Sequential weak continuity of null Lagrangians at the boundary

We show weak* in measures on (Omega) over bar /weak-L-1 sequential continuity of u bar right arrow f (x, del u) : W-1,W- p(Omega; R-m) -> L-1(Omega), where f (x, center dot) is a null Lagrangian for x is an element of Omega, it is a null Lagrangian at the boundary for x is an element of partial derivative Omega and vertical bar f (x, A)vertical bar = C(1 + vertical bar A vertical bar(p)). We also give a precise characterization of null Lagrangians at the boundary in arbitrary dimensions. Our results explain, for instance, why u bar right arrow det del u : W-1,W- n(Omega; R-n) -> L-1(Omega) 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 by Muller (Bull. Am. Math. Soc. New Ser. 21(2): 245-248, 1989) 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.