# Interactions beyond nearest neighbours and rigidity of discrete energies: a compactness result and an application to dimension reduction

created by alicandr on 21 Jan 2016
modified by lazzaroni on 30 Nov 2016

[BibTeX]

Preprint

Inserted: 21 jan 2016
Last Updated: 30 nov 2016

Year: 2016
Notes:

This work has been split into two parts, see

http:$//$cvgmt.sns.it$/$paper$/$3247$/$

http:$//$cvgmt.sns.it$/$paper$/$3248$/$

Abstract:

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.