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Mathematical tools for studying oscillations and concentrations: From Young measures to \(H\)-measures and their variants. (English) Zbl 1015.35001

Antonić, Nenad (ed.) et al., Multiscale problems in science and technology. Challenges to mathematical analysis and perspectives. Proceedings of the conference on multiscale problems in science and technology, Dubrovnik, Croatia, September 3-9, 2000. Berlin: Springer. 1-84 (2002).
This work comprises the six lectures which the author gave with the purpose to retrace the evolution of the subject over the last twenty years. The contents of the lectures is the following. Lecture 1: A little history of the subject. Suitable weak topologies. On a formula of Landau and Lifschitz. On Hashin-Shtrikman bounds. Homogenisation and periodicity assumption. Lecture 2: Study of oscillations. Problem modelled on the Navier-Stokes system. What are the \(H\)-measures? Young measures and compensated compactness. \(H\)-measures and compensated compactness. Pseudodifferential calculus. Examples of \(H\)-measures. Localisation principle. Lecture 3: Problem modelled on the Navier-Stokes system (revisited). Small amplitude homogenisation. Propagation of oscillations and concentration effects. Lecture 4: Propagation for the wave equation. Geometric theory of diffraction. Quantum mechanics and electrodynamics. Compactness by integration. Variants of \(H\)-measures: One characteristic length. Lecture 5: Semi-classical measures: One characteristic length. Wigner transform and two-point correlations. Examples with one characteristic length. Semi-classical measures: Two characteristic lengths. Lecture 6: Variants of \(H\)-measures: Two characteristic lengths. Another look at problem modelled on Navier-Stokes’ system. Relations between Young measures and \(H\)-measures. Small amplitude homogenisation (revisited). Nonlocal effects in homogenisation.
For the entire collection see [Zbl 0989.00039].

MSC:

35-02 Research exposition (monographs, survey articles) pertaining to partial differential equations
35B27 Homogenization in context of PDEs; PDEs in media with periodic structure
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