A multiple state optimal design problem with presence of uncertainty on the right-hand side is considered, in the context of stationary diffusion with two isotropic phases. A similar problem with one state equation has already been considered by Buttazzo and Maestre (2011). We shall address the question of relaxation by the homogenization method and necessary conditions of optimality. The case of discrete probability space leads to another multiple state problem (possibly with an infinite number of states), which could be treated by similar techniques to those presented in Allaire (2002) and Vrdoljak (2010). The relaxation can be expressed in a simpler form for problems with spherical symmetry in the case of minimization (or maximization) of averaged energy, and we present an example which can be solved explicitly.
Submitted April 28, 2017. Published March 2, 2018.
Math Subject Classifications: 49K35, 49K20, 49J20, 80M40
Key Words: Stationary diffusion; optimal design; homogenization; random perturbation; optimality conditions.
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| Marko Vrdoljak |
Department of Mathematics
Faculty of Science
University of Zagreb, Croatia
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