Minimum variational principles for time-harmonic waves in a dissipative medium and associated variational principles of Hashin–Shtrikman type

Abstract

Minimization variational principles for linear elastodynamic, acoustic or electromagnetic time-harmonic waves in dissipative media were obtained by Milton et al. (Milton et al. 2009 Proc. R. Soc. A 465 , 367–396 ( doi:10.1098/rspa.2008.0195 )), generalizing the quasistatic variational principles of Cherkaev and Gibiansky (Cherkaev & Gibiansky 1994 J. Math. Phys. 35 , 127–145 ( doi:10.1063/1.530782 )). 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.