Manuscript XII-D@1 · XII-D.tex · theorem · 308a2080

Theorem: Actual response law

D5: the exact signed plateau variation › The finite cubic identity and its sign · lines 1139–1164 · proof included or linked

XII-D@1#thm:response-law, XII-D@1#eq:response-law

The prescribed response obeys

u_{D} [γ + K ξ] - u_{D} [γ] = - s_{ξ} [γ] + K ρ_{ξ} [γ] - P E_{0}^{R} L_{ξ}^{dens} (A γ) .

Proof. Apply $G_{gr}^{R}$ to (eq:finite-source). For compact $s$ , the identities in the input sheet imply

G_{gr}^{R} A s = G_{gr}^{R} (L - 2 D^{*} D) s = s - K E_{0}^{R} D s .

The term with $L_{ξ}^{dens} (A γ)$ is $- P E_{0}^{R} L_{ξ}^{dens} (A γ)$ . Substitution gives (eq:response-law).

Displayed formulas in this passage

XII-D@1#eq:response-law · line 1142
$u_{D} [γ + K ξ] - u_{D} [γ] = - s_{ξ} [γ] + K ρ_{ξ} [γ] - P E_{0}^{R} L_{ξ}^{dens} (A γ) .$
unlabeled · line 1156
$G_{gr}^{R} A s = G_{gr}^{R} (L - 2 D^{*} D) s = s - K E_{0}^{R} D s .$