CorollaryStatement: 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.

Poisson supremum estimate from a quadratic barrier

Statement

Let $n \geq 2$ and let $Ω \subseteq R^{n}$ be bounded, nonempty and open, with $Ω \subset {x : 0 < x_{1} < d}$ for some $d > 0$ . If $u \in C^{2} (Ω) \cap C (\overline{Ω})$ and $L = \sup_{Ω} ∣ Δ u ∣ < \infty$ , then $\sup_{\overline{Ω}} ∣ u ∣ \leq \sup_{\partial Ω} ∣ u ∣ + \frac{d^{2}}{2} L .$

Facts & Assumptions

Given: The objects and hypotheses in the statement.

[F1]

Classical comparison holds on bounded nonempty open sets for $C^{2}$ functions continuous on the closure. (Comparison principle for classical subharmonic functions).

Proof

technique · direct

1.1

Put $b = \max_{\partial Ω} ∣ u ∣$ and $v (x) = b + \frac{L}{2} (d^{2} - x_{1}^{2})$ . Since $0 \leq x_{1} \leq d$ on the closure, $v \geq b$ there, and $Δ v = - L \leq Δ u, Δ (- u)$ .

givenalgebra

2.1

Both $u$ and $- u$ are at most $v$ on the boundary, so comparison gives $∣ u ∣ \leq v \leq b + L d^{2} / 2$ on the closure.

F1step 1.1∎

Depends on

Comparison principle for classical subharmonic functions

Used by

Nothing in the library uses this result yet.

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

Gantumur, Harmonic functions (standard reference, not scraped)