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On the continuity of the Nemitsky operator induced by a Lipschitz continuous map

Author: Roberta Musina
Journal: Proc. Amer. Math. Soc. 111 (1991), 1029-1041
MSC: Primary 58C07; Secondary 46E99, 47H99, 90C25, 90C48
MathSciNet review: 1039260
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Abstract: Let $ f \in {\mathbb{R}^N} \to {\mathbb{R}^k}$ be a Lipschitz continuous function, and let $ \Omega $ be a bounded domain in the Euclidean space $ {\mathbb{R}^n}$. For every exponent $ p \in [1, + \infty [$ the composite map $ {T_f} = f \circ u$ maps the Sobolev space $ {W^{1,p}}(\Omega ,{\mathbb{R}^N})$) into $ {W^{1,p}}(\Omega ,{\mathbb{R}^k})$). In the scalar case, namely, when $ N = 1$, the operator $ {T_f}$ is continuous from $ {W^{1,p}}(\Omega ,{\mathbb{R}^N})$ into $ {W^{1,p}}(\Omega ,{\mathbb{R}^k})$. In this paper we illustrate a counterexample to the continuity of the operator $ {T_f}$ in the case where $ N > 1$. In the last part of the paper we give some sufficient conditions for the continuity of $ {T_f}$, and we conclude with some examples.

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Article copyright: © Copyright 1991 American Mathematical Society

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