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Weak compactness of Fréchet-derivatives: application to composition operators

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

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
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Abstract

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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

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