LemmaStatement: AI-adaptedProof: 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.

Strict subharmonic perturbation

Statement

Let $n \geq 2$ , $Ω \subseteq R^{n}$ be open, and $u \in C^{2} (Ω)$ be subharmonic. For $ε > 0$ , put $u_{ε} (x) = u (x) + ε ∣ x ∣^{2}$ . Then $Δ u_{ε} \geq 2 n ε > 0$ , and $u_{ε}$ has no interior local maximum.

Facts & Assumptions

Given: The objects and hypotheses in the statement.

[F1]

Subharmonicity of a real $C^{2}$ function means $Δ u \geq 0$ . (Subharmonic and superharmonic functions in rn).

Proof

technique · direct

1.1

$\partial_{i i} ∣ x ∣^{2} = 2$ for each of the $n$ coordinates. Therefore $Δ u_{ε} = Δ u + 2 n ε \geq 2 n ε > 0$ .

F1givenalgebra

2.1

At an interior local maximum $q$ , each restriction $t \mapsto u_{ε} (q + t e_{i})$ would have second derivative at zero at most zero. Summing would give $Δ u_{ε} (q) \leq 0$ , incompatible with the strict positive value already obtained. Hence no such maximum exists.

step 1.1algebra∎

Depends on

Subharmonic and superharmonic functions in rn

Used by

Weak maximum principle for the laplacian Theorem

Dependency tree · two levels

2 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)