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 L2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^2$$\end{document}-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 [11,18]), 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 [14] 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 [12,15] 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 5.8, is the extension of the result in [15] to the smooth anisotropic setting. More precisely, we show that at the maximal existence time of the geometric solution (see Definition 2.8), 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 5.3).
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 [15] 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 [6,16] 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 [2,10] the authors discuss existence of global weak solutions for the evolutions of embedded networks by anisotropic curvature flow.
In [12,15] the authors proved, in the isotropic case, the long-time existence for the evolution of a network of three curves and the convergence to the minimal Steiner configuration, under the assumption that the length of each curve is bounded away from zero. The main difficulty in extending such a result to the anisotropic setting is the lack of a monotonicity formula as in [9] (see [15,Proposition 6.4] for its adaptation to the case of a network), which in turns prevents a characterization of the parabolic blow-up of the evolution at singular times. This is a challenging and very interesting research direction. Vol. 89 (2021) Anisotropic Curvature Flow of Immersed Networks 149 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 [5,15]. 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.
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.

Anisotropic Scalar Curvature and Anisotropic Curve Shortening Flow
When u is smooth and the anisotropy ϕ is smooth and elliptic the classical formulation of the anisotropic curvature flow is given by the equation (see [1]) where the scalar anisotropic curvature is given by with N = Dϕ • (ν) the Cahn-Hoffman vector. Thus Clearly, boundary and initial conditions (and compatibility conditions) have to be specified as well, but for the moment we neglelct those and focus only on the evolution equation. By setting a straightforward calculation gives so that we can rewrite the flow (2.5) as where κ is the Euclidean curvature and Note that by (2.4), the ellipticity of ϕ implies uniform bounds for ψ, i.e., In the following we shall admit tangential components to the flow, therefore we will consider evolution equations of type for some sufficiently smooth scalar function λ. Vol. 89 (2021) Anisotropic Curvature Flow of Immersed Networks 151 Figure 1 Network with one triple point O and three endpoints P 1 , P 2 , P 3 .
Definition 2.4. The special anisotropic curve shortening flow is defined through a specific choice of tangential term, namely we take (2.12). Thus, the special anisotropic curve shortening flow is given by Next we derive the evolution laws of relevant geometric quantities.

The Geometric Problem
For basic definitions of networks see for instance [14, § 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 ( Fig. 1).

(2.19)
A solution to such problem is called geometric solution.
(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 2.4 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: 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) for every i, j ∈ {1, 2, 3}. Multiplying by Dϕ • (ν i ), summing over i, and using (2.20) gives In the isotropic case this amounts to 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 [14, §3]. The above system can be written as ⎛ 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 2.5 (in particular also (2.15)) we observe that w := V 2 satisfies (cp. with [15, page 263] for the isotropic case) Note that N vanishes in the isotropic case. Bringing N to the left-hand side and multiplying both sides of the equation by e −2 ln ψ(θ) we obtain 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] w(t, ·) in an interior point. By Hamilton's trick ( [13, 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 2.10 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 We assume that σ i ∈ C 2,α ([0, 1], R 2 ), i = 1, 2, 3, are regular maps fulfilling the following compatibility conditions: 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 short-time existence for the original nonlinear 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 [17, page 66 and 91]) and fix some notation.
For a function v : For α ∈ (0, 1) and k ∈ N 0 we define C . We adopt the following conventions: • whenever clear from the context we shall not write the set of the parabolic Hölder spaces, that is, we simply write v ,k+α -norm of a vector-valued map is the sum of the norms of its components.
• for C k,α -Hölder norms on spaces in only one variable we always write the set and use the notation k + α,

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, a similar reasoning as in [7, Lemma 3.1] (using now Lemma 6.3) yields that it is possible to choose T = T (M, δ, σ) so small in the definition of X i above 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 [17]. 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 [17, 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 [17, p. 11]. First of all we consider the system of boundary conditions at the junction point at x = 0. Here the system reads Bu = with all coefficients evaluated at x = 0. Therefore we obtain x | 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 iθ p with −π/2 θ p π/2 and |p| = 0 we may write Following [17, p. 11] we set By [17, 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 w T · A(x = 0, t, iτ, p) = (0, 0, 0, 0, 0, 0) mod M + then w = 0. 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 Vol. 89 (2021) Anisotropic Curvature Flow of Immersed Networks 161 Since (τ − τ + 1 ) 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 simpler 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. [17, 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 . (3.11)

