Minimization to the Zhang's energy on $BV(\Omega)$ and sharp affine Poincar\'e-Sobolev inequalities

TL;DR

This paper establishes the existence of minimizers for certain variational problems involving Zhang's affine energy on BV spaces, leading to extremal functions for affine Poincaré-Sobolev inequalities, with implications for non-coercive functional analysis.

Contribution

It introduces new existence results for minimizers in BV spaces under affine energy constraints, advancing the understanding of affine inequalities and their extremals.

Findings

Existence of minimizers for constrained variational problems involving affine energy.

Existence of extremal functions for affine Poincaré-Sobolev inequalities.

Analysis of non-coercive geometry and weak* topology properties in BV spaces.

Abstract

We prove the existence of minimizers for some constrained variational problems on $BV(\Omega)$, under subcritical and critical restrictions, involving the affine energy introduced by Zhang in \cite{Z}. Related functionals have non-coercive geometry and properties like semicontinuity and affine compactness are deeper in the weak* topology. As a by-product of the theory, extremal functions are shown to exist for various affine Poincar\'e-Sobolev type inequalities.