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

TL;DR

This paper proves the existence of extremal functions for sharp Poincare-Sobolev inequalities involving functions of bounded variation on smooth domains and certain Riemannian manifolds, under specific curvature conditions.

Contribution

It establishes the achievement of sharp constants in Poincare-Sobolev inequalities for BV functions on domains and manifolds, extending previous results to new geometric settings.

Findings

Sharp constants are achieved for BV functions on smooth domains.

Existence of extremals on manifolds depends on scalar curvature conditions.

Results extend classical inequalities to functions of bounded variation.

Abstract

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.