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

TL;DR

This paper investigates Poincaré trace inequalities for BV functions in the unit ball, identifying extremal functions under nonstandard normalization conditions and characterizing sharp constants via isoperimetric inequalities.

Contribution

It introduces new extremal functions for Poincaré trace inequalities with nonstandard mean or median constraints and links sharp constants to isoperimetric inequalities in arbitrary domains.

Findings

Extremal functions depend on the imposed normalization constraint.

Unusually shaped extremal functions arise under median constraints.

Sharp constants are characterized via isoperimetric inequalities.

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 $\Omega$.