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A chain rule formula for the composition of a vector-valued function by a piecewise smooth function. (English) Zbl 1179.46037

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.

MSC:

46G05 Derivatives of functions in infinite-dimensional spaces
26B35 Special properties of functions of several variables, Hölder conditions, etc.
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