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

Abstract

AbstractThe paper deals with the comparison in dimension two between the strong Jacobian determinant $$\textrm{det}\hspace{0.56905pt}$$ det and the weak (or distributional) Jacobian determinant $$\textrm{Det}\hspace{0.56905pt}$$ 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 $$BV(\Omega )^2$$ B V ( Ω ) 2 such that the equality $$\textrm{det}\hspace{0.56905pt}(\nabla U)=\textrm{Det}\hspace{0.56905pt}(\nabla U)$$ det ( ∇ U ) = Det ( ∇ U ) holds either in the distributional sense on $$\Omega $$ Ω , or almost-everywhere in $$\Omega $$ Ω when U is in $$W^{1,1}(\Omega )^2$$ W 1 , 1 ( Ω ) 2 . The key-assumption of the result is the regularity of the Jacobian matrix-valued $$\nabla U$$ ∇ U along the direction of a given non vanishing vector field $$b\in C^1(\Omega )^2$$ b ∈ C 1 ( Ω ) 2 , i.e.$$\nabla U\, b$$ ∇ U b is assumed either to belong to $$C^0(\Omega )^2$$ C 0 ( Ω ) 2 with one of its coordinates in $$C^1(\Omega )$$ C 1 ( Ω ) , or to belong to $$C^1(\Omega )^2$$ 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 $$BV(\Omega )^2$$ B V ( Ω ) 2 , assuming that the vector field b and one of the coordinates of $$\nabla U\, b$$ ∇ U b lie in a compact set of regular vector-valued functions.