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

1.1F1givenalgebra

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

2.1step 1.1algebra∎

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.

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)