Super regularity for Beltrami systems

TL;DR

This paper proves that solutions to certain nonlinear Beltrami equations exhibit higher regularity under specific conditions, advancing understanding of their smoothness properties in planar domains.

Contribution

It establishes higher regularity results for solutions to nonlinear elliptic Beltrami equations when the coefficient function is linear at infinity, a novel condition in this context.

Findings

Solutions are in W^{2,2+ε}_{loc} when initially in W^{1,1}_{loc}

Higher regularity depends explicitly on ellipticity bounds

The linear at infinity condition is shown to be necessary

Abstract

We prove a surprising higher regularity for solutions to the nonlinear elliptic autonomous Beltrami equation in a planar domain $\Omega$, \[ f_\zbar = {\cal A}(f_z) \hskip15pt a.e.\;\; z\in \Omega, \] when ${\cal A}$ is linear at $\infty$. Namely $W^{1,1}_{loc}(\Omega)$ solutions are $W^{2,2+\epsilon}_{loc}(\Omega)$. Here $\epsilon>0$ depends explicitly on the ellipticity bounds of ${\cal A}$. The condition ``is linear at $\infty$'' is necessary - the result is false for the equation $f_\zbar = k|f_z|$, for any $0<k<1$, ($k=0$ is Weyl's lemma). We discuss the subsequent higher regularity implications for fully non-linear Beltrami systems.