Super regularity for Beltrami systems

We prove a surprising higher regularity for solutions to the nonlinear elliptic autonomous Beltrami equation in a planar domain $\mathrm{\Omega }$, $$f_{\text{zbar}}=\mathcal{A}(f_z)a\mathrm{.}e\mathrm{.}z\in \mathrm{\Omega },$$ when $\mathcal{A}$ is linear at $\mathrm{\infty }$. Namely $W_{loc}^{1,1}(\mathrm{\Omega })$ solutions are $W_{loc}^{2,2+ϵ}(\mathrm{\Omega })$. Here $ϵ>0$ depends explicitly on the ellipticity bounds of $\mathcal{A}$. The condition ``is linear at $\mathrm{\infty }$’’ is necessary - the result is false for the equation $f_{\text{zbar}}=k\mathrm{\mid }{f}_z\mathrm{\mid }$, for any $0<k<1$, ($k=0$ is Weyl’s lemma). We discuss the subsequent higher regularity implications for fully non-linear Beltrami systems.