Elastic curves and phase transitions

TL;DR

This paper studies elastic curves with fixed endpoints, exploring a straightening limit and establishing new uniqueness results without symmetry assumptions, using a novel singular perturbation approach related to phase transition energies.

Contribution

It introduces a foundational singular perturbation theory for elastic energy and proves a first uniqueness result for least energy solutions without symmetry.

Findings

Established a straightening limit for elastic curves.

Developed a singular perturbation theory for the energy.

Proved a uniqueness result for solutions without symmetry.

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.