On distributional adjugate and derivative of the inverse

Abstract: Let $\Omega\subset\er^3$ be a domain and let $f\colon\Omega\to\er^3$ be a bi-$BV$ homeomorphism. Very recently in \cite{HKL} it was shown that the distributional adjugate of $Df$ (and thus also of $Df^{-1}$) is a matrix-valued measure. In the present paper we show that the components of $\Adj Df$ are equal to components of $Df^{-1}(f(U))$ as measures and that the absolutely continuous part of the distributional adjugate $\Adj Df$ equals to the pointwise adjugate $\adj Df(x)$ a.e. We also show the equivalence of several approaches to the definition of the distributional adjugate.