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Optimization of the principal eigenvalue for elliptic operators

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Abstract

Maximization and minimization problems of the principle eigenvalue for divergence form second order elliptic operators with the Dirichlet boundary condition are considered. The principal eigen map of such elliptic operators is introduced and some basic properties of this map, including continuity, concavity, and differentiability with respect to the parameter in the diffusibility matrix, are established. For maximization problem, the admissible control set is convexified to get the existence of an optimal convexified relaxed solution. Whereas, for minimization problem, the relaxation of the problem under H-convergence is introduced to get an optimal H-relaxed solution for certain interesting special cases. Some necessary optimality conditions are presented for both problems and a couple of illustrative examples are presented as well.

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Acknowledgements

The authors would like to thank the referee for his/her comments and suggestions.

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Authors and Affiliations

  1. School of Mathematical Sciences, Fudan University, Shanghai, 200433, China

    Hongwei Lou

  2. Department of Mathematics, University of Central Florida, Orlando, FL, 32816, USA

    Jiongmin Yong

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  1. Hongwei Lou
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  2. Jiongmin Yong
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Correspondence to Jiongmin Yong.

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Communicated by F. H. Lin.

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This work is supported in part by NSFC Grant 11771097 and by NSF Grant DMS-1812921.

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Lou, H., Yong, J. Optimization of the principal eigenvalue for elliptic operators. Calc. Var. 60, 139 (2021). https://doi.org/10.1007/s00526-021-02011-8

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  • DOI: https://doi.org/10.1007/s00526-021-02011-8

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