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WEAK CONVERGENCE OF JACOBIAN DETERMINANTS Weak convergence of Jacobian determinants under asymmetric assumptions

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Author(s): Teresa Alberico | Costantino Capozzoli

Journal: Le Matematiche
ISSN 0373-3505

Volume: 67;
Issue: 1;
Start page: 3;
Date: 2012;
Original page

Keywords: Convergence in the sense of measures | Jacobian determinant | distributional Jacobian determinant | Orlicz-Sobolev spaces

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 $W^{1,2}(Omega, R^2)$. 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.$$u_k ightharpoonup u ;;mbox{weakly in} ;; W^{1,2}(Omega) qquad , v_k ightharpoonup v ;;mbox{weakly in} ;; W^{1,q}(Omega)$$for some $qin(1,2)$, thenegin{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 andif 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 $qin(1,2)$, thenegin{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 $alpha >1$.    
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