Two-dimensional Jacobians det0.56905pt and Det0.56905pt for bounded variation functions and applications

The paper deals with the comparison in dimension two between the strong Jacobian determinant det0.56905pt det and the weak (or distributional) Jacobian determinant Det0.56905pt 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(Ω )² B V ( Ω ) 2 such that the equality det0.56905pt(∇ U)=Det0.56905pt(∇ U) det ( ∇ U ) = Det ( ∇ U ) holds either in the distributional sense on Ω Ω , or almost-everywhere in Ω Ω when U is in W1,1(Ω )² W 1 , 1 ( Ω ) 2 . The key-assumption of the result is the regularity of the Jacobian matrix-valued ∇ U ∇ U along the direction of a given non vanishing vector field b∈ C¹(Ω )² b ∈ C 1 ( Ω ) 2 , i.e. ∇ U\, b ∇ U b is assumed either to belong to C⁰(Ω )² C 0 ( Ω ) 2 with one of its coordinates in C¹(Ω ) C 1 ( Ω ) , or to belong to C¹(Ω )² C 1 ( Ω ) 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(Ω )² B V ( Ω ) 2 , assuming that the vector field b and one of the coordinates of ∇ U\, b ∇ U b lie in a compact set of regular vector-valued functions.