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

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

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