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

Abstract

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

arXiv comment12 pages, incorporated changes requested by the referee