Weak Regularity of the Inverse Under Minimal Assumptions

Abstract

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