The Framed Beltrami-Vekua Normal Form and its Pseudo-Analytic Mass

Abstract

We normalize a first-order real planar elliptic system, by pointwise algebra, to a framed Beltrami-Vekua equation $\Phi(w_{\bar z} - \mu w_z) + \Psi(\overline{w_z} - \mu\,\overline{w_{\bar z}}) + \mathfrak{a} w + \mathfrak{b} \bar w = \mathfrak{f}$, with $|\mu| < 1$ and $|\Phi| > |\Psi|$, and compute the closed transformation laws of its data under the recombination of unknowns $w \mapsto \varphi w + \psi \bar w$ and under orientation-preserving $C^1$ changes of variables. The 2-form $\Theta = \frac{\bigl|\,\Phi\,\mathfrak{b} - \Psi\,\mathfrak{a} - (\Phi\, L\Psi - \Psi\, L\Phi)\,\bigr|^2}{\bigl(|\Phi|^2 - |\Psi|^2\bigr)^2\,\bigl(1 - |\mu|^2\bigr)}\; dx\, dy$, with $L = \bar\partial - \mu\,\partial$, is invariant under the recombination and covariant under the changes of variables. The total mass $\mathcal{M} = \int_\Omega \Theta$ is therefore an invariant of the equivalence class. One recombination and one scaling carry any framed equation, in closed form, onto the trivial-frame slice - a Beltrami-Vekua equation over the same $\mu$ - there identifying $\Theta$ with the pseudo-analytic mass density of the unframed equation. We then show all of this persists at measurable regularity: it suffices that $\mu$ be measurable and locally elliptic and that the frame lie in $W^{1,2}_{\mathrm{loc}} \cap L^\infty_{\mathrm{loc}}$, the changes of variables then being quasiconformal homeomorphisms. In that class every equation with $\|\mu\|_\infty < 1$ is quasiconformally equivalent, of equal mass, to one over $\mu = 0$.