### Abstract

We consider a functional related with phase transition models in the Heisenberg group framework. We prove that level sets of local minimizers satisfy some density estimates, that is, they behave as "codimension one" sets. We thus deduce a uniform convergence property of these level sets to interfaces with minimal area. These results are then applied in the construction of (quasi)periodic, plane-like minimizers, i.e. minimizers of our functional whose level sets are contained in a spacial slab of universal size in a prescribed direction. As a limiting case, we obtain the existence of hypersurfaces contained in such a slab which minimize the surface area with respect to a given periodic metric.

Original language | English |
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Pages (from-to) | 671-719 |

Number of pages | 49 |

Journal | Communications in Contemporary Mathematics |

Volume | 10 |

Issue number | 5 |

DOIs | |

Publication status | Published - 1 Oct 2008 |

Externally published | Yes |

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*Communications in Contemporary Mathematics*,

*10*(5), 671-719. https://doi.org/10.1142/S0219199708002946

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*Communications in Contemporary Mathematics*, vol. 10, no. 5, pp. 671-719. https://doi.org/10.1142/S0219199708002946

**The Ginzburg-Landau equation in the Heisenberg group.** / Birindelli, Isabeau; Valdinoci, Enrico.

Research output: Contribution to journal › Article

TY - JOUR

T1 - The Ginzburg-Landau equation in the Heisenberg group

AU - Birindelli, Isabeau

AU - Valdinoci, Enrico

PY - 2008/10/1

Y1 - 2008/10/1

N2 - We consider a functional related with phase transition models in the Heisenberg group framework. We prove that level sets of local minimizers satisfy some density estimates, that is, they behave as "codimension one" sets. We thus deduce a uniform convergence property of these level sets to interfaces with minimal area. These results are then applied in the construction of (quasi)periodic, plane-like minimizers, i.e. minimizers of our functional whose level sets are contained in a spacial slab of universal size in a prescribed direction. As a limiting case, we obtain the existence of hypersurfaces contained in such a slab which minimize the surface area with respect to a given periodic metric.

AB - We consider a functional related with phase transition models in the Heisenberg group framework. We prove that level sets of local minimizers satisfy some density estimates, that is, they behave as "codimension one" sets. We thus deduce a uniform convergence property of these level sets to interfaces with minimal area. These results are then applied in the construction of (quasi)periodic, plane-like minimizers, i.e. minimizers of our functional whose level sets are contained in a spacial slab of universal size in a prescribed direction. As a limiting case, we obtain the existence of hypersurfaces contained in such a slab which minimize the surface area with respect to a given periodic metric.

KW - Allen-Cahn-Ginzburg-Landau-type functionals

KW - Geometric properties of minimizers

KW - Subelliptic operators and minimal surfaces on the Heisenberg group

UR - http://www.scopus.com/inward/record.url?scp=51449101940&partnerID=8YFLogxK

U2 - 10.1142/S0219199708002946

DO - 10.1142/S0219199708002946

M3 - Article

VL - 10

SP - 671

EP - 719

JO - Communications in Contemporary Mathematics

JF - Communications in Contemporary Mathematics

SN - 0219-1997

IS - 5

ER -