Abstract or Additional Information
A classical result of H.Weyl states that a function with integrable weak derivatives satisfiing the Cauchy Riemann equations is automatically complex analytic. Recently, Gaven Martin proved that this remains true solutions to autonomous nonlinear Beltrami equations $\partial_\overline{z} f=\mathcal{H}(\partial_z f)$ provided the nonlinearity $\mathcal{H}$ is asymptotically linear. In contrast we find a class of nonlinearities (including real-analytic ones) for which such the Weyl exponent coincides with that of $K$ quasiregular maps. The proof makes essential use of staircase laminates and convex integration for unbounded sets.
We will revisit the non linear Beltrami equation, highlighting the open questions, and discuss a variant of a recent formalism developed by Kleiner, Muller, Szekelyhidi and Xie relating Staircase Laminates and Convex Integration adapted to our setting.
This is a joint work with K.Astala (Helsinki) A.Clop (Barcelona), J.Jääskeläinen (Helsinki), A.Koski (Helsinki) and G.Martin (Auckland).