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The critical points of the elastic energy among curves pinned at endpoints

In this paper we find curves minimizing the elastic energy among curves whose length is fixed and whose ends are pinned. Applying the shooting method, we can identify all critical points explicitly and determine which curve is the global minimizer. As a result we show that the critical points consist of wavelike elasticae and the minimizers do not have any loops or interior inflection points.


Introduction
Let γ be a smooth planar curve. Then elastic energy for γ is given by where s denotes the arc length parameter and κ denotes the curvature. The minimization problem for W is called Euler's elastica problem and has been studied due to not only mathematical interest but also the importance of applications. See e.g., [1,24,27,41] for more details of the history and see e.g., [7,8,14,31] for applications.
For given constants 0 < l < L, we define A l,L := γ ∈ C ∞ [0, 1]; R 2 γ(0) = (0, 0), γ(1) = (0, l), L(γ) = L, γ ′ (t) = 0 , where L(γ) denotes the length of γ. Hereafter, we use both the original parameter t ∈ [0, 1] and the arc length parameterization s ∈ [0, L]. For a curve γ, we denote its arc length reparameterization byγ. In this paper we are interested in the minimization problem for W among curves belonging to A l,L . According to [2], critical points of W in A l,L are called pinned elasticae and a way to approximate pinned elasticae by a numerical procedure is demonstrated. However, as mentioned in [2], (A) whether one can construct all critical points explicitly or not is an open problem. Although Linnér [26] obtained some results toward (A), one needs to solve a complicated system so as to obtain curves explicitly. Furthermore, to the best of the author's knowledge, the following problems are also open: (C) Which of the critical points is the global minimizer?
The aim of this paper is to give answers to problems (A), (B) and (C). Let K(p) and E(p) be the complete elliptic integrals of the first and second kind, respectively (p ∈ (0, 1) is the modulus) and let cn be the Jacobi elliptic function (see Section 2.2 for definitions). Theorem 1.1 is concerned with problem (A): Theorem 1.1. The set of critical points of W in A l,L is γ ∈ A l,L γ isγ + n orγ − n orγ + n orγ − n for some n ∈ N ∪ {0} .
Here We remark thatγ − n andγ − n are obtained by the reflection ofγ + n andγ + n across the x-axis, respectively. The number n ∈ N ∪ {0} in Theorem 1.1 is the number of inflection points ofγ ± n orγ ± n in 0 < x < l (see Figure 1). In particular, from the formulas in Theorem 1.1, we infer that the curvesγ ± n have (n + 1)-loops. Moreover, we find that whether or notγ ± n can be represented as a graph depends on the ratio l/L (see Section 3 for these characterizations).
Hence by Theorem 1.1, not only the formulas but also the shapes of all critical points of W in A l,L are deduced. Furthermore, since we can compare the energy of each critical point (see Section 3.3), we have: Theorem 1.2. Let 0 < l < L be arbitrary. For the curvesγ ± n andγ ± n defined by Theorem 1.1, it holds that Moreover, we also obtain |W(γ + 0 ) − W(γ + 0 )| → 0 as l → 0. Finally, problems (B) and (C) are solved as a corollary of Theorem 1.2: Theorem 1.3. For each 0 < l < L, the global minimizer of W in A l,L isγ ± 0 , which implies (i) the uniqueness of global minimizers holds up to reflection; According to [40], these curve are called wavelike elasticae.
(ii) the global minimizer has no loop and the only inflection points are its endpoints.
The minimization problem for W has been studied among various classes. Let γ be a critical point of W under the assumption that the length is prescribed. Then, by the classical Lagrange multiplier method, the curvature κ of γ satisfies for some λ ∈ R. Euler [19] derived the equation (1.1) in 18th century and the curves whose curvature solves (1.1) are called Euler's elasticae. The shapes of Euler's elasticae are well known, see e.g., [24, Figure 11]. The patterns of elasticae among closed curves are well understood. It is shown in [3,23,37] that any critical point is an n-fold circle or an n-fold figure-of-eight, any local minimizer is an n-fold circle or the 1-fold figure-of-eight, and a global minimizer is the 1-fold circle. Moreover, critical points have been studied not only in R 2 but also in higher-dimensional spaces, other manifolds and the hyperbolic space (e.g. [20,21,22,40]). In the case of open curves, however, there have been less results than the case of closed curves. For example, clamped boundary value problems (the tangents at endpoints are prescribed) are considered: Watanabe [42] deduced the representation formulas for a special case; Miura [30] revealed the shapes of minimizers in view of the phase transitions. However, some problems such as uniqueness (B) are still open. One of difficulties for open curves is the treatment of boundary conditions. Furthermore, another difficulty arises from the multiplier λ. In the case of closed curves, representation formulas have been obtained, taking into account the multiplier (e.g., [32,33,43]). In the case of open curves, the equation (1.1) under Navier boundary conditions (the curvatures at endpoints are prescribed) has been solved in the case of λ = 0, see e.g. [16,17,28]. To the best of the author's knowledge, however, there are no results obtaining representation formulas for open curves with non-zero λ, except for [42].
