Abstract
Let \(N\ge 3\). We construct a homeomorphism \(f\) in the Sobolev space \(W^{1,1}((0,1)^N,(0,1)^N)\) such that \(f^{-1}\in W^{1,1}((0,1)^N,(0,1)^N)\), \(J_f=0\) a.e. and \(J_{f^{-1}}=0\) a.e. It follows that \(f\) maps a set of full measure to a null set and a remaining null set to a set of full measure.We also show that such a pathological homeomorphism cannot exist in dimension \(N=2\) or with higher regularity \(f\in W^{1,N-1}\).
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Acknowledgments
The authors would like to thank Carlo Sbordone for pointing their interest to the problem. They would also like to thank to the De Giorgi Center where the part of our research was done.
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Communicated by L.Ambrosio.
S. Hencl was supported by the ERC CZ grant LL1203 of the Czech Ministry of Education.
To Carlo Sbordone on his 65th birthday.
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D’Onofrio, L., Hencl, S. & Schiattarella, R. Bi-Sobolev homeomorphism with zero Jacobian almost everywhere. Calc. Var. 51, 139–170 (2014). https://doi.org/10.1007/s00526-013-0669-6
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DOI: https://doi.org/10.1007/s00526-013-0669-6