Manuscript IX-B-ST@1 · IX-B-ST.tex · definition · f6af1523

Definition: Supported developing laboratory

Prescribed cone and supported regular laboratory · lines 303–338

IX-B-ST@1#def:support

Fix a product-adapted isometric developing embedding

ι : U = (- δ, H) \times B_{R} ⟶ M_{δ_{c}}, t = c (T - T_{Σ}),

with developed metric

η = \diag (- 1, 1, 1, 1) .

Choose $T_{0}, r_{0} > 0$ and $R > r_{0} + T_{0}$ , and set

V = (0, T_{0}) \times B_{r_{0}}, Σ = {0} \times B_{R}, K_{*} = [0, T_{0}] \times B_{r_{0} + T_{0}} ⋐ U .

All observation tests are supported in $V$ . The compact analytic and operational laboratory is contained in $ι (U)$ , has closure disjoint from $W$ , and contains a neighbourhood of $ι (K_{*})$ .

For $f \in D (V)$ , the full truncated developed past

D_{f} = J_{η}^{-} (supp f) \cap {t \geq 0}

is retained. It is contained in $K_{*}$ . The response prescription has zero value and zero first $t$ -derivative at $t = 0$ , with no additional incoming homogeneous or lateral-boundary data.

For each fixed compact test family, the permitted positive smoothing widths are small enough that the smoothed field kernels remain inside $U$ .

Displayed formulas in this passage

unlabeled · line 306
$ι : U = (- δ, H) \times B_{R} ⟶ M_{δ_{c}}, t = c (T - T_{Σ}),$
unlabeled · line 312
$η = \diag (- 1, 1, 1, 1) .$
unlabeled · line 316
$V = (0, T_{0}) \times B_{r_{0}}, Σ = {0} \times B_{R}, K_{*} = [0, T_{0}] \times B_{r_{0} + T_{0}} ⋐ U .$
unlabeled · line 327
$D_{f} = J_{η}^{-} (supp f) \cap {t \geq 0}$

Definition: Supported developing laboratory

Displayed formulas in this passage

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