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 $Φ(w_{\bar z} - μw_z) + Ψ(\overline{w_z} - μ\,\overline{w_{\bar z}}) + \mathfrak{a} w + \mathfrak{b} \bar w = \mathfrak{f}$, with $|μ| < 1$ and $|Φ| > |Ψ|$, and compute the closed transformation laws of its data under the recombination of unknowns $w \mapsto φw + ψ\bar w$ and under orientation-preserving $C^1$ changes of variables. The 2-form $Θ= \frac{\bigl|\,Φ\,\mathfrak{b} - Ψ\,\mathfrak{a} - (Φ\, LΨ- Ψ\, LΦ)\,\bigr|^2}{\bigl(|Φ|^2 - |Ψ|^2\bigr)^2\,\bigl(1 - |μ|^2\bigr)}\; dx\, dy$, with $L = \bar\partial - μ\,\partial$, is invariant under the recombination and covariant under the changes of variables. The total mass $\mathcal{M} = \int_ΩΘ$ 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 $μ$ - there identifying $Θ$ with the pseudo-analytic mass density of the unframed equation. We then show all of this persists at measurable regularity: it suffices that $μ$ 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 $\|μ\|_\infty < 1$ is quasiconformally equivalent, of equal mass, to one over $μ= 0$.