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 converยญ 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 underยญ 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 oscillaยญ 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 seยญ 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 nonlinยญ 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
CONTENT
1. Introduction -- 2. Some Variational Problems -- 3. The Calculus of Variations under Convexity Assumptions -- 4. Nonconvexity and Relaxation -- 5. Phase Transitions and Microstructure -- 6. Parametrized Measures -- 7 Analysis of Parametrized Measures -- 8. Analysis of Gradient Parametrized Measures -- 9. Quasiconvexity and Rank-one Convexity -- 10. Analysis of Divergence-Free Parametrized Measures
SUBJECT
Mathematics
Mathematical analysis
Analysis (Mathematics)
Applied mathematics
Engineering mathematics
Mathematics
Analysis
Appl.Mathematics/Computational Methods of Engineering
