Manuscript V@1 · V.tex · theorem · e7ddf37d

Theorem: Restricted positive extension

Restricted compact representatives on a two-dimensional TT plane › Compact curvature currents · lines 1817–1841 · proof included or linked

V@1#thm:planeextension, V@1#eq:planeJ

For every fixed $f, ϕ$ above,

J_{c} = b_{1} J_{1} + b_{2} J_{2} \in E

satisfies

B_{f} (k, ϕ) = F_{J_{c}} (k) (k \in V_{fin}) .

Its support is contained in the fixed compact set $supp ρ$ , inside the open response slab.

Proof. For $k = λ_{1} v_{1} + λ_{2} v_{2}$ ,

F_{J_{c}} (k) = a, c \sum b_{a} λ_{c} δ_{a c} = λ_{1} B_{f} (v_{1}, ϕ) + λ_{2} B_{f} (v_{2}, ϕ) .

This is the actual response by linearity, using (eq:ba). Derivatives in (eq:curvaturecurrent) do not enlarge support.

Displayed formulas in this passage

V@1#eq:planeJ · line 1820
$J_{c} = b_{1} J_{1} + b_{2} J_{2} \in E$
unlabeled · line 1825
$B_{f} (k, ϕ) = F_{J_{c}} (k) (k \in V_{fin}) .$
unlabeled · line 1834
$F_{J_{c}} (k) = a, c \sum b_{a} λ_{c} δ_{a c} = λ_{1} B_{f} (v_{1}, ϕ) + λ_{2} B_{f} (v_{2}, ϕ) .$