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Elastic curves and phase transitions

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This paper is devoted to a classical variational problem for planar elastic curves of clamped endpoints, so-called Euler’s elastica problem. We investigate a straightening limit that means enlarging the distance of the endpoints, and obtain several new results concerning properties of least energy solutions. In particular we reach a first uniqueness result that assumes no symmetry. As a key ingredient we develop a foundational singular perturbation theory for the modified total squared curvature energy. It turns out that our energy has almost the same variational structure as a phase transition energy of Modica–Mortola type at the level of a first order singular limit.

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Acknowledgements

The author would like to thank Professor Yoshikazu Giga, Professor Yasuhito Miyamoto, Professor Michiaki Onodera, and Dr. Olivier Pierre-Louis for their helpful comments and discussion. In particular, Professor Miyamoto indicated to the author that our study is related to the works by Ni and Takagi. The author learned of Audoly and Pomeau’s book from Dr. Pierre-Louis, and found a description related to our study. The author would also like thank anonymous referees for their careful reading and useful comments. This work is mainly carried out at the University of Tokyo and partly at the Max Planck Institute for Mathematics in the Sciences. This work is partially supported by JSPS KAKENHI Grant Numbers JP15J05166, JP18J30004, and also the Program for Leading Graduate Schools, MEXT, Japan.

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    Present address: Department of Mathematics, Tokyo Institute of Technology, 2-12-1 Ookayama, Meguro, Tokyo, 152-8551, Japan

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    Tatsuya Miura

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Appendices

A Existence of minimizers

Fix \(l>0\) and \(\theta _0,\theta _1\in [-\pi ,\pi ]\). We say that \(\gamma \in H^2(I;{\mathbb {R}}^2)\subset C^1({\bar{I}};{\mathbb {R}}^2)\) is \(H^2\)-admissible if \(\gamma \) is of constant speed and satisfying the boundary condition (2.1). We denote the set of \(H^2\)-admissible curves by \({\mathcal {X}}\). Note that the \(H^2\)-weak topology is stronger than \(C^1\)-topology; hence, in particular, the set \({\mathcal {X}}\) is \(H^2\)-weakly closed in \(H^2(I;{\mathbb {R}}^2)\).

In this \(H^2\)-framework we have an existence theorem of standard type: Let \({\mathcal {X}}'\subset {\mathcal {X}}\) be an \(H^2\)-weakly closed subset. Then the functional \({\mathcal {E}}_\varepsilon =\varepsilon ^2{\mathcal {B}}+{\mathcal {L}}\) defined on \({\mathcal {X}}'\) attains its minimum in \({\mathcal {X}}'\).

The proof is straightforward. Since any \(\gamma \in {\mathcal {X}}'\) is of constant speed, we have the following representations:

$$\begin{aligned} {\mathcal {L}}[\gamma ]\equiv |{\dot{\gamma }}|\ge l, \quad {\mathcal {B}}[\gamma ]=\frac{1}{{\mathcal {L}}[\gamma ]^3}\int _I|\ddot{\gamma }(t)|^2dt. \end{aligned}$$

By the above relations and the boundary condition, we find that a minimizing sequence is \(H^2\)-bounded. Since \({\mathcal {E}}_\varepsilon \) is lower semicontinuous with respect to the \(H^2\)-weak topology, a standard direct method implies the existence of a minimizer, completing the proof.

Moreover, if \({\mathcal {X}}'\) admits any local perturbation, then we find that any minimizer \(\gamma \in {\mathcal {X}}'\) is of class \(C^\infty \) by a bootstrap argument. In particular, the problem (2.2) admits a smooth minimizer. Using the Lagrange multiplier method to modify the length constraint, we find that the problem (2.4) also admits a smooth minimizer. One may also refer to [52, Theorem 2.2] for a different argument.

