Quasiharmonic Fields and Beltrami Operators

A quasiharmonic field is a pair $\Cal{F} = [B,E]$ of vector fields verifying $div B=0$, $curl E=0$, and coupled by a distorsion inequality. For a given $\Cal {F}$, we construct a matrix field $\Cal{A}=\Cal{A}[B,E]$ such that $\Cal{A} E=B$. This remark in particular shows that the theory of quasiharmonic fields is equivalent (at least locally) to that of elliptic PDEs. Here we stress some properties of our operator $\Cal {A}[B,E]$ and find applications of them to the study of regularity of solutions to elliptic PDEs, and to some questions of G-convergence.