DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-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.

Interior sphere condition and sphere normal

Definition

Let $n \geq 2$ , let $Ω \subseteq R^{n}$ be open, and let $p \in \partial Ω$ . An interior tangent ball at $p$ is a ball $B_{R} (a) \subset Ω$ with $R > 0$ and $∣ p - a ∣ = R$ . Existence of such a ball is the interior sphere condition. Its supplied outward sphere direction is $ν = (p - a) / R$ .

For a real function defined on $B_{R} (a) \cup {p}$ , the outward directional derivative, when the following finite limit exists, is $\partial_{ν} u (p) = \lim_{t ↓ 0} \frac{u (p) - u (p - t ν)}{t} .$ The points $p - t ν$ are in the ball for $0 < t < 2 R$ . This definition uses a specified sphere, without assuming a differentiable boundary or a normal field on all of $\partial Ω$ .

Used by

Hopf lemma needs a boundary geometry hypothesis Counterexample
Interior sphere barrier for the laplacian Lemma

Dependency tree · 0 levels

Nothing. This result depends on no other item in the library.

Sources

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