential component λ i at the triple junction
At the triple junction beside the concurrency condition we impose that the velocity be the same for all curves involved, hence we impose or equivalently (after rotation by π/2) Multiplying with Dϕ • (ν i ), summing over i, and using (2.20) gives In the isotropic case this amounts to 3 i=1 κ i = 0 = 3 i=1 λ i . On the other hand, starting from (2.23) and taking the inner product with appropriate normals and tangents we get (with the convention that the superscripts are considered "modulus 3") For the isotropic case where all constants and coefficients can be given explicitly see [13, §3]. The above system can be written as Writing α = (ν i+1 · ν i ), β = (τ i+1 · ν i ), γ = (ν i−1 · ν i ), δ = (τ i−1 · ν i ) we see that above matrix has determinant equal to det = (α 2 + β 2 )(δ 2 + γ 2 ), which can never be zero since α and β, respectively δ and γ, can not vanish simultaneously. Thus we obtain From the first two equations we infer that if β = 0 or δ = 0 then we can express λ i as a linear combination of ψ(θ i )κ i and ψ(θ i±1 )κ i±1 . By (2.21) we know that in fact |β| and |δ| are bounded from below. In particular we obtain that at the triple junction with C = C(a 0 ) depending on the anisotropy. For the analysis that follows we will also need expressions for the time derivative λ i t . Using (2.26) we can write with C = C(a 0 ) depending on the anisotropy. Proof. The statement follows by adding the contribution of each curve as computed in (2.17), using (2.25) at the triple junction, and the fact that λ i = 0 = κ i at the fixed points P i , i = 1, 2, 3 (this follows from (2.12) and ∂ t u i = 0 at P i ).

Special Flow: behavior of λ i in the interior points
In the following we assume that (2.13) holds for every curve of the network and that we have a uniform bound on the curvatures, namely Since the following considerations hold for any curve of the network we drop the indices for simplicity of notation. Upon recalling (2.12) let us denote with V the length of the velocity vector. Then Using Lemma 2.1 (in particular also (2.15)) we observe that w := V 2 satisfies (cp. with [14, page 263] for the isotropic case) Note that N vanishes in the isotropic case. Bringing N to the left-hand side and multiplying both side of the equation with e −2 ln ψ(θ) we obtain (we −2 ln ψ(θ) ) t = ψ(θ)e −2 ln ψ(θ) w ss − λe −2 ln ψ(θ) w s + 2ψ(θ)κ 2 we −2 ln ψ(θ) If w(t, ·) = V 2 (t, ·) ≥ 0 does not take its maximum at the boundary (where κ and hence λ, recall (2.27), are controlled by assumption) then it achieves its maximum w max (t) = max [0,1] is an interior point where w(t, ·) assumes its maximum. Then where C depends on C 0 and on the anisotropy map (recall (2.11)). Gronwall's inequality yields It follows that V i and λ i are uniformly bounded on [0, T ) for i = 1, 2, 3.

Short-time existence for the Special Flow
The aim of this section is to establish a short time existence result for the special anisotropic curve shortening flow (recall Definition 2.7 and (2.13)). More precisely, given an initial network σ := (σ 1 , σ 2 , σ 3 ) of sufficiently smooth regular curves satisfying appropriate boundary conditions (see below) we look for T > 0 and u i : with initial datum u i (0, ·) = σ i (·) and boundary conditions We assume that σ i ∈ C 2,α ([0, 1], R 2 ), i = 1, 2, 3, are regular maps fulfilling the following compatibility conditions: as well as ).
Existence and uniqueness in the isotropic case have been shown in Bronsard and Reitich [5]. There the short-time existence proof is carried out in three steps: first a linearization around the initial data is performed, second the classical theory for parabolic system is used to prove existence for the linearized system, third a fixed-point argument is applied to obtain shorttime existence for the original non-linear problem. Due to the presence of the anisotropy map the problem is now clearly highly nonlinear and some details require attention. In the following we provide the main arguments. With respect to [5] one striking difference consists in the treatment of the boundary condition at the triple junction. In the isotropic case (2.20) yields τ 1 + τ 2 + τ 3 = 0, which gives an angle condition described in [5, eq.(28)] as τ 1 · τ 2 = cos(2π/3) = τ 2 · τ 3 . The latter two equations are then accordingly linearized around the initial datum. Here we need to work with (2.20) directly, since ϕ • is a given arbitrary (smooth and elliptic) anisotropy map.
Function spaces and notation. For the convenience of the reader let us recall the definition of the parabolic Hölder spaces (recall [16, page 66 and 91]) and fix some notation.
For a function v : . We adopt the following conventions: • in the proofs, and whenever clear from the context, we do not write the set of the parabolic Hölder spaces. In other words we simply write v • for Hölder norms on spaces in only one variable we always write the set, for instance in • the C k+α 2 ,k+α -norm of a vector-valued map is the sum of the norms of its components.
Useful lemmas for parabolic Hölder spaces are collected in Appendix A.

