If $\Omega \subset \R^n$ is a smooth bounded domain and $q \in (0, \frac{ n}{ n-1})$ we consider the Poincare-Sobolev inequality \[ c \Bigl(\int_{ \Omega} \abs{ u}^\frac{ n}{ n-1}\Bigr)^{ 1-\frac{ 1}{ n}} \le \int_{ \Omega} \abs{ Du}, \] for every $u \in \mathrm{ BV}(\Omega)$ such that $\int_{ \Omega} \abs{ u}^{ q-1} u = 0$. 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.
Categories: math.AP