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.

Subharmonic and superharmonic functions in rn

Definition

Let $n \geq 2$ , let $Ω \subseteq R^{n}$ be open, and let $u : Ω \to R$ belong to $C^{2} (Ω)$ . With the Laplacian of The Laplacian of a $C^{2}$ function and of a $C^{2}$ vector field, $u$ is subharmonic when $Δ u \geq 0$ at every point, and superharmonic when $Δ u \leq 0$ . Thus $u$ is harmonic exactly when both conditions hold, and $u$ is superharmonic exactly when $- u$ is subharmonic. Here a domain means a nonempty connected open set. This is the classical $C^{2}$ convention; the equivalence with local mean inequalities is established in the mean-inequality lemma.

Subharmonic and superharmonic functions in rn

Definition

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