Euler's elastica functional as a large mass limit of a two-dimensional non-local isoperimetric problem

TL;DR

This paper studies the large mass limit of a non-local isoperimetric problem with Yukawa potential in 2D, showing the energy converges to a sum involving perimeter and Euler's elastica functional, with minimizers being disks or annuli.

Contribution

It establishes the $ ext{Gamma}$-limit of the energy as a weighted sum of perimeter and elastica functional, linking non-local isoperimetric problems to elastica theory.

Findings

Energy minimizers converge to disks or annuli.

The $ ext{Gamma}$-limit involves perimeter and elastica functional.

Existence of minimizers in the critical regime.

Abstract

We consider a large mass limit of the non-local isoperimetric problem with a repulsive Yukawa potential in two space dimensions. In this limit, the non-local term concentrates on the boundary, resulting in the existence of a critical regime in which the perimeter and the non-local terms cancel each other out to leading order. We show that under appropriate scaling assumptions the next-order $\Gamma$-limit of the energy with respect to the $L^1$ convergence of the rescaled sets is given by a weighted sum of the perimeter and Euler's elastica functional, where the latter is understood via the lower-semicontinuous relaxation and is evaluated on the system of boundary curves. As a consequence, we prove that in the considered regime the energy minimizers always exist and converge to either disks or annuli, depending on the relative strength of the elastica term.