Weak convergence of Jacobian determinants under asymmetric assumptions

Abstract

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 and
if 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 $\alpha >1$.