Manuscript XI-C@1 · XI-C.tex · section · acec1422

Slab, ramp, and admitted classical parameters

Copied handoff and frozen conventions › Slab, ramp, and admitted classical parameters · lines 334–370

Use

M_{T} = [0, T] \times R^{3}, M_{T}^{\circ} = (0, T) \times R^{3}, Σ = {t = 0},

η = diag (1, - 1, - 1, - 1), □ = \partial_{t}^{2} - Δ, P_{m} = □ + m^{2}, m \geq 0.

All classical configurations are smooth on a neighbourhood of the closed slab. Parentheses include the factor $1/2$ .

Fix $0 < ε < T$ , with the same smooth ramp $σ_{ε} (t) = χ (t / ε)$ as XI-A and XI-B:

χ (s) = ⎩ ⎨ ⎧ 0, \frac{e ^{- 1/ s}}{e ^{- 1/ s} + e ^{- 1/ (1 - s)}}, 1, s \leq 0, 0 < s < 1, s \geq 1.

Fix $g \in C_{c}^{\infty} (M_{T}^{\circ}; [0, 1])$ , with nonempty plateau $P_{g} = int {g = 1}$ , and

U \subset P_{g} \cap {ε < t < T}, g_{U} = C_{c}^{\infty} (U; T M) .

The observation test is $f \in C_{c}^{\infty} (M_{T}^{\circ}; R)$ , and $K = supp f$ . No limit of $ε$ is taken.

Displayed formulas in this passage

unlabeled · line 337
$M_{T} = [0, T] \times R^{3}, M_{T}^{\circ} = (0, T) \times R^{3}, Σ = {t = 0},$
unlabeled · line 342
$η = diag (1, - 1, - 1, - 1), □ = \partial_{t}^{2} - Δ, P_{m} = □ + m^{2}, m \geq 0.$
unlabeled · line 352
$χ (s) = ⎩ ⎨ ⎧ 0, \frac{e ^{- 1/ s}}{e ^{- 1/ s} + e ^{- 1/ (1 - s)}}, 1, s \leq 0, 0 < s < 1, s \geq 1.$
unlabeled · line 363
$U \subset P_{g} \cap {ε < t < T}, g_{U} = C_{c}^{\infty} (U; T M) .$