CounterexampleConstruction: AI-adaptedVerification: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-05

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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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The elliptic-parabolic-hyperbolic trichotomy is not a global taxonomy

Statement refuted

Every partial differential equation belongs globally to exactly one of the three classes elliptic, parabolic, or hyperbolic.

Facts & Assumptions

Given: The Tricomi operator and the fourth-order biharmonic operator $Δ^{2}$ .

[L1]

The Tricomi equation changes type across $y = 0$ (The Tricomi equation changes type across y = 0).

[L2]

The threefold classification on this page is only for scalar real second-order principal symbols (Limits of the elliptic-parabolic-hyperbolic trichotomy).

Counterexample

technique · direct

1.1

By [L1], the single operator $y u_{x x} + u_{y y}$ is elliptic on one open region, hyperbolic on another, and degenerate on the separating line, so even within second order one operator need not belong globally to just one of the three names.

2.1

The biharmonic equation $Δ^{2} u = 0$ has order $4$ , so [L2] says the second-order trichotomy does not classify it at all; the claimed global taxonomy therefore fails both because some operators change type from point to point and because others lie outside the stated second-order scope.

L2step 1.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

5 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

Victor Ivrii, Partial Differential Equations (standard reference, not scraped)