Harmonic functions via restricted mean-value theorems

2007/09/20 by Mohammad Javaheri, Javaheri, Mohammad
Mathematics · #34K10 #35J25 #65N99 #Analysis of PDEs (math.AP) #FOS: Mathematics #Functional Equations Stability Results #Mathematical Dynamics and Fractals #advanced mathematical theories #math.AP #msc:34K10 #msc:35J25 #msc:65N99

paper · pdf · doi:10.48550/arxiv.0709.3311

9 pages

arxiv created 2007/09/20 · openalex publication_date 2007/09/20 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let f be a function on a bounded domain Ω⊆ ℝⁿ and δ be a positive function on Ω such that B(x,δ(x))⊆ Ω. Let σ(f)(x) be the average of f over the ball B(x,δ(x)). The restricted mean-value theorems discuss the conditions on f,δ, and Ω under which σ(f)=f implies that f is harmonic. In this paper, we study the stability of harmonic functions with respect to the map σ. One expects that, in general, the sequence σⁿ(f) converges to a harmonic function. Among our results, we show that if Ω is strongly convex (respectively C^2,α-smooth for some α∈ [0,1]), the function δ(x) is continuous, and f∈ C⁰( Ω) (respectively, f∈ C^2,α( Ω)), then σⁿ(f) converges to a harmonic function uniformly on Ω.

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