On distributional adjugate and derivative of the inverse

Abstract

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