Invertible harmonic mappings, beyond Kneser

We prove necessary and sufficient criteria of invertibility for planar harmonic mappings which generalize a classical result of H. Kneser, also known as the Rad\'{o}-Kneser-Choquet theorem.Comment: One section added. 15 page

Quantitative estimates on Jacobians for hybrid inverse problems

We consider $\sigma$-harmonic mappings, that is mappings $U$ whose components $u_i$ solve a divergence structure elliptic equation ${\rm div} (\sigma \nabla u_i)=0$, for $i=1,\ldots,n$. We investigate whether, with suitably prescribed Dirichlet data, the Jacobian determinant can be bounded away from zero. Results of this sort are required in the treatment of the so-called hybrid inverse problems, and also in the field of homogenization studying bounds for the effective properties of composite materials.Comment: 15 pages, submitte

Estimates for the dilatation of σ\sigma-harmonic mappings

We consider planar $\sigma$-harmonic mappings, that is mappings $U$ whose components $u^1$ and $u^2$ solve a divergence structure elliptic equation ${\rm div} (\sigma \nabla u^i)=0$, for $i=1,2$. We investigate whether a locally invertible $\sigma$-harmonic mapping $U$ is also quasiconformal. Under mild regularity assumptions, only involving $\det \sigma$ and the antisymmetric part of $\sigma$, we prove quantitative bounds which imply quasiconformality.Comment: 8 pages, to appear on Rendiconti di Matematica e delle sue applicazion

Gradient integrability and rigidity results for two-phase conductivities in two dimensions

This paper deals with higher gradient integrability for σ-harmonic functions u with discontinuous coefficients σ, i.e. weak solutions of div(σ∇u)=0 in dimension two. When σ is assumed to be symmetric, then the optimal integrability exponent of the gradient field is known thanks to the work of Astala and Leonetti and Nesi. When only the ellipticity is fixed and σ is otherwise unconstrained, the optimal exponent is established, in the strongest possible way of the existence of so-called exact solutions, via the exhibition of optimal microgeometries. We focus also on two-phase conductivities, i.e., conductivities assuming only two matrix values, σ1 and σ2, and study the higher integrability of the corresponding gradient field |∇u| for this special but very significant class. The gradient field and its integrability clearly depend on the geometry, i.e., on the phases arrangement described by the sets Ei=σ−1(σi). We find the optimal integrability exponent of the gradient field corresponding to any pair {σ1,σ2} of elliptic matrices, i.e., the worst among all possible microgeometries. We also treat the unconstrained case when an arbitrary but finite number of phases are present

Breaking through borders with \u3c3-harmonic mappings

We consider mappings U=(u1,u2), whose components solve an arbitrary elliptic equation in divergence form in dimension two, and whose respective Dirichlet data \u3c61,\u3c62 constitute the parametrization of a simple closed curve \u3b3. We prove that, if the interior of the curve \u3b3 is not convex, then we can find a parametrization \u3a6=(\u3c61,\u3c62) such that the mapping U is not invertible

Breaking through borders with σ-harmonic mappings

We consider mappings U=(u1,u2), whose components solve an arbitrary elliptic equation in divergence form in dimension two, and whose respective Dirichlet data φ1,φ2 constitute the parametrization of a simple closed curve γ. We prove that, if the interior of the curve γ is not convex, then we can find a parametrization Φ=(φ1,φ2) such that the mapping U is not invertible

Elliptic systems and material interpenetration

We classify the second order, linear, two by two systems for which the two fundamental theorems for planar harmonic mappings, the Rado'-Kneser-Choquet Theorem and the H. Lewy Theorem, hold. They are those which, up to a linear change of variable, can be written in diagonal form with the same operator on both diagonal blocks. In particular, we prove that the aforementioned Theorems cannot be extended to solutions of either the Lame' system of elasticity, or of elliptic systems in diagonal form, even with just slightly different operators for the two components.Comment: 10 pages, two figures, submitte