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Divergence-free concentrations come from vanishing sequences
Abstract
A vanishing sequence Vn of divergence-free matrix fields is one that is bounded in L1 and is carried by open sets An of vanishing volume. Recently established, Bouchitté's vanishing mass conjecture says that the directions such a sequence can carry are rigidly constrained: their limiting distribution must be a superposition of microstructures whose barycenters are singular matrices. We prove the converse in the non-symmetric setting: every such superposition is attained by a vanishing sequence. In fact, we are able to construct divergence-free fields Vn $supported$ on An, whose relative boundary is a smooth compact manifold.
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BibTeX
@misc{arroyorabasa2026divergencefree,
title = {{Divergence-free concentrations come from vanishing sequences}},
author = {Adolfo Arroyo-Rabasa and Francesco Nobili and Ivan Yuri Violo},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.24767},
eprint = {2609.24767},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.24767},
}
RIS
TY - UNPB
TI - Divergence-free concentrations come from vanishing sequences
AU - Adolfo Arroyo-Rabasa
AU - Francesco Nobili
AU - Ivan Yuri Violo
PY - 2026
DA - 2026/09/21
PB - arXiv
DO - 10.48550/arxiv.2609.24767
UR - https://arxiv.org/abs/2609.24767
AB - A vanishing sequence V_n of divergence-free matrix fields is one that is bounded in L^1 and is carried by open sets A_n of vanishing volume. Recently established, Bouchitté's vanishing mass conjecture says that the directions such a sequence can carry are rigidly constrained: their limiting distribution must be a superposition of microstructures whose barycenters are singular matrices. We prove the converse in the non-symmetric setting: every such superposition is attained by a vanishing sequence. In fact, we are able to construct divergence-free fields V_n $supported$ on A_n, whose relative boundary is a smooth compact manifold.
ER -