Let $n\in\mathbb N$, let $U\subset\mathbb R^n$ be open, and let $(u_k)_{k=1}^{\infty}$ be a sequence in $W^{1,n}(U;\mathbb R^n)$. Suppose that $u\in W^{1,n}(U;\mathbb R^n)$ and $u_k\rightharpoonup u$ in $W^{1,n}(U;\mathbb R^n)$. For each $k\in\mathbb N$, write $\nabla u_k:U\to\mathbb R^{n\times n}$ for the weak Jacobian matrix of $u_k$, and write $\nabla u:U\to\mathbb R^{n\times n}$ for the weak Jacobian matrix of $u$, represented by their almost-everywhere defined $L^n(U;\mathbb R^{n\times n})$ functions. Let $\operatorname{cof}:\mathbb R^{n\times n}\to\mathbb R^{n\times n}$ denote the cofactor map, with the convention that $A(\operatorname{cof}A)^\top=(\det A)I_n$ for every $A\in\mathbb R^{n\times n}$, and let $\det:\mathbb R^{n\times n}\to\mathbb R$ denote the determinant map. If $n>1$, then $\operatorname{cof}\nabla u_k\rightharpoonup \operatorname{cof}\nabla u$ in the local Lebesgue space with exponent

\begin{align*} \frac{n}{n-1} \end{align*}

that is, in the space

\begin{align*} L^{n/(n-1)}_{\mathrm{loc}}(U;\mathbb R^{n\times n}). \end{align*}

Moreover, $\det\nabla u_k\rightharpoonup \det\nabla u$ in the sense of distributions on $U$, meaning that for every $\phi\in C_c^\infty(U)$,

\begin{align*} \lim_{k\to\infty}\int_U \phi(x)\det\nabla u_k(x)\,d\mathcal L^n(x)=\int_U \phi(x)\det\nabla u(x)\,d\mathcal L^n(x). \end{align*}