English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

In this paper, we focus on finite Bravais lattice energies per point in twodimensions. We compute the first and second derivatives of these energies. Weprove that the Hessian at the square and the triangular lattice are diagonaland we give simple sufficient conditions for the local minimality of theselattices. Furthermore, we apply our result to Lennard--Jones type interactingpotentials that appear to be accurate in many physical and biological models.The goal of this investigation is to understand how the minimum of theLennard--Jones lattice energy varies with respect to the density of the points.Considering the lattices of fixed area A, we find the maximal open set to whichA must belong so that the triangular lattice is a minimizer (resp. a maximizer)among lattices of area A. Similarly, we find the maximal open set to which Amust belong so that the square lattice is a minimizer (resp. a saddle point).Finally, we present a complete conjecture, based on numerical investigationsand rigorous results among rhombic and rectangular lattices, for the minimalityof the classical Lennard--Jones energy per point with respect to its area. Inparticular, we prove that the minimizer is a rectangular lattice if the area issufficiently large.

Local variational study of 2d lattice energies and application to Lennard-Jones type interactions