Ebook: Parametrized Measures and Variational Principles

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.

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The purpose of this book is to present a new approach to fundamental questions concerning the calculus of variations (and more generally to the theory of optimization) based on a systematic analysis of Young measures. Weak lower semicontinuity and relaxation are main areas of concentration in this work. The unified treatment of scalar and vector cases developed here is suitable also for more general situations. Applications to problems in continuum mechanics and nonlinear elasticity are analyzed in some depth. Other problems are included to motivate interest in Young measures and to emphasize the wide range of applicability of these techniques (to, for example, optimal control, optimal design and turbulence). Researchers and graduate students in applied mathematics and mechanics will benefit from this general approach to variational principles.

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The purpose of this book is to present a new approach to fundamental questions concerning the calculus of variations (and more generally to the theory of optimization) based on a systematic analysis of Young measures. Weak lower semicontinuity and relaxation are main areas of concentration in this work. The unified treatment of scalar and vector cases developed here is suitable also for more general situations. Applications to problems in continuum mechanics and nonlinear elasticity are analyzed in some depth. Other problems are included to motivate interest in Young measures and to emphasize the wide range of applicability of these techniques (to, for example, optimal control, optimal design and turbulence). Researchers and graduate students in applied mathematics and mechanics will benefit from this general approach to variational principles.
Content:
Front Matter....Pages i-xi
Introduction....Pages 1-24
Some Variational Problems....Pages 25-41
The Calculus of Variations under Convexity Assumptions....Pages 43-60
Nonconvexity and Relaxation....Pages 61-70
Phase Transitions and Microstructure....Pages 71-94
Parametrized Measures....Pages 95-114
Analysis of Parametrized Measures....Pages 115-131
Analysis of Gradient Parametrized Measures....Pages 133-159
Quasiconvexity and Rank-one Convexity....Pages 161-177
Analysis of Divergence-Free Parametrized Measures....Pages 179-191
Back Matter....Pages 193-212

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The purpose of this book is to present a new approach to fundamental questions concerning the calculus of variations (and more generally to the theory of optimization) based on a systematic analysis of Young measures. Weak lower semicontinuity and relaxation are main areas of concentration in this work. The unified treatment of scalar and vector cases developed here is suitable also for more general situations. Applications to problems in continuum mechanics and nonlinear elasticity are analyzed in some depth. Other problems are included to motivate interest in Young measures and to emphasize the wide range of applicability of these techniques (to, for example, optimal control, optimal design and turbulence). Researchers and graduate students in applied mathematics and mechanics will benefit from this general approach to variational principles.
Content:
Front Matter....Pages i-xi
Introduction....Pages 1-24
Some Variational Problems....Pages 25-41
The Calculus of Variations under Convexity Assumptions....Pages 43-60
Nonconvexity and Relaxation....Pages 61-70
Phase Transitions and Microstructure....Pages 71-94
Parametrized Measures....Pages 95-114
Analysis of Parametrized Measures....Pages 115-131
Analysis of Gradient Parametrized Measures....Pages 133-159
Quasiconvexity and Rank-one Convexity....Pages 161-177
Analysis of Divergence-Free Parametrized Measures....Pages 179-191
Back Matter....Pages 193-212
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