On the shape of minimizers for the periodic nonlocal perimeter in $\mathbb{R}^2$
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866911459345170432 |
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| author | Bruera, Renzo |
| author_facet | Bruera, Renzo |
| contents | In this paper, we study planar nonlocal Delaunay sets. That is, open sets in $\mathbb{R}^2$ with constant nonlocal mean curvature that are periodic in $x_1$, and even in $x_1$ and in $x_2$. Using bifurcation analysis and fine explicit computations, we prove that every sufficiently $C^{1,β}$-flat nonlocal Delaunay set in $\mathbb{R}^2$ that is not a straight band is unstable with respect to volume-preserving periodic variations.
Our results support the conjecture that, as in the local case, in the range of large areas, minimizers of the periodic nonlocal isoperimetric problem -- also known as the nonlocal liquid drop problem with prescribed area between two parallel hyperplanes -- are all straight bands. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_18215 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On the shape of minimizers for the periodic nonlocal perimeter in $\mathbb{R}^2$ Bruera, Renzo Analysis of PDEs In this paper, we study planar nonlocal Delaunay sets. That is, open sets in $\mathbb{R}^2$ with constant nonlocal mean curvature that are periodic in $x_1$, and even in $x_1$ and in $x_2$. Using bifurcation analysis and fine explicit computations, we prove that every sufficiently $C^{1,β}$-flat nonlocal Delaunay set in $\mathbb{R}^2$ that is not a straight band is unstable with respect to volume-preserving periodic variations. Our results support the conjecture that, as in the local case, in the range of large areas, minimizers of the periodic nonlocal isoperimetric problem -- also known as the nonlocal liquid drop problem with prescribed area between two parallel hyperplanes -- are all straight bands. |
| title | On the shape of minimizers for the periodic nonlocal perimeter in $\mathbb{R}^2$ |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2602.18215 |