From $\mathrm{BV}^\mathcal A$ to $\mathrm{BV}$: An endpoint Korn estimate

RESUMO

In this short note we prove that, given a $\mathbb{C}$-elliptic operator $\mathcal{A}$ and a map $u\in \mathrm{BV}^{\mathcal{A}}(\Omega;V)$ satisfying \( \nabla_{\mathrm{ap}}u\in L^1(\Omega;V\otimes\mathbb{R}^d), \) then $u\in \mathrm{BV}(\Omega;V)$. 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 such $u\in \mathrm{BV}^{\mathcal{A}}(\Omega;V)$. As a consequence, we obtain the characterization \[ \mathrm{BV}^{\mathcal{A}}(\Omega;V)\setminus \mathrm{BV}(\Omega;V) = \left\{ u\in \mathrm{BV}^{\mathcal{A}}(\Omega;V): \nabla_{\mathrm{ap}}u\notin L^1(\Omega;V\otimes\mathbb{R}^d) \right\}. \] Generative AI has been exploited. The usage is detailed in a specific Section.