We consider two-phase multiple state optimal design problems for stationary diffusion equation. Both phases are taken to be isotropic, and the goal is to find the optimal distribution of materials within domain, with prescribed amounts, that minimizes a weighted sum of energies. In the case of one state equation, it is known that the proper relaxation of the problem via the homogenization theory is equivalent to a simpler relaxed problem, stated only in terms of the local proportion of given materials.
We prove an analogous result for multiple state problems if the number of states is less than the space dimension. In spherically symmetric case, the result holds for arbitrary number of states, and the optimality conditions of a simpler relaxation problem, which are necessary and sufficient, enable us to explicitly calculate the unique solution of proper relaxation for some examples. In contrary to maximization problems, these solutions are not classical.
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This work has been supported in part by Croatian Science Foundation under the project 9780 WeConMApp.
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Burazin, K., Vrdoljak, M. Exact Solutions in Optimal Design Problems for Stationary Diffusion Equation. Acta Appl Math 161, 71–88 (2019). https://doi.org/10.1007/s10440-018-0204-z