In order to overcome these difficulties, we apply the shooting method to the problem. The shooting method enables us to deal with the boundary conditions and the multiplier simultaneously (see Section 3). Consequently we obtain the formula, properties of curves (such as inflection points, the number of loops) and the uniqueness of minimizers, as we state in Theorems 1.1 and 1.3.
Another emphatic point which the formulas give is the relation between minimizers and the number of inflection points (points where the curvature changes sign). It is already known that the global minimizer has at most two inflection points by different methods (see [5,35,36,38]). Hence Theorem 1.3 yields another point of view on the relation between minimizers and inflection points. The author expects that the method developed in this paper is applicable to the analysis of (1.1) with Navier and clamped boundary conditions. The elastic energy can be regarded as the one-dimensional version of the Willmore functional. The Willmore functional is the integral of the squared mean curvature of surfaces, and its critical points are called Willmore surfaces. See e.g. [9,34,39] for boundary value problems for Willmore surface and see e.g. [4,10,11,12,18,29] for Willmore surfaces of revolution. Recently, Mandel [29] obtained explicit formulas for Willmore surfaces of revolution (with no constraints). Our strategy for dealing with the Lagrange multiplier may be useful for extending the results of [29] to some constrained problems.
This paper is organized as follows. In Sect. 2 we shall consider the Euler-Lagrange equation for W and collect the notations and facts related to the Jacobi Elliptic function. In Sect. 3 we apply the shooting method to the Euler-Lagrange equation so that Theorems 1.1 and 1.2 are proved. Definition 2.1. We call a γ ∈ A l,L critical curve if there exists a λ ∈ R such that the curvature κ of γ satisfies We remark here that it is not restrictive to consider smooth curves only: As the admissible set, seems more suitable than A l,L . Nevertheless we choose A l,L as the admissible set since the regularity of γ ∈ H l,L satisfying (2.1) can be always improved, up to a reparameterization (for details, see Appendix A).
Hereafter we shall show that (2.1) holds if and only if γ is a critical point of W in A l,L and shall deduce the related formulas. Let γ be a curve belonging to A l,L andγ(s) = (X(s), Y (s)) be its arc length parameterization. To this end, consider the variation γ ε for ε > 0, where We note here that s does not generally give the arc length parameter for γ ε . Setting we notice that gives the arc length parameter of γ ε . We denote the unit tangent vector t and unit normal vector n of γ ε by respectively. Hence the curvature of γ ε is given by where ′ denotes the derivative with respect to the parameter s. Therefore we can identify the elastic energy of γ ε with and it holds that where we used integration by parts and ∂ ε φ = ∂ ε ψ = 0 on {0, L}. On the other hand, the first variation formula of L is given by According to the Lagrange multiplier method, γ is a critical point of W in A l,L if and only if there exists some λ ∈ R such that By these formulas for the first variations of W and L, and by recalling that ℓ(0, s) ≡ 1 and m(0, s) = κ(s), we infer from (2.3) that where ′ denotes the derivative with respect to the parameter s. Moreover, since γ is regular, it follows from the third and fourth term of the right hand side of (2.2) that m(0, 0) = m(0, L) = 0, i.e., Solving the above equations for X ′ and Y ′ respectively, we obtain Since s denotes the arc length parameter of γ, it follows that X ′ (s) 2 + Y ′ (s) 2 = 1 and hence we obtain Here it follows that c 2 1 + c 2 2 > 0. In fact, if c 2 1 + c 2 2 = 0 holds, we infer from (2.7) that (κ 2 −λ) 2 +(2κ ′ ) 2 ≡ 0. Then we notice that κ 2 ≡ λ, which implies γ is a line segment or a circle. This contradicts γ ∈ A l,L and hence c 2 1 + c 2 2 > 0 follows. Together with (2.7) this implies that any point s The former contradicts γ ∈ A l,L by the same argument as for c 2 1 + c 2 2 = 0 and hence (κ 2 − λ) 2 + (2κ ′ ) 2 ≡ c 2 1 + c 2 2 holds. Combining this with (2.6), we have Since γ ∈ A l,L implies that Y (0) = Y (L) = 0, integrating (2.9) and using (2.5), we obtain which gives c 2 = 0. In fact, if not, then L 0 (κ(s) 2 −λ)ds = 0 holds. By (2.8) it holds that X(L) = 0, which contradicts γ ∈ A l,L . Therefore for any critical point γ of W in A l,L , its arc length reparameterizationγ(s) = (X(s), Y (s)) can be represented by This together with Y ′′ = κX ′ implies that Therefore any critical point γ ∈ A l,L satisfies (2.1).

