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arXiv · 2508.18894

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

Cyrill B. Muratov·Matteo Novaga·Theresa M. Simon
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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 $Γ$-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.

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Cyrill B. Muratov, Matteo Novaga, Theresa M. Simon. 2025-08-26. Euler's elastica functional as a large mass limit of a two-dimensional non-local isoperimetric problem. https://arxiv.org/abs/2508.18894

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