Univalentσ-harmonic mappings: applications to composites

Abstract

Abstract. This paper is part of a larger project initiated with [2]. The final aim of the present paper is to give bounds for the homogenized (or effective) conductivity in two dimensional linear conductivity. The main focus is therefore the periodic setting. We prove new variational principles that are shown to be of interest in finding bounds on the homogenized conductivity. Our results unify previous approaches by the second author and make transparent the central role of quasiconformal mappings in all the two dimensional G-closure problems in conductivity.

Is it wise to keep laminating?

“…This result has been proved in [1] (see [2] for application to composites of the latter result). 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].…”

Invertible harmonic mappings, beyond Kneser

“…We recall that this theorem had a remarkable impact in the development of the theory of minimal surfaces, see for instance [17]. Its influence appears also in other areas of mathematics, let us mention here homogenization and effective properties of materials [2,3,5], inverse boundary value problems [1,10,11] and, quite recently, variational problems for maps of finite distortion [4]. See also, as general references, and for many interesting related results, the book by Duren [9] and the review article by Bshouty and Hengartner [6].…”

Estimates for the dilatation of $σ$-harmonic mappings

“…Next, since we know from [7], as already remarked, that in order to have σ ∈ Σ qc , it suffices to exhibit a single σ-harmonic mapping whose dilatation is bounded, we focus on the special class of so-called periodic σ-harmonic mappings, which are of special relevance in homogenization and which have beee analyzed in depth in [5] and in [6]. More precisely, denoting by W 1,2 ♯ (R 2 , R 2 ) the space of W 1,2 loc mappings whose components are 1-periodic in each variable, and assuming that σ ∈ M(α, β, Ω) is also 1-periodic, in each variable, we consider…”