Weak convergence of minors

Using calculus of variations techniques, it is shown that if u_n ∈ W^1,N(Ω;R^N) converge to a function u in L¹(Ω;R^N), where Ω is an open, bounded subset of R^N, if the sequence of all minors {M(𝛻u_n)} is equi-bounded in L¹, and if det𝛻u_n converge to a function f weakly in L¹(Ω), then f = det𝛻u     a.e. 𝜘 ∈ Ω.

This result was previously obtained by Giaquinta, Modica and Soucek using tools from Geometric Measure Theory, and it is sharp. In particular, for all q ≥ 1, 1 ≤ p < N - 1, for all f ∈ L^q(Ω), and for every u ∈ W^1,p(Ω;R^N)

This work in collaboration with Jan Malý was initiated during his visit to the Max Planck Institute in March-April 1998.