On Born's conjecture about optimal distribution of charges for an infinite ionic crystal

32 pages. 3 Figures. To appear in Journal of Nonlinear Science. DOI :10.1007/s00332-018-9460-3We study the problem for the optimal charge distribution on the sites of a fixed Bravais lattice. In particular, we prove Born's conjecture about the optimality of the rock-salt alternate distribution of charges on a cubic lattice (and more generally on a d-dimensional orthorhombic lattice). Furthermore, we study this problem on the two-dimensional triangular lattice and we prove the optimality of a two-component honeycomb distribution of charges. The results holds for a class of completely monotone interaction potentials which includes Coulomb type interactions. In a more general setting, we derive a connection between the optimal charge problem and a minimization problem for the translated lattice theta function