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

2012/11/13 by Shu-Cheng Chang, Jui-Tang Chen, Chang, Shu-Cheng +3 · 1 citation
Mathematics · #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.1211.2899

openalex publication_date 2012/11/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 a Kato type inequality, 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, where the range for q contains p. As an application, we extend this theorem to some p-harmonic maps such as p-harmonic morphisms and conformal maps between Riemannian manifolds.

Citations

• C1 + α local regularity of weak solutions of degenerate elliptic equations

Cited by

• Refined Kato type inequalities and new vanishing theorems on complete Kähler and quaternionic Kähler manifolds

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