Let $n\geq 3$, $\Omega$, $\Omega'\subset\er^n$ be bounded domains and let $f_m\colon\Omega\to\Omega'$ be a sequence of homeomorphisms that satisfy the Lusin $(N)$ condition with prescribed Dirichlet boundary condition and either $\sup_m \int_{\Omega}(|Df_m|^{n-1}+A(\adj Df_m)+\phi(J_f))<\infty$ (where $A$ satisfies $\lim_{t\to\infty}\frac{A(t)}{t}=\infty$) or $\sup_m \int_{\Omega}(|Df_m|^{n-1}+1/(J_f)^a)<\infty$ for $a=(n-1)/(n^2-3n+1)$.
Let $f$ be a weak limit of $f_m$ in $W^{1,n-1}$. We show that $f$ satisfies the $\INV$ condition of Conti and De Lellis and therefore it satisfies the Lusin $(N)$ condition and the polyconvex energy functional is lower semicontinuous.
This is a joint work with A. Dole\v{z}alov\'{a}, J. Mal\'y and A. Molchanova.