Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

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
DOI: https://doi.org/10.1090/S0002-9939-1991-1039260-X
MathSciNet review: 1039260
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 58C07, 46E99, 47H99, 90C25, 90C48

Retrieve articles in all journals with MSC: 58C07, 46E99, 47H99, 90C25, 90C48


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1991-1039260-X
Article copyright: © Copyright 1991 American Mathematical Society

American Mathematical Society