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A Halfspace Theorem for Global Anisotropic Perimeter Minimizers
Abstract
We prove an arbitrary-dimensional halfspace theorem for global minimizers of a smooth uniformly elliptic anisotropic perimeter: if the nonempty boundary of a minimizing set is contained in a halfspace, then the set itself is a halfspace. No evenness of the integrand or regularity of the minimizing boundary is assumed. The proof combines wall contact with a plane-peeling argument for the complementary phase defect and a simultaneous second blow-down. We also give a direct, self-contained exterior-barrier proof of the two-dimensional stationary anisotropic statement. Its rigidity conclusion is already covered by Bergner's earlier halfspace theorem; the proof is retained because it works directly with the anisotropic Euler-Lagrange operator.
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BibTeX
@misc{yang2026halfspace,
title = {{A Halfspace Theorem for Global Anisotropic Perimeter Minimizers}},
author = {Yang Yang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.03916},
eprint = {2609.03916},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.03916},
}
RIS
TY - UNPB
TI - A Halfspace Theorem for Global Anisotropic Perimeter Minimizers
AU - Yang Yang
PY - 2026
DA - 2026/09/04
PB - arXiv
DO - 10.48550/arxiv.2609.03916
UR - https://arxiv.org/abs/2609.03916
AB - We prove an arbitrary-dimensional halfspace theorem for global minimizers of a smooth uniformly elliptic anisotropic perimeter: if the nonempty boundary of a minimizing set is contained in a halfspace, then the set itself is a halfspace. No evenness of the integrand or regularity of the minimizing boundary is assumed. The proof combines wall contact with a plane-peeling argument for the complementary phase defect and a simultaneous second blow-down. We also give a direct, self-contained exterior-barrier proof of the two-dimensional stationary anisotropic statement. Its rigidity conclusion is already covered by Bergner's earlier halfspace theorem; the proof is retained because it works directly with the anisotropic Euler-Lagrange operator.
ER -