Manuscript III@2 · III.tex · section · 82dad3e5

A fixed compact-data background

The C1 triangular Cauchy quotient › A fixed compact-data background · lines 1625–1666

III@2#eq:background, III@2#eq:plateau

Take $A_{⋆} > 0$ , $R > 0$ , and real $χ \in C_{c}^{\infty} (R^{3})$ with $χ = 1$ on $∣ x ∣ \leq R$ . Let $τ = t - t_{0}$ and define

P Φ = 0, Φ (t_{0}, x) = A_{⋆} χ (x), \partial_{t} Φ (t_{0}, x) = 0.

Then

Φ (t) = A_{⋆} cos (τ Ω_{m}) χ, Ω_{m} = - Δ + m^{2} .

Finite propagation and local uniqueness give

Φ (t, x) = A_{⋆} cos (m τ) if ∣ x ∣ + ∣ τ ∣ < R .

The C1 equations are

G^{(1)} (h) = 0, P φ = - L_{h} Φ.

Their coupled transformation is

(h, φ) ⟼ (h + K ξ, φ + X_{ξ} Φ) .

Indeed,

- L_{h + K ξ} Φ = - L_{h} Φ + [P, X_{ξ}] Φ = P (φ + X_{ξ} Φ) .

It preserves harmonic representatives when $□_{η} ξ = 0$ .

Unlike the coupled jet, this solution space does not vary $Φ$ . It is linear in $(h, φ)$ .

Displayed formulas in this passage

III@2#eq:background · line 1630
$P Φ = 0, Φ (t_{0}, x) = A_{⋆} χ (x), \partial_{t} Φ (t_{0}, x) = 0.$
III@2#eq:plateau · line 1642
$Φ (t, x) = A_{⋆} cos (m τ) if ∣ x ∣ + ∣ τ ∣ < R .$
unlabeled · line 1637
$Φ (t) = A_{⋆} cos (τ Ω_{m}) χ, Ω_{m} = - Δ + m^{2} .$
unlabeled · line 1649
$G^{(1)} (h) = 0, P φ = - L_{h} Φ.$
unlabeled · line 1653
$(h, φ) ⟼ (h + K ξ, φ + X_{ξ} Φ) .$
unlabeled · line 1657
$- L_{h + K ξ} Φ = - L_{h} Φ + [P, X_{ξ}] Φ = P (φ + X_{ξ} Φ) .$