VARIATIONAL-PRINCIPLES FOR COMPLEX CONDUCTIVITY, VISCOELASTICITY, ANDSIMILAR PROBLEMS IN MEDIA WITH COMPLEX MODULI

Linear processes in media with dissipation arising in conductivity, op tics, viscoelasticity, etc. are considered. Time-periodic fields in su ch media are described by linear differential equations for complex-va lued potentials. The properties of the media are characterized by comp lex valued tensors, for example, by complex conductivity or complex el asticity tensors. Variational formulations are suggested for such prob lems: The functionals whose Euler equations coincide with the original ones are constructed. Four equivalent variational principles are obta ined: two minimax and two minimal ones. The functionals of the obtaine d minimal variational principles are proportional to the energy dissip ation averaged over the period of oscillation. The last principles can be used in the homogenization theory to obtain the bounds on the effe ctive properties of composite materials with complex valued properties tensors.