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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This work was supported in part by Croatian Science Foundation under the Project 9780 WeConMApp.
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
- Stationary diffusion
- Optimal design
- Optimality conditions
Mathematics Subject Classification