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Zur Invarianz von \(W^ 1_ p\)-Räumen gegenüber Lipschitz-stetigen Abbildungen. (The invariance of \(W^ 1_ p\)-spaces with respect to Lipschitz continuous maps). (German) Zbl 0605.46026

The proof of the following theorem is sketched:
Let \(D\subseteq {\mathbb{R}}^ n\), \(n\in {\mathbb{N}}\), a bounded domain for which a linear continuous extension operator \(F_ D\in {\mathcal L}(W^ 1_ p(D),W^ 1_ p({\mathbb{R}}^ n))\), \(1\leq p<\infty\), exists. Let \(\ell \in C^{0,1}({\mathbb{R}})\). Then with u the superposition \(\ell \circ u\) belongs to \(W^ 1_ p(D)\) and almost everywhere in D the chain rule \((\ell \circ u)_{x_ i}=(\ell '\circ u)u_{x_ i}\), \(1\leq i\leq n\), is valid.

MSC:

46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
46E15 Banach spaces of continuous, differentiable or analytic functions
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