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 6.1 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 6.1 and of Lemmas 6.1, 6.2, 6.4, 6.5, and 6.6 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.
3 j=1 X j be two solutions of the linear problem (LP). Set w = (w 1 , w 2 , w 3 ) with w j = u j − v j , j = 1, 2, 3. 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 . (3.15) 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 possibly choosing an even smaller T , 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.
Observe that by the construction of φ 0 the compatibility conditions of order zero are fulfilled. Instead of solving the PDE for φ, it is convenient to work with the inverse diffeomorphism η = η(t, x), such that φ(t, η(t, x)) = x, and derive its existence first (as proposed in [8]). Indeed we see that η must solve the linear PDE follows from standard theory [17]. Possibly making the time interval smaller we can ensure that η(t, ·) is a diffeomorphism. Finally we take φ(t, ·) = η −1 (t, ·).
From Lemma 4.1, Theorem 3.1 and Corollary 3.2 we directly obtain the following result. 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 6.7, 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 reparametrized network σ i ε satisfies 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 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 2.3). We shall always assume that the flow is smooth up to the initial time t = 0, which is not restrictive in view of Theorem 4.2.
We start with a general lemma.
Also we recall some useful interpolation estimates (here we cite [15, Proposition 3.11, Remark 3.12]): Then, for any m ≥ 1 and p ∈ [2, +∞], we have the estimates Vol. 89 (2021) Anisotropic Curvature Flow of Immersed Networks 169 5.1. 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 2.5 we compute (here and below we write w ⊥ = (w · ν)ν for the normal component of a vector w ∈ R 2 ) 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 H. Kröner et al. Vol. 89 (2021) which yields Derivation in time of (2.20) gives (at the junction) After multiplication by 0 −1 1 0 , 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

(5.4)
A more geometrical interpretation of the above expression is discussed in Remark 5.4 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 5.2 it follows Vol. 89 (2021) Anisotropic Curvature Flow of Immersed Networks 171 Moreover, for the boundary term we use (2.27) and Proposition 5.2 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 Sect. 2.3: 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 5.4. 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
Note that, due to the smoothness assumptions on the anisotropy map ϕ • we will be able to bound uniformly from above all the coefficient maps For this reason we treat these maps as coefficients and refer to them as such. More precisely the maps C( 1 ψ , ψ, . . . , ∂ h θ ψ) are assumed to be sums of rational functions of type polynomial with constant coefficients in the variables ψ(θ), . . . , ∂ h θ ψ(θ) ψ r (θ) for some r ∈ N 0 and, as a consequence, the following rule applies which is obtained by differentiating the left-hand side and recalling that θ s = κ = (l + 1)β l = σ.
We exemplify the notation just introduced in the next lemma (which is partially the anisotropic counterpart of [15,Lemma 3.7] and) which will be used subsequently.
Using Lemma 5.6, interpolation estimates, and Hölder inequality as in [15, p. 262] we obtain From (5.11), (5.12), (5.13), choosing appropriately, integrating in time and using (2.11) we obtain . By Lemma 5.6, and together again with interpolation and Hölder inequalities (cp. with [15, p263]) we obtain that at the junction point we have, for any time t, so that we finally infer and C depends on (2.11), the anisotropy map, and the bound on the lengths of the curves. Upon recalling that κ ϕ = 1 ϕ • (ν) (ψ(θ)κ) and interpolation inequalities from Proposition 5.2 we can summarize our above findings as follows: hold for a solution of the geometric problem (cf. Section 2.3). The constant C depends on δ, C K , T , the initial data (ψκ i ) ss L 2 (0) for i = 1, 2, 3, m, M (recall (2.11)), a 0 (recall (2.21)), and on vw
Vol. 89 (2021) Anisotropic Curvature Flow of Immersed Networks 183 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 , σ i C 2+α ([0,1]) ) as well as ).
We conclude the Appendix with a repametrization result used in the proof of ), and the curveγ = γ•φ satisfies the required properties.