DefinitionDefinition: Literature-sourcedProof: Not applicableaudited 2026-09-05

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Characteristic covectors, hypersurfaces, and noncharacteristic data

Definition

Let $p_{m} (x, ξ)$ be the principal symbol of an order- $m$ scalar equation.

A nonzero covector $ξ \in T_{x}^{*} Ω$ is characteristic at $x$ when

$p_{m} (x, ξ) = 0.$

If $Σ \subseteq Ω$ is a $C^{1}$ hypersurface locally written as $Σ \cap U = {ϕ = 0}$ with $d ϕ \neq 0$ on $Σ \cap U$ , then $Σ$ is characteristic at $x \in Σ$ when its conormal $d ϕ (x)$ is characteristic, and noncharacteristic at $x$ otherwise.

Accordingly, Cauchy data prescribed on $Σ$ are noncharacteristic near $x$ when the defining conormal is nowhere characteristic there.

Depends on

Principal part and principal symbol of a scalar PDE

Used by

Dependency tree · two levels

3 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)