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Minimum variational principles for time-harmonic waves in a dissipative medium and associated variational principles of Hashin-Shtrikman type. (English) Zbl 1211.74180
Summary: Minimization variational principles for linear elastodynamic, acoustic or electromagnetic time-harmonic waves in dissipative media were obtained by the first author, P. Seppecher and G. BouchittĂ© [Proc. R. Soc. Lond., Ser. A, Math. Phys. Eng. Sci. 465, No. 2102, 367–396 (2009; Zbl 1186.74044)], generalizing the quasistatic variational principles of A. V. Cherkaev and L. V. Gibiansky [J. Math. Phys. 35, No. 1, 127–145 (1994; Zbl 0805.49028)]. Here, a further generalization is made to allow for a much wider variety of boundary conditions, and in particular Dirichlet and Neumann boundary conditions. In addition minimization or maximization principles of the Hashin-Shtrikman type, incorporating ‘polarization fields’, are developed. The corresponding principles for static problems have found substantial use in bounding the effective static properties of composite materials. The new dynamical principles offer the prospect of developing bounds on the effective dynamic properties of such materials.

MSC:
74Q10 Homogenization and oscillations in dynamical problems of solid mechanics
74E99 Material properties given special treatment
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