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

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

2010-01-26

arxiv.org/abs/1001.4651v2

Nonlinear Elliptic Partial Differential Equations, Amer. Math. Soc., Contemporary Mathematics, No. 240, 2011, 47-58

Mathematics

Analysis of PDEs

12 pages, incorporated changes requested by the referee

Scientific paper

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.

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