Lattices with finite renormalized coulombian interaction energy in the plane

Abstract

We present criteria for a certain coulombian interaction energy of infinitely many points in ${ℝ}^d$, $d\ge 1$, with a uniformly charged background, to be finite, as well as examples. We also show that in this unbounded setting, it is not always possible to project an $L_{loc}^2$ vector field onto the set of gradients in a way that reduces its average $L^2$ norm on large balls.