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?)

  • [1] L. Ambrosio and G. Dal Maso, A general chain rule for distributional derivatives, Proc. Amer. Math. Soc. 108 (1990), 691-702. MR 969514 (90j:26019)
  • [2] J. P. Aubin and I. Ekeland, Applied nonlinear analysis, John Wiley & Sons, New York, 1984. MR 749753 (87a:58002)
  • [3] L. Boccardo and F. Murat, Remarques sur l'homogeineisation de certains problêmes quasilineaires, Portugal Math. 41 (1982), 535-562. MR 766874 (86a:35022)
  • [4] G. Buttazzo and A. Leaci, A continuity theorem for operators from $ {W^{1,q}}(\Omega )$ into $ {L^q}(\Omega )$, J. Funct. Anal. 58 (1984), 216-224. MR 757996 (85k:49036)
  • [5] G. Dal Maso and R. Musina, An approach to the thin obstacle problem for variational functions depending on vector valued functions, Comm. Partial Differential Equations 14 (1989), 1717-1743. MR 1039915 (91e:35107)
  • [6] H. Federer, Geometric measure theory, Springer, Berlin, 1969. MR 0257325 (41:1976)
  • [7] M. Marcus and V. J. Mizel, Absolute continuity on traks and mappings of Sobolev spaces, Arch. Rational Mech. Anal. 45 (1972), 294-320. MR 0338765 (49:3529)
  • [8] -, Every superposition operator mapping one Sobolev space into another is continuous, J. Funct. Anal. 33 (1979), 217-229. MR 546508 (80h:47039)
  • [9] C. B. Morrey, Multiple integrals in the calculus of variations, Springer, Berlin, 1966.
  • [10] K. Steffen, Isoperimetric inequalities and the problem of plateau, Math. Ann. 222 (1976), 97-144. MR 0417903 (54:5951)

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