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arXiv · 2406.19834

A Bourgain-Brezis-Mironescu -type characterization for Sobolev differential forms

Ilmari Kangasniemi
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Abstract

Given a bounded domain $Ω\subset \mathbb{R}^n$, a result by Bourgain, Brezis, and Mironescu characterizes when a function $f \in L^p(Ω)$ is in the Sobolev space $W^{1,p}(Ω)$ based on the limiting behavior of its Besov seminorms. We prove a direct analogue of this result which characterizes when a differential $k$-form $ω\in L^p(\wedge^k T^* Ω)$ has a weak exterior derivative $dω\in L^p(\wedge^{k+1} T^* Ω)$, where the analogue of the Besov seminorm that our result uses is based on integration over simplices.

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Ilmari Kangasniemi. 2025-05-15. A Bourgain-Brezis-Mironescu -type characterization for Sobolev differential forms. https://arxiv.org/abs/2406.19834

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