An example concerning the zero set of the Jacobian (Q819676)

The author considers homeomorphisms \(f \in W^{1,1}(\Omega, \mathbb{R}^n)\) of finite distortion \(K \,.\) It is known that \(K^{1/(n-1)} \in L^1(\Omega) \) implies that the Jacobian \(J_f \) of \(f\) is positive almost everywhere in \(\Omega\,.\) The author shows that this integrability requirement is sharp. The construction make use of regular Cantor sets of positive and zero measure.