On distributional adjugate and derivative of the inverse

Abstract

Let $\mathrm{\Omega }\subset {\mathbf{R}}^3$ be a domain and let $f:\mathrm{\Omega }\to {\mathbf{R}}^3$ be a bi-$BV$ homeomorphism. Very recently in [16] it was shown that the distributional adjugate of $Df$ (and thus also of $D{f}^{-1}$) is a matrix-valued measure. In the present paper we show that the components of Adj $Df$ coincide with the components of $D{f}^{-1}(f(U))$ 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.