Vincenzo Nesi

Univalent σ-harmonic mappings

Abstract: We study mappings from ℝ2 into ℝ2 whose components are weak solutions to the elliptic equation in divergence form, div (σ∇u)= 0, which we call σ-harmonic mappings. We prove sufficient conditions for the univalence, i.e., injectivity, of such mappings. Moreover we prove local bounds in BMO on the logarithm of the Jacobian determinant of such univalent mappings, thus obtaining the a.e. nonvanishing of their Jacobian. In particular, our results apply to σ-harmonic mapping associated with any periodic structure and therefore they play an important role in homogenization.

Polycrystalline configurations that maximize electrical resistivity

Abstract A lower bound on the effective conductivity tensor of polycrystalline aggregates formed from a single basic crystal of conductivity σ was recently established by Avellaneda. Cherkaev, Lurie and Milton. The bound holds for any basic crystal, but for isotropic aggregates of a uniaxial crystal, the bound is achieved by a sphere assemblage model of Schulgasser. This left open the question of attainability of the bound when the crystal is not uniaxial. The present work establishes that the bound is always attained by a rather large class of polycrystalline materials. These polycrystalline materials, with maximal electrical resistivity, are constructed by sequential lamination of the basic crystal and rotations of itself on widely separated length scales. The analysis is facilitated by introducing a tensor s = 0(0I + σ)−1 where 0 > 0 is chosen so that Tr s = 1. This tensor s is related to the electric field in the optimal polycrystalline configurations.

Dielectric breakdown: optimal bounds

We study dielectric breakdown for composites made of two isotropic phases. We show that Sachss bound is optimal. This simple example is used to illustrate a variational principle which departs from the traditional one. We also derive the usual variational principle by elementary means without appealing to the technology of convex duality.

Improved bounds for the yield stress of a model polycrystalline material

Anti-plane deformation of a model two-dimensional polycrystal composed of rigid-perfectly plastic grains is considered. Each grain has two orthogonal slip planes, with different Schmid stresses M1 and M2. New upper bounds for the overall, or effective, yield stress are derived, in the case that the microgeometry of the polycrystal is isotropic. One is obtained by application of the translation method, and this is further refined by use of a method that combines the Talbot–Willis variational approach with the translation method. The new bounds always improve on the Taylor–Bishop–Hill bound and also improve on a bound recently established by R.V. Kohn and T.D. Little for a range of values of M1/M2. The “refined” bound also improves on the bound obtained by direct use of the Talbot–Willis machinery, but it was obtained only after a relatively difficult computation.

Quasiconformal mappings as a tool to study certain two-dimensional G-closure problems

A recent theorem due to Astala establishes the best exponent for the area distortion of planar K-quasiconformal mappings. We use a refinement of Astalas theorem due to Eremenko and Hamilton to prove new bounds on the effective conductivity of two-dimensional composites. The bounds are valid for composites made of an arbitrary finite number n of possibly anisotropic phases in prescribed volume fractions. For n= 2 we prove the optimality of the bounds under certain additional assumptions on the G-closure parameters.

Improved bounds for composites and rigidity of gradient fields

We determine an improved lower bound for the conductivity of three-component composite materials. Our bound is strictly larger than the well-known Hashin–Shtrikman bound outside the regime where the latter is known to be optimal. The main ingredient of our result is a new quantitative rigidity estimate for gradient fields in two dimensions.

Rigidity and lack of rigidity for solenoidal matrix fields

The problem of finding a curl–free matrix–valued field E with values in an assigned set of matricesКhas received considerable attention. Given a bounded connected open set Ω, and a compact set Кof m × n matrices, in this paper we establish existence or non–existence results for the following problem: find B ∈ L∞(Ω, М m × n) such that Div B = 0 in Ω in the sense of distributions under the constraint that B(x) ∈ К almost everywhere in Ω. We consider the case of К={A,B}, rank(A−B) = n, and we establish non–existence both for the case of exact solutions described above and for the case of approximate solutions described in §1. We also prove existence of approximate solutions for a suitably chosen triple {A1,A2,A3,} of matrices with rank{Ai −Aj} =n, i≠jandi,j = 1,2,3. We give examples when the differential constraints are of a different type and present some applications to composites.

Translation and related bounds for the response of a nonlinear composite conductor

Lower bounds for the overall energy functions of a class of nonlinear composite conductors are established by two different methods: the translation method, and the use of a comparison linear composite. When the properties of the comparison linear composite are themselves estimated by the translation method, the resulting comparison bound is demonstrated never to be superior to the bound obtained by direct application of the translation method to the nonlinear composite. Examples show, however, that the bounds obtained by these two methods in fact often coincide; when they do, the use of the comparison medium permits the development of more refined bounds, by exploiting better bounds for the linear composite. Particular emphasis is placed on a limiting case of material behaviour for which explicit results can be obtained: a composite dielectric that displays breakdown when the local electric field exceeds a critical value. The formalism delivers an upper bound for the value of the mean field in the composite which induces breakdown in the composite. Bounds are also established for a composite with a power-law relation between current and electric field.

Area Formulas for σ-Harmonic Mappings

The goal of the present paper is two-fold. First, we review some recent progress concerning generalizations of various classical results, such as sufficient conditions to guarantee univalence of harmonic mappings in dimension two, to certain pairs of elliptic partial differential equations with measurable coefficients. Second, we apply these results to prove new area formulas which are valid for a large class of mappings arising as solutions of these pairs of elliptic partial differential equations. Finally, we briefly discuss some applications to homogenized constants in the context of G-closure problems. To Professor Olga A. Ladyzhenskaya with our deep admiration

A Class of Optimal Two-Dimensional Multimaterial Conducting Laminates

We introduce a family of optimal anisotropic two-dimensional multimaterial laminate composites which correspond to extreme overall conductivity. These laminates attain the translation bounds and generalize all previously known constructions for these bounds. The method of construction is based on the analysis of the fields in optimal structures.