Parametrized Measures and Variational Principles

• Pablo Pedregal
Book

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

1. Front Matter
Pages i-xi
2. Pablo Pedregal
Pages 1-24
3. Pablo Pedregal
Pages 25-41
4. Pablo Pedregal
Pages 43-60
5. Pablo Pedregal
Pages 61-70
6. Pablo Pedregal
Pages 71-94
7. Pablo Pedregal
Pages 95-114
8. Pablo Pedregal
Pages 115-131
9. Pablo Pedregal
Pages 133-159
10. Pablo Pedregal
Pages 161-177
11. Pablo Pedregal
Pages 179-191
12. Back Matter
Pages 193-212

Introduction

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.

Keywords

Applied Mathematics Calculus of variations Optimization PDE's calculus mathematics mechanics

Authors and affiliations

• Pablo Pedregal
• 1

Bibliographic information

• DOI https://doi.org/10.1007/978-3-0348-8886-8
• Copyright Information Birkhäuser Verlag, P.O. Box 133, CH-4010 Basel, Switzerland 1997
• Publisher Name Birkhäuser, Basel
• eBook Packages
• Print ISBN 978-3-0348-9815-7
• Online ISBN 978-3-0348-8886-8
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