Manuscript Em06-B@1 · Em06-B-v1.tex · assumption · 61e38859

Model specification: Common classical data

Frozen conventions and the classical object › Background, supports, and input topologies · lines 171–209

Em06-B@1#spec:data

Fix

M = R^{1 + 3}, η = diag (1, - 1, - 1, - 1), Σ_{0} = {t = 0},

a mass $m > 0$ , and $T > 0$ . Set

□ = \partial_{t}^{2} - Δ, P_{m} = □ + m^{2} .

Fix compact sets

K, L \subset (0, T) \times B_{R}

for some $R > 0$ , where $B_{R}$ is the spatial ball of radius $R$ . Fix an open set $U$ compactly contained in $(0, T) \times B_{R}$ .

The allowed observation tests and switches are

f \in C_{K}^{\infty} (M; R), χ \in C_{L}^{\infty} (M; R), 0 \leq χ \leq 1,

with $χ = 1$ on a neighborhood of $U$ . We assume this switch class is nonempty.

The metric coefficient $h$ is any real smooth symmetric covariant tensor on $M$ . Its relevant control region is

E = [0, T] \times B_{R + T + 2} .

The admitted gauge parameters are exactly

g_{U} = C_{c}^{\infty} (U; T M) .

All support sets, the initial surface, $m$ , and $T$ remain fixed.

Displayed formulas in this passage

unlabeled · line 174
$M = R^{1 + 3}, η = diag (1, - 1, - 1, - 1), Σ_{0} = {t = 0},$
unlabeled · line 180
$□ = \partial_{t}^{2} - Δ, P_{m} = □ + m^{2} .$
unlabeled · line 184
$K, L \subset (0, T) \times B_{R}$
unlabeled · line 191
$f \in C_{K}^{\infty} (M; R), χ \in C_{L}^{\infty} (M; R), 0 \leq χ \leq 1,$
unlabeled · line 201
$E = [0, T] \times B_{R + T + 2} .$
unlabeled · line 205
$g_{U} = C_{c}^{\infty} (U; T M) .$