  In this paper, we focus on finite Bravais lattice energies per point in two
dimensions. We compute the first and second derivatives of these energies. We
prove that the Hessian at the square and the triangular lattice are diagonal
and we give simple sufficient conditions for the local minimality of these
lattices. Furthermore, we apply our result to Lennard--Jones type interacting
potentials that appear to be accurate in many physical and biological models.
The goal of this investigation is to understand how the minimum of the
Lennard--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 which
A 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 A
must belong so that the square lattice is a minimizer (resp. a saddle point).
Finally, we present a complete conjecture, based on numerical investigations
and rigorous results among rhombic and rectangular lattices, for the minimality
of the classical Lennard--Jones energy per point with respect to its area. In
particular, we prove that the minimizer is a rectangular lattice if the area is
sufficiently large.
