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The periodic Wulff problem in two dimensions
Abstract
We characterize the minimizers of the anisotropic perimeter in the flat torus under an area constraint. We show that, depending on the area, the periodic global minimizer undergoes two phase transitions, changing from a droplet (a rescaled Wulff shape) to a flat band wrapping around the torus in a preferred direction determined by the anisotropy (lamellar phase), and finally to a bubble (the complement of a rescaled Wulff shape). For certain non-symmetric surface tensions, the lamellar phase is absent altogether; for other non regular surface tensions, there exist infinitely many minimizing configurations coexisting with the lamellar phase.
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BibTeX
@misc{bonacini2026periodic,
title = {{The periodic Wulff problem in two dimensions}},
author = {Marco Bonacini and Alberto Fiorini},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.06584},
eprint = {2609.06584},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.06584},
}
RIS
TY - UNPB
TI - The periodic Wulff problem in two dimensions
AU - Marco Bonacini
AU - Alberto Fiorini
PY - 2026
DA - 2026/09/06
PB - arXiv
DO - 10.48550/arxiv.2609.06584
UR - https://arxiv.org/abs/2609.06584
AB - We characterize the minimizers of the anisotropic perimeter in the flat torus under an area constraint. We show that, depending on the area, the periodic global minimizer undergoes two phase transitions, changing from a droplet (a rescaled Wulff shape) to a flat band wrapping around the torus in a preferred direction determined by the anisotropy (lamellar phase), and finally to a bubble (the complement of a rescaled Wulff shape). For certain non-symmetric surface tensions, the lamellar phase is absent altogether; for other non regular surface tensions, there exist infinitely many minimizing configurations coexisting with the lamellar phase.
ER -