Optimal condition for the weak convergence of the jacobian determinant

Question

Whenever $n<q,$ it is known that given a sequence $\{ u_{k} \}$ which is weakly convergent in $W^{1,q}(U)$ one has that the Jacobian determinants $\text{det} Du_{k}$ converge weakly in $L^{q/n}(U).$

Together with polyconvexity, this serves to prove weak lower semicontinuity of some functionals whose lagrangians fail to be convex.

Motivated by this fact (proving weak lower semicontinuity of some functionals) I wish to find out whether the above result can be improved. In other words, can one have $q<n$ and still have weak convergence of the jacobians? I know one cannot lower $q$ too much, but still one would expect to be able to lower it a bit.

The reason I think this might not be true is that in the proof one crucially needs Morrey's inequality, which of course only holds when $n<q,$ but hopefully there is some way around this difficulty (perhaps by imposing an extra condition on the matrix of cofactors?)

Thank you for your time

Answer 1

If $q=n,$ then $\operatorname*{det}\nabla u \in L^1,$ but we have only convergence in distributions, i.e $$\operatorname*{det} \nabla u_{k} \rightharpoonup \operatorname*{det} \nabla u \quad \text{ in } \mathcal{D}'(U). $$ but not in $L^1,$ i.e $$\operatorname*{det} \nabla u_{k} \not\rightharpoonup \operatorname*{det} \nabla u \quad \text{ in } L^1(U). $$ If $q< n$, then $\operatorname*{det}\nabla u$ is not defined as a 'function' in $L^1$, thus pointwise Jacobian determinant does not make sense. However, it is possible to define weak or distributional jacobian, called $\operatorname*{Det} \nabla u ,$ when $u\in W^{1, \frac{n^2}{n+1}}.$ However, the weak continuity property, i.e $$ \operatorname*{Det} \nabla u_{k} \rightharpoonup \operatorname*{Det} \nabla u \quad \text{ in } \mathcal{D}'(U), $$ is only true if $q> \frac{n^2}{n+1}.$ If $q = \frac{n^2}{n+1},$ then it is possible to prove $$ \operatorname*{Det} \nabla u_{k} \rightharpoonup \operatorname*{Det} \nabla u + \mu\quad \text{ in } \mathcal{D}'(U), $$ where the distribution $\mu$ is non-zero in general.

See Dacorogna-Murat, "On the optimality of certain Sobolev exponents for the weak continuity of determinants" in Journal of Functional Analysis, 1992.