Sharp Holder regularity for autonomous Beltrami equations

Abstract

Let $Ω\subset\mathbb C$ be a domain and let $\mathcal H:\mathbb C\to\mathbb C$ be Lipschitz with constant $0<k<1$. We prove that every weak $W^{1,2}_{\mathrm{loc}}$ solution of the autonomous Beltrami equation $f_{\bar z}=\mathcal H(f_z)$ belongs to $C^{1,γ}_{\mathrm{loc}}$, where $γ=3/(2K+1)$ and $K=(1+k)/(1-k)$. The exponent is optimal, resolving the conjecture of Astala, Clop, Faraco, Jääskeläinen and Koski.