Nonlocal-to-local variational limits of energies on partitions

Abstract

We study, through a Γ-convergence procedure, the asymptotic behaviour of a general class of nonlocal energies defined on functions taking values in a finite set Y={y₁,…,y_N}⊂ℝ^m. These energies are written in terms of double integrals accounting for pairwise interactions in an open set Ω, weighted by a kernel K_ε. If the latter is controlled from above and below by suitable non-negative radial kernels with unit mass concentrating at the origin, then, up to subsequences, the Γ-limit has domain BV(Ω;Y) and is given by a local surface energy. We show that the locality of the Γ-limit is not guaranteed if the concentration of mass fails. For a class of reference energies, we obtain a more explicit representation of the limit energy, also for non-radial kernels, recovering the limits of isotropic and anisotropic fractional perimeters. We also prove a homogenization result for periodic kernels.

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BibTeX RIS

BibTeX

@misc{alicandro2026nonlocaltolocal,
  title = {{Nonlocal-to-local variational limits of energies on partitions}},
  author = {Roberto Alicandro and Giuliana Fusco and Chiara Leone},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.23389},
  eprint = {2610.23389},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.23389},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - Nonlocal-to-local variational limits of energies on partitions
AU  - Roberto Alicandro
AU  - Giuliana Fusco
AU  - Chiara Leone
PY  - 2026
DA  - 2026/10/16
PB  - arXiv
DO  - 10.48550/arxiv.2610.23389
UR  - https://arxiv.org/abs/2610.23389
AB  - We study, through a Γ-convergence procedure, the asymptotic behaviour of a general class of nonlocal energies defined on functions taking values in a finite set Y={y_1,…,y_N}⊂ℝ^m. These energies are written in terms of double integrals accounting for pairwise interactions in an open set Ω, weighted by a kernel K_ε. If the latter is controlled from above and below by suitable non-negative radial kernels with unit mass concentrating at the origin, then, up to subsequences, the Γ-limit has domain BV(Ω;Y) and is given by a local surface energy. We show that the locality of the Γ-limit is not guaranteed if the concentration of mass fails. For a class of reference energies, we obtain a more explicit representation of the limit energy, also for non-radial kernels, recovering the limits of isotropic and anisotropic fractional perimeters. We also prove a homogenization result for periodic kernels.
ER  -

Abstract

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