Hostname: page-component-586b7cd67f-dsjbd Total loading time: 0 Render date: 2024-12-01T00:20:14.395Z Has data issue: false hasContentIssue false
  • English
  • Français

Weak compactness of Fréchet-derivatives: application to composition operators

Part of: Linear function spaces and their duals Calculus on manifolds; nonlinear operators Nonlinear operators and their properties

Published online by Cambridge University Press:  09 April 2009

Peter Dierolf  and
Jürgen Voigt
Peter Dierolf
Affiliation:
Mathematisches Institut der Universität München Theresienstr. 39 D-8000 München 2 Bundesrepublik Deutschland
Jürgen Voigt
Affiliation:
Mathematisches Institut der Universität München Theresienstr. 39 D-8000 München 2 Bundesrepublik Deutschland
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We prove a result on compactness properties of Fréchet-derivatives which implies that the Fréchet-derivative of a weakly compact map between Banach spaces is weakly compact. This result is applied to characterize certain weakly compact composition operators on Sobolev spaces which have application in the theory of nonlinear integral equations and in the calculus of variations.

MSC classification

Secondary: 58C20: Differentiation theory (Gateaux, Fréchet, etc.) 46E35: Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems 47H99: None of the above, but in this section
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 29 , Issue 4 , June 1980 , pp. 399 - 406
Copyright
Copyright © Australian Mathematical Society 1980

References

Adams, R. A. (1975), Sobolev spaces (Academic Press, New York).Google Scholar
Batt, J. (1970), ‘Nonlinear compact mappings and their adjoints’, Math. Annalen 189, 5–25.CrossRefGoogle Scholar
Floret, K. (1978), Lectures on weakly compact sets (State University of New York at Buffalo).Google Scholar
Krasnosel'skii, M. A. (1964), Topological methods in the theory of nonlinear integral equations (Macmillan, New York).Google Scholar
Michlin, S. G. (1972), Lehrgang der mathematischen Physik (Akademie-Verlag, Berlin).Google Scholar
Schwartz, J. T. (1965), Nonlinear functional analysis (Courant Institute of Mathematical Sciences, New York).Google Scholar
Vainberg, M. M. (1964), Variational methods for the study of nonlinear operators (Holden-Day, Inc., San Francisco).Google Scholar
Yamamuro, S. (1974), Differential calculus in topological linear spaces (Lecture Notes in Mathematics, 374, Springer-Verlag, Berlin).CrossRefGoogle Scholar