On distributional adjugate and derivative of the inverse

TL;DR

This paper investigates the properties of the distributional adjugate of the derivative of bi-$BV$ homeomorphisms, establishing measure-theoretic identities and equivalences among different definitions.

Contribution

It proves that the components of the distributional adjugate match the measures derived from the inverse function and clarifies the relation between the absolutely continuous part and the pointwise adjugate.

Findings

Components of the distributional adjugate equal measures from the inverse function.

The absolutely continuous part of the adjugate equals the pointwise adjugate almost everywhere.

Multiple definitions of the distributional adjugate are shown to be equivalent.

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.