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The Asymptotic Theory of Ginzburg--Landau Critical Points on Domains with Corners

  • Yucen Han
  • Apala Majumdar
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Abstract

We study the critical points Uε of the two-dimensional Ginzburg--Landau energy on domains with corners Ω and imposed Dirichlet boundary condition which vanishes precisely at the vertices V. We isolate the forced scalar boundary layer ρε by the weighted Lassoued--Mironescu splitting. Assuming the reduced logarithmic energy bound and vertex logarithmic tightness, we prove that after passing to a subsequence, the normalized maps vε = Uε/ρε converges in Ckloc(Ω∖ A) ∀ k≥0 away from a finite set of interior vortices A. The limiting map is a canonical S1-valued harmonic map u* containing integer interior vortex factors and fixed fractional corner factors. The renormalized energy is defined by removing discs around the interior vortices and sectors around the vertices and subtracting the corresponding logarithmic divergences. The interior vortex locations of the limiting map are critical points of this renormalized energy.

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Cite

BibTeXRIS
BibTeX
@misc{han2026asymptotic,
  title = {{The Asymptotic Theory of Ginzburg--Landau Critical Points on Domains with Corners}},
  author = {Yucen Han and Apala Majumdar},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.01410},
  eprint = {2610.01410},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.01410},
}
RIS
TY  - UNPB
TI  - The Asymptotic Theory of Ginzburg--Landau Critical Points on Domains with Corners
AU  - Yucen Han
AU  - Apala Majumdar
PY  - 2026
DA  - 2026/10/01
PB  - arXiv
DO  - 10.48550/arxiv.2610.01410
UR  - https://arxiv.org/abs/2610.01410
AB  - We study the critical points U_ε of the two-dimensional Ginzburg--Landau energy on domains with corners Ω and imposed Dirichlet boundary condition which vanishes precisely at the vertices V. We isolate the forced scalar boundary layer ρ_ε by the weighted Lassoued--Mironescu splitting. Assuming the reduced logarithmic energy bound and vertex logarithmic tightness, we prove that after passing to a subsequence, the normalized maps v_ε = U_ε/ρ_ε converges in C^k_loc(Ω∖ A) ∀ k≥0 away from a finite set of interior vortices A. The limiting map is a canonical S^1-valued harmonic map u_* containing integer interior vortex factors and fixed fractional corner factors. The renormalized energy is defined by removing discs around the interior vortices and sectors around the vertices and subtracting the corresponding logarithmic divergences. The interior vortex locations of the limiting map are critical points of this renormalized energy.
ER  -