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Elastic curves and phase transitions

  • Tatsuya MiuraEmail author
Article
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Abstract

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.

Mathematics Subject Classification

49Q10 53A04 

Notes

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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Authors and Affiliations

  1. 1.Graduate School of Mathematical SciencesThe University of TokyoTokyoJapan
  2. 2.Department of MathematicsTokyo Institute of TechnologyTokyoJapan

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