Univalent solutions of an elliptic system of partial differential equations arising in homogenization

Is it wise to keep laminating?

“…Similarly if Y is replaced by any simply connected domain which is convex and the solution is searched for in H 1 0 rather than in H 1 , the same conclusion hold [1]. A weaker but equally useful result was previously proved in [3]. A corollary of any of these results can be obtained thank to the work of Tartar and Murat [17] yielding that for any sequence σ ε satisfying (1.5) and for which the H-limit σ 0 is constant, the corrector matrix converges to a constant matrix P 0 and it satisfies the following property in dimension d = 2.…”

“…Definition 1.1. Assume that for any σ 3 ∈ [σ 2 , ∞) there exists M = M (σ 3 ) ∈ G(σ 3 ) and there exists a bound B(σ 3 ) ⊇ G(σ 3 ) such that i) the sets B(σ 3 ) converge in the sense of Kuratowski to a set B(∞), namely any convergent sequence (as σ 3 tends to infinity) P (σ 3 ) ∈ B(σ 3 ) converges to a point of B(∞) and any point of B(∞) is the limit of such a sequence; ii) the sequence of matrices M (σ 3 ) converge to a matrix…”

“…Now we choose ξ 1 as the unit-norm eigenvector associated to the eigenvalue λ 3 . By the previous estimates we have…”

Univalentσ-harmonic mappings: applications to composites

“…This has been made by using (apparently) different approaches. In the first of such results [42], use has been made of the fact (proved in [10]) that suitably chosen σ-harmonic mappings U are sense preserving (det DU ≥ 0 a.e. ).…”

“…where ∂ ξ denotes directional derivative in the direction ξ, and that, for a given K ≥ 1, a non constant f ∈ W 1,2 loc (Ω, R 2 ) is said to be a (sense preserving) K-quasiregular mapping if 10) where Df denotes the Jacobian matrix of f . A mapping f will be said K-quasiconformal if in addition it is injective.…”