Super regularity for Beltrami systems

Abstract

We prove a surprising higher regularity for solutions to the nonlinear elliptic autonomous Beltrami equation in a planar domain $\mathrm{\Omega }$, $$f_{\bar{z}}=\mathcal{A}(f_z)a.e.z\in \mathrm{\Omega },$$ when $\mathcal{A}$ is linear at $\mathrm{\infty }$. Namely $W_{\mathrm{loc}}^{1,1}(\mathrm{\Omega })$ solutions are $W_{\mathrm{loc}}^{2,2+ϵ}(\mathrm{\Omega })$. 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 $f_{\bar{z}}=k|f_z|$, for any $0<k<1$, ($k=0$ is Weyl's lemma) and the improved regularity is sharp, but can be further improved if, for instance, $\mathcal{A}$ is smooth. We also discuss the subsequent higher regularity implications for fully non-linear Beltrami systems $$f_{\bar{z}}=\mathcal{A}(z,f_z)a.e.z\in \mathrm{\Omega }.$$ There the condition "linear at $\mathrm{\infty }$" also implies improved regularity for $W_{\mathrm{loc}}^{1,1}(\mathrm{\Omega })$ solutions.