An approximation of the Jacobian determinant (Q2467899)

The authors consider a mapping \(u:\Omega\subset\mathbb{R}^n\to\mathbb{R}^n\), \(n\geq 2\) and then, by denoting the differential of \(u\) by \(Du\), the Jacobian determinant of \(Du\) by \(J_u=\det Du\) and the matrix of cofactors of \(Du\) by \(D^{\#}u\), they prove that there exists a sequence \(\{u_h\}\) of mappings in the Sobolev space \(W^{1,n}(B;\mathbb{R}^n)\), where \(B\) denotes the unit ball in \(\mathbb{R}^n\), such that \(u_h\to u\) in \(W^{1,n-1}(B;\mathbb{R}^n)\) and \(J_{u_h}\to J_u\) in \({\mathcal L}^1(B)\), whenever \(D^{\#}u\in{\mathcal L}^{\frac{n}{n-1}}(B;\mathbb R^{n\times n})\). This means that \(Du\) need not be \({\mathcal L}^n\)-integrable and yet an approximation of the Jacobian can be found.