arXiv subjects
Publications and source records attributed to Mariapia Palombaro.
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.
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.
We propose nonlinear semi-discrete and discrete models for the elastic energy induced by a finite systems of edge dislocations in two dimensions. Within the dilute regime, we analyze the asymptotic behavior of the nonlinear elastic energy, as the core-radius (in the semi-discrete model) and the lattice spacing (in the purely discrete one) vanish. Our analysis passes through a linearization procedure within the rigorous framework of Gamma-convergence.
We consider a discrete model of planar elasticity where the particles, in the reference configuration, sit on a regular triangular lattice and interact through nearest neighbor pairwise potentials, with bonds modeled as linearized elastic springs. Within this framework we introduce plastic slip fields, whose discrete circulation around each triangle detects the possible presence of an edge dislocation. We provide a $Γ$-convergence analysis, as the lattice spacing tends to zero, of the elastic energy induced by edge dislocations in the energy regime corresponding to a finite number of geometrically necessary dislocations.
In this paper we show the emergence of polycrystalline structures as a result of elastic energy minimisation. For this purpose, we introduce a variational model for two-dimensional systems of edge dislocations, within the so-called core radius approach, and we derive the $Γ$-limit of the elastic energy functional as the lattice space tends to zero. In the energy regime under investigation, the symmetric and skew part of the strain become decoupled in the limit, the dislocation measure being the curl of the skew part of the strain. The limit energy is given by the sum of a plastic term, acting on the dislocation density, and an elastic term, which depends on the symmetric strains. Minimisers under suitable boundary conditions are piece-wise constant antisymmetric strain fields, representing in our model a polycrystal whose grains are mutually rotated by infinitesimal angles.
We study the higher gradient integrability of distributional solutions $u$ to the equation $div(σ\nabla u) = 0$ in dimension two, in the case when the essential range of $σ$ consists of only two elliptic matrices, i.e., $σ\in\{σ_1, σ_2\}$ a.e. in $Ω$. In [4], for every pair of elliptic matrices $σ_1$ and $σ_2$, exponents $p_{σ_1,σ_2}\in(2,+\infty)$ and $q_{σ_1,σ_2}\in (1,2)$ have been characterised so that if $u\in W^{1,q_{σ_1,σ_2}}(Ω)$ is solution to the elliptic equation then $\nabla u\in L^{p_{σ_1,σ_2}}_{\rm weak}(Ω)$ and the optimality of the upper exponent $p_{σ_1,σ_2}$ has been proved. In this paper we complement the above result by proving the optimality of the lower exponent $q_{σ_1,σ_2}$. Precisely, we show that for every arbitrarily small $δ$, one can find a particular microgeometry, i.e., an arrangement of the sets $σ^{-1}(σ_1)$ and $σ^{-1}(σ_2)$, for which there exists a solution $u$ to the corresponding elliptic equation such that $\nabla u \in L^{q_{σ_1,σ_2}-δ}$, but $\nabla u \notin L^{q_{σ_1,σ_2}}.$ The existence of such optimal microgeometries is achieved by convex integration methods, adapting to the present setting the geometric constructions provided in [2] for the isotropic case.
We analyse the rigidity of discrete energies where at least nearest and next-to-nearest neighbour interactions are taken into account. Our purpose is to show that interactions beyond nearest neighbours have the role of penalising changes of orientation and, to some extent, they may replace the positive-determinant constraint that is usually required when only nearest neighbours are accounted for.} In a discrete to continuum setting, we prove a compactness result for a surface-scaled energy and we give bounds on its possible Gamma-limit. In the second part of the paper we follow the approach developed in the first part to study a discrete model for (possibly heterogeneous) nanowires. In the heterogeneous case, by applying the compactness result shown in the first part of the paper, we obtain an estimate on the minimal energy spent to match different equilibria. This gives insight into the nucleation of dislocations in epitaxially grown heterostructured nanowires.
We propose and analyze a simple variational model for dislocations at semi-coherent interfaces. The energy functional describes the competition between two terms: a surface energy induced by dislocations that compensate the lattice misfit at the interface, and a far field elastic energy, spent to decrease the amount of needed dislocations. We prove that the former scales like the surface area of the interface, the latter like its diameter. The proposed continuum model is deduced from some rigorous derivation from the semi-discrete theory of dislocations. Even if we deal with finite elasticity, linearized elasticity naturally emerges in our analysis since the far field strain vanishes as the interface size increases.
