Homogenization of Two–phase Metrics and Applications

We consider two–phase metrics of the form φ(x, ξ) := αχBα(x) |ξ|+βχBα(x) |ξ|, where α,β are fixed positive constants, and Bα, Bβ are disjoint Borel sets whose union is R , and we prove that they are dense in the class of symmetric Finsler metrics φ satisfying α|ξ| ≤ φ(x, ξ) ≤ β|ξ| on R × R . Then we study the closure Cl(M θ ) of the class M α,β θ of two–phase periodic metrics with prescribed volume fraction θ of the phase α. We have not a complete answer to this problem at the moment: we give upper and lower bounds for the class Cl(M θ ), and we localize the problem, generalizing the bounds to the non–periodic setting. Finally, we apply our results to study the closure, in terms of Γ–convergence, of two–phase gradient-constraints in composites of the type f(x,∇u) ≤ C(x), with C(x) ∈ {α, β} for almost every x.