On Unique Solutions of Multiple-State Optimal Design Problems on an Annulus

Abstract

We study the uniqueness and explicit derivation of the relaxed optimal solutions, corresponding to the minimization of weighted sum of potential energies for a mixture of two isotropic conductive materials on an annulus. Recently, it has been shown by Burazin and Vrdoljak that even for multiple-state problems, if the domain is spherically symmetric, then the proper relaxation of the problem by the homogenization method is equivalent to a simpler relaxed problem, stated only in terms of local proportions of given materials. This enabled explicit calculation of a solution on a ball, while problems on an annulus appeared to be more tedious. In this paper, we discuss the uniqueness of a solution of this simpler relaxed problem, when the domain is an annulus and we use the necessary and sufficient conditions of optimality to present a method for explicit calculation of the unique solution of this simpler proper relaxation, which is demonstrated on an example.

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Acknowledgements

This work was supported in part by Croatian Science Foundation under the Project 9780 WeConMApp.

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Correspondence to Krešimir Burazin.

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Communicated by Gregoire Allaire.

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Burazin, K. On Unique Solutions of Multiple-State Optimal Design Problems on an Annulus. J Optim Theory Appl 177, 329–344 (2018). https://doi.org/10.1007/s10957-018-1284-7

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Keywords

  • Stationary diffusion
  • Optimal design
  • Homogenization
  • Optimality conditions

Mathematics Subject Classification

  • 80A20
  • 49J20
  • 80M40
  • 49K35