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

TL;DR

This paper extends a classical characterization of Sobolev functions to differential forms, using simplex-based seminorms to identify when a form has a weak exterior derivative in $L^p$ spaces.

Contribution

It provides a new Bourgain-Brezis-Mironescu-type characterization for Sobolev differential forms, linking the behavior of simplex-based seminorms to the existence of weak exterior derivatives.

Findings

Characterization of Sobolev differential forms via simplex-based seminorms.

Extension of Bourgain-Brezis-Mironescu results to differential forms.

Identification of conditions for weak exterior derivatives in $L^p$ spaces.

Abstract

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