Sequential weak continuity of null Lagrangians at the boundary

We show weak* in measures on $\bar{\text{O}}$/ weak-$L^1$ sequential continuity of $u\mapsto f(x,\mathrm{\nabla }u):W^{1,p}(\text{O};{\mathbb{R}}^m)\to L^1(\text{O})$, where $f(x,\cdot )$ is a null Lagrangian for $x\in \text{O}$, it is a null Lagrangian at the boundary for $x\in \mathrm{\partial }\text{O}$ and $\mathrm{\mid }f(x,A)\mathrm{\mid }\le C(1+\mathrm{\mid }A{\mathrm{\mid }}^p)$. We also give a precise characterization of null Lagrangians at the boundary in arbitrary dimensions. Our results explain, for instance, why $u\mapsto \det\nabla u:W^{1,n}(\O;\R^n)\to L^1(\O)$ fails to be weakly continuous. Further, we state a new weak lower semicontinuity theorem for integrands depending on null Lagrangians at the boundary. The paper closes with an example indicating that a well-known result on higher integrability of determinant \cite{Mue89a} need not necessarily extend to our setting. The notion of quasiconvexity at the boundary due to J.M. Ball and J. Marsden is central to our analysis.