Manuscript VII-A@1 · VII-A.tex · section · 11da3fe6

The switched solution with fixed free input

The specified switched scalar action › The switched solution with fixed free input · lines 665–701

VII-A@1#eq:phigexp, VII-A@1#eq:phi1, VII-A@1#eq:phi2

Define

ϕ_{g} [g; ϕ_{0}] = ϕ_{0} + κ ϕ_{1, g} + κ^{2} ϕ_{2, g} + O (κ^{3})

D_{g} [g] ϕ_{g} = 0 (mod κ^{3})

and by requiring all positive-order scalar coefficients to have zero Cauchy data. Coefficient comparison gives

ϕ_{1, g} ϕ_{2, g} = - E_{m} V_{g} [h] ϕ_{0}, = - E_{m} (V_{g} [h] ϕ_{1, g} + W_{g} [h, k] ϕ_{0}) = E_{m} V_{g} [h] E_{m} V_{g} [h] ϕ_{0} - E_{m} W_{g} [h, k] ϕ_{0} .

Each source to which $E_{m}$ is applied in these formulas is smooth and compactly supported, because the coefficients of $V_{g}$ and $W_{g}$ are compactly supported. Thus these formulas define smooth classical coefficients without any product of singular distributions.

Displayed formulas in this passage

VII-A@1#eq:phigexp · line 668
$ϕ_{g} [g; ϕ_{0}] = ϕ_{0} + κ ϕ_{1, g} + κ^{2} ϕ_{2, g} + O (κ^{3})$
VII-A@1#eq:phi1 · line 681
$ϕ_{1, g} ϕ_{2, g} = - E_{m} V_{g} [h] ϕ_{0}, = - E_{m} (V_{g} [h] ϕ_{1, g} + W_{g} [h, k] ϕ_{0}) = E_{m} V_{g} [h] E_{m} V_{g} [h] ϕ_{0} - E_{m} W_{g} [h, k] ϕ_{0} .$
VII-A@1#eq:phi2 · line 681
$ϕ_{1, g} ϕ_{2, g} = - E_{m} V_{g} [h] ϕ_{0}, = - E_{m} (V_{g} [h] ϕ_{1, g} + W_{g} [h, k] ϕ_{0}) = E_{m} V_{g} [h] E_{m} V_{g} [h] ϕ_{0} - E_{m} W_{g} [h, k] ϕ_{0} .$
unlabeled · line 676
$D_{g} [g] ϕ_{g} = 0 (mod κ^{3})$

Nearby in this section

lemma: Lemma: Uniqueness of the switched scalar jet