Elastic curves and phase transitions

Abstract

This paper is devoted to classical variational problems 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.

40 pages, 14 figures

References in corpus (4)

• Singular perturbation by bending for an adhesive obstacle problem
• Phase field models for thin elastic structures with topological constraint
• The elastic trefoil is the twice covered circle
• Overhanging of membranes and filaments adhering to periodic graph substrates

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• A new energy method for shortening and straightening complete curves
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