The divergence of continuously differentiable vector fields
Abstract
For m≥ 2, we give a complete characterisation of the continuous functions on ℝm that are divergences of continuously differentiable vector fields. The characterisation is formulated in terms of distributions and, equivalently, in terms of the continuous dual of a suitably topologised subspace of the Arens--Eells space. The two implications yield contrasting applications. First, there exists a compactly supported continuous function which is the divergence of a bounded Lipschitz vector field but not of any continuously differentiable vector field. Second, every continuous function whose distributional partial derivatives are locally m-integrable (the critical exponent for Sobolev embedding) is the divergence of a continuously differentiable vector field.
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BibTeX
@misc{pauw2026divergence,
title = {{The divergence of continuously differentiable vector fields}},
author = {Thierry De Pauw},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.35452},
eprint = {2610.35452},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.35452},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - The divergence of continuously differentiable vector fields
AU - Thierry De Pauw
PY - 2026
DA - 2026/10/26
PB - arXiv
DO - 10.48550/arxiv.2610.35452
UR - https://arxiv.org/abs/2610.35452
AB - For m≥ 2, we give a complete characterisation of the continuous functions on ℝ^m that are divergences of continuously differentiable vector fields. The characterisation is formulated in terms of distributions and, equivalently, in terms of the continuous dual of a suitably topologised subspace of the Arens--Eells space. The two implications yield contrasting applications. First, there exists a compactly supported continuous function which is the divergence of a bounded Lipschitz vector field but not of any continuously differentiable vector field. Second, every continuous function whose distributional partial derivatives are locally m-integrable (the critical exponent for Sobolev embedding) is the divergence of a continuously differentiable vector field.
ER -