Linearized Problem
For some 0 < T < 1 and M > 0 to be chosen later on (cf. (3.12)) define by extending it as a constant function in time, similar reasoning as in [7, Lemma 3.1] (using now Lemma A.3) yields that it is possible to choose T = T (M, δ, σ) so small in the definition of X i above so that any map v ∈ X i is regular for all times. From now on we assume that T is fixed in such a way that the regularity of the curves is guaranteed, that is for anyū i ∈ X i , i = 1, 2, 3. As in [5] we seek a fixed point of the map where u solves the following linearized system, which we refer to as the linear problem.

Solution of the linear problem (LP)
As in [5] we follow the theory developed in [16]. The above system can be written as for l ∈ {1, 2, 3} and j ∈ {1, 2}. Since many terms coincide in the following we simply write As in [5] we note that the parabolicity condition [16, p. 8] is fulfilled since for any i = 1, 2, 3 we have that where m is as in (2.11).
At the boundary we need to check the so-called complementary conditions [16, p. 11]. First of all we consider the system of boundary conditions at the junction point at x = 0. Here the where each block entry is a (2 × 2) matrix with with all coefficients evaluated at x = 0. Therefore we obtain with all expressions evaluated at x = 0. In the isotropic case b 51 = b 53 = b 55 = 0 and Next note that as a function of τ the polynomial L(x, t, iτ, p) has six roots with positive imaginary parts and six roots with negative imaginary parts provided Re(p) ≥ 0 and p = 0. More precisely writing p = |p|e iθp with −π/2 θ p π/2 and |p| = 0 we may write Following [16, p. 11] we set By [16, p. 11] the complementary condition at x = 0 is satisfied if the rows of the matrix are linearly independent modulo M + whereby p = 0, Re(p) ≥ 0. Therefore we need to verify that if there exists w ∈ R 6 such that This gives the six equations Using the fact that A i and M + have many factors in common, we infer that the first equation in equivalent to can not divide p 1 (τ ) then τ + 1 must be a root of the remaning linear factor. Reasoning in a similar way for the other five equations we obtain that w must satisfy the system It follows that w = 0 and the complementary condition at x = 0 is fulfilled. Checking the complementary condition at x = 1 is done in a similar way, but here computations are much simplier since B(x = 1, t, iτ, p) is given by the identity matrix.
Finally we observe that at t = 0 the initial condition is given by the system Cu = σ where C ∈ R 6×6 is the identity matrix. The complementary condition here (cf. [16, p 12]) requires that the rows of the matrix D(x, p) = C ·L(x, 0, 0, p) are linearly independent modulo p 6 at each point x ∈ (0, 1). This is readily checked.
Using (3.3), (3.4), (3.5) and the definition of the spaces X j we also observe that the linear problem fulfills the compatibility conditions of order zero (cf. [16, p. 98]). Application of [16,Thm. 4.9] yields the existence of a unique solution u

Fixed point argument
be the solution of the linear problem (LP). We would like to verify the self-map and self-contraction property of the operator R (recall (3.8)). To that end we employ (3.11).
Self-map property We need to estimate the right-hand side in (3.11). For j = 1, 2, 3 and using the definition of X j as well as Lemma A.1 we compute Writing out the expressions of type ψ(θ) in terms of tangents and normals (recall (2.10), (2.8)), manipulating them appropriately into products of differences (similarly to what we have done above) and application of Remark A.1 and of Lemmas A.1, A.2, A.4, A.5, and A.6 yields . Putting all estimates together we derive from (3.11) Hence choosing and taking T < 1 so that 3C 0 (C 1 + C 2 )T α 2 M/2 we infer that R maps X 1 × X 2 × X 3 into itself. This will be assumed henceforth.

