Manuscript IX-D@3 · IX-D-v3.tex · lemma · 3ad4586e

Lemma: Smooth comparison sign

D1: domains and separate causal interfaces › Subtraction and the smooth sign comparison · lines 956–972 · proof included or linked

IX-D@3#lem:sign

Under Assumption ass:smooth, setting $u_{f} = - G_{A} f$ gives the smeared comparison test $q = + Q [h] u_{f}$ .

Proof. The adjoint identities give

⟨ f, ϕ_{1} ⟩ = - ⟨ G_{A} f, Q [h] ϕ_{0} ⟩ = ⟨ u_{f}, Q [h] ϕ_{0} ⟩ = ⟨ Q [h] u_{f}, ϕ_{0} ⟩ .

An additional minus in the last test would contradict this declared Duhamel convention.

Displayed formulas in this passage

unlabeled · line 964
$⟨ f, ϕ_{1} ⟩ = - ⟨ G_{A} f, Q [h] ϕ_{0} ⟩ = ⟨ u_{f}, Q [h] ϕ_{0} ⟩ = ⟨ Q [h] u_{f}, ϕ_{0} ⟩ .$