# American Institute of Mathematical Sciences

September  2018, 8(3&4): 855-877. doi: 10.3934/mcrf.2018038

## Minimization of the elliptic higher eigenvalues for multiphase anisotropic conductors

 1 School of Mathematical Sciences, and LMNS, Fudan University, Shanghai 200433, China 2 School of Mathematical Sciences, and SCMS, Fudan University, Shanghai 200433, China

Dedicated to Professor Jiongmin Yong on the Occasion of His 60th Birthday

Received  August 2017 Revised  February 2018 Published  September 2018

Fund Project: This work was supported in part by NSFC Grant 11771097.

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.

Citation: Hongwei Lou, Xueyuan Yin. Minimization of the elliptic higher eigenvalues for multiphase anisotropic conductors. Mathematical Control & Related Fields, 2018, 8 (3&4) : 855-877. doi: 10.3934/mcrf.2018038
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