The best-on-average quasiconformal mappings

We seek the best-on-average quasiconformal mappings as the extremals of the functional equal to the integral of the conformality squared and multiplied by a particular weight. The inverse mapping to an extremal is an extremal of the same functional. We obtain necessary and sufficient conditions for the ellipticity in the sense of Petrovskii of the system of Euler equations for the extremal. We prove the local unique solvability of boundary problems for this system in the two-dimensional case. In the general case we prove the unique solvability of boundary problems for the system linearized at the identity mapping.