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arXiv · 2601.08677

Non-local planelike minimizers and $\Gamma$-convergence of periodic energies to a local anisotropic perimeter

Serena Dipierro·Matteo Novaga·Enrico Valdinoci·Riccardo Villa
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Abstract

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 $\Gamma$-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.

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Serena Dipierro, Matteo Novaga, Enrico Valdinoci, Riccardo Villa. 2026-01-13. Non-local planelike minimizers and $\Gamma$-convergence of periodic energies to a local anisotropic perimeter. https://arxiv.org/abs/2601.08677

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