Jacobi elliptic functions.
Here we collect the notations and facts related to the Jacobi elliptic functions. Let K(p) and E(p) be the complete elliptic integral of the first and second kind, i.e., Proposition 2.3. The function p → K(p) is strictly increasing and p → E(p) is strictly decreasing. Moreover, it holds that d dp for p ∈ (0, 1).
The above formulas are well known so we omit the proof (see e.g. [6, p.282]).
By Proposition 2.3, we obtain the derivative of the right hand side of (2.13) as follows. Since the straightforward calculation yields Lemma 2.6, we omit the proof. Lemma 2.6. d dp p 2 K(p) − K(p) + E(p) = p · K(p) > 0 for p ∈ (0, 1).

Representation formulas
In this section we shall deduce the representation formulas for critical curves. To this end, we consider two cases in subsections 3.1 and 3.2 respectively and then Theorem 1.1 is shown. Next we discuss about formulas of elastic energy in subsection 3.3 and then Theorem 1.2 is shown.
By the argument in subsection 2.1, critical curves satisfy (2.1) and (2.10) so we focus on them hereafter. In order to solve the boundary value problem (2.1), we employ the shooting method, i.e., we consider the following initial value problem: where b, λ ∈ R. Let κ b,λ be the unique solution of (3.1) for each b, λ ∈ R. Let (X b,λ , Y b,λ ) be the curve whose curvature is κ b,λ and parameterized by the arc length s. If (X b,λ , Y b,λ ) ∈ A l,L is critical, then it follows from (2.10) that for some c ∈ R and hence we can replace c with ± √ λ 2 + 4b 2 . According to [25], κ b,λ is given by We shall find b ∈ R and λ ∈ R satisfying When b = 0, by (3.2)-(3.5) λ is equal to 0 and the solution of (3.1) is κ ≡ 0. Namely, if b = 0, then any critical curve is a line segment, which does not satisfy l < L. Therefore hereafter we eliminate the case b = 0. Since κ ′ (0) = b, it follows from κ(0) = 0 and the symmetry of cn that which in combination with (3.7) yields Therefore it suffices to consider either b > 0 or b < 0. As discussed below, the sign of c plays an important role in the shapes of curves.
3.1. Critical concave curves. In this subsection we consider (3.1) under c > 0. As mentioned earlier, it is sufficient to consider b < 0 for a while. Recalling (2.11), we find that β in (3.2) is either Then since sn(±K(p), p) = ±1 and dn(±K(p), p) = 1 − p 2 , by (3.3) we have Moreover, it follows from (3.2)-(3.6) and (3.9) that A, α and p satisfy (3.11) If the above equation holds, we find that Y b,λ (L) = 0 since where we used (3.12) and the relation 2α 2 = √ λ 2 + 4b 2 in (3.10). The integral in (3.14) is reduced to where we used the periodicity of cn in the last equality. Since A 2 = 4α 2 p 2 , it follows from Lemma 2.5 and (3.11) that (3.15) Therefore we can rewrite (3.14) into which in combination with (3.10) and (3.11) gives Therefore p ∈ [0, 1] must satisfy Lemma 2.4 implies that such p is uniquely determined and letp denote the solution of (3.16). Then pluggingp into (3.10) and αL = 2(n + 1)K(p), we notice that b and λ satisfying (3.6) are given by (b, λ) = (b n ,λ n ), wherê Since (X n , −Ŷ n ) is also critical by considering (b, λ) = (−b n ,λ n ), we can summarize the above arguments as follows: Theorem 3.1. Let γ be a curve which belongs to A l,L and satisfies ∂ t κ(0) < 0. Then γ is a critical curve if and only if there exists n ∈ N ∪ {0} such that its arc length parametrizationγ is (X n ,Ŷ n ) or (X n , −Ŷ n ), wherê By the periodicity, it suffices to consider s ∈ [0, L/2(n + 1)]. Then we notice that X ′ n (s) is increasing in [0, L/2(n + 1)] and satisfieŝ Recalling thatp is defined by (3.16), we obtain the following: where R * is given by Moreover, sinceŶ ′ n (s) ≥ 0 holds for s ∈ [0, L/2(n + 1)] (the equality holds if s = L/2(n + 1)), the curveγ n can be