On the continuity of the Nemitsky operator induced by a Lipschitz continuous map
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- by Roberta Musina PDF
- Proc. Amer. Math. Soc. 111 (1991), 1029-1041 Request permission
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.References
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Additional Information
- © Copyright 1991 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 111 (1991), 1029-1041
- MSC: Primary 58C07; Secondary 46E99, 47H99, 90C25, 90C48
- DOI: https://doi.org/10.1090/S0002-9939-1991-1039260-X
- MathSciNet review: 1039260