WEAK CONVERGENCE OF JACOBIAN DETERMINANTS

Let $\Om$ be a bounded open set in $\R^2$ sufficiently smooth and $f_k=(u_k,v_k)$ and $f=(u,v)$ mappings belong to the Sobolev space $W^{1,2}(\Om,\R^2)$. We prove that if the sequence of Jacobians $J_{f_k}$ converges to a measure $\mu$ in sense of measures andif one allows different assumptions on the two components of $f_k$ and $f$, e.g.$$u_k \rightharpoonup u \;\;\mbox{weakly in} \;\; W^{1,2}(\Om) \qquad \, v_k \rightharpoonup v \;\;\mbox{weakly in} \;\; W^{1,q}(\Om)$$for some $q\in(1,2)$, then\begin{equation}\label{0}d\mu=J_f\,dz.\end{equation}Moreover, we show that this result is optimal in the sense that conclusion fails for $q=1$.On the other hand, we prove that \eqref{0} remains valid also if one considers the case $q=1$, but it is necessary to require that $u_k$ weakly converges to $u$ in a Zygmund-Sobolev space with a slightly higher degree of regularity than $W^{1,2}(\Om)$ and precisely$$ u_k \rightharpoonup u \;\;\mbox{weakly in} \;\; W^{1,L^2 \log^\alpha L}(\Om)$$for some $\alpha >1$. Let $\Om$ be a bounded open set in $\R^2$ sufficiently smooth and$f_k=(u_k,v_k)$ and $f=(u,v)$ mappings belong to the Sobolev space $W^{1,2}(\Omega, \R^2)$. We prove that if the sequence of Jacobians $J_{f_k}$ converges to a measure $\mu$ in sense of measures andif one allows different assumptions on the two components of $f_k$ and $f$, e.g.$$u_k \rightharpoonup u \;\;\mbox{weakly in} \;\; W^{1,2}(\Omega) \qquad \, v_k \rightharpoonup v \;\;\mbox{weakly in} \;\; W^{1,q}(\Omega)$$for some $q\in(1,2)$, then\begin{equation}\label{0}d\mu=J_f\,dz.\end{equation}Moreover, we show that this result is optimal in the sense that conclusion fails for $q=1$.On the other hand, we prove that \eqref{0} remains valid also if one considers the case $q=1$, but it is necessary to require that $u_k$ weakly converges to $u$ in a Zygmund-Sobolev space with a slightly higher degree of regularity than $W^{1,2}(\Omega)$ and precisely$$u_k \rightharpoonup u \;\;\mbox{weakly in} \;\; W^{1,L^2 \log^\alpha L}(\Omega)$$for some $\alpha > 1$.