Extremal functions in Poincare-Sobolev inequalities for functions of bounded variation

If $\mathrm{\Omega }\subset {\mathbb{R}}^n$ is a smooth bounded domain and $q \in (0, \frac{n}{n-1})$ we consider the Poincare-Sobolev inequality $$c({\int }_{\mathrm{\Omega }}\text{abs}{u}^{\frac{n}{n-1}}{)}^{1-\frac{1}{n}}\le {\int }_{\mathrm{\Omega }}\text{abs}Du,$$ for every $u\in \mathrm{B}\mathrm{V}(\mathrm{\Omega })$ such that ${\int }_{\mathrm{\Omega }}\text{abs}{u}^{q-1}u=0$. We show that the sharp constant is achieved. We also consider the same inequality on an $n$–dimensional compact Riemannian manifold $M$. When $n\ge 3$ and the scalar curvature is positive at some point, then the sharp constant is achieved. In the case $n\ge 2$, we need the maximal scalar curvature to satisfy some strict inequality.