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 $w\mapsto \phi w+\psi \bar{w}$ and under orientation-preserving $C^1$ changes of variables. The 2-form $\Theta =\frac{|\Phi \mathfrak{b}-\Psi \mathfrak{a}-(\Phi L\Psi -\Psi L\Phi ){|}^2}{(|\Phi {|}^2-|\Psi {|}^2{)}^2(1-|\mu {|}^2)}dxdy$, 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 $W_{\mathrm{loc}}^{1,2}\cap L_{\mathrm{loc}}^{\infty }$, the changes of variables then being quasiconformal homeomorphisms. In that class every equation with $\Vert \mu {\Vert }_{\infty }<1$ is quasiconformally equivalent, of equal mass, to one over $\mu =0$.

19 pages