zbMATH — the first resource for mathematics

\(H\)-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
Full Text: DOI
[1] Milton, Homogenization and Effective Moduli of Materials and Media pp 150– (1986) · doi:10.1007/978-1-4613-8646-9_7
[2] Murat, Ann. Scuola Norm. Sup. Pisa Cl. Sci. 5 pp 489– (1978)
[3] DiPerna, Directions in Partial Differential Equations pp 43– (1987) · doi:10.1016/B978-0-12-195255-6.50010-6
[4] DOI: 10.1007/BF01206047 · Zbl 0533.76071 · doi:10.1007/BF01206047
[5] DOI: 10.1007/BF00251724 · Zbl 0519.35054 · doi:10.1007/BF00251724
[6] David, C. R. Acad. Sci. Paris, Sir. I. Math.. 296 pp 761– (1983)
[7] Coifman, Astirisque 57 pp 1– (1978)
[8] DOI: 10.1073/pnas.53.5.1092 · Zbl 0151.16901 · doi:10.1073/pnas.53.5.1092
[9] Young, Lectures on the Calculus of Variation and Optimal Control Theory (1969) · Zbl 0177.37801
[10] DOI: 10.1007/BF00279992 · Zbl 0368.73040 · doi:10.1007/BF00279992
[11] Tartar, Homogenization and Effective Moduli of Materials and Media (1986)
[12] DOI: 10.1137/0147082 · Zbl 0632.73079 · doi:10.1137/0147082
[13] Tartar, Nonlinear Systems of Partial Differential Equations in Applied Mathematics pp 243– (1986) · Zbl 0602.35009
[14] DOI: 10.1002/cpa.3160400502 · Zbl 0629.73010 · doi:10.1002/cpa.3160400502
[15] Tartar, Ennio de Giorgi Colloquium pp 168– (1985)
[16] Tartar, Systems of Nonlinear Partial Differential Equations pp 263– (1983) · doi:10.1007/978-94-009-7189-9_13
[17] Landau, Electrodynamics of Continuous Media (1984)
[18] Kohn, Homogenization and Effective Moduli of Materials and Media pp 97– (1986) · doi:10.1007/978-1-4613-8646-9_5
[19] Höormander., The Analysis of Linear Partial Differenital Operators (1983)
[20] DOI: 10.1007/BF00280908 · Zbl 0604.73013 · doi:10.1007/BF00280908
[21] DOI: 10.1002/cpa.3160400304 · Zbl 0850.76730 · doi:10.1002/cpa.3160400304
[22] DOI: 10.1007/BF01214424 · Zbl 0626.35059 · doi:10.1007/BF01214424
[23] DOI: 10.1002/cpa.3160420603 · Zbl 0698.35128 · doi:10.1002/cpa.3160420603
[24] DOI: 10.2307/1971423 · Zbl 0698.45010 · doi:10.2307/1971423
[25] Tartar, Nonlinear Analysis and Mechanics, Heriot-Watt Symposium pp 136– (1979)
[26] DOI: 10.1007/BFb0063632 · doi:10.1007/BFb0063632
[27] Tartar, Singular Perturbations and Boundary Layer Theory pp 474– (1976)
[28] Lions, Rev. Mat. Iberoamericana 1 pp 145– (1985) · Zbl 0704.49005 · doi:10.4171/RMI/6
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.