Weak convergence of minors

Using calculus of variations techniques, it is shown that if $u_n\in w^{1,N}(\mathrm{\Omega };R^N)$ converge to a function $u$ in $L^1(\mathrm{\Omega };R^N)$, where $\mathrm{\Omega }$; is an open, bounded subset of $R^N$, if the sequence of all minors $\{M(\mathrm{\nabla }{u}_n)\}$ is equi-bounded in $L^1$, and if $det\mathrm{\nabla }{u}_n$ converge to a function $f$ weakly in $L^1(\mathrm{\Omega })$, then $$f=det\mathrm{\nabla }u$$ a.e. $$x\in \mathrm{\Omega }$$.This result was previously obtained by Giaquinta, Modica and Soucek using tools from Geometric Measure Theory, and it is sharp. In particular, for all $q\ge 1$, $1\le p<N-1$, for all $f\in L^q(\mathrm{\Omega })$, and for every $u\in W^{1,p}(\mathrm{\Omega };R^N)$ $$\underset{\{u_n\}}{inf}\{\underset{n\to 0}{lim inf}\int |det\mathrm{\nabla }{u}_n-f{|}^{q}dx:u_n\in W^{1,N}(\mathrm{\Omega };R^N),u_n\rightharpoonup uin{W}^{1,p}(\mathrm{\Omega };R^N)\}=0$$This work in collaboration with Jan Malý was initiated during his visit to the Max Planck Institute in March-April 1998.