### Abstract

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 language | English (US) |
---|---|

Pages (from-to) | 2994-3034 |

Number of pages | 41 |

Journal | SIAM Journal on Mathematical Analysis |

Volume | 48 |

Issue number | 4 |

DOIs | |

State | Published - 2016 |

### Fingerprint

### Keywords

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

### ASJC Scopus subject areas

- Analysis
- Computational Mathematics
- Applied Mathematics

### Cite this

*SIAM Journal on Mathematical Analysis*,

*48*(4), 2994-3034. https://doi.org/10.1137/15M1028960

**Branched microstructures in the Ginzburg-Landau model of type-I superconductors.** / Conti, Sergio; Otto, Felix; Serfaty, Sylvia.

Research output: Contribution to journal › Article

*SIAM Journal on Mathematical Analysis*, vol. 48, no. 4, pp. 2994-3034. https://doi.org/10.1137/15M1028960

}

TY - JOUR

T1 - Branched microstructures in the Ginzburg-Landau model of type-I superconductors

AU - Conti, Sergio

AU - Otto, Felix

AU - Serfaty, Sylvia

PY - 2016

Y1 - 2016

N2 - 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.

AB - 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.

KW - Calculus of variations

KW - Ginzburg-landau model

KW - Pattern formation

KW - Superconductivity

UR - http://www.scopus.com/inward/record.url?scp=84984993643&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84984993643&partnerID=8YFLogxK

U2 - 10.1137/15M1028960

DO - 10.1137/15M1028960

M3 - Article

VL - 48

SP - 2994

EP - 3034

JO - SIAM Journal on Mathematical Analysis

JF - SIAM Journal on Mathematical Analysis

SN - 0036-1410

IS - 4

ER -