Homogenization of two-phase metrics and applications

Abstract

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

Optimal bounds for periodic mixtures of nearest-neighbour ferromagnetic interactions

“…The previous proposition, together with (20) and the trivial bound (11) gives the bounds in the statement of Theorem 4. We now prove their optimality.…”

“…Even though curves and boundaries of sets have some topological differences, it has been shown in [18], that in the periodic setting the homogenized energy densities can be computed by optimal paths (curves) on the dual lattice. The problem on curves has been studied in [22], where it is shown that homogenized metrics satisfy α|ν| ≤ ϕ(ν) ≤ (θβ + (1 − θ)α)|ν|, but the optimality of such bounds is not proved. That result provides bounds also for the 'dual' equivalent formulation in dimension 2 of the homogenization of periodic perimeter functionals of the formˆ∂ with the same type of a as above (see [5,7]).…”

“…The continuous counterpart of this problem is the determination of optimal bounds for Finsler metrics obtained from the homogenization of periodic Riemannian metrics (see [1,7,6]) of the form b a a(u(t))|u | 2 dt, and a(u) is a periodic function in R 2 taking only the values α and β. This problem has been studied in [11], where it is shown that homogenized metrics satisfy α ≤ ϕ(ν) ≤ (θβ + (1 − θ)α), but the optimality of such bounds is not proved. A 'dual' equivalent formulation in dimension 2 is obtained by considering the homogenization of periodic perimeter functionals of the form ∂A a(x) dH 1 (x) with the same type of a as above (see [3,4]).…”

L∞-variational problems associated to measurable Finsler structures

“…The proof is based on a personal communication with Professor Andrea Davini. In particular, he draws our attention on the useful reference [13] and carefully explains its relation with [12]. We would like to express our gratitude here for his kind help.…”

Design of lattice surface energies

“…Even though curves and boundaries of sets have some topological differences, it has been shown in [19], that in the periodic setting the homogenized energy densities can be computed by optimal paths (curves) on the dual lattice. The problem on curves has been studied in [23], where it is shown that homogenized metrics satisfy α|ν| ≤ ϕ(ν) ≤ (θβ + (1 − θ)α)|ν|, but the optimality of such bounds is not proved. That result provides bounds also for the 'dual' equivalent formulation in dimension 2 of the homogenization of periodic perimeter functionals of the formˆ∂…”