Alphabeta Math
CounterexampleConstruction: AI-adaptedVerification: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-07
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Liouville needs one sided boundedness

Statement refuted

For n≥2, an entire real harmonic function need not be constant without a one-sided bound. The coordinate function u(x)=x1 is a counterexample.

Facts & Assumptions

Given: The objects and hypotheses in the statement refuted.

[F1]

An entire real harmonic function is constant if it is bounded above or bounded below. (Liouville theorem for bounded harmonic functions).

Counterexample

technique · direct
1.1

All second derivatives vanish, so u is entire harmonic, and u(e1)=1≠0=u(0).

givenalgebra
2.1

As t→∞, u(te1)=t→∞ and u(−te1)=−t→−∞. Thus neither global one-sided boundedness hypothesis of Liouville is present, and constancy fails.

F1step 1.1algebra∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

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Sources