arXiv subjects

Vincenzo Nesi

Publications and source records attributed to Vincenzo Nesi.

11 recordsLinked to original sources

arXiv2026

Generic configurations in 2D strongly competing systems

We study a problem modelling segregation of an arbitrary number of competing species in planar domains. The solutions give rise to a well known free boundary problem with the domain partitioning itself into subdomains occupied by different species. In principle, several of them can coexist in a neighborhood of any point. However, we show that {\it generically} the domain partitions into subdomains with only triple junctions, meaning that at most three populations meet at the free boundary. Our main tools are the use of the formalism of harmonic maps into singular spaces and the introduction of a complex structure via the Hopf differential.

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arXiv2025

Stability under lamination and polycrystalline effective conductivity

We prove the stability under lamination of a set of real, symmetric 3$\times$3 matrices that can be viewed as a subset of the effective conductivities of a polycrystal. Constructed in a companion paper, such set in combination with several previous constructions provides the best inner bound known so far on the $G$-closure of a three dimensional polycrystal.

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arXiv2025

Differential inclusions and polycrystals

We study the differential inclusion $Du\in K$, where $K$ is an unbounded and rotationally invariant subset of the real symmetric $3\times 3$ matrices. We exhibit a subset of all possible average fields. The corresponding microgeometries are laminates of infinite rank. The problem originated in the search for the effective conductivity of polycrystalline composites. In the latter context, our result is an improvement of the previously known bounds established by Nesi $\&$ Milton, hence proving the optimality of a new full-measure class of microgeometries.

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arXiv2019

Globally diffeomorphic $σ$--harmonic mappings

Given a two--dimensional mapping $U$ whose components solve a divergence structure elliptic equation, we give necessary and sufficient conditions on the boundary so that $U$ is a global diffeomorphism.

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arXiv2018

Locally invertible $σ$-harmonic mappings

We extend a classical theorem by H. Lewy to planar $σ$-harmonic mappings, that is mappings $U$ whose components $u^1$ and $u^2$ solve a divergence structure elliptic equation ${\rm div} (σ\nabla u^i)=0$ , for $i=1,2$. A similar result is established for pairs of solutions of certain second order non--divergence equations.

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arXiv2015

Quantitative estimates on Jacobians for hybrid inverse problems

We consider $σ$-harmonic mappings, that is mappings $U$ whose components $u_i$ solve a divergence structure elliptic equation ${\rm div} (σ\nabla u_i)=0$, 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.

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arXiv2014

Estimates for the dilatation of $σ$-harmonic mappings

We consider planar $σ$-harmonic mappings, that is mappings $U$ whose components $u^1$ and $u^2$ solve a divergence structure elliptic equation ${\rm div} (σ\nabla u^i)=0$, for $i=1,2$. We investigate whether a locally invertible $ σ$-harmonic mapping $U$ is also quasiconformal. Under mild regularity assumptions, only involving $\det σ$ and the antisymmetric part of $σ$, we prove quantitative bounds which imply quasiconformality.

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arXiv2012

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

This paper deals with higher gradient integrability for $σ$-harmonic functions $u$ with discontinuous coefficients $σ$, i.e. weak solutions of $÷(σ\nabla u) = 0$. We focus on two-phase conductivities, and study the higher integrability of the corresponding gradient field $|\nabla u|$. The gradient field and its integrability clearly depend on the geometry, i.e., on the phases arrangement. We find the optimal integrability exponent of the gradient field corresponding to any pair $\{σ_1,σ_2\}$ of positive definite matrices, i.e., the worst among all possible microgeometries. We also show that it is attained by so-called exact solutions of the corresponding PDE. Furthermore, among all two-phase conductivities with fixed ellipticity, we characterize those that correspond to the worse integrability.

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arXiv2008

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.

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arXiv2008

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 Radó-Kneser-Choquet theorem.

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arXiv2007

Beltrami operators, non--symmetric elliptic equations and quantitative Jacobian bounds

In recent studies on the G-convergence of Beltrami operators, a number of issues arouse concerning injectivity properties of families of quasiconformal mappings. Bojarski, D'Onofrio, Iwaniec and Sbordone formulated a conjecture based on the existence of a so-called primary pair. Very recently, Bojarski proved the existence of one such pair. We provide a general, constructive, procedure for obtaining a new rich class of such primary pairs. This proof is obtained as a slight adaptation of previous work by the authors concerning the nonvanishing of the Jacobian of pairs of solutions of elliptic equations in divergence form in the plane. It is proven here that the results previously obtained when the coefficient matrix is symmetric also extend to the non-symmetric case. We also prove a much stronger result giving a quantitative bound for the Jacobian determinant of the so-called \emph{periodic} $σ$-harmonic sense preserving homeomorphisms of $\mathbb C$ onto itself.

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