Euler's elastica functional as a large mass limit of a two-dimensional non-local isoperimetric problem
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| Format: | Preprint |
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2025
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| _version_ | 1866911121715232768 |
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| author | Muratov, Cyrill B. Novaga, Matteo Simon, Theresa M. |
| author_facet | Muratov, Cyrill B. Novaga, Matteo Simon, Theresa M. |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2508_18894 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Euler's elastica functional as a large mass limit of a two-dimensional non-local isoperimetric problem Muratov, Cyrill B. Novaga, Matteo Simon, Theresa M. Analysis of PDEs 49Q10, 49Q20, 49S05 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. |
| title | Euler's elastica functional as a large mass limit of a two-dimensional non-local isoperimetric problem |
| topic | Analysis of PDEs 49Q10, 49Q20, 49S05 |
| url | https://arxiv.org/abs/2508.18894 |