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

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.