Non-local planelike minimizers and $Γ$-convergence of periodic energies to a local anisotropic perimeter
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| Main Authors: | , , , |
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| Format: | Preprint |
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2026
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| _version_ | 1866909991905001472 |
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| author | Dipierro, Serena Novaga, Matteo Valdinoci, Enrico Villa, Riccardo |
| author_facet | Dipierro, Serena Novaga, Matteo Valdinoci, Enrico Villa, Riccardo |
| contents | We investigate a homogenization problem related to a non-local interface energy with a periodic forcing term. We show the existence of planelike minimizers for such energy. Moreover, we prove that, under suitable assumptions on the non-local kernel and the external field, the sequence of rescaled energies $Γ$-converges to a suitable local anisotropic perimeter, where the anisotropy is defined as the limit of the normalized energy of a planelike minimizer in larger and larger cubes (i.e., what is called in jargon "stable norm"). To obtain this, we also establish several auxiliary results, including: the minimality of the level sets of the minimizers, explicit bounds on the oscillations of the minimizers, density estimates for almost minimizers, and non-local perimeter estimates in the large. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_08677 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Non-local planelike minimizers and $Γ$-convergence of periodic energies to a local anisotropic perimeter Dipierro, Serena Novaga, Matteo Valdinoci, Enrico Villa, Riccardo Analysis of PDEs Dynamical Systems We investigate a homogenization problem related to a non-local interface energy with a periodic forcing term. We show the existence of planelike minimizers for such energy. Moreover, we prove that, under suitable assumptions on the non-local kernel and the external field, the sequence of rescaled energies $Γ$-converges to a suitable local anisotropic perimeter, where the anisotropy is defined as the limit of the normalized energy of a planelike minimizer in larger and larger cubes (i.e., what is called in jargon "stable norm"). To obtain this, we also establish several auxiliary results, including: the minimality of the level sets of the minimizers, explicit bounds on the oscillations of the minimizers, density estimates for almost minimizers, and non-local perimeter estimates in the large. |
| title | Non-local planelike minimizers and $Γ$-convergence of periodic energies to a local anisotropic perimeter |
| topic | Analysis of PDEs Dynamical Systems |
| url | https://arxiv.org/abs/2601.08677 |