Branched microstructures in the Ginzburg-Landau model of type-I superconductors

Sergio Conti, Felix Otto, Sylvia Serfaty

Research output: Contribution to journalArticlepeer-review


We consider the Ginzburg-Landau energy for a type-I superconductor in the shape of an infinite three-dimensional slab, with two-dimensional periodicity, with an applied magnetic field which is uniform and perpendicular to the slab. We determine the optimal scaling law of the minimal energy in terms of the parameters of the problem when the applied magnetic field is sufficiently small and the sample sufficiently thick. This optimal scaling law is proven via ansatz-free lower bounds and an explicit branching construction which refines further and further as one approaches the surface of the sample. Two different regimes appear, with different scaling exponents. In the first regime, the branching leads to an almost uniform magnetic field pattern on the boundary; in the second one the inhomogeneity survives up to the boundary.

Original languageEnglish (US)
Pages (from-to)2994-3034
Number of pages41
JournalSIAM Journal on Mathematical Analysis
Issue number4
StatePublished - 2016


  • Calculus of variations
  • Ginzburg-landau model
  • Pattern formation
  • Superconductivity

ASJC Scopus subject areas

  • Analysis
  • Computational Mathematics
  • Applied Mathematics


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