Manuscript XI-B@1 · XI-B.tex · section · 063b989e

Geometry and the plateau

The XI-A handoff and convention sheet › Geometry and the plateau · lines 317–359

Retain

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

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

Fields are smooth on a neighbourhood of the closed slab. Indices are raised and lowered with $η$ ; parentheses contain the factor $1/2$ .

The switch satisfies

g \in C_{c}^{\infty} (M_{T}^{\circ}; [0, 1]), P_{g} = int {g = 1} \neq = \emptyset .

The fixed ramp is

σ_{ε} (t) = χ (t / ε), 0 < ε < T,

where

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

Choose $U$ open and relatively compact with

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

Neither $g$ nor $σ_{ε}$ occurs in the free kinetic operator.

Displayed formulas in this passage

unlabeled · line 320
$M_{T} = [0, T] \times R^{3}, M_{T}^{\circ} = (0, T) \times R^{3}, Σ_{0} = {t = 0},$
unlabeled · line 325
$η = diag (1, - 1, - 1, - 1), □ = \partial_{t}^{2} - Δ, P_{m} = □ + m^{2}, m \geq 0.$
unlabeled · line 335
$g \in C_{c}^{\infty} (M_{T}^{\circ}; [0, 1]), P_{g} = int {g = 1} \neq = \emptyset .$
unlabeled · line 340
$σ_{ε} (t) = χ (t / ε), 0 < ε < T,$
unlabeled · line 344
$χ (s) = ⎩ ⎨ ⎧ 0, \frac{e ^{- 1/ s}}{e ^{- 1/ s} + e ^{- 1/ (1 - s)}}, 1, s \leq 0, 0 < s < 1, s \geq 1.$
unlabeled · line 354
$U \subset P_{g} \cap {ε < t < T}, g_{U} = C_{c}^{\infty} (U; T M) .$