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Linear processes in media with dissipation arising in conductivity, optics, viscoelasticity, etc. are considered. Time-periodic fields in such media are described by linear differential equations for complex-valued potentials. The properties of the media are characterized by complex valued tensors, for example, by complex conductivity or complex elasticity tensors. Variational formulations are suggested for such problems: The functionals whose Euler equations coincide with the original ones are constructed. Four equivalent variational principles are obtained: two minimax and two minimal ones. The functionals of the obtained minimal variational principles are proportional to the energy dissipation averaged over the period of oscillation. The last principles can be used in the homogenization theory to obtain the bounds on the effective properties of composite materials with complex valued properties tensors.
Cherkaev et al. (Sat,) studied this question.