Variational method for quasiconformal mappings (Q1091505)

The authors consider the quasilinear Beltrami equation \[ (1)\quad f_{\bar z} = \nu (z,f)f_ z \] for a bounded coefficient \(\nu\) (z,w) satisfying \[ (2)\quad | \nu (z,w)| \leq q(z,w)\leq q<1. \] Here q(z,w): \({\mathbb{C}}\times {\mathbb{C}}\to [0,q]\), \(q<1\), satisfying the Carathéodory conditions; q(\(\cdot,w)\) is measurable as a function of z for fixed w in \({\mathbb{C}}\) and q(z,\(\cdot)\) is continuous for almost all z in \({\mathbb{C}}\). Let \(\Sigma_{Q(z,w)}\), where \[ Q(z,w) = (1+q(z,w)/(1- q(z,w)), \] denote the set of homeomorphisms f of the plane which are solutions of (1) for some coefficient \(\nu\) (z,w) satisfying (2) and the Carathéodory conditions and for which f(z) is asymptotic to z as z approaches infinity. The principle of the new result in the paper is that the method of constructing a variation which is standard in the linear Beltrami equation can be extended to the quasilinear Beltrami equation. The authors also obtain necessary conditions for a functional on the space \(\Sigma_{Q(z,w)}\) to be extremal expressed in terms of the Gateaux derivative of the functional.