Homogenization of two-phase metrics and applications

Abstract

We consider two-phase metrics of the form ϕ(x, ξ) ≔ $\alpha \chi _{B_\alpha } (x)\left| \xi \right| + \beta \chi _{B_\alpha } (x)\left| \xi \right|$, where α,β are fixed positive constants and B _α, B _β are disjoint Borel sets whose union is ℝ^N, and prove that they are dense in the class of symmetric Finsler metrics ϕ satisfying

$$\alpha \left| \xi \right| \leqslant \varphi (x,\xi ) \leqslant \beta \left| \xi \right| on \mathbb{R}^N \times \mathbb{R}^N $$

. Then we study the closure $Cl(\mathcal{M}_\theta ^{\alpha ,\beta } )$ of the class $\mathcal{M}_\theta ^{\alpha ,\beta }$ of two-phase periodic metrics with prescribed volume fraction θ of the phase α. We give upper and lower bounds for the class $Cl(\mathcal{M}_\theta ^{\alpha ,\beta } )$ and 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.