Murat, François; Trombetti, Cristina A chain rule formula for the composition of a vector-valued function by a piecewise smooth function. (English) Zbl 1179.46037 Boll. Unione Mat. Ital., Sez. B, Artic. Ric. Mat. (8) 6, No. 3, 581-595 (2003). Summary: 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. Cited in 8 Documents MSC: 46G05 Derivatives of functions in infinite-dimensional spaces 26B35 Special properties of functions of several variables, Hölder conditions, etc. PDFBibTeX XMLCite \textit{F. Murat} and \textit{C. Trombetti}, Boll. Unione Mat. Ital., Sez. B, Artic. Ric. Mat. (8) 6, No. 3, 581--595 (2003; Zbl 1179.46037) Full Text: EuDML