Sequential weak continuity of null Lagrangians at the boundary

Abstract

We show weak* in measures on $\barØ$/ weak-$L^1$ sequential continuity of $u\mapsto f(x,\nabla u):W^{1,p}(Ø;\R^m)\to L^1(Ø)$, where $f(x,\cdot)$ is a null Lagrangian for $x\inØ$, it is a null Lagrangian at the boundary for $x\in\partialØ$ and $|f(x,A)|\le C(1+|A|^p)$. We also give a precise characterization of null Lagrangians at the boundary in arbitrary dimensions. Our results explain, for instance, why $u\mapsto \det\nabla u:W^{1,n}(Ø;\R^n)\to L^1(Ø)$ 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.