Geodesic
A chain rule formula for the composition of a vector-valued function by a piecewise smooth function
Bollettino della Unione matematica italiana, Série 8, 6B (2003) no. 3, pp. 581-595

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We state and prove a chain rule formula for the composition $T(u)$ of a vector-valued function $u\in W^{1, r}(\Omega;\mathbb{R}^{M})$ by a globally Lipschitz-continuous, piecewise $C^{1}$ function $T$. We also prove that the map $u \to T(u)$ is continuous from $W^{1, r}(\Omega;\mathbb{R}^{M})$ into $W^{1,r}(\Omega)$ for the strong topologies of these spaces. 
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     author = {Murat, Fran\c{c}ois and Trombetti, Cristina},
     title = {A chain rule formula for the composition of a vector-valued function by a piecewise smooth function},
     journal = {Bollettino della Unione matematica italiana},
     pages = {581--595},
     publisher = {mathdoc},
     volume = {Ser. 8, 6B},
     number = {3},
     year = {2003},
     zbl = {1179.46037},
     mrnumber = {MR2014820},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/BUMI_2003_8_6B_3_a5/}
}
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Murat, François; Trombetti, Cristina. A chain rule formula for the composition of a vector-valued function by a piecewise smooth function. Bollettino della Unione matematica italiana, Série 8, 6B (2003) no. 3, pp. 581-595. http://geodesic.mathdoc.fr/item/BUMI_2003_8_6B_3_a5/