Liouville properties for p-harmonic maps with finite q-energy

Shu Cheng Chang, Jui-Tang Chen, Shihshu Walter Wei

Research output: Contribution to journalArticle

9 Citations (Scopus)

Abstract

We introduce and study an approximate solution of the p-Laplace equation and a linearlization ℒ ϵ of a perturbed p-Laplace operator. By deriving an ℒ ϵ -type Bochner’s formula and Kato type inequalities, we prove a Liouville type theorem for weakly p-harmonic functions with finite p-energy on a complete noncompact manifold M which supports a weighted Poincaré inequality and satisfies a curvature assumption. This nonexistence result, when combined with an existence theorem, yields in turn some information on topology, i.e. such an M has at most one p-hyperbolic end. Moreover, we prove a Liouville type theorem for strongly p-harmonic functions with finite q-energy on Riemannian manifolds. As an application, we extend this theorem to some p-harmonic maps such as p-harmonic morphisms and conformal maps between Riemannian manifolds. In particular, we obtain a Picard-type theorem for p-harmonic morphisms.

Original languageEnglish
Pages (from-to)787-825
Number of pages39
JournalTransactions of the American Mathematical Society
Volume368
Issue number2
DOIs
Publication statusPublished - 2016 Feb 1

Fingerprint

P-harmonic Maps
P-harmonic
Harmonic functions
Harmonic Morphisms
Liouville Type Theorem
Laplace equation
Energy
Harmonic Functions
Riemannian Manifold
Topology
P-Laplace Operator
P-Laplace Equation
Conformal Map
Weighted Inequalities
Noncompact Manifold
Theorem
Existence Theorem
Nonexistence
Approximate Solution
Curvature

Keywords

  • Liouville type properties
  • Perturbed p-Laplace operator
  • Weakly p-harmonic function
  • p-harmonic map
  • p-hyperbolic end

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics

Cite this

Liouville properties for p-harmonic maps with finite q-energy. / Chang, Shu Cheng; Chen, Jui-Tang; Wei, Shihshu Walter.

In: Transactions of the American Mathematical Society, Vol. 368, No. 2, 01.02.2016, p. 787-825.

Research output: Contribution to journalArticle

Chang, Shu Cheng ; Chen, Jui-Tang ; Wei, Shihshu Walter. / Liouville properties for p-harmonic maps with finite q-energy. In: Transactions of the American Mathematical Society. 2016 ; Vol. 368, No. 2. pp. 787-825.
@article{d80403838f634863aea9596e24eed340,
title = "Liouville properties for p-harmonic maps with finite q-energy",
abstract = "We introduce and study an approximate solution of the p-Laplace equation and a linearlization ℒ ϵ of a perturbed p-Laplace operator. By deriving an ℒ ϵ -type Bochner’s formula and Kato type inequalities, we prove a Liouville type theorem for weakly p-harmonic functions with finite p-energy on a complete noncompact manifold M which supports a weighted Poincar{\'e} inequality and satisfies a curvature assumption. This nonexistence result, when combined with an existence theorem, yields in turn some information on topology, i.e. such an M has at most one p-hyperbolic end. Moreover, we prove a Liouville type theorem for strongly p-harmonic functions with finite q-energy on Riemannian manifolds. As an application, we extend this theorem to some p-harmonic maps such as p-harmonic morphisms and conformal maps between Riemannian manifolds. In particular, we obtain a Picard-type theorem for p-harmonic morphisms.",
keywords = "Liouville type properties, Perturbed p-Laplace operator, Weakly p-harmonic function, p-harmonic map, p-hyperbolic end",
author = "Chang, {Shu Cheng} and Jui-Tang Chen and Wei, {Shihshu Walter}",
year = "2016",
month = "2",
day = "1",
doi = "10.1090/tran/6351",
language = "English",
volume = "368",
pages = "787--825",
journal = "Transactions of the American Mathematical Society",
issn = "0002-9947",
publisher = "American Mathematical Society",
number = "2",

}

TY - JOUR

T1 - Liouville properties for p-harmonic maps with finite q-energy

AU - Chang, Shu Cheng

AU - Chen, Jui-Tang

AU - Wei, Shihshu Walter

PY - 2016/2/1

Y1 - 2016/2/1

N2 - We introduce and study an approximate solution of the p-Laplace equation and a linearlization ℒ ϵ of a perturbed p-Laplace operator. By deriving an ℒ ϵ -type Bochner’s formula and Kato type inequalities, we prove a Liouville type theorem for weakly p-harmonic functions with finite p-energy on a complete noncompact manifold M which supports a weighted Poincaré inequality and satisfies a curvature assumption. This nonexistence result, when combined with an existence theorem, yields in turn some information on topology, i.e. such an M has at most one p-hyperbolic end. Moreover, we prove a Liouville type theorem for strongly p-harmonic functions with finite q-energy on Riemannian manifolds. As an application, we extend this theorem to some p-harmonic maps such as p-harmonic morphisms and conformal maps between Riemannian manifolds. In particular, we obtain a Picard-type theorem for p-harmonic morphisms.

AB - We introduce and study an approximate solution of the p-Laplace equation and a linearlization ℒ ϵ of a perturbed p-Laplace operator. By deriving an ℒ ϵ -type Bochner’s formula and Kato type inequalities, we prove a Liouville type theorem for weakly p-harmonic functions with finite p-energy on a complete noncompact manifold M which supports a weighted Poincaré inequality and satisfies a curvature assumption. This nonexistence result, when combined with an existence theorem, yields in turn some information on topology, i.e. such an M has at most one p-hyperbolic end. Moreover, we prove a Liouville type theorem for strongly p-harmonic functions with finite q-energy on Riemannian manifolds. As an application, we extend this theorem to some p-harmonic maps such as p-harmonic morphisms and conformal maps between Riemannian manifolds. In particular, we obtain a Picard-type theorem for p-harmonic morphisms.

KW - Liouville type properties

KW - Perturbed p-Laplace operator

KW - Weakly p-harmonic function

KW - p-harmonic map

KW - p-hyperbolic end

UR - http://www.scopus.com/inward/record.url?scp=84951937520&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84951937520&partnerID=8YFLogxK

U2 - 10.1090/tran/6351

DO - 10.1090/tran/6351

M3 - Article

VL - 368

SP - 787

EP - 825

JO - Transactions of the American Mathematical Society

JF - Transactions of the American Mathematical Society

SN - 0002-9947

IS - 2

ER -