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A standard assumption in the geometric theory of deformations is that the mappings that are the object of study have non-negative Jacobian determinant. This is rather obvious when the deformations in question are diffeomorphisms (the condition that the deformation preserves sense). A natural question arises: how much topological information does this condition carry for weakly differentiable homeomorphisms? One can ask a more specific question: does a sense preserving, weakly differentiable homeomorphism (of an \(n\)-dimensional cube) necessarily have non-negative Jacobian? A positive answer has been given recently (by Hencl and Maly) for \(W^{1,p}\) homeomorphisms with \(p>n/2\) (and for \(p=1, \, 2\), \(n=3\)). It has been conjectured that a likewise answer holds for all \(W^{1,1}\) homeomorphisms.
In a joint work with Piotr Hajłasz (University of Pittsburgh), we provide sort of a counterexample to the conjecture, although the homeomorphism of the \(n\)-dimensional cube we construct is not in \(W^{1,1}\). It is, however, Hölder continuous (i.e. in a fractional Sobolev space), a.e. approximately differentiable, equals identity on the cube's boundary (and thus is sense-preserving), is measure-preserving (in particular has Lusin's property), and its approximate Jacobian determinant is equal to \(-1\) a.e. Moreover, it is a uniform limit of measure preserving diffeomorphisms.
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