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Jacobians of \(W^{1,p}\) homeomorphisms, case \(p=[n/2]\)

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We investigate a known problem whether a Sobolev homeomorphism between domains in \(\mathbb {R}^n\) can change sign of the Jacobian. The only case that remains open is when \(f\in W^{1,[n/2]}\), \(n\ge 4\). We prove that if \(n\ge 4\), and a sense-preserving homeomorphism f satisfies \(f\in W^{1,[n/2]}\), \(f^{-1}\in W^{1,n-[n/2]-1}\) and either f is Hölder continuous on almost all spheres of dimension [n / 2], or \(f^{-1}\) is Hölder continuous on almost all spheres of dimensions \(n-[n/2]-1\), then the Jacobian of f is non-negative, \(J_f\ge 0\), almost everywhere. This result is a consequence of a more general result proved in the paper. Here [x] stands for the greatest integer less than or equal to x.

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We would like to thank Jan Malý for providing us with a beautiful proof of Proposition 28. A few days before completion of this work we learned the sad news that Professor Bogdan Bojarski had passed away. He was the Ph.D. advisor of Piotr Hajłasz and an inspiration for both of us. We mourn his passing, and we dedicate this paper with deep respect to his memory.

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Correspondence to Piotr Hajłasz.

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Communicated by L.Ambrosio.

In memoriam: Bogdan Bojarski (1931–2018).

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P.G. was partially supported by National Science Center Grant No 2012/05/E/ST1/03232.

P.H. was supported by NSF Grant DMS-1800457.

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Goldstein, P., Hajłasz, P. Jacobians of \(W^{1,p}\) homeomorphisms, case \(p=[n/2]\). Calc. Var. 58, 122 (2019).

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