Contraction property
Then the w j 's satisfy x | and note that here ψ(θ(u)) is given by (2.8) and (2.10) with tangent and normal vector of the curve u) This is again a linear parabolic system and it satisfies the complementary and compatibility conditions. In particular it satisfies the Schauder-type estimate .
Using the lemmas from the Appendix A, the definition of X j , and arguments similar to those employed in the verification of the self-map property we compute for j = 1, 2, 3 . Thus, by choosing T possibly even smaller, we obtain and the contraction property of R is established.
Finally application of the Banach's fixed point theorem yields the existence of a unique map u ∈ 3 j=1 X j with u = R(u), that is a solution to (3.1), (3.2). In particular we can state the following theorem.
From Lemma 4.1, Theorem 3.1 and Corollary 3.1 we directly obtain the following result.
Theorem 4.1. Let α ∈ (0, 1), P i ∈ R 2 , i = 1, 2, 3, be given points and σ i , i = 1, 2, 3, as in Definition 2.5. Then there exists T > 0 and regular maps 1], in the sense of Definition 2.6 (i.e., up to reparametrization of the given initial data). Moreover, the solutions u i are unique up to reparametrization, that is, they parametrize a geometrically unique evolving network.
We eventually show that at the maximal existence time either the length of one curve goes to zero or the H 1 -norm of the curvature blows up. Proof. Assume by contradiction that L(u i (t)) ≥ δ and κ i ϕ H 1 (I) ≤ C, for all i = 1, 2, 3 and t ∈ [0, T ), and for some δ, C > 0. By Lemma A.7, for any ε ∈ (0, T ) we can reparametrize the admissible network u i (·, T − ε), i = 1, 2, 3, in such a way that the reparametrizated network σ i ε satisfy the compatibility conditions (3.3), (3.4), (3.5) and moreover where the constant C > 0 depends only on δ and C. Indeed, we can first reparametrize u i (·, T − ε) by constant speed. Then we notice that for the so obtained parametrization v i the uniform bound on the (anisotropic) curvature yields that For the compatibility conditions (3.3), (3.4), (3.5) to hold we need now to reparametrize v i again (as explained in (4.3) with v instead of u, so that |v x | = 1/L(v i ) and v xx · v x = 0). As appropriate diffeomorphisms φ i we take now suitable perturbations near the junction of the identity map such that (φ i ) (0) = (φ i ) (1) = 1, (φ i ) > 0 on [0, 1], (4.3) holds, and the φ i C 2,1/2 -norm is uniformly bounded by a constant depending only on C, δ, L(u i (T −ε)), and the anisotropy map (see (4.3) and recall (2.27), (2.11), Lemma A.7). The maps σ i = v i (φ i ) satisfy the claims.
Then, by Theorem 3.1 there exist solutions u i ε to the special flow starting from σ i ε at T − ε, defined on the time interval [T − ε, T − ε + τ ), where τ > 0 depends only on δ and C (in particular it is independent of ε). By choosing ε small enough we then have T − ε + τ > T . Notice that, by Lemma 4.1 (see also Corollary 3.1) there exist smooth give rise to geometric solution defined on the time interval [0, T − ε + τ ), contradicting the maximality of T .

Integral estimates and main result
In this section we derive integral estimates for a solution of the geometric problem (recall Section 2.0.3). We shall always assume that the flow is smooth up to the initial time t = 0, which is not restrictive in view of Theorem 4.1.
We start with a general lemma.
Lemma 5.1. Let u : I → R 2 satisfy (2.12) for some smooth map λ. Let S : I → R 2 be a normal vector field along the curve u, that is (S · τ ) ≡ 0. Then d dt Proof. Since (ds) t = (λ s − (κν · u t ))ds = (λ s − ψ(θ)κ 2 )ds a direct computation gives d dt Using the expression for θ t from Lemma 2.1 we observe and the claim follows.
Also we recall some useful interpolation estimates (here we cite [14, Proposition 3.11, Remark 3.12]): Proposition 5.1. Let u be a smooth regular curve in R 2 with finite length L. If f is a smooth function defined on u and m ≥ 1, p ∈ [2, +∞], we have the estimates for every n ∈ {0, . . . , m − 1} where σ = n+1/2−1/p m and the constants C n,m,p , B n,m,p are independent of u. In particular

