A maximum principle on unbounded domains and a Liouville theorem for fractional p-harmonic functions

2019/05/24 by Wenxiong Chen, Leyun Wu, Chen, Wenxiong +1 · 1 citation
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Differential Equations Analysis #Nonlinear Partial Differential Equations #math.AP

paper · pdf · doi:10.48550/arxiv.1905.09986

arxiv created 2019/05/24 · openalex publication_date 2019/05/24 · arxiv updated 2019/05/27 · openalex created_date 2019/05/29 · openalex updated_date 2026/07/28

Abstract

In this paper, we establish the following Liouville theorem for fractional p-harmonic functions. \em Assume that u is a bounded solution of (-\lap)^s_p u(x) = 0, x ∈ ℝⁿ, with 0<s<1 and p ≥ 2. Then u must be constant. A new idea is employed to prove this result, which is completely different from the previous ones in deriving Liouville theorems. For any given hyper-plane in ℝⁿ, we show that u is symmetric about the plane. To this end, we established a \em maximum principle for anti-symmetric functions on any half space. We believe that this \em maximum principle, as well as the ideas in the proof, will become useful tools in studying a variety of problems involving nonlinear non-local operators.

Citations

• Differential equations on riemannian manifolds and their geometric applications
• Multidimensional nonlinear diffusion arising in population genetics
• Liouville-type theorems and harnack-type inequalities for semilinear elliptic equations
• Classification of solutions for an integral equation

Cited by

• Hopf's lemmas for parabolic fractional Laplacians and parabolic fractional p-Laplacians

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