An example concerning the zero set of the Jacobian

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Let f ∈ W 1 , 1 ( Ω , R n ) be a homeomorphism of finite distortion K. It is known that if K 1 / ( n − 1 ) ∈ L 1 ( Ω ) , then the Jacobian J f 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 lim t → ∞ α ( t ) = ∞ , then there exists f such that K 1 / ( n − 1 ) / α ( K ) ∈ L 1 ( Ω ) and J f vanishes in a set of positive measure.