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

Abstract

In this short note we prove that, given a $\mathbb{C}$-elliptic operator $\mathcal{A}$ and a map $u\in \mathrm{BV}^{\mathcal{A}}(Ω;V)$ satisfying \( \nabla_{\mathrm{ap}}u\in L^1(Ω;V\otimes\mathbb{R}^d), \) then $u\in \mathrm{BV}(Ω;V)$. The result is quantitative and follows from the Korn-type estimate \[ |Du|(Ω) \leq C_{d,\mathcal{A}}\left( \|\nabla_{\mathrm{ap}}u\|_{\mathrm{L}^1(Ω)} + |\mathcal{A}u|(Ω) \right), \] valid for every such $u\in \mathrm{BV}^{\mathcal{A}}(Ω;V)$. As a consequence, we obtain the characterization \[ \mathrm{BV}^{\mathcal{A}}(Ω;V)\setminus \mathrm{BV}(Ω;V) = \left\{ u\in \mathrm{BV}^{\mathcal{A}}(Ω;V): \nabla_{\mathrm{ap}}u\notin L^1(Ω;V\otimes\mathbb{R}^d) \right\}. \] Generative AI has been exploited. The usage is detailed in a specific Section.