TheoremStatement: Literature-sourcedProof: AI-adaptedPipeline-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.

Strong maximum principle for harmonic functions

Statement

Let $n \geq 2$ , let $Ω \subseteq R^{n}$ be a domain, and let $u \in C^{2} (Ω)$ be harmonic. If $u$ attains a global maximum or a global minimum at a point of $Ω$ , it is constant.

Facts & Assumptions

Given: The objects and hypotheses in the statement.

[F1]

A classical harmonic function equals its ball average on each compactly contained ball. (Ball mean-value property for harmonic functions).

[F2]

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

Proof

technique · direct

1.1

For an attained global maximum $M$ , the set $E = {u = M}$ is nonempty and relatively closed. For $x \in E$ take ${\overline{B}}_{r} (x) \subset Ω$ . The ball mean property makes $\int_{B_{r} (x)} (M - u) = 0$ .

F1given

2.1

The integrand is nonnegative and continuous. A positive value would make its integral positive on a small ball; hence it vanishes everywhere on this ball. Thus $E$ is open, and connectedness gives $E = Ω$ .

F2step 1.1

3.1

If the attained extremum is a minimum, apply the preceding argument to the harmonic function $- u$ . Constancy of $- u$ is constancy of $u$ .

step 2.1algebra∎

Depends on

Used by

Nonnegative harmonic function with an interior zero vanishes Corollary
Strong maximum principle needs connectedness Counterexample
Subharmonic quartic and harmonic saddle Example
Maximum principles need domain and boundary hypotheses Remark

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)