Proposition: Explicit edge-zone defect

Let $0\ne \alpha \in {C_c}^{\infty }((0,\epsilon ))$ be nonnegative, and choose $\beta \in {C_c}^{\infty }({\mathbb{R}}^3)$ with $\int \beta {\mathrm{d}}^{3}x=1$. Set

For every $t\in [\epsilon ,T_{\ast }]$,

This parameter is outside ${\mathfrak{g}}_p$.

Proof. The source for $w[K\xi ]$ is compact, so its solution has compact spatial support on each bounded time slice. Put $W^{\mu }(t)=\int w^{\mu }[K\xi ](t,\mathbf{x}){\mathrm{d}}^{3}x$ and ${\Xi }^{\mu }(t)=\int {\xi }^{\mu }(t,\mathbf{x}){\mathrm{d}}^{3}x$. Spatial divergences integrate to zero. From

one obtains

Thus

For $t\ge \epsilon$, integration by parts gives (eq:edge). Its strict sign follows from ${\sigma }^{\prime }>0$ in the ramp and $\alpha \ge 0$, $\alpha \ne 0$. The support of $\xi$ is disjoint from the prescribed post-ramp $U$.