CounterexampleConstruction: Literature-sourcedVerification: 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.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Weak maximum principle needs boundedness or control at infinity

Statement refuted

Without boundedness or control at infinity, a harmonic function continuous on the closure of an open set and zero on its nonempty boundary need not be nonpositive inside. For $n \geq 2$ , take $Ω = {x \in R^{n} : x_{n} > 0}$ and $u (x) = x_{n}$ .

Facts & Assumptions

Given: The objects and hypotheses in the statement refuted.

[F1]

The weak maximum principle assumes a bounded nonempty open set and a subharmonic $C^{2}$ function continuous on the closure. (Weak maximum principle for the laplacian).

Counterexample

technique · direct

1.1

The half-space is open, connected and unbounded, with nonempty boundary ${x_{n} = 0}$ . The affine function is smooth on all of $R^{n}$ , has Laplacian zero, and vanishes on that boundary.

givenalgebra

2.1

For every $t > 0$ , $u (t e_{n}) = t > 0$ , and these values tend to infinity as $t \to \infty$ . Hence the proposed interior bound fails. The bounded-open-set hypothesis of the weak maximum principle does not hold for this example.

F1step 1.1algebra∎

Depends on

Weak maximum principle for the laplacian

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

4 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

Thomas Schmidt, Lectures Notes, PDE (standard reference, not scraped)