  Let $\Omega$, $\Omega'\subset\mathbb{R}^n$ be bounded domains and let
$f_m\colon\Omega\to\Omega'$ be a sequence of homeomorphisms with positive
Jacobians $J_{f_m} >0$ a.e. and prescribed Dirichlet boundary data. Let all
$f_m$ satisfy the Lusin (N) condition and $\sup_m
\int_{\Omega}(|Df_m|^{n-1}+A(|\text{cof} Df_m|)+\phi(J_f))<\infty$, where $A$
and $\varphi$ are positive convex functions. Let $f$ be a weak limit of $f_m$
in $W^{1,n-1}$. Provided certain growth behaviour of $A$ and $\varphi$, we show
that $f$ satisfies the (INV) condition of Conti and De Lellis, the Lusin (N)
condition, and polyconvex energies are lower semicontinuous.
