Harmonic functions via restricted mean-value theorems

Let f be a function on a bounded domain Ω ⊆ R 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 σ(f) converges to a harmonic function. Among our results, we show that if Ω is strongly convex (respectively Csmooth for some α ∈ [0, 1]), the function δ(x) is continuous, and f ∈ C(Ω) (respectively, f ∈ C(Ω)), then σ(f) converges to a harmonic function uniformly on Ω.