Balls minimize trace constants in BV

TL;DR

This paper proves that among all Euclidean domains, balls have the smallest optimal constants in Poincaré type boundary trace inequalities for BV functions with zero median or mean.

Contribution

It establishes the minimality of balls for trace constants in BV spaces, extending understanding of geometric optimization in functional inequalities.

Findings

Balls minimize trace constants among Euclidean domains.

Optimal constants are achieved by balls in Poincaré type inequalities.

Results apply to BV functions with zero median or mean value.

Abstract

Balls are shown to have the smallest optimal constant, among all admissible Euclidean domains, in Poincar\'e type boundary trace inequalities for functions of bounded variation with vanishing median or mean value.