Mariapia (2018) Derivation of a linearised elasticity model from singularly perturbed multiwell energy functionals.Abstract. Linear elasticity can be rigorously derived from finite elasticity under the assumption of small loadings in terms of Gamma-convergence. This was first done in the case of one-well energies with super-quadratic growth and later generalised to different settings, in particular to the case of multi-well energies where the distance between the wells is very small (comparable to the size of the load). In this paper we study the case when the distance between the wells is independent of the size of the load. In this context linear elasticity can be derived by adding to the multi-well energy a singular higher order term which penalises jumps from one well to another. The size of the singular term has to satisfy certain scaling assumptions whose optimality is shown in most of the cases. Finally, the derivation of linear elasticty from a two-well discrete model is provided, showing that the role of the singular perturbation term is played in this setting by interactions beyond nearest neighbours.

Abstract. 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.

Abstract. We study, for times of order 1/ε, solutions of wave equations which are O(ε 2 ) modulations of an ε periodic wave equation. The solutions are of slowly varying amplitude type built on Bloch plane waves with wavelength of order ε. 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 at the plane wave. A ray average hypothesis of small divisor type guarantees stability. We introduce techniques related to those developed in nonlinear geometric optics which lead to new results even on times scales t = O(1). A pair of asymptotic solutions yield accurate approximate solutions of oscillatory initial value problems. The leading term yields H 1 asymptotics when the envelopes are only H 1 .

In the context of nanowire heterostructures we perform a discrete to continuum limit of the corresponding free energy by means of Γ-convergence techniques. Nearest neighbours are identified by employing the notions of Voronoi diagrams and Delaunay triangulations. The scaling of the nanowire is done in such a way that we perform not only a continuum limit but a dimension reduction simultaneously. The main part of the proof is a discrete geometric rigidity result that we announced in an earlier work and show here in detail for a variety of three-dimensional lattices. We perform the passage from discrete to continuum twice: once for a system that compensates a lattice mismatch between two parts of the heterogeneous nanowire without defects and once for a system that creates dislocations. It turns out that we can verify the experimentally observed fact that the nanowires show dislocations when the radius of the specimen is large. 1

We study the homogenization and singular perturbation of the wave equation in a periodic media for long times of the order of the inverse of the period. We consider initial data that are Bloch wave packets, i.e., that are the product of a fast oscillating Bloch wave and of a smooth envelope function. We prove that the solution is approximately equal to two waves propagating in opposite directions at a high group velocity with envelope functions which obey a Schrödinger type equation. Our analysis extends the usual WKB approximation by adding a dispersive, or diffractive, effect due to the non uniformity of the group velocity which yields the dispersion tensor of the homogenized Schrödinger equation.