Let $m,n\in\mathbb N$, let $U\subset\mathbb R^n$ be a bounded Lipschitz [open set](/page/Open%20Set), let $1<p<\infty$, and let $\mathcal A\subset W^{1,p}(U;\mathbb R^m)$ be sequentially weakly closed. Let $S\subset \{1,\dots,\min\{m,n\}\}$ be a finite set. For each $k\in S$, let $N_k$ be the number of $k\times k$ minors of an $m\times n$ matrix, and let $M_k:\mathbb R^{m\times n}\to\mathbb R^{N_k}$ denote the map whose components are all $k\times k$ minors. Define $M_S:\mathbb R^{m\times n}\to\prod_{k\in S}\mathbb R^{N_k}$ by $M_S(A)=(M_k(A))_{k\in S}$. Assume that $k\le p$ for every $k\in S$.

Let $G:U\times\mathbb R^m\times\prod_{k\in S}\mathbb R^{N_k}\to\mathbb R\cup\{+\infty\}$ be a normal integrand. Assume that, for $\mathcal L^n$-a.e. $x\in U$, the map $(y,z)\mapsto G(x,y,z)$ is lower semicontinuous on $\mathbb R^m\times\prod_{k\in S}\mathbb R^{N_k}$ and, for each fixed $y\in\mathbb R^m$, the map $z\mapsto G(x,y,z)$ is convex. Define $W:U\times\mathbb R^m\times\mathbb R^{m\times n}\to\mathbb R\cup\{+\infty\}$ by $W(x,y,A)=G(x,y,M_S(A))$. Define $I:\mathcal A\to\mathbb R\cup\{+\infty\}$ by

\begin{align*} I[u]=\int_U W(x,u(x),\nabla u(x))\,d\mathcal L^n(x). \end{align*}

Suppose there exist $c>0$ and $a\in L^1(U)$ such that, for $\mathcal L^n$-a.e. $x\in U$ and every $(y,A)\in\mathbb R^m\times\mathbb R^{m\times n}$,

\begin{align*} W(x,y,A)\ge c|A|^p-a(x). \end{align*}

Finally, assume that whenever $(u_j)_{j=1}^{\infty}$ is a sequence in $\mathcal A$ such that $u_j\rightharpoonup u$ in $W^{1,p}(U;\mathbb R^m)$ and $\sup_j I[u_j]<\infty$, the family $(M_k(\nabla u_j))_{j=1}^{\infty}$ is uniformly integrable in $L^1(U;\mathbb R^{N_k})$ for every $k\in S$ with $k=p$.

Then $I$ is sequentially weakly lower semicontinuous on $\mathcal A$.