### Abstract

We study the Ginzburg-Landau functional I_{ε}(u):= 1/ln(1/ε) ∫_{U} 1/2|∇u|^{2} + 1/4∈^{2} (1- |u|^{2})^{2} dx, for u ∈ H^{1} (U; ℝ^{2}), where U is a bounded, open subset of R^{2}. We show that if a sequence of functions u^{ε} satisfies sup I_{ε}(u^{ε}) < ∞, then their Jacobians Ju^{ε} are precompact in the dual of C_{c}^{0,α} for every α ∈ (0, 1]. Moreover, any limiting measure is a sum of point masses. We also characterize the Γ-limit I(·) of the functionals I_{ε} (·), in terms of the function space B2V introduced by the authors in [16, 17]: we show that I(u) is finite if and only if u ∈ B2V(U; S^{1}), and for u ∈ B2V(U; S^{1}), I(u) is equal to the total variation of the Jacobian measure Ju. When the domain U has dimension greater than two, we prove if I_{ε} (u^{ε}) ≤ C then the Jacobians Ju^{ε} are again precompact in (C_{c}^{0,α})* for all α ∈ (0, 1], and moreover we show that any limiting measure must be integer multiplicity rectifiable. We also show that the total variation of the Jacobian measure is a lower bound for the Γ limit of the Ginzburg-Landau functional.

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

Pages (from-to) | 151-191 |

Number of pages | 41 |

Journal | Calculus of Variations and Partial Differential Equations |

Volume | 14 |

Issue number | 2 |

DOIs | |

State | Published - Mar 1 2002 |

Externally published | Yes |

### All Science Journal Classification (ASJC) codes

- Analysis
- Applied Mathematics

## Fingerprint Dive into the research topics of 'The Jacobian and the Ginzburg-Landau energy'. Together they form a unique fingerprint.

## Cite this

*Calculus of Variations and Partial Differential Equations*,

*14*(2), 151-191. https://doi.org/10.1007/s005260100093