Weak regularity of the inverse under minimal assumptions

TL;DR

This paper proves that if a homeomorphism in three dimensions has a finite measure distributional adjugate, then its inverse is also of bounded variation, establishing a necessary and sufficient condition for weak regularity.

Contribution

It introduces a minimal assumption involving the distributional adjugate that guarantees the inverse of a homeomorphism is of bounded variation, and proves this condition is both necessary and sufficient.

Findings

Inverse has bounded variation if the distributional adjugate is finite measure.

The finite measure condition on the adjugate is necessary and sufficient for inverse regularity.

Provides a characterization of inverse regularity under minimal assumptions.

Abstract

Let $\Omega\subset\mathbb{R}^3$ be a domain and let $f\in BV_{\operatorname{loc}}(\Omega,\mathbb{R}^3)$ be a homeomorphism such that its distributional adjugate is a finite Radon measure. We show that its inverse has bounded variation $f^{-1}\in BV_{\operatorname{loc}}$. The condition that the distributional adjugate is finite measure is not only sufficient but also necessary for the weak regularity of the inverse.