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

TL;DR

This paper extends variational principles for time-harmonic waves in dissipative media, allowing for diverse boundary conditions and introducing Hashin-Shtrikman type principles with polarization fields.

Contribution

It generalizes existing variational principles to include a broader range of boundary conditions and develops Hashin-Shtrikman type principles with polarization fields.

Findings

Extended variational principles to Dirichlet and Neumann boundary conditions.

Developed Hashin-Shtrikman type principles with polarization fields.

Provided a unified framework for dissipative media wave problems.

Abstract

Minimization variational principles for linear elastodynamic, acoustic, or electromagnetic time-harmonic waves in dissipative media were obtained by Milton, Seppecher and Bouchitt\'e generalizing the quasistatic variational principles of Cherkaev and Gibiansky. 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.