represented as the graph of a function if and only if l/L < R * (see Figure 2). Remark 3.2. As explained in [30, Fig.14],λ n plays the role of "tension", that is, critical curves cannot be represented as the graph of any function whenλ n > 0 holds. The values ofλ n are given by (3.17), which implies that the tension value is determined only by the ratio l/L. and hence the necessary and sufficient condition for X(L) = 0 is that Therefore p ∈ [0, 1] needs to satisfy We remark here that we have considered the cases into c > 0 and c < 0 separately due to the difference between (3.16) and (3.24). Lemma 2.4 implies that such a p is uniquely determined and hence we denote byp the solution of (3.24). Hence pluggingp into (3.10) and αL = 2(n + 1)K(p), we notice that b and λ satisfying (3.6) are given by (b, λ) = (b n ,λ n ), wherě for n ∈ N ∪ {0}. Consequently we obtain the following: Theorem 3.3. Let γ be a curve which belongs to A l,L and satisfies ∂ t κ(0) > 0. Then γ is a critical curve if and only if there exists n ∈ N ∪ {0} such that its arc length parametrizationγ is (X n ,Y n ) or (X n , −Y n ), wherě for s ∈ [0, L]. Herep is uniquely determined by the solution of (3.24).
proof of Theorem 1.1. We obtain the conclusion from Theorems 3.1 and 3.3.
Remark 3.4. In the case l = 0 we can infer from [13] that the minimizer is the half figure-eight and is unique (although the setting is bit different between this paper and [13], and [13] considers a more general case). Letting l → 0, we find that curvesγ 0 andγ 0 tend to the curves obtained in [13].
3.3. Comparison among critical curves. In this subsection we turn to the proof of Theorem 1.2 and consider the case that l/L is sufficiently close to 1.
This completes the proof.
We are ready to determine which curve is the global minimizer: proof of Theorem 1.3. The existence of global minimizers of W in A l,L follows by the direct method in the calculus of variations (see Appendix A). Since minimizers are also critical points of W in A l,L , in order to identify minimizers it suffices to find critical points whose energy is least. By Theorem 1.2,γ + 0 = (X 0 ,Ŷ 0 ) attains the minimum of W among critical points and so doesγ − 0 = (X 0 , −Ŷ 0 ). Moreover, it has been shown that the other critical points do not attain the minimum. This implies that minimizers are nothing butγ ± 0 .
By (3.16) and (3.26), the smaller the ratio l/L is, the less the elastic energy of γ n is. Conversely, it follows from (3.27) and (3.16) that the larger the ratio l/L is, the less the elastic energy ofγ n is. In order to investigate W(γ 0 ) when l/L is close to 1, we will require the following lemma: Proposition 3.5. Let K(p) and E(p) be the complete elliptic integral. Then it holds that Proof. To begin with, we notice that for any p ∈ (0, 1) Hence it follows that This together with E(1) = 1 and lim p↑1 K(p) = ∞ yields which completes the proof.
Remark 3.6. Fix L > 0 arbitrarily. It follows from the previous argument that • when l is sufficiently close to L, W(γ 0 ) > W(γ 1 ) holds.
In fact, since bothp andp tend to p 0 as l ↓ 0, W(γ n ) is almost equal to W(γ n ) for each n ∈ N. This observation implies that l/L determines which of the curvesγ 1 andγ 0 has the second smallest elastic energy.

Appendix A. Existence of minimizers
Here we shall prove the existence and the smoothness of minimizers of W in A l,L via a direct method in the calculus of variations. Hereafter W ′ (γ)(ϕ) and L ′ (γ)(ϕ) denote the first variation of W and L at γ in direction ϕ, respectively. That is, For γ parameterized by the arc length, it holds that where s is the arc length parameter of γ (for (A.1), see e.g. [15]).
Furthermore, since γ is a minimizer in H l,L , we can deduce thatγ is also a minimizer in A l,L .