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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

1.1given

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 $d ψ (x) = h (x) d ϕ (x)$ .

2.1L1L2step 1.1∎

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.

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)