Mariapia Palombaro

A Variational Model for Dislocations at Semi-coherent Interfaces

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 and a bulk elastic energy, spent to decrease the amount of dislocations needed to compensate the lattice misfit. We prove that, for minimizers, the former scales like the surface area of the interface, the latter like its diameter. The proposed continuum model is built on some explicit computations done in the framework of 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.

On a differential inclusion related to the Born--Infeld equations

We study a partial differential relation that arises in the context of the Born--Infeld equations (an extension of Maxwells equations) by using Gromovs method of convex integration in the setting of divergence-free fields.

Rank-(n – 1) convexity and quasiconvexity for divergence free fields

Abstract We prove that rank-(n – 1) convexity does not imply quasiconvexity with respect to divergence free fields (so-called S-quasiconvexity) in for m > n, by adapting the well-known Šveráks counterexample to the solenoidal setting. On the other hand, we also remark that rank-(n – 1) convexity and S-quasiconvexity turn out to be equivalent in the space of n × n diagonal matrices.

Derivation of a rod theory from lattice systems with interactions beyond nearest neighbours

We study continuum limits of discrete models for (possibly heterogeneous) nanowires. The lattice energy includes at least nearest and next-to-nearest neighbour interactions: the latter have the role of penalising changes of orientation. In the heterogeneous case, 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.

Interfacial Energies of Systems of Chiral Molecules

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.

Optimal lower exponent for the higher gradient integrability of solutions to two-phase elliptic equations in two dimensions

We study the higher gradient integrability of distributional solutions u to the equation

Diffractive behavior of the wave equation in periodic media: weak convergence analysis

{{\mathrm{div}}}(\sigma \nabla u) = 0

Existence of minimizers for a polyconvex energy in a crystal with dislocations

div(σ∇u)=0 in dimension two, in the case when the essential range of