On reduced Beltrami equations and linear families of quasiregular mappings (Q2856654)

The author studies reduced Beltrami equations NEWLINE\[NEWLINE \frac{\partial f}{\partial \bar{z}}=\lambda(z)\Im\left(\frac{\partial f}{\partial {z}}\right),\quad |\lambda(z)|\leq k<1. NEWLINE\]NEWLINE A solution \(f\) to such an equation is called a reduced quasiregular mapping and the quantity \(\Im\left({\partial f}/{\partial {z}}\right)\) is analogous to the Jacobian determinant.NEWLINENEWLINEIn particular, the author proves the following conjecture due to \textit{K. Astala} et al. [Elliptic partial differential equations and quasiconformal mappings in the plane. Princeton, NJ: Princeton University Press (2009; Zbl 1182.30001)]. If \(\Omega\) is a domain in the complex plane and \(f:\Omega\to \mathbb{C}\) is a function of class \(W_{\text{loc}}^{1,2}(\Omega)\) that is a solution to a reduced Beltrami equation, then either \(\partial f/\partial z\) is constant, or else \(\Im\left({\partial f}/{\partial {z}}\right)\neq0\) for almost every \(z\in\Omega\). This extends earlier work of \textit{F. Giannetti}, \textit{T. Iwaniec}, \textit{G. Moscariello}, \textit{C. Sbordone} and \textit{L. Kovalev} [``On G-compactness of the Beltrami operators'', in: Nonlinear homogenization and its applications to composites, polycrystals and smart materials, NATO Sci. Ser. II Math. Phys. Chem. 170, Dordrecht: Kluwer Acad. Publ., 107--138, (2004)] and \textit{G. Alessandrini} and \textit{V. Nesi} [Ann. Acad. Sci. Fenn., Math. 34, No. 1, 47--67 (2009; Zbl 1177.30019)].