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

1.1givenalgebra

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.

2.1F1step 1.1∎

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

Comparison principle for classical subharmonic functions

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources