Weak regularity of the inverse under minimal assumptions

Abstract

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

Second version fixes a gap in the original proof of Proposition 4.2