%0 Journal Article
%T WEAK CONVERGENCE OF JACOBIAN DETERMINANTS Weak convergence of Jacobian determinants under asymmetric assumptions
%A Teresa Alberico
%A Costantino Capozzoli
%J Le Matematiche
%D 2012
%I University of Catania
%X 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 and if one allows different assumptions on the two components of $f_k$ and $f$, e.g. $$ u_k ightharpoonup u ;;mbox{weakly in} ;; W^{1,2}(Omega) qquad , v_k ightharpoonup v ;;mbox{weakly in} ;; W^{1,q}(Omega) $$ for some $qin(1,2)$, then egin{equation}label{0} dmu=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 $$ u_k ightharpoonup u ;;mbox{weakly in} ;; W^{1,L^2 log^alpha L}(Omega)$$ 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 $W^{1,2}(Om,R^2)$. 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. $$ u_k ightharpoonup u ;;mbox{weakly in} ;; W^{1,2}(Om) qquad , v_k ightharpoonup v ;;mbox{weakly in} ;; W^{1,q}(Om) $$ for some $qin(1,2)$, then egin{equation}label{0} dmu=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 $$ u_k ightharpoonup u ;;mbox{weakly in} ;; W^{1,L^2 log^alpha L}(Om)$$ for some $alpha >1$.
%K Convergence in the sense of measures
%K Jacobian determinant
%K distributional Jacobian determinant
%K Orlicz-Sobolev spaces
%U http://www.dmi.unict.it/ojs/index.php/lematematiche/article/view/769