Manuscript XI-E@1 · XI-E.tex · section · 46e55514

The actual fixed-width kernels

Route 1: membership, change of product data, and the loop stop › The actual fixed-width kernels · lines 832–855

XI-E@1#eq:bf

Let $G_{0}^{R} (x - z)$ be the full retarded wave kernel and put

a^{α μ, ab} = \frac{1}{2} η^{α μ} η^{ab} - η^{α (a} η^{b) μ} .

Then

Λ_{ε}^{μ, ab} (x, z) = σ_{ε} (z) a^{α μ, ab} \partial_{α}^{z} G_{0}^{R} (x - z),

and the mixed insertion kernel is

b_{f}^{ab} (x, z) = \partial_{μ}^{x} (f (x) Λ_{ε}^{μ, ab} (x, z)) .

Its support is compact: $x \in supp f$ and $z \in J^{-} (x) \cap {t \geq 0}$ . The ramp is flat at $t = 0$ , so extension to the ambient space adds no boundary conormal.

Displayed formulas in this passage

XI-E@1#eq:bf · line 847
$b_{f}^{ab} (x, z) = \partial_{μ}^{x} (f (x) Λ_{ε}^{μ, ab} (x, z)) .$
unlabeled · line 835
$a^{α μ, ab} = \frac{1}{2} η^{α μ} η^{ab} - η^{α (a} η^{b) μ} .$
unlabeled · line 841
$Λ_{ε}^{μ, ab} (x, z) = σ_{ε} (z) a^{α μ, ab} \partial_{α}^{z} G_{0}^{R} (x - z),$

The actual fixed-width kernels

Displayed formulas in this passage

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