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

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.

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Principal part and principal symbol of a scalar PDE

Definition

Let

$L u = \sum_{∣ α ∣ \leq m} a_{α} (x) D^{α} u$

be a linear scalar differential operator of order $m$ on an open $Ω \subseteq R^{n}$ .

Its principal part is the homogeneous order- $m$ operator

$L_{m} u : = \sum_{∣ α ∣ = m} a_{α} (x) D^{α} u .$

Its principal symbol is the homogeneous polynomial in the covector $ξ \in (R^{n})^{*}$ given by

$p_{m} (x, ξ) : = \sum_{∣ α ∣ = m} a_{α} (x) ξ^{α} .$

If the equation is quasilinear, first freeze the lower jet $(u, D u, \dots, D^{m - 1} u)$ at the point under discussion; the resulting linear operator in the highest derivatives has a principal part and principal symbol defined by the same formula.

Depends on

Used by

Dependency tree · one level

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Sources

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