Two-dimensional Jacobians \(\textrm{det}\hspace{0.56905pt}\) and \(\textrm{Det}\hspace{0.56905pt}\) for bounded variation functions and applications

The paper deals with the comparison in dimension two between the strong Jacobian determinant \(\textrm{det}\hspace{0.56905pt}\) $\text{det}$ and the weak (or distributional) Jacobian determinant \(\textrm{Det}\hspace{0.56905pt}\) $\text{Det}$ . Restricting ourselves to dimension two, we extend the classical results of Ball and Müller as well as more recent ones to bounded variation vector-valued functions, providing a sufficient condition on a vector-valued U in \(BV(\Omega )^2\) $BV{(\mathrm{\Omega })}^2$ such that the equality \(\textrm{det}\hspace{0.56905pt}(\nabla U)=\textrm{Det}\hspace{0.56905pt}(\nabla U)\) $\text{det}(\mathrm{\nabla }U)=\text{Det}(\mathrm{\nabla }U)$ holds either in the distributional sense on  \(\Omega \) $\mathrm{\Omega }$ , or almost-everywhere in \(\Omega \) $\mathrm{\Omega }$ when U is in \(W^{1,1}(\Omega )^2\) $W^{1,1}{(\mathrm{\Omega })}^2$ . The key-assumption of the result is the regularity of the Jacobian matrix-valued \(\nabla U\) $\mathrm{\nabla }U$ along the direction of a given non vanishing vector field \(b\in C^1(\Omega )^2\) $b\in C^1{(\mathrm{\Omega })}^2$ , i.e. \(\nabla U\, b\) $\mathrm{\nabla }Ub$ is assumed either to belong to \(C^0(\Omega )^2\) $C^0{(\mathrm{\Omega })}^2$ with one of its coordinates in \(C^1(\Omega )\) $C^1(\mathrm{\Omega })$ , or to belong to \(C^1(\Omega )^2\) $C^1{(\mathrm{\Omega })}^2$ . Two examples illustrate this new notion of two-dimensional distributional determinant. Finally, we prove the lower semicontinuity of a polyconvex energy defined for vector-valued functions U in \(BV(\Omega )^2\) $BV{(\mathrm{\Omega })}^2$ , assuming that the vector field b and one of the coordinates of \(\nabla U\, b\) $\mathrm{\nabla }Ub$ lie in a compact set of regular vector-valued functions.