The Absorption Theorem for the Beltrami-Vekua Normal Form

Abstract

The Beltrami-Vekua normal form assigns to every smooth first-order real planar elliptic system a complex equation $w_{\bar{z}}-\mu w_z+\mathcal{A}w+\mathcal{B}\bar{w}=\mathcal{F}$ by an explicit pipeline. A companion paper showed that the density $\Theta =|\mathcal{B}{|}^2/(1-|\mu {|}^2)dxdy$ and its total mass are invariants under multiplicative gauges $w\mapsto \varphi w$ and orientation-preserving diffeomorphisms. The real system carries a larger symmetry: its unknowns may be recombined by any pointwise invertible real-linear substitution $w=\phi v^{\prime }+\psi {\bar{v}}^{\prime }$, the complex gauges being the case $\psi \equiv 0$. We prove the absorption theorem: re-normalizing through the pipeline after any such substitution returns to the gauge orbit of the original equation, with a universal explicit gauge $\tilde{\phi }=-i\lambda /(\phi -\psi )$, where $\lambda$ is the spectral root of the structure polynomial.

arXiv comment12 pages