Manuscript VII-A-ST@1 · VII-A-ST.tex · proposition · c39a67a3

Proposition: Exact defect for the larger compact class

First-order repair and the larger-support residual · lines 817–838 · proof included or linked

VII-A-ST@1#prop:defect, VII-A-ST@1#eq:defect

Let $ξ \in C_{c}^{\infty} (D_{+}^{\circ}; T D)$ , without the plateau support restriction. Under the unchanged-free-input comparison $(h, ϕ_{0}) \mapsto (h + K ξ, ϕ_{0})$ ,

A_{1}^{(g)} [h + K ξ; ϕ_{0}] - A_{1}^{(g)} [h; ϕ_{0}] = E_{m} Q_{(1 - g) K ξ}^{(m)} ϕ_{0} .

Proof. The difference is

- E_{m} Q_{g K ξ}^{(m)} ϕ_{0} - X_{ξ} ϕ_{0} .

The zero-data identity and (eq:QK-shell) give $X_{ξ} ϕ_{0} = - E_{m} Q_{K ξ}^{(m)} ϕ_{0}$ . Linearity in the tensor argument proves (eq:defect).

Displayed formulas in this passage

VII-A-ST@1#eq:defect · line 823
$A_{1}^{(g)} [h + K ξ; ϕ_{0}] - A_{1}^{(g)} [h; ϕ_{0}] = E_{m} Q_{(1 - g) K ξ}^{(m)} ϕ_{0} .$
unlabeled · line 832
$- E_{m} Q_{g K ξ}^{(m)} ϕ_{0} - X_{ξ} ϕ_{0} .$

Proposition: Exact defect for the larger compact class

Displayed formulas in this passage

Nearby in this section