We show that in the euclidean space Rn it is possible to construct a homeomorphism in the Sobolev space such that its Jacobian vanishes almost everywhere. It follows that we can find a set of measure zero N whose image has full measure, and the complement of N has full measure but it is mapped to a set of zero measure. We also discuss the optimal Sobolev regularity of such pathological homeomorphisms or the Sobolev regularity of the inverse mapping.
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