Alphabeta Math
CounterexampleConstruction: AI-adaptedVerification: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-07
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  • 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.
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Maximum principle fails for superharmonic maxima

Statement refuted

The assertion that every superharmonic u∈C2(B1)∩C(B‾1) has its maximum on ∂B1 is false, for every n≥2. A witness is u(x)=−∣x∣2.

Facts & Assumptions

Given: The objects and hypotheses in the statement refuted.

[F1]

Superharmonic means a real C2 function with Δu≤0. (Subharmonic and superharmonic functions in rn).

Counterexample

technique · direct
1.1F1givenalgebra

Δu=−2n<0, so u is superharmonic, with the required regularity.

2.1step 1.1algebra∎

Its unique maximum on the closure is u(0)=0, while it is −1 at every boundary point. Thus a superharmonic maximum can be strictly interior.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

2 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