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Parametrized Measures and Variational Principles

  • Book
  • © 1997

Overview

Authors:
  1. Pablo Pedregal

    1. E.T.S.I. Industriales, Universidad de Castilla-La Mancha, Ciudad Real, Spain

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Part of the book series: Progress in Nonlinear Differential Equations and Their Applications (PNLDE, volume 30)

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Table of contents (10 chapters)

  1. Front Matter

    Pages i-xi
    Download chapter PDF

  2. Introduction

    • Pablo Pedregal
    Pages 1-24

  3. Some Variational Problems

    • Pablo Pedregal
    Pages 25-41

  4. The Calculus of Variations under Convexity Assumptions

    • Pablo Pedregal
    Pages 43-60

  5. Nonconvexity and Relaxation

    • Pablo Pedregal
    Pages 61-70

  6. Phase Transitions and Microstructure

    • Pablo Pedregal
    Pages 71-94

  7. Parametrized Measures

    • Pablo Pedregal
    Pages 95-114

  8. Analysis of Parametrized Measures

    • Pablo Pedregal
    Pages 115-131

  9. Analysis of Gradient Parametrized Measures

    • Pablo Pedregal
    Pages 133-159

  10. Quasiconvexity and Rank-one Convexity

    • Pablo Pedregal
    Pages 161-177

  11. Analysis of Divergence-Free Parametrized Measures

    • Pablo Pedregal
    Pages 179-191

  12. Back Matter

    Pages 193-212
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Keywords

About this book

Weak convergence is a basic tool of modern nonlinear analysis because it enjoys the same compactness properties that finite dimensional spaces do: basically, bounded sequences are weak relatively compact sets. Nonetheless, weak conver­ gence does not behave as one would desire with respect to nonlinear functionals and operations. This difficulty is what makes nonlinear analysis much harder than would normally be expected. Parametrized measures is a device to under­ stand weak convergence and its behavior with respect to nonlinear functionals. Under suitable hypotheses, it yields a way of representing through integrals weak limits of compositions with nonlinear functions. It is particularly helpful in comprehending oscillatory phenomena and in keeping track of how oscilla­ tions change when a nonlinear functional is applied. Weak convergence also plays a fundamental role in the modern treatment of the calculus of variations, again because uniform bounds in norm for se­ quences allow to have weak convergent subsequences. In order to achieve the existence of minimizers for a particular functional, the property of weak lower semicontinuity should be established first. This is the crucial and most delicate step in the so-called direct method of the calculus of variations. A fairly large amount of work has been devoted to determine under what assumptions we can have this lower semicontinuity with respect to weak topologies for nonlin­ ear functionals in the form of integrals. The conclusion of all this work is that some type of convexity, understood in a broader sense, is usually involved.

Authors and Affiliations

  • E.T.S.I. Industriales, Universidad de Castilla-La Mancha, Ciudad Real, Spain

    Pablo Pedregal

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