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Differentiability in the Sobolev space \(W^{1,n-1}\)

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Abstract

Let \(\Omega \subset {\mathbb {R}}^{n}\) be a domain, \(n \ge 2\). We show that a continuous, open and discrete mapping \(f \in W_{\mathrm{loc }}^{1,n-1}(\Omega , {\mathbb {R}}^{n})\) with integrable inner distortion is differentiable almost everywhere on \(\Omega \). As a corollary we get that the branch set of such a mapping has measure zero.

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Acknowledgments

The author was supported by the Academy of Finland. Part of the research was done while the author was visiting at the University of Michigan. He would like to thank the department for its hospitality. The author would like to thank Jani Onninen and Kai Rajala for their many useful comments on the manuscript. The author thanks the referee for very careful reading of the paper and many useful suggestions.

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  1. Department of Mathematics and Statistics, University of Jyväskylä, P.O. Box 35 (MaD), 40014 , University of Jyväskylä, Finland

    Ville Tengvall

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  1. Ville Tengvall
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Communicated by J. Ball.

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Tengvall, V. Differentiability in the Sobolev space \(W^{1,n-1}\) . Calc. Var. 51, 381–399 (2014). https://doi.org/10.1007/s00526-013-0679-4

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