Preprint
Inserted: 11 sep 2026
Last Updated: 11 sep 2026
Year: 2026
Abstract:
In this short note we prove that, given a $\mathbb{C}$-elliptic operator $\mathcal{A}$ and a map
$u\in BV^{\mathcal{A}}(\Omega;V)$ satisfying
\(
\nabla_{\mathrm{ap}}u\in
L^1(\Omega;V\otimes\mathbb{R}^d),
\)
then $u\in BV(\Omega;V)$. The result is quantitative and follows from
the Korn-type estimate
\[
Du
(\Omega)
\leq
C_{d,\mathcal{A}}\left(
\
\nabla_{\mathrm{ap}}u\
_{\mathrm{L}^1(\Omega)}
+
\mathcal{A} u
(\Omega)
\right),
\]
valid for every $u\in BV^\mathcal{A}(\Omega;V)$. As a consequence, we obtain the characterization
\[
BV^\mathcal{A}(\Omega;V)\setminus BV(\Omega;V)
=
\left\{
u\in BV^\mathcal{A}(\Omega;V):
\nabla_{\mathrm{ap}}u\notin
L^1(\Omega;V\otimes\mathbb{R}^d)
\right\}.
\]
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