Minimization of the elliptic higher eigenvalues for multiphase anisotropic conductors

Hongwei Lou, ,School of Mathematical Sciences, and LMNS, Fudan University, Shanghai 200433, China, Xueyuan Yin, ,School of Mathematical Sciences, and SCMS, Fudan University, Shanghai 200433, China
2018 Mathematical Control and Related Fields  
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 , ν
more » ... ]) × L ∞ (Ω; M (µ 0 , µ 1 )). 2010 Mathematics Subject Classification. Primary: 35J15, 47A75; Secondary: 49J20, 49K20.
doi:10.3934/mcrf.2018038 fatcat:txdn6nx2njggrar4s7mlm2r5n4