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.

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

Weak minimum principle for the laplacian

Statement

For $n \geq 2$ , a bounded nonempty open $Ω \subseteq R^{n}$ and $u \in C^{2} (Ω) \cap C (\overline{Ω})$ with $Δ u \leq 0$ satisfy $\min_{\overline{Ω}} u = \min_{\partial Ω} u$ .

Facts & Assumptions

Given: The objects and hypotheses in the statement.

[F1]

On a bounded nonempty open set, a subharmonic $C^{2}$ function continuous on the closure has its closure maximum on the boundary. (Weak maximum principle for the laplacian).

Proof

technique · direct

1.1givenalgebra

Put $v = - u$ . It has the same regularity and $Δ v = - Δ u \geq 0$ .

2.1F1step 1.1algebra∎

Apply the weak maximum principle to $v$ and multiply its equality by $- 1$ . Maxima of $- u$ are negatives of minima of $u$ , which proves the claimed equality.

Depends on

Weak maximum principle for the laplacian

Used by

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

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