CorollaryStatement: AI-adaptedProof: AI-adaptedPipeline-generatedprecheck passverified 2026-09-23 (gpt-6-sol)

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
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Entire harmonic functions with bounded gradient are affine

Statement

Assume the Axiom of Countable Choice ${A C}_{ω}$ . Let $n \geq 2$ . If $u : R^{n} \to R$ is harmonic and $\sup_{x} ∣ \nabla u (x) ∣ < \infty$ , then $u (x) = b + c \cdot x$ for some $b \in R$ and $c \in R^{n}$ .

Facts & Assumptions

Given: Countable Choice and the objects and hypotheses in the statement (The Axiom of Countable Choice ( ${A C}_{ω}$ )).

[F1]

Partial derivatives of smooth harmonic functions are smooth harmonic. (Derivatives of harmonic functions are harmonic).

[F2]

An entire real harmonic function with a one-sided bound is constant. (Liouville theorem for bounded harmonic functions).

[F3]

Harmonic functions have the ball mean-value property. (Ball mean-value property for harmonic functions under Countable Choice).

[F4]

A continuous function with the ball mean-value property is smooth harmonic. (Continuous ball-mean-value functions are harmonic).

Proof

technique · direct

1.1F1F2F3F4given

The ball mean property and the continuous mean-value theorem give smoothness of $u$ . Each $\partial_{i} u$ is therefore an entire harmonic function; boundedness of the gradient bounds its absolute value. Liouville makes it a constant $c_{i}$ .

2.1step 1.1algebra∎

For fixed $x$ , the fundamental theorem along the segment $t \mapsto t x$ gives $u (x) - u (0) = \int_{0}^{1} \nabla u (t x) \cdot x d t = c \cdot x$ . Set $b = u (0)$ .

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

21 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

Hunter, Notes on Partial Differential Equations (standard reference, not scraped)