DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-07

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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.

Depends on

The Laplacian of a $C^2$ function and of a $C^2$ vector field

Used by

Maximum principle fails for superharmonic maxima Counterexample
Subharmonic quartic and harmonic saddle Example
Classical subharmonic mean value inequalities Lemma
Interior sphere barrier for the laplacian Lemma
Strict subharmonic perturbation Lemma

Dependency tree · two levels

5 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)