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$$H$$-measures, a new approach for studying homogenisation, oscillations and concentration effects in partial differential equations. (English) Zbl 0774.35008
The author describes a new approach for studying oscillation and concentration effects in partial differential equations, based on the theory of so called $$H$$-measures. Applications to transport properties and homogenization are given. The proposed method is a deeper development of the known author’s results on compensated compactness and its applications to homogenization theory. Let $$u_ \varepsilon$$ be a sequence of vector functions, defined on an open set $$\Omega \subset \mathbb{R}^ n$$ and converging weakly to zero in $$(L^ 2(\Omega))^ p$$. The basic idea is to extract a subsequence (denoted again by $$u_ \varepsilon$$ so that for every $$i,j=1,2,\dots,p$$ one can define a complex Radon measure $$\mu_{ij}$$ in $$\Omega \times S$$ $$(S$$ denotes the unit sphere in $$\mathbb{R}^ N)$$ by: $\langle \mu_{ij},\varphi_ 1\varphi^*_ 2\otimes \psi \rangle=\lim_{\varepsilon \to 0}\int_{R^ N}[F(\varphi_ 1u_{i \varepsilon})(\xi)][F(\varphi_ 2u_{j \varepsilon})(\xi )]^* \psi \left({\xi \over | \xi |}\right)d \xi$ for every $$\psi \in C(S)$$ and $$\varphi_ 1,\varphi_ 2 \in C_ c(\Omega)$$; $$F$$ denotes the Fourier transform operator. The test function $$\psi$$ is used in order to localize in the direction of the dual variable $$\xi$$; functions $$\varphi_ 1$$ and $$\varphi_ 2$$ have the same purpose of localizing in the space variable $$x$$. The measures $$\mu_{ij}$$ give a useful description of oscillation and concentration effects of the sequence $$u_ \varepsilon$$ in particular their propagation, through some kind of microlocal $$H$$- calculus enabling us to use the balance equation in a better way.
After investigation of the existence and basic properties of $$H$$- measures, the author explains the connection of the new notion with the compensated compactness method. The main part of the paper is dedicated to the transport properties of $$H$$-measures.

##### MSC:
 35B27 Homogenization in context of PDEs; PDEs in media with periodic structure 35Q30 Navier-Stokes equations 35L99 Hyperbolic equations and hyperbolic systems
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