1. Motivation and Framework

It is not difficult to motivate, from a practical point of view, the kind of situations we would like to deal with and analyze. We have selected a typical example in heat conduction, but many other examples are as valid as this one. Suppose we have two very different materials at our disposal: the first, with conductivity α ₁ = 1, is a good and expensive conductor; the other is a cheap material, almost an insulator with conductivity coefficient α ₀ = 0. 001. These two materials are to be used to fill up a given design domain Q, which we assume to be a unit square for simplicity (Fig. 1.1), in given proportions t ₁, t ₀, with t ₁ + t ₀ = 1. Typically, t ₁ < t ₀ given that the first material is much more expensive than the second. We will take, for definiteness, t ₁ = 0. 4, t ₀ = 0. 6. The thermal device is isolated all over ∂ Q, except for a small sink Γ ₀ at the middle of the left side where we normalize temperature to vanish, and there is a uniform source of heat all over Q of size unity. The mixture of the two materials is to be decided so that the dissipated energy is as small as possible.