TheoremStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passverified 2026-09-23 (gpt-6-sol)

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Liouville theorem for bounded harmonic functions

Statement

Assume the Axiom of Countable Choice ${A C}_{ω}$ . Let $n \geq 2$ and $u : R^{n} \to R$ be harmonic. If $u$ is bounded above or bounded below on all of $R^{n}$ , then $u$ is constant.

Facts & Assumptions

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

[F1]

Under Countable Choice, nonnegative harmonic functions on $B_{R} (y)$ satisfy $v (x) \leq (R / (R - ∣ x - y ∣))^{n} v (y)$ (Harnack inequality on a ball).

Proof

technique · direct

1.1givenalgebra

If $u \geq b$ , set $v = u - b$ ; if $u \leq b$ , set $v = b - u$ . In either situation $v$ is nonnegative and entire harmonic.

2.1F1step 1.1algebra∎

Fix $x, y \in R^{n}$ . For every $R > ∣ x - y ∣$ , ball Harnack at center $y$ gives $v (x) \leq (R / (R - ∣ x - y ∣))^{n} v (y)$ . Let $R \to \infty$ to obtain $v (x) \leq v (y)$ . Interchanging $x, y$ gives equality. Therefore $v$ , and hence $u$ , is constant.

Depends on

Used by

Entire harmonic functions with bounded gradient are affine Corollary
Positive entire harmonic functions are constant Corollary
Poisson kernel and bounded Dirichlet problem on a half-space Theorem

Dependency tree · two levels

9 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

Gantumur, Harmonic functions (standard reference, not scraped)