Homogenization of two-phase metrics and applications

We consider two-phase metrics of the form phi( x, xi) := alpha chi B-alpha(x) vertical bar xi vertical bar + beta chi B-alpha(x) |xi|, where alpha, beta are fixed positive constants and B-alpha, B-beta are disjoint Borel sets whose union is R-N, and prove that they are dense in the class of symmetric Finsler metrics phi satisfying alpha vertical bar xi vertical bar <= phi(x,xi) <= beta vertical bar xi vertical bar on R-N x R-N. Then we study the closure Cl(M-theta(alpha, beta)) of the class M-theta(alpha, beta) of two-phase periodic metrics with prescribed volume fraction theta of the phase alpha. We give upper and lower bounds for the class Cl(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 Gamma-convergence, of two- phase gradient- constraints in composites of the type f (x, del u) <= C(x), with C(x) is an element of {alpha, beta} for almost every x.