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Higher eigenvalues of composite materials for anisotropic conductors are considered. To get the existence result for minimizing problems, relaxed problems are introduced by the homogenization method. Then, necessary conditions for minimizers are yielded. Based on the necessary conditions, it is shown that in some cases, optimal conductivities of relaxed minimizing problems can be replaced equivalently by a weighted harmonic mean of conductivities. M (ν 0 , ν 1 ; µ 0 , µ 1 ) := L ∞ (Ω; [ν 0 , νdoi:10.3934/mcrf.2018038 fatcat:txdn6nx2njggrar4s7mlm2r5n4