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.2010 Mathematics Subject Classification. 49Q10, and 53A04.

Some minimization problems for planar networks of elastic curves

“…If τ 0 and τ 1 are in the opposite half-plane with respect to the line that pass through the points P 0 and P 1 we can call this problem the minimal elastic lens problem (see Figure 2). There are also other possible choices, motivated by the study of boundary value problems, see for instance [9,10,[16][17][18]20]. Recently also the associated obstacle problem has been studied, see [4] and [19].…”

Asymmetric equilibria of two nested elastic rings

“…Further, beyond the elastic regime, thin sheets can develop intricate ridge networks when folded and crumpled [1]. More recently, the problem of confined elastic curves has gained interest amongst mathematicians [9,15,18,7] and control engineers [2].…”