Lattices with finite renormalized coulombian interaction energy in the plane

Abstract

. We present criteria for the coulombian interaction energy of infinitely many points in the plane with a uniformly charged background introduced in [5] to be finite, as well as examples. We also show that in this unbounded setting, it is not always possible to project an L 2loc vector field onto the set of gradients in a way that reduces its average L 2 norm on large balls.