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Control in the coefficients of linear hyperbolic equations via spacio-temporal components. (English) Zbl 1035.78021

Berdichevsky, V. (ed.) et al., Homogenization. In memory of Serguei Kozlov. Singapore: World Scientific (ISBN 981-02-3096-6/hbk). Ser. Adv. Math. Appl. Sci. 50, 285-315 (1999).
From the text: In this important paper, the author gives illustrations of some specific features of control over the solutions of linear hyperbolic equations produced by the variation of their senior coefficients.
The paper consists of two parts. In Part I, the author explores the spatial temporal laminates – examples of chattering microstructures in space-time. These laminates are introduced in connection with the simplest hyperbolic equation \[ (\rho v_t)_t-(kv_z)_z=0\tag{1} \] with variable coefficients \(\rho=\rho(t,z)\), \(k=k(t,z)\) taking values from an admissible set \(U\). An elliptic counterpart of this situation is given by the equation \[ (\lambda w_x)_x+(\lambda w_y)_y=0\tag{2} \] with \(\lambda= \lambda (x,y)\); the spatial variability of coefficients is then particularly implemented in the purely spatial laminates.
Homogenization procedure applied to (1) generates formulas for the effective material parameters of a hyperbolic system. These parameters participate in the characterization of a hyperbolic \(G\)-closure ([cf. K. A. Lurie and A. V. Cherkaev, Prog. Nonlinear Differ. Equ. Appl. 31, 175–258 (1997; Zbl 0927.74056)] where this notion is discussed for an elliptic case) of the original set \(U\) formed by the admissible values of \(\rho\) and \(k\). The laminates constitute a part of the \(G\)-closure, but, in contrast to an elliptic case [loc. cit.], it is still unknown whether this entire set may ever be covered by laminates alone. The description of a hyperbolic \(G\)-closure is significant for optimization purposes. Unfortunately, however, for Eq. (1) such a description is still inaccessible, and both Part I and Part II of the paper contain information that may help to build it.
\(G\)-closure is an invariant object, and its appropriate definition requires a covariant formulation of the basic equations, particularly, of Eq. (1). In Part II he shows that to obtain such a formulation, one should perceive Eq. (1) as a part of the Maxwell system formulated for the relevant material continuum. In this interpretation, \(\rho\) may be treated as the magnetic permeability \(\mu\) of the medium whereas \(k\) is interpreted as its inverse dielectric permittivity \(1/\varepsilon\).
Maxwell’s system incorporates the fundamental relativistic invariant properties of space-time and, therefore, provides the required covariant description. Particularly, the material tensor \(s\) of the medium is defined by parameters \(\mu\) and \(\varepsilon\). Maxwell’s system is of course formulated for the most general three-dimensional case. In the case of one spatial dimension illustrated by Eq. (1), the \(G\)-closure possesses a special property: if the elements of \(U\) have the same value of det \(s\), then det\(s_0=\det\,s\), where \(s_0\) is an effective tensor of the medium after homogenization. This is a hyperbolic analogue of the conservation law that holds for a two-dimensional polycrystal appearing in the relevant problems for Eq. (2) (loc. cit.).
For the entire collection see [Zbl 0924.00012].

MSC:

78M40 Homogenization in optics and electromagnetic theory
35B27 Homogenization in context of PDEs; PDEs in media with periodic structure
49K20 Optimality conditions for problems involving partial differential equations
74Q05 Homogenization in equilibrium problems of solid mechanics

Citations:

Zbl 0927.74056
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