OP-00658 Solution claimed

Global optimality of the Abrikosov triangular lattice

Let $\mathcal{A}_1$ be the set of vector fields $E$ on $\mathbb{R}^2$, locally square-integrable away from a locally finite set $\Lambda \subset \mathbb{R}^2$ with $\sup_{R \ge 1} \#(\Lambda \cap [-R,R]^2)/R^2 < \infty$, that satisfy in the sense of distributions $$\operatorname{div} E = 2\pi\Bigl(\sum_{p \in \Lambda} \delta_p - 1\Bigr), \qquad \operatorname{curl} E = 0.$$ Fix smooth cutoffs $0 \le \chi_R \le 1$, equal to one on $[-R+1, R-1]^2$, supported in $[-R, R]^2$, with uniformly bounded gradients, and let $$W(E) = \limsup_{R \to \infty} \frac{1}{4R^2} \lim_{\eta \downarrow 0} \Bigl[\frac12 \int_{\mathbb{R}^2 \setminus \bigcup_{p \in \Lambda} B(p, \eta)} \chi_R |E|^2\,dx + \pi \log \eta \sum_{p \in \Lambda} \chi_R(p)\Bigr].$$ Let $\Lambda_\triangle = \sqrt{2/\sqrt3}\,\bigl[\mathbb{Z}(1,0) + \mathbb{Z}(\tfrac12, \tfrac{\sqrt3}{2})\bigr]$, the triangular lattice of covolume one, and let $E_\triangle$ be the mean-zero periodic field in $\mathcal{A}_1$ associated with it. This is Sandier and Serfaty's Coulomb renormalized energy, which they derived as a limit of the Ginzburg–Landau energy for vortices in superconductors. In their ball-averaged formulation they proved that its minimum over the admissible class is attained and is approached by fields periodic with respect to square lattices, and that among lattice configurations the triangular lattice is the unique minimizer up to rotation.

Problem. Is $W(E) \ge W(E_\triangle)$ for every $E \in \mathcal{A}_1$? No periodicity is required of competitors.

Source: Sandier and Serfaty (2012), Conjecture 1, for their ball-averaged renormalized energy. Stated by AIM — Open Applied Problems with square cutoffs and unit density, in the normalization of Rota Nodari and Serfaty (2015).

Imported from github.com/MColbrook/AIM.

Classification

Primary: AP — Analysis of PDEs
Secondary: MP, OC
Keywords: Abrikosov lattice, AIM Open Applied Problems, Coulomb gases, renormalized energy

References

@article{sandier2012,
  author        = {Sandier, Etienne and Serfaty, Sylvia},
  title         = {From the {G}inzburg--{L}andau model to vortex lattice problems},
  journal       = {Comm. Math. Phys.},
  volume        = {313},
  number        = {3},
  year          = {2012},
  pages         = {635--743},
  doi           = {10.1007/s00220-012-1508-x},
  eprint        = {1011.4617},
  archivePrefix = {arXiv},
}
@article{serfaty2014,
  author  = {Serfaty, Sylvia},
  title   = {Ginzburg--{L}andau vortices, {C}oulomb gases, and {A}brikosov lattices},
  journal = {C. R. Phys.},
  volume  = {15},
  number  = {6},
  year    = {2014},
  pages   = {539--546},
  doi     = {10.1016/j.crhy.2014.06.001},
}
@article{rotanodari2015,
  author        = {Rota Nodari, Simona and Serfaty, Sylvia},
  title         = {Renormalized energy equidistribution and local charge balance in {2D} {C}oulomb systems},
  journal       = {Int. Math. Res. Not. IMRN},
  volume        = {2015},
  number        = {11},
  year          = {2015},
  pages         = {3035--3093},
  doi           = {10.1093/imrn/rnu031},
  eprint        = {1307.3363},
  archivePrefix = {arXiv},
}
@misc{aim-abrikosov-global-crystallization,
  author       = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
  title        = {{Global optimality of the Abrikosov triangular lattice}},
  howpublished = {AIM --- Open Applied Problems, GitHub repository},
  year         = {2026},
  url          = {https://github.com/MColbrook/AIM/blob/1b7c7fd251c1643bb41f0e3cef38704bcef410fd/problems/234-abrikosov-global-crystallization.md},
  note         = {Entry 234 at commit 1b7c7fd, 2026-10-13},
}

Resolution claims

Triangular minimality for planar Coulomb renormalized energy v1 GitHub ↗

OpenAI

We prove the Sandier--Serfaty conjecture: the triangular lattice of covolume one minimizes planar Coulomb renormalized energy over all admissible curl-free fields with a unit uniform background. Combined with Bétermin and Sandier's asymptotic formula, this also proves the Brauchart--Hardin--Saff conjecture for the linear term of optimal ordered-pair logarithmic energy on the unit two-sphere. The proof uses a direct Voronoi-cell comparison with …

Full resolution · not verified · fidelity not established · posted on GitHub 6 Oct 2026

Status history

When	Change	By	Reason
2026-10-16	Needs verification → Solution claimed	—	Recorded here: math @ 1581d4c, build/main.tex.

Citing this problem

This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.

OP-00658, "Global optimality of the Abrikosov triangular lattice", clopen.solutions.