Super regularity for Beltrami systems

Gaven J. Martin¹

¹ Massey University, Institute for Advanced Study

Annales Fennici Mathematici, Tome 46 (2021) no. 1, pp. 59-65.

Résumé

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.  

Détail

Keywords: Beltrami systems, quasiconformal, higher regularity