An example concerning the zero set of the Jacobian

Let f ∈W1,1(Ω,Rn) be a homeomorphism of finite distortion K. It is known that if K1/(n−1) ∈ L1(Ω), then the Jacobian Jf of f is positive almost everywhere in Ω. We will show that this integrability assumption on K is sharp in any Orlicz-scale: if α is increasing function (satisfying minor technical assumptions) such that limt→∞α(t) = ∞, then there exists f such that K1/(n−1)/α(K) ∈ L1(Ω) and Jf vanishes in a set of positive measure.  2005 Elsevier Inc. All rights reserved.