In addition, it is also proved that there are infinitely many local minimizers with different winding numbers in a sense. Here \(\gamma \in {\mathcal {X}}\) is a local minimizer of the energy \({\mathcal {E}}_\varepsilon \) if there is \(\delta >0\) such that \({\mathcal {E}}_\varepsilon [\gamma ]\le {\mathcal {E}}_\varepsilon [\gamma ']\) for any \(\gamma '\in {\mathcal {X}}\) with \(\Vert \gamma -\gamma '\Vert _{H^2}\le \delta \). To state the above fact, we use a kind of winding number; for \(\gamma \in {\mathcal {X}}\) we define \({\mathcal {N}}[\gamma ]\in {\mathbb {Z}}\) as

$$\begin{aligned} {\mathcal {N}}[\gamma ]=\frac{1}{2\pi }\left( \int _{\gamma }\kappa ds + \theta _0-\theta _1\right) , \end{aligned}$$

where \(\kappa \) is the counterclockwise signed curvature (\(\kappa =\partial _s\vartheta _{{\tilde{\gamma }}}\)). We notice that the functional \({\mathcal {N}}\) is \({\mathbb {Z}}\)-valued and continuous with respect to the \(H^2\)-weak and -strong topologies. Thus for any \(m\in {\mathbb {Z}}\) the set \({\mathcal {X}}_m=\{\gamma \in {\mathcal {X}} \mid {\mathcal {N}}[\gamma ]=m \}\) is open and closed in \({\mathcal {X}}\) both weakly and strongly. Since \({\mathcal {X}}_m\) is weakly closed, the energies \({\mathcal {E}}_\varepsilon \) defined on \({\mathcal {X}}_m\) and \({\mathcal {B}}\) defined on \({\mathcal {X}}_m\cap {\mathcal {X}}^L\) attain their minimizers, where \(L>l\) and \({\mathcal {X}}^L=\{\gamma \in {\mathcal {X}}\mid {\mathcal {L}}[\gamma ]=L \}\). Moreover, the set \({\mathcal {X}}_m\) is strongly open, and hence such minimizers are local minimizers on \({\mathcal {X}}\) or \({\mathcal {X}}^L\), respectively.

B Uniqueness of minimizers for well-prepared boundary conditions

In this section we briefly show that the uniqueness of global minimizers is easily proved or disproved for some special parameters of the boundary condition, which possess well-prepared symmetry.

We observe that under the generic boundary angle condition (2.3), if circular arcs are admissible, then global minimizers are unique. For the extensible problem with any fixed \(\varepsilon >0\), the inequality \(X^2+Y^2\ge 2XY\) leads to

$$\begin{aligned} \varepsilon ^2\int _\gamma \kappa ^2 ds + \int _\gamma ds \ge 2\varepsilon \int _\gamma |\kappa | ds, \end{aligned}$$

where the equality is attained if and only if \(|\kappa |=1/\varepsilon \). In addition we easily observe that the right-hand side attains its minimum among admissible curves \(\gamma \in {\mathcal {A}}_{\theta _0,\theta _1,l}\) if and only if a convex curve of rotation angle \(|\theta _0|+|\theta _1|\) is admissible. Therefore, if a circular arc of radius \(1/\varepsilon \) is admissible, i.e., \(\theta _0=-\theta _1\) and \(l=2\cos \theta _0/\varepsilon \), then the circular arc of radius \(1/\varepsilon \) is a unique global minimizer. For the inextensible problem with fixed \(L>0\), the Cauchy–Schwarz inequality leads to

$$\begin{aligned} \int _\gamma \kappa ^2 ds \ge \frac{1}{L}\left( \int _\gamma |\kappa | ds\right) ^2, \end{aligned}$$

where the equality is attained if and only if \(|\kappa |\) is constant. Then a similar consideration implies that if a circular arc is admissible, i.e., \(\theta _0=-\theta _1\) and \(l=L\sin \theta _0/\theta _0\), then the circular arc of radius \(L/2\theta _0\) is a unique global minimizer.

We shall finally think of some particular critical cases. Both for the extensible problem with fixed \(\varepsilon >0\) and the inextensible problem with fixed \(L>0\), the most trivial case is that \((l,\theta _0,\theta _1)=(l,0,0)\) with \(l>0\); in this case the segment is a unique minimizer. If \((l,\theta _0,\theta _1)=(0,0,0)\) or \((l,|\theta _0|,|\theta _1|)=(0,\pi ,\pi )\), a consideration as in the above paragraph implies there are only two minimizers of suitable circle, thus being not unique in the strict sense but unique up to symmetry. In other critical cases, we are often able to ensure some nonuniqueness by a simple argument on symmetry, but up-to-symmetry uniqueness is a delicate issue (cf. [76]).

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Miura, T. Elastic curves and phase transitions. Math. Ann. 376, 1629–1674 (2020). https://doi.org/10.1007/s00208-019-01821-8

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