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In recent studies on the G-convergence of Beltrami operators, a number ofissues arouse concerning injectivity properties of families of quasiconformalmappings. Bojarski, D'Onofrio, Iwaniec and Sbordone formulated a conjecturebased on the existence of a so-called primary pair. Very recently, Bojarskiproved the existence of one such pair. We provide a general, constructive,procedure for obtaining a new rich class of such primary pairs. This proof isobtained as a slight adaptation of previous work by the authors concerning thenonvanishing of the Jacobian of pairs of solutions of elliptic equations indivergence form in the plane. It is proven here that the results previouslyobtained when the coefficient matrix is symmetric also extend to thenon-symmetric case. We also prove a much stronger result giving a quantitativebound for the Jacobian determinant of the so-called \emph{periodic}$\sigma$-harmonic sense preserving homeomorphisms of $\mathbb C$ onto itself.

Beltrami operators, non--symmetric elliptic equations and quantitative Jacobian bounds