Disseminating quality controlled scientific knowledge

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

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\$.