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We study the two dimensional Lennard-Jones energy per particle of latticesand we prove that the minimizer among Bravais lattices with sufficiently largedensity is triangular and that is not the case for sufficiently small density.We give other results about the global minimizer of this energy. Moreover westudy the energy per particle stemming from Thomas-Fermi model in twodimensions and we prove that the minimizer among Bravais lattices with fixeddensity is triangular. We use a result of Montgomery from Number Theory aboutthe minimization of Theta functions in the plane.

Minimization of energy per particle among Bravais lattices in R^2 : Lennard-Jones and Thomas-Fermi cases