CorollaryStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-07

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

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.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

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Nonnegative harmonic function with an interior zero vanishes

Statement

Let $n \geq 2$ and let $u \geq 0$ be harmonic on a domain $Ω \subseteq R^{n}$ . Then either $u \equiv 0$ or $u (x) > 0$ for every $x \in Ω$ .

Facts & Assumptions

Given: The objects and hypotheses in the statement.

[F1]

A harmonic function on a domain attaining an interior global maximum or minimum is constant. (Strong maximum principle for harmonic functions).

Proof

technique · direct

1.1

If $u (a) = 0$ at some $a \in Ω$ , it attains its global minimum there. The strong harmonic maximum/minimum principle makes it constant, hence identically zero.

F1given

2.1

If there is no such point, nonnegativity forces $u (x) > 0$ at every point. These alternatives exhaust the possibilities.

step 1.1given∎

Depends on

Strong maximum principle for harmonic functions

Used by

Harnack inequality on compact subsets Corollary

Dependency tree · two levels

3 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)