Weak limit of homeomorphisms in $W^{1,n-1}$: invertibility and lower semicontinuity of energy

Let $\mathrm{\Omega }$, ${\mathrm{\Omega }}^{\mathrm{\prime }}\subset {\mathbb{R}}^n$ be bounded domains and let $f_m\text{\hspace{0.17em}⁣}:\mathrm{\Omega }\to {\mathrm{\Omega }}^{\mathrm{\prime }}$ 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 ${\sup }_m{\int }_{\mathrm{\Omega }}(\mathrm{\mid }D{f}_m{\mathrm{\mid }}^{n-1}+A(\mathrm{\mid }\text{cof}D{f}_m\mathrm{\mid })+\phi (J_f))<\mathrm{\infty }$, where $A$ and $\phi$ 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 $\phi$, we show that $f$ satisfies the (INV) condition of Conti and De Lellis, the Lusin (N) condition, and polyconvex energies are lower semicontinuous.