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

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