Archive for Rational Mechanics and Analysis

, Volume 231, Issue 3, pp 1531–1614 | Cite as

Three Dimensional Vortex Approximation Construction and \({\varepsilon}\)-Level Estimates for the Ginzburg–Landau Functional

  • Carlos RománEmail author


We provide a quantitative three dimensional vortex approximation construction for the Ginzburg–Landau functional. This construction gives an approximation of vortex lines coupled to a lower bound for the energy, optimal to leading order, analogous to the two dimensional ones, and valid for the first time at the \({\varepsilon}\)-level. These tools allow for a new approach to analyzing the behavior of global minimizers for the Ginzburg–Landau functional below and near the first critical field in three dimensions, followed in Román (On the first critical field in the three dimensional Ginzburg–Landau model of superconductivity, 2018). In addition, they allow one to obtain an \({\varepsilon}\)-quantitative product estimate for the study of Ginzburg–Landau dynamics.


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I would like to warmly thank my Ph.D. advisors Etienne Sandier and Sylvia Serfaty for suggesting the problem, for their helpful advice, careful reading, and useful comments. I also thank the anonymous referee for helpful comments that led to improvements in the presentation of the appendices of the paper. This work was supported by a public grant overseen by the French National Research Agency (ANR) as part of the “Investissements d’Avenir” program (reference: ANR-10-LABX-0098, LabEx SMP).

Conflict of interest

The author declares that they have no conflict of interest.


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Sorbonne Universités, UPMC Univ Paris 06, CNRS, UMR 7598, Laboratoire Jacques-Louis LionsParisFrance
  2. 2.Mathematisches InstitutUniversität LeipzigLeipzigGermany

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