On determining a mapping from its normalized Jacobian (Q1204767)

This article deals with the special equation \[ f'(x)= K(x) (\text{det } J(x,f))^{1/n} \] with respect to unknown \(f(x): {\mathcal D}\to \mathbb{R}^ n\) from \(\mathbb{W}^ 1_{\text{loc}}({\mathcal D})\) (\({\mathcal D}\) is a domain in \(\mathbb{R}^ n\)), \(J(\cdot, f)\) is the Jacobi matrix of \(f(\cdot)\); \(K(x)\) is a known mapping. The article presents some condition for \(K(x)\) that is necessary for solvability of this equation and under some additional assumptions turns out to be sufficient.