Existence, regularity and structure of confined elasticae

TL;DR

This paper investigates the properties of elastic curves confined within a domain, establishing existence, regularity, and structure of minimizers, including convexity in convex domains and examples with self-intersections.

Contribution

It proves the existence and regularity of minimizers of elastic energy under confinement, and characterizes their structure, especially convexity in convex sets.

Findings

Minimizers exist for the elastic energy problem in confined domains.

Minimizers are regular and possess specific structural properties.

In convex domains, minimizers are necessarily convex curves.

Abstract

We consider the problem of minimizing the bending or elastic energy among Jordan curves confined in a given open set $\Omega$. We prove existence, regularity and some structural properties of minimizers. In particular, when $\Omega$ is convex we show that a minimizer is necessarily a convex curve. We also provide an example of a minimizer with self-intersections.