Lattice tilings minimizing nonlocal perimeters
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2023
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| _version_ | 1866910571317690368 |
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| author | Cesaroni, Annalisa Fragalà, Ilaria Novaga, Matteo |
| author_facet | Cesaroni, Annalisa Fragalà, Ilaria Novaga, Matteo |
| contents | We prove the existence of periodic tessellations of $\mathbb{R}^N$ minimizing a general nonlocal perimeter functional, defined as the interaction between a set and its complement through a nonnegative kernel, which we assume to be either integrable at the origin, or singular, with a fractional type singularity. We reformulate the optimal partition problem as an isoperimetric problem among fundamental domains associated with discrete subgroups of $\mathbb{R}^N$ , and we provide the existence of a solution by using suitable concentrated compactness type arguments and compactness results for lattices. Finally, we discuss the possible optimality of the hexagonal tessellation in the planar case. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_01054 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Lattice tilings minimizing nonlocal perimeters Cesaroni, Annalisa Fragalà, Ilaria Novaga, Matteo Analysis of PDEs 52C07, 52C22, 53A10, 58E12 We prove the existence of periodic tessellations of $\mathbb{R}^N$ minimizing a general nonlocal perimeter functional, defined as the interaction between a set and its complement through a nonnegative kernel, which we assume to be either integrable at the origin, or singular, with a fractional type singularity. We reformulate the optimal partition problem as an isoperimetric problem among fundamental domains associated with discrete subgroups of $\mathbb{R}^N$ , and we provide the existence of a solution by using suitable concentrated compactness type arguments and compactness results for lattices. Finally, we discuss the possible optimality of the hexagonal tessellation in the planar case. |
| title | Lattice tilings minimizing nonlocal perimeters |
| topic | Analysis of PDEs 52C07, 52C22, 53A10, 58E12 |
| url | https://arxiv.org/abs/2310.01054 |