Weak continuity and lower semicontinuity results for determinants

Weak continuity properties of minors and lower semicontinuity properties of functionals with polyconvex integrands are addressed in this paper. In particular, it is shown that if {u(n)} is bounded in W-1,W- N - 1 (Omega; R-N), {adj del u(n)} subset of L n/N - 1 (Omega; R-N x N), and if u is an element of BV (Omega; R-N) are such that u(n) -> u in L-1 (Omega; R-N) and det del u(n) ->* mu in the sense of measures, then for L-N a.e. x is an element of Omega, det del u (x) = d mu/dL(N) (x). This result is sharp, and counterexamples are provided in the cases where the regularity of {u(n)} or the type of weak convergence is weakened.