  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.
