Classical Optimal Designs for Stationary Diffusion with Multiple Phases
We study optimal design problems for stationary diffusion involving one or more state equations and mixtures of an arbitrary number of anisotropic materials. Since such problems typically do not admit classical solutions, we adopt a homogenization-based relaxation framework. The objective considered is the maximization of a weighted sum of the energies associated with each state equation, with particular emphasis on identifying cases in which the optimal design is classical, that is, of bang-bang type, composed solely of the original pure materials. Such cases provide valuable benchmarks for numerical methods in optimal design. A simplified optimization problem expressed in terms of local material proportions is analyzed through a dual formulation in terms of fluxes. Using a saddle-point characterization, we establish a complete description of its optimal solutions. The proposed approach is applied in detail to spherically symmetric problems. In the case of a ball, the method yields explicit classical solutions of the homogenization-based relaxation problem.