We consider a simple model for the assembly of chiral molecules in two dimensions driven by maximization of the contact area. We derive a macroscopic model described by a parameter taking nine possible values corresponding to the possible minimal microscopic patterns and modulated phases of the chiral molecules. We describe the overall behaviour by means of an interaction energy of perimeter type between such phases. This energy is a crystalline perimeter energy, highlighting preferred directions for the interfaces between ensembles of molecules labelled by different values of the parameter.
We study the homogenization of a Schrödinger equation in a locally periodic medium. For the time and space scaling of semi-classical analysis we consider well-prepared initial data that are concentrated near a stationary point (with respect to both space and phase) of the energy, i.e. the Bloch cell eigenvalue. We show that there exists a localized solution which is asymptotically given as the product of a Bloch wave and of the solution of an homogenized Schrödinger equation with quadratic potential.
We study, for times of order 1/h, solutions of Maxwell's equations in an O(h^2) modulation of an h-periodic medium. The solutions are of slowly varying amplitude type built on Bloch plane waves with wavelength of order h. We construct accurate approximate solutions of three scale WKB type. The leading profile is both transported at the group velocity and dispersed by a Schrödinger equation given by the quadratic approximation of the Bloch dispersion relation. A weak ray average hypothesis guarantees stability. Compared to earlier work on scalar wave equations, the generator is no longer elliptic. Coercivity holds only on the complement of an infinite dimensional kernel. The system structure requires many innovations.
Epitaxially grown heterogeneous nanowires present dislocations at the interface between the phases if their radius is big. We consider a corresponding variational discrete model with quadratic pairwise atomic interaction energy. By employing the notion of Gamma-convergence and a geometric rigidity estimate, we perform a discrete to continuum limit and a dimension reduction to a one-dimensional system. Moreover, we compare a defect-free model and models with dislocations at the interface and show that the latter are energetically convenient if the thickness of the wire is sufficiently large.
We study a partial differential relation that arises in the context of the Born-Infeld equations (an extension of the Maxwell's equations) by using Gromov's method of convex integration in the setting of divergence free fields.
This short note is a sequel to our previous papers on the asymptotic behavior of Bloch wave packet solutions of the wave equation in periodic media. The purpose is to prove that the group velocity for these Bloch wave packets is bounded by the maximal speed of propagation for the original wave equation.
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.
Starting from three-dimensional elasticity we derive a rod theory for biphase materials with a prescribed dislocation at the interface. The stored energy density is assumed to be non-negative and to vanish on a set consisting of two copies of SO(3). First, we rigorously justify the assumption of dislocations at the interface. Then, we consider the typical scaling of multiphase materials and we perform an asymptotic study of the rescaled energy, as the diameter of the rod goes to zero, in the framework of $Γ$-convergence.
We prove that rank-$(n-1)$ convexity does not imply ${\mathcal S}$-quasiconvexity (i.e., quasiconvexity with respect to divergence free fields) in ${\mathbb M}^{m\times n}$ for $m>n$, by adapting the well-known Sverak's counterexample [5] to the solenoidal setting. On the other hand, we also remark that rank-$(n-1)$ convexity and ${\mathcal S}$-quasiconvexity turn out to be equivalent in the space of $n\times n$ diagonal matrices. This follows by a generalization of Mueller's work [4].
We prove that for any connected open set $Ω\subset \R^n$ and for any set of matrices $K=\{A_1,A_2,A_3\}\subset M^{m\times n}$, with $m\ge n$ and rank$(A_i-A_j)=n$ for $i\neq j$, there is no non-constant solution $B\in L^{\infty}(Ω,M^{m\times n})$, called exact solution, to the problem Div B=0 \quad \text{in} D'(Ω,\R^m) \quad \text{and} \quad B(x)\in K \text{a.e. in} Ω. In contrast, A. Garroni and V. Nesi \cite{GN} exhibited an example of set $K$ for which the above problem admits the so-called approximate solutions. We give further examples of this type. We also prove non-existence of exact solutions when $K$ is an arbitrary set of matrices satisfying a certain algebraic condition which is weaker than simultaneous diagonalizability.