#### Vol. 3, No. 1, 2021

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Lattices with finite renormalized coulombian interaction energy in the plane

### Yuxin Ge and Etienne Sandier

Vol. 3 (2021), No. 1, 93–120
##### 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.

##### Keywords
coulombian renormalized energy, Fekete points, jellium, quasiperiodic lattices
Primary: 58E20