The elliptic prescription and its boundary condition

Use

for $f\in {C_c}^{\infty }({\mathbb{R}}^3)$. Then $\Delta G_{\Delta }f=f$, and $G_{\Delta }f$ decays at infinity.

For an induced metric ${\gamma }_e=a^2(\delta +ec+\frac{1}{2}{e}^{2}d)$ with compactly supported coefficient perturbations, expand its Laplacian as

The rod equations recursively read

Their right-hand sides are spatially compact: the perturbation operators have compact coefficient support. Application of $G_{\Delta }$ gives smooth decaying coefficients. A smooth harmonic difference tending to zero at infinity vanishes by the maximum principle on expanding balls. This proves coefficientwise existence and uniqueness for the rod prescriptions used below.