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 \(\Omega\), \[f_\overline{z} = \mathcal{A}(f_z)\ a.e.\ z\in\Omega,\] when \(\mathcal{A}\) is linear at \(\infty\). Namely \(W^{1,1}_{\operatorname{loc}}(\Omega)\) solutions are \(W^{2,2+\epsilon}_{\operatorname{loc}}(\Omega)\). Here \(\epsilon>0\) depends explicitly on the ellipticity bounds of \(\mathcal{A}\). The condition "is linear at \(\infty\)" is necessary - the result is false for the equation \(f_\overline{z} = k|f_z|\), 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 \[f_\overline{z} = \mathcal{A}(z, f_z)\ a.e.\ z\in\Omega.\] There the condition "linear at \(\infty\)" also implies improved regularity for \(W^{1,1}_{\operatorname{loc}}(\Omega)\) solutions.