Asymmetric affine Poincaré–Sobolev–Wirtinger inequalities on BV(Ω) and characterization of extremizers in one-dimension

The present work deals with sharp asymmetric Poincaré–Sobolev–Wirtinger inequalities involving the Zhang’s energy on the space of bounded variation functions $BV\left(\Omega \right)$ for any bounded domain $\Omega \subset R^n$ in any dimension $n\ge 1$. We establish the existence of a curve of optimal constants along with several of its properties such as attainability, symmetry, monotonicity, positivity, continuity and also asymptotic ones. Moreover, for $n=1$, our approach allows to exhibit its precise shape and to characterize all extremizers.