Poincar\'e trace inequalities in $BV(\mathbb B^n)$ with nonstandard normalization

Abstract

Extremal functions are exhibited in Poincar\'e trace inequalities for functions of bounded variation in the unit ball ${\mathbb B}^n$ of the $n$-dimensional Euclidean space ${\mathbb R}^n$. Trial functions are subject to either a vanishing mean value condition, or a vanishing median condition in the whole of ${\mathbb B}^n$, instead of just on $\partial {\mathbb B}^n$, as customary. The extremals in question take a different form, depending on the constraint imposed. In particular, under the latter constraint, unusually shaped extremal functions appear. A key step in our approach is a characterization of the sharp constant in the relevant trace inequalities in any admissible domain $\Omega \subset {\mathbb R}^n$, in terms of an isoperimetric inequality for subsets of $\Omega$.