TheoremStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-07

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Continuous ball-mean-value functions are harmonic

Statement

If $u \in C (Ω)$ has the ball mean-value property, then $u \in C^{\infty} (Ω)$ and $Δ u = 0$ .

Proof

Given: $u \in C (Ω)$ has the ball mean-value property.

1.1

Integrating the ball identity in radius gives the spherical identity; Radial mollification fixes local mean-value functions then gives $u = u * ρ_{ε}$ on $Ω_{ε} : = {x : B_{ε} (x) ⋐ Ω}$ [given].

2.1

Thus $u$ is smooth locally. Taylor's formula Second-order Taylor expansion $f (a + h) = f (a) + \nabla f (a) \cdot h + \frac{1}{2} h^{T} H_{f} (a) h + o (∥ h ∥^{2})$ and rotational symmetry give $A_{u} (x, r) = u (x) + r^{2} Δ u (x) / (2 (n + 2)) + o (r^{2})$ [step 1.1].

3.1

The ball identity and division by $r^{2}$ force $Δ u (x) = 0$ for every $x \in Ω$ [step 2.1]. ∎

Depends on

Used by

A pointwise local ball mean property is enough Corollary
Locally uniform limits of harmonic functions are harmonic Theorem

Dependency tree · two levels

16 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

PDE source treatment (standard reference, not scraped)