Concentration effects and $\Gamma$-limit for the elastica functional for open and closed curves

TL;DR

This paper analyzes the $ ext{Gamma}$-convergence of elastica energies on planar curves, revealing how curvature concentration leads to singularities and a simplified limiting energy depending on the number of concentration points.

Contribution

It characterizes the first-order $ ext{Gamma}$-limit for both open and closed curves, linking curvature concentration to atomic measures and energy contributions.

Findings

Curvature concentrates at finite points as $ ext{Epsilon} o 0^+$.

Limiting energy depends on the number of concentration points, each contributing multiples of $2 extpi$.

Rescaled energies resemble one-dimensional Modica--Mortola functionals.

Abstract

We study the $\Gamma$-convergence of a class of elastica-type energies defined on immersed planar curves and depending on a small positive parameter $\epsilon$. As $\epsilon\to 0^+$, sequences with equibounded energy develop concentration phenomena in the curvature, leading to the emergence of singularities described by atomic measures. This naturally gives rise to a limiting framework in terms of pointed curves, consisting of a curve together with a measure encoding curvature concentration. We characterize the first-order $\Gamma$-limit in two settings: for immersed open curves with fixed endpoints and boundary conditions on the tangents, and for immersed closed curves of prescribed length. In both cases, the limiting energy depends only on the number of concentration points and is expressed as a sum of contributions, each given by an integer multiple of $2\pi$. A key feature of the problem is that the rescaled energies exhibit a structure closely related to one-dimensional Modica--Mortola type functionals.