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Characteristic hypersurfaces are independent of the defining function

Statement

Let $Σ \cap U = {ϕ = 0} = {ψ = 0}$ be a $C^{1}$ hypersurface with $d ϕ \neq 0$ and $d ψ \neq 0$ on $Σ \cap U$ . If $p_{m}$ is the principal symbol of an order- $m$ scalar operator, then

$p_{m} (x, d ϕ (x)) = 0 ⟺ p_{m} (x, d ψ (x)) = 0$

for every $x \in Σ \cap U$ .

Facts & Assumptions

Given: Two defining functions $ϕ, ψ$ for the same hypersurface $Σ$ , and the principal symbol $p_{m}$ .

[L1]

Characteristic covectors and characteristic hypersurfaces are defined by vanishing of the principal symbol on the conormal (Characteristic covectors, hypersurfaces, and noncharacteristic data).

[L2]

The principal symbol is homogeneous of degree $m$ in the covector variable (Principal part and principal symbol of a scalar PDE).

Proof

technique · direct

1.1

Fix $x \in Σ \cap U$ . Any tangent vector $v \in T_{x} Σ$ is the velocity of a $C^{1}$ curve in $Σ$ , so $d ϕ (x) (v) = d ψ (x) (v) = 0$ because both defining functions vanish on $Σ$ ; thus $d ϕ (x)$ and $d ψ (x)$ have the same kernel, namely the tangent hyperplane $T_{x} Σ$ . Since both covectors are nonzero, the annihilator of that hyperplane is one-dimensional, so there is a unique scalar $h (x) \neq 0$ with $d ψ (x) = h (x) d ϕ (x)$ .

given

2.1

By [L2], homogeneity gives $p_{m} (x, d ψ (x)) = p_{m} (x, h (x) d ϕ (x)) = h (x)^{m} p_{m} (x, d ϕ (x))$ for every $x \in Σ \cap U$ ; since $h (x) \neq 0$ , these values vanish together, so [L1] shows that the characteristic property is independent of the chosen defining function.

L1L2step 1.1∎

Depends on

Used by

In two variables, type and characteristic directions are coordinate invariant Theorem

Cited to discharge well-definedness by Characteristic covectors, hypersurfaces, and noncharacteristic data.

Dependency tree · two levels

4 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)