\(H\)-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.


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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