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

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Strong maximum principle for classical subharmonic functions

Statement

Assume countable choice for the mean-inequality input. Let $n \geq 2$ , let $Ω \subseteq R^{n}$ be a domain, and let $u \in C^{2} (Ω)$ satisfy $Δ u \geq 0$ . If there is $a \in Ω$ with $u (x) \leq u (a)$ for every $x \in Ω$ , then $u$ is constant.

Facts & Assumptions

Given: The objects and hypotheses in the statement.

[F1]

Under countable choice, classical subharmonic functions lie below their ball averages on compactly contained balls. (Classical subharmonic mean value inequalities).

[F2]

A connected space admits no partition into two nonempty disjoint open subsets. (Separation of a topological space, connected and disconnected spaces, clopen sets, and connected subsets).

Proof

technique · direct

1.1

Put $M = u (a)$ and $E = {x \in Ω : u (x) = M}$ . This is nonempty and relatively closed by continuity.

given

2.1

For $x \in E$ choose $r > 0$ with ${\overline{B}}_{r} (x) \subset Ω$ . The ball mean inequality gives $\int_{B_{r} (x)} (M - u) \leq 0$ , while the integrand is nonnegative. If it were positive at a point, continuity would give a positive lower bound on a smaller ball of positive volume, contradicting that integral inequality. Thus $u = M$ throughout $B_{r} (x)$ , and $E$ is open.

F1step 1.1

3.1

If $Ω ∖ E$ were nonempty, it and $E$ would separate $Ω$ into disjoint nonempty relatively open sets. Connectedness therefore gives $E = Ω$ .

F2step 2.1∎

Remarks

The alternative Hopf argument is recorded with its proof after the boundary-point lemma. The present proof uses only the mean inequality and connectedness.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

12 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)