Coarea formulae and chain rules for the Jacobian determinant in fractional Sobolev spaces

Gladbach, Peter;Olbermann, Heiner
(2020) Journal of Functional Analysis — Vol. 278, n° 2, p. 108312 (2020)

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Abstract
We prove weak and strong versions of the coarea formula and the chain rule for distributional Jacobian determinants Ju for functions u in fractional Sobolev spaces Ws,p(Ω), where Ω is a bounded domain in Rn with smooth boundary. The weak forms of the formulae are proved for the range sp>n−1, s≥ [Formula presented], while the strong versions are proved for the range sp>n, s≥ [Formula presented]. We also provide a chain rule for the distributional Jacobian determinant of Hölder functions and point out its relation to two open problems in geometric analysis.
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Gladbach, P., & Olbermann, H. (2020). Coarea formulae and chain rules for the Jacobian determinant in fractional Sobolev spaces. Journal of Functional Analysis, 278(2), 108312. https://doi.org/10.1016/j.jfa.2019.108312 (Original work published 2020)