On Some Properties of Homogenised Coefficients for Stationary Diffusion Problem

We consider optimal design of stationary diffusion problems for two-phase materials. Such problems usually have no solution. A relaxation consists in introducing the notion of composite materials, as fine mixtures of different phases, mathematically described by the homogenisation theory. The problem can be written as an optimisation problem over К(θ), the set of all possible composite materials with given local proportion θ. Tartar and Murat (1985) described the set К(θ)e, for some vector e, and used this result to replace the optimisation over the complicated set К(θ) by a much simpler one. Analogous characterisation holds even for the case of mixing more than two materials (possibly anisotropic), where the set К(θ) is not effectively known (Tartar, 1995). We address the question of describing the set {(Ae, Af : A ∊ К(θ)} (for given e and f), which is important for optimal design problems with multiple state equations (different right-hand sides). In other words, we are interested in describing two columns of matrices in К(θ). In two dimensions we describe this set in appropriate coordinates and give some geometric interpretation. For the three-dimensional case we consider the set {Af : A ∊ К(θ), Ae = t}, for a fixed t, and show how it can be reduced to a two-dimensional one, albeit through tedious computations.