TheoremStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-07

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Comparison principle for classical subharmonic functions

Statement

Let $n \geq 2$ , let $Ω \subseteq R^{n}$ be bounded, nonempty and open, and let $u, v \in C^{2} (Ω) \cap C (\overline{Ω})$ . If $Δ u \geq Δ v$ in $Ω$ and $u \leq v$ on $\partial Ω$ , then $u \leq v$ on $\overline{Ω}$ .

Facts & Assumptions

Given: The objects and hypotheses in the statement.

[F1]

The weak maximum principle applies to bounded nonempty open sets and subharmonic $C^{2}$ functions continuous on their closures. (Weak maximum principle for the laplacian).

Proof

technique · direct

1.1

The function $w = u - v$ is continuous on the closure, belongs to $C^{2} (Ω)$ , and satisfies $Δ w \geq 0$ and $w \leq 0$ on the boundary.

givenalgebra

2.1

The weak maximum principle gives $\max_{\overline{Ω}} w = \max_{\partial Ω} w \leq 0$ , exactly the required comparison.

F1step 1.1∎

Depends on

Weak maximum principle for the laplacian

Used by

Poisson supremum estimate from a quadratic barrier Corollary
Removable singularity for bounded harmonic functions Corollary
Uniqueness for the classical dirichlet problem Corollary
Subharmonic quartic and harmonic saddle Example

Dependency tree · two levels

4 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)