The Ginzburg-Landau equation in the Heisenberg group

Isabeau Birindelli, Enrico Valdinoci

Research output: Contribution to journalArticle

10 Citations (Scopus)

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 languageEnglish
Pages (from-to)671-719
Number of pages49
JournalCommunications in Contemporary Mathematics
Volume10
Issue number5
DOIs
Publication statusPublished - 1 Oct 2008
Externally publishedYes

Fingerprint

Ginzburg-Landau Equation
Heisenberg Group
Level Set
Phase transitions
Minimizer
Density Estimates
Transition Model
Local Minimizer
Uniform convergence
Surface area
Convergence Properties
Codimension
Hypersurface
Deduce
Phase Transition
Limiting
Minimise
Metric

Cite this

@article{c3fe8a7acbd245789a5a061c65b1b643,
title = "The Ginzburg-Landau equation in the Heisenberg group",
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.",
keywords = "Allen-Cahn-Ginzburg-Landau-type functionals, Geometric properties of minimizers, Subelliptic operators and minimal surfaces on the Heisenberg group",
author = "Isabeau Birindelli and Enrico Valdinoci",
year = "2008",
month = "10",
day = "1",
doi = "10.1142/S0219199708002946",
language = "English",
volume = "10",
pages = "671--719",
journal = "Communications in Contemporary Mathematics",
issn = "0219-1997",
publisher = "World Scientific Publishing",
number = "5",

}

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

In: Communications in Contemporary Mathematics, Vol. 10, No. 5, 01.10.2008, p. 671-719.

Research output: Contribution to journalArticle

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 -