Lattice tilings minimizing nonlocal perimeters

In this paper, we prove the existence of periodic tessellations of ℝN minimizing a general nonlocal perimeter functional, defined as the interaction between a set and its complement through a nonnegative kernel, which we assume to be either integrable at the origin, or singular, with a fractional type singularity. We reformulate the optimal partition problem as an isoperimetric problem among fundamental domains associated with discrete subgroups of ℝN, and we provide the existence of a solution by using suitable concentrated compactness type arguments and compactness results for lattices. Finally, we discuss the possible optimality of the hexagonal tessellation in the planar case.