Two-dimensional Jacobians det and Det for bounded variation functions and applications

The paper deals with the comparison in dimension two between the strong Jacobian determinant det and the weak (or distributional) Jacobian determinant Det. Restricting ourselves to dimension two, we extend the classical results of Ball and Muller as well as more recent ones to bounded variation vector-valued functions, providing a sufficient condition on a vector-valued U in (Omega)(2) such that the equality det(del)=Det(del) holds either in the distributional sense on Omega, or almost-everywhere in Omega when U is in (1,1)(Omega)(2). The key-assumption of the result is the regularity of the Jacobian matrix-valued del along the direction of a given non vanishing vector field is an element of (1)(Omega)(2), i.e. del is assumed either to belong to (0)(Omega)(2) with one of its coordinates in (1)(Omega), or to belong to (1)(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 (Omega)(2), assuming that the vector field b and one of the coordinates of del lie in a compact set of regular vector-valued functions.