Estimates on κ L 2 and κ ϕ L 2
We now apply the Lemma 5.1 for the special choice of S = ψ(θ)κν, which is the normal component of the velocity vector. Using Lemma 2.1 we compute (here and below we write w ⊥ = (w · ν)ν for the normal component of a vector w ∈ R 2 ) S = (u t ) ⊥ = ψ(θ)κν, as well as Therefore the integral terms appearing in the right hand-side of (5.1) amount to In particular notice that with S t −ψ(θ)S ss high order terms disappear. Equation (5.1) becomes d dt For the boundary term we notice that This motivates the choice of S since at the boundary the velocity u t is either zero (at the fixed boundary point) or coincides with the velocity of the other curves meeting at the triple junction. We can then lower the order of the terms at the moving boundary point by exploiting the boundary conditions. More precisely we write Derivation in time of (2.20) gives (at the junction) , which rotates vectors by π/2, we finally infer that holds at the junction point. In particular, since u 1 t = u 2 t = u 3 t at the junction point, we infer Summing (5.2) for every curve in the network, we therefore obtain A more geometrical interpretation of the above expression is discussed in Remark 5.1 below. Using (2.11) as well as C −1 ϕ • (ν) C and |Dϕ • (ν)| C (recall that Dϕ • (ν) lies on the Wulff shape) we can write Next we apply interpolation estimates, under the assumption that we have a uniform control (from below) of the lengths of the curves composing the network. By Proposition 5.1 it follows Moreover, for the boundary term we use (2.27) and Proposition 5.1 to infer where for the last step, we have used several times the Young-inequality. Putting all estimates together and choosing appropriately we infer where C depends on the anisotropy (precisely (2.11), a 0 as in (2.21), as well as C −1 ϕ • (ν) C and |Dϕ • (ν)| C) and on the uniform bound from below on the lengths of the curves. Note also that so far only information of λ i at the boundary has played a role. Recalling that κ ϕ = ψ(θ) ϕ • (ν) κ and integration in time for 0 t 1 < t 2 yields In particular if, for 0 < T < ∞, there exists a sequence of times t j → T , for j → ∞, such that then we obtain for any t ∈ [0, T ) that Therefore we can conclude with the following statement, that is valid for a solution to the geometric problem posed in Section 2.0.3: Lemma 5.2. If for 0 < T < ∞, the lengths of the curves are uniformly bounded from below L(u i (t)) ≥ δ > 0, i = 1, 2, 3, for any t ∈ [0, T ) and there exists a sequence of times t j → T , for j → ∞, such that (5.5) holds, then there exists a positive constant C such that where the constant C > 0 depends on δ and ϕ • (namely m, M (recall (2.11)), a 0 (recall (2.21)), C −1 ϕ • (ν) C and |Dϕ • (ν)| C). In particular, taking t = 0 we have Remark 5.1. Upon recalling that κ ϕ = ψ(θ) ϕ • (ν) κ, observe that (5.4) can also be written as where in the integration by parts we have used the fact that the velocities and hence the curvatures vanish at the fixed boundary points. On the other hand note that and therefore

It follows then
For the second last integral on the right-hand side note that and so it can be nicely absorbed. It follows then
We exemplify the notation just introduced in the next lemma (which is partially the anisotropic counterpart of [14,Lemma 3.7] and) which will be used subsequently.
Moreover using Lemma 5.3 with j = 2, and Lemma 2.1 we can write Therefore and we obtain as well as Plugging the above expression into (5.1) yields d dt With help of Young inequality and using (2.11) we achieve d dt To treat the boundary term it is imperative to be able to lower the order of the term with three spacial derivatives. Note that the λ-term is of type To handle the term (S · S s ) observe that by Lemma 2.1 and (5.7) we can write Next, using (5.9) and the expression derived above for S s , we observe that where we have used Lemma 5.3 in the second last equality. Hence so far we have shown that To handle the last term we use the boundary conditions: twice derivation in time of (2.20) gives (at the junction point) Since here u 1 tt = u 2 tt = u 3 tt , we obtain Ru 1 tt = Ru 2 tt = Ru 3 tt with R = 0 −1 1 0 which rotates vectors by π/2, and hence (recall (5.9))

It follows that
The expression above together with (5.8) and (5.10) yields Finally we apply interpolation inequalities. Using Proposition 5.1 and Hölder inequality as demonstrated and carefully explained in [14, p.260-261] we obtain that where the constants depends on (2.11), the anisotropy map, and the bounds of the lengths of the curves. At the triple junction recall that we can write λ i in terms of (ψ(θ j )κ j ) for j = i. In particular, we have that (2.27) holds. Together with (2.28), Lemma 2.1 and Lemma 5.3 we infer that Lemma 5.4. We have that at the junction point there holds where C depends on the anisotropy map and where the polynomials on the right-hand side now contains derivatives of (ψ(θ j )κ j ) for the three different curves.

Main result
From Lemma 5.5, Theorem 4.1 and Proposition 4.1 we finally obtain our main result on the behavior of a geometric solution at the maximal existence time.
Theorem 5.1. Let α ∈ (0, 1), σ i be as in Definition 2.5, and u i ∈ C A Some useful results The following remark and the next three lemmas are a straight forward adaptation to the present setting of the lemmas presented in [7,Appendix B]. .
Next we provide a list of results that are useful in the contraction argument in the proof of the short-time existence. In the following lemma we use that, given σ i ∈ C 2,α ([0, 1]), then σ i ∈ C for some universal constant C. Moreover, for T < 1 we have that , σ i C 2,α ([0,1]) ) as well as ).

Proof. It follows by writing every equation in the form
and using the previous Lemmas A. 1 We conclude the Appendix with a repametrization result used in the proof of Proposition 4.1. Finally we have φ C 2,1/2 ≤ C(L, f C 1 2 ), and the curveγ = γ • φ satisfies the required properties.