Manuscript X-G-ST@1 · X-G-ST.tex · section · 3c97567f

A compact background inside the onset

Stage 3: variational reciprocity and the positive witness › A compact background inside the onset · lines 1400–1437

X-G-ST@1#eq:witnessdata

A nonconstant smooth onset with the stated endpoint extensions has $σ_{ε}^{''} \neq \equiv 0$ on $(0, ε)$ . Otherwise it would be affine there, and smoothness at the endpoints would force zero slope.

Choose

0 \neq = ζ \in C_{c}^{\infty} ((0, ε)), ζ \geq 0, supp ζ \subset {σ_{ε}^{''} \neq = 0}, ψ = ζ σ_{ε}^{''} .

Choose

χ_{0} \in C_{c}^{\infty} (R^{3}), χ_{0} \geq 0, M_{χ} = \int χ_{0} d^{3} x > 0.

Set

ξ_{on} = ψ (t) χ_{0} (x) \partial_{t}, w = K ξ_{on}, u = e_{00}, v = t e_{00},

where $e_{00}$ has only covariant component $00$ equal to one. Both $u, v$ solve $P_{gr} u = P_{gr} v = 0$ .

The vector field $ξ_{on}$ parametrizes an allowed background. It is not asserted to lie in the plateau algebra $g_{p}$ .

Displayed formulas in this passage

X-G-ST@1#eq:witnessdata · line 1423
$ξ_{on} = ψ (t) χ_{0} (x) \partial_{t}, w = K ξ_{on}, u = e_{00}, v = t e_{00},$
unlabeled · line 1409
$0 \neq = ζ \in C_{c}^{\infty} ((0, ε)), ζ \geq 0, supp ζ \subset {σ_{ε}^{''} \neq = 0}, ψ = ζ σ_{ε}^{''} .$
unlabeled · line 1417
$χ_{0} \in C_{c}^{\infty} (R^{3}), χ_{0} \geq 0, M_{χ} = \int χ_{0} d^{3} x > 0.$

A compact background inside the onset

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