Abstract
We consider the Sobolev space \(X = W^{s,p} \left( {\mathbb{S}^m ;\mathbb{S}^{k - 1} } \right)\). We prove the existence of a robust distributional Jacobian Ju for u ∈ X, provided sp ≥ k − 1; this generalizes a result of Bourgain, Brezis, and the second author [10] dealing with the case m = k. We identify the image of the map X ϶ u ↦ Ju in the critical case sp = k − 1. This extends a result of Alberti, Baldo, and Orlandi [2] for s = 1 and p = k − 1. We also present a new, analytical, dipole construction method.
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Bousquet, P., Mironescu, P. Prescribing the Jacobian in critical spaces. JAMA 122, 317–373 (2014). https://doi.org/10.1007/s11854-014-0010-0
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DOI: https://doi.org/10.1007/s11854-014-0010-0