Beltrami equations and mappings with asymptotic homogeneity at infinity

TL;DR

This paper investigates the asymptotic behavior of solutions to degenerate Beltrami equations, establishing criteria for their existence with specific homogeneity properties, with applications in fluid mechanics in complex media.

Contribution

It introduces new criteria for the existence of solutions with asymptotic homogeneity at infinity for degenerate Beltrami equations, linking BMO conditions to solution behavior.

Findings

BMO condition implies asymptotic homogeneity at the origin

Criteria established for solutions with asymptotic homogeneity at infinity

Applications to fluid mechanics in anisotropic media

Abstract

First of all, we prove that the BMO condition by John-Nirenberg leads in the natural way to the asymptotic homogeneity at the origin of regular homeomorphic solutions of the degenerate Beltrami equations. Then on this basis we establish a series of criteria for the existence of regular homeomorphic solutions of the degenerate Beltrami equations in the whole complex plane with asymptotic homogeneity at infinity. These results can be applied to the fluid mechanics in strictly anisotropic and inhomogeneous media because the Beltrami equation is a complex form of the main equation of hydromechanics.