Manuscript XIII-A@1 · XIII-A.tex · definition · f56e91c7

Definition: Fixed restriction data

One frozen region and the observation-qualified defect set · lines 542–564

XIII-A@1#def:domains, XIII-A@1#eq:W, XIII-A@1#eq:defect-set

Fix $f, g, m, ε, T$ and a nonempty open set $W$ satisfying

W \subset P_{g} \cap {ε < t < T} .

This choice precedes all parameter selections. Since $g$ is compactly supported, $W$ is compact. Define

g_{W} = C_{c}^{\infty} (W; T M), C_{W} = {u \in C_{sol} : D^{*} b = 0 on W},

and

C_{def} (W; f, g) = {u \in C_{sol} : B_{ξ} (u) = 0 for every ξ \in g_{W}} .

The dependence on $W, f, g$ is suppressed only after those data are frozen.

Displayed formulas in this passage

XIII-A@1#eq:W · line 545
$W \subset P_{g} \cap {ε < t < T} .$
XIII-A@1#eq:defect-set · line 557
$C_{def} (W; f, g) = {u \in C_{sol} : B_{ξ} (u) = 0 for every ξ \in g_{W}} .$
unlabeled · line 551
$g_{W} = C_{c}^{\infty} (W; T M), C_{W} = {u \in C_{sol} : D^{*} b = 0 on W},$