On one extremal problem for nonlinear Cauchy-Riemann-Beltrami systems

Abstract

The study of nonlinear Cauchy--Riemann--Beltrami systems is conditioned study of certain problems of hydrodynamics and gas dynamics, in which there is an inhomogeneity of media and a certain singularity. The paper considers a nonlinear Cauchy--Riemann--Beltrami type system in the polar coordinate system in which the radial derivative is expressed through the complex coefficient, the angular derivative and its m-degree module. In particular, if m is equal to zero, then this system of equations is reduced to the ordinary linear system of Beltrami equations. Note that general first-order systems were used by M.А. Lavrentyev to define quasiconformal mappings on the plane, see \cite{L}. The problem of area distortion under quasi-conformal mappings is due to the work of B. Boyarsky, see \cite{Bo}. For the first time, the upper estimate of the area of the disk image under quasi-conformal mappings was obtained by M.А. Lavrentyev, see \cite{L}. A refinement of the Lavrentyev inequality in terms of the angular dilatation was obtained in the monograph \cite{BGMR}, see Proposition 3.7. In the present paper, it is found an exact upper estimate of the area of the image of the disk, which is analogous to the known result by Lavrentyev. Also, we find here a mapping on which the estimate is achieved. Thus, the work solves the extreme problem for the area functional of the image of disks under a certain class of regular homeomorphic solutions of nonlinear systems of the Cauchy--Riemann--Beltrami type with generalized derivatives integrated with a square. The work uses p-angular dilatation. In the conformal case, angular dilatation is important in the theory of quasi-conformal mappings and nondegenerate Beltrami equations. Proof of the main result of the article is based on the differential relation for the area function of the image of disks of arbitrary radii, which was established in the previous work of the authors for regular homeomorphisms with